Masters Thesis: Deconvolution Filters for Dynamic Rocket ... · DeconvolutionFiltersfor...

Aerospace Engineering

Deconvolution Filters forDynamic Rocket ThrustMeasurements

Richard B.H. Ahlfeld

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Deconvolution Filters forDynamic Rocket Thrust

Measurements

Master of Science Thesis

For the degree of Master of Science in Aerodynamics at DelftUniversity of Technology

Richard B.H. Ahlfeld

August 19, 2014

Faculty of Aerospace Engineering · Delft University of Technology

The work in this thesis was carried out at Airbus Defence and Space in Ottobrunn, Germany.

Copyright © Aerodynamics GroupAll rights reserved.

DELFT UNIVERSITY OF TECHNOLOGYDEPARTMENT OF AERODYNAMICS

The undersigned hereby certify that they have read and recommend to the Faculty of AerospaceEngineering for acceptance the thesis entitled “Deconvolution Filters for Dynamic RocketThrust Measurements” by Richard B.H. Ahlfeld in fulfilment of the requirements forthe degree of Master of Science.

Dated: August 19, 2014

Supervisors:

ii

Richard B.H. Ahlfeld Master of Science Thesis

Abstract

Every spacecraft carries small rocket engines called thrusters for the purpose of orbit andposition corrections in space. A precise operation of the thrusters saves fuel and a lowerfuel consumption can extend the service life of the spacecraft. Precise thrust levels can bebetter obtained by pulsed firing than by continuous firing of the thrusters. To guarantee aprecise operation, thrusters are tested extensively in a vacuum chamber beforehand on earthto measure their thrust. Measurement errors always impede the accurate determination ofthe rocket thrust. The measurement of pulsed rocket thrust is especially difficult, if usingstrain gage type thrust stands typical for the space industry. Their low first natural frequencyresults in a dominant transient response of the thrust stand structure, which interferes withthe thrust signal. In this thesis, an augmented state-space Kalman Filter is proposed as alow-cost and easy-to-implement solution to deconvolute the thrust signal from the transientthrust stand response. The typical weakness of the Kalman Filter - the complicated determi-nation of its process and measurement noise covariance matrices - is overcome by presentingseveral methods of choosing the matrices for pulsed thrust measurements. The measurementstate covariance matrix of the Deconvolution Kalman Filter is used to obtain an estimate ofthe uncertainty attached to the deconvoluted thrust signal. Using several test cases, the De-convolution Kalman Filter’s superiority to evaluate pulsed thrust measurements is proven bycomparing it with Infinite Impulse Response filters commonly used in the industry. Finally,a sequential Monte Carlo method is applied to the dynamic thrust measurement problem forthe first time to show that while precise thrust levels may be better obtained by pulsed firing,the attached uncertainty can become up to three times higher.

Master of Science Thesis Richard B.H. Ahlfeld

iv Abstract


Table of Contents

Abstract iii

Acknowledgements xiii

1 Introduction 1

2 Literature Review 72-1 Rocket Thrust Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2-1-1 Thrust Stands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72-1-2 Steady-State and Pulsed Thrust Measurements . . . . . . . . . . . . . . 8

2-2 The Dynamic Thrust Measurement Problem . . . . . . . . . . . . . . . . . . . . 92-2-1 Analytical Calculation of the System Response . . . . . . . . . . . . . . . 102-2-2 Building a Distortionless Measurement System . . . . . . . . . . . . . . . 132-2-3 Experimental Determination of the System Response . . . . . . . . . . . 14

2-3 Digital Deconvolution Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152-3-1 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162-3-2 Ill-Posedness of the Deconvolution Problem . . . . . . . . . . . . . . . . 172-3-3 Thrust Compensation Filters . . . . . . . . . . . . . . . . . . . . . . . . 192-3-4 Low-Pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2-4 Infinite Impulse Response (IIR) Deconvolution Filters . . . . . . . . . . . . . . . 232-4-1 Finite Impulse Response (FIR) and IIR Filters . . . . . . . . . . . . . . . 232-4-2 Autoregressive-Moving-Average Models . . . . . . . . . . . . . . . . . . 252-4-3 Discrete-Time Linear Least Squares System Identification for IIR Filters . 262-4-4 Stabilisation of IIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 282-4-5 Approximation Error and Error Statistics . . . . . . . . . . . . . . . . . . 29

2-5 The Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312-5-1 Advantages of the Kalman Filter for the Dynamic Thrust Measurement

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31


vi Table of Contents

2-5-2 State-Space Representation . . . . . . . . . . . . . . . . . . . . . . . . . 322-5-3 Kalman Filter Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 342-5-4 IIR Deconvolution from Noisy Observations Using the Kalman Filter . . . 392-5-5 Determination of Q and R . . . . . . . . . . . . . . . . . . . . . . . . . 41

2-6 Uncertainty Evaluation for Deconvolution Filters . . . . . . . . . . . . . . . . . . 422-6-1 Monte Carlo Uncertainty Quantification for Thrust Deconvolution Methods 422-6-2 Sequential Monte Carlo Method . . . . . . . . . . . . . . . . . . . . . . 442-6-3 Setting up the Filter Covariance Matrix Uba . . . . . . . . . . . . . . . . 45

2-7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462-8 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Airbus DS Thrust Measurement Equipment and Evaluation Method 493-1 Measurement Evaluation Procedure . . . . . . . . . . . . . . . . . . . . . . . . 493-2 Measurement Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3-2-1 Thrust Stand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523-2-2 Thrust Force Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533-2-3 Thruster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3-3 The Airbus DS Thrust Compensation Method . . . . . . . . . . . . . . . . . . . 563-3-1 The Digital Filter and its History . . . . . . . . . . . . . . . . . . . . . . 563-3-2 Linearity Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573-3-3 High-Pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583-3-4 IIR System Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 583-3-5 Filter Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603-3-6 Gain Change Caused by Order Reduction . . . . . . . . . . . . . . . . . 633-3-7 Second Order System Low-Pass Filter . . . . . . . . . . . . . . . . . . . 633-3-8 Design of the Compensation Filter . . . . . . . . . . . . . . . . . . . . . 653-3-9 Error Analysis for Pulse Mode Firing (PMF) Measurements . . . . . . . . 65

3-4 The MATLAB Edition of the TDMS Method . . . . . . . . . . . . . . . . . . . 673-4-1 Lower Identification Order . . . . . . . . . . . . . . . . . . . . . . . . . 673-4-2 Step-Function as Identification Input . . . . . . . . . . . . . . . . . . . . 683-4-3 Zero-Point Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693-4-4 Results Obtained by the Matlab Edition . . . . . . . . . . . . . . . . . 70

3-5 Improvement Possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723-5-1 Inverse Transfer Function Method . . . . . . . . . . . . . . . . . . . . . 723-5-2 System Identification with Firing Signal . . . . . . . . . . . . . . . . . . 733-5-3 Filter Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753-5-4 Low-Pass Filter Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 783-5-5 Phase Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803-5-6 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823-5-7 Design of Inverse Compensation Filter . . . . . . . . . . . . . . . . . . . 82


Table of Contents vii

3-6 Uncertainty Analysis for the Airbus DS Thrust Measurements . . . . . . . . . . . 843-6-1 Electronic Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853-6-2 Identification Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873-6-3 Non-Repeatability Force Transducer . . . . . . . . . . . . . . . . . . . . 883-6-4 Firing Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893-6-5 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3-7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4 Kalman Filter for Deconvolution 934-1 Preliminary Conditions for the Use of the Kalman Filter . . . . . . . . . . . . . . 934-2 Test of Bora’s Deconvolution Kalman Filter . . . . . . . . . . . . . . . . . . . . 94

4-2-1 Set-Up of Test Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . 944-2-2 Removal of Initialisation Error . . . . . . . . . . . . . . . . . . . . . . . 954-2-3 Determination of Q and R . . . . . . . . . . . . . . . . . . . . . . . . . 96

4-3 Deconvolution Kalman Filter for Dynamic Thrust Measurements . . . . . . . . . 994-3-1 Choosing the Integrated Low-Pass Filter . . . . . . . . . . . . . . . . . . 994-3-2 Heuristic Tuning of Q and R . . . . . . . . . . . . . . . . . . . . . . . . 1034-3-3 Rational Tuning of Q and R . . . . . . . . . . . . . . . . . . . . . . . . 1064-3-4 Automatic Tuning of Q and R . . . . . . . . . . . . . . . . . . . . . . . 107

4-4 Uncertainty Analysis for the Deconvolution Kalman Filter . . . . . . . . . . . . . 1084-4-1 Uncertainty Analysis Using the Deconvolution Kalman Filter . . . . . . . 1084-4-2 Uncertainty Analysis Using an Efficient Implementation of the Monte Carlo

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5 Test Results 1155-1 Performance of Deconvolution Filters in Test Scenarios . . . . . . . . . . . . . . 115

5-1-1 Comparison of Deconvolution Filters . . . . . . . . . . . . . . . . . . . . 1185-1-2 White Gaussian Random Noise . . . . . . . . . . . . . . . . . . . . . . . 1225-1-3 Low-Pass Filter Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5-2 Realistic Test Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265-3 Computational Effort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285-4 Summary of Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6 Conclusion 1316-1 Further Research Possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Bibliography 137

Glossary 143List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143


viii Table of Contents


List of Figures

1-1 ATV firing its thrusters [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11-2 Hot firing test of an AHRES engine [2] . . . . . . . . . . . . . . . . . . . . . . . 21-3 Airbus DS Bipropellant Thrusters [3] . . . . . . . . . . . . . . . . . . . . . . . . 3

2-1 Typical thrust stand configuration [4] . . . . . . . . . . . . . . . . . . . . . . . . 82-2 Illustration of Steady-State Firing (SSF) and PMF . . . . . . . . . . . . . . . . . 92-3 Dynamic thrust measurement example . . . . . . . . . . . . . . . . . . . . . . . 102-4 Analytical multi-degree of freedom thrust stand model [4] . . . . . . . . . . . . . 112-5 Normed step response of a thrust stand behaving like a system of order 2. . . . . 112-6 Analytical single-degree-of-freedom thrust stand model [4]. . . . . . . . . . . . . 122-7 Frequency response of the PMF measurement platform designed by Gao [7]. . . . 142-10 White noise cancellation performance of deadbeat and non-deadbeat digital com-

pensator [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202-11 Ideal frequency response for a thrust compensation low-pass filter according to [4]. 212-12 Schematic frequency response of an elliptic low-pass filter. . . . . . . . . . . . . 212-13 Comparison of the amplitude characteristics of Butterworth of order 3, Chebychev

and digitalised Bessel filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222-14 Illustration of the working principle of FIR and IIR filters. . . . . . . . . . . . . . 242-15 Idea of Kalman filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322-16 Kalman filtering to recover a noise corrupted input signal [9] . . . . . . . . . . . 392-17 Performance of the Kalman Filter for different noise covariance matrices [10]. . . 412-18 Relation between Number of Monte Carlo Trials and Digit Accuracy from [11] . . 43

3-1 Flowchart illustrating the steps of a thrust measurement evaluation. . . . . . . . 503-2 Measurement signal of thruster showing constructive interference. . . . . . . . . 51


x List of Figures

3-3 Offset of zero point during longer SSF measurement . . . . . . . . . . . . . . . 513-4 Thrust stand sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523-5 Characteristics of SM S-Type Load Cell [12]. . . . . . . . . . . . . . . . . . . . . 533-6 External influences on the Thrust Measurement Bridge (TMB). . . . . . . . . . 533-7 Thruster characteristics [13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543-8 Isp for SSF and PMF [13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543-9 Ibit repeatability and test range [13] . . . . . . . . . . . . . . . . . . . . . . . . 553-10 The algorithm of the TDMS method. . . . . . . . . . . . . . . . . . . . . . . . 563-11 Band-stop filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563-12 Check of linearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573-13 Frequency spectrum of PMF test showing low frequency noise. . . . . . . . . . . 583-14 Calibration with impulse hammer. . . . . . . . . . . . . . . . . . . . . . . . . . 593-15 Time series used for identification. . . . . . . . . . . . . . . . . . . . . . . . . . 593-16 System identification flow chart . . . . . . . . . . . . . . . . . . . . . . . . . . . 603-17 Frequency spectra of two tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . 613-18 Example for frequency, damping, amplitude and phase of a transfer function of

order 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623-19 Frequency response and reconstructed thrust input . . . . . . . . . . . . . . . . 633-20 Frequency response of inverted, reduced transfer function . . . . . . . . . . . . . 643-21 Frequency response of low-pass filter Second Order System (SOS) . . . . . . . . 653-22 Frequency response of the Airbus DS thrust compensation filter. . . . . . . . . . 653-23 Frequency, damping, amplitude and phase for a transfer function of order 60 . . 673-24 Frequency spectrum of measurement signal . . . . . . . . . . . . . . . . . . . . 683-25 Impulse and step-function input used for system identification. . . . . . . . . . . 693-26 Zero point correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693-27 Thrust signal validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703-28 Two exceptional test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713-29 Algorithm of the Inverse Transfer Function method. . . . . . . . . . . . . . . . . 723-30 Firing signal used as input for identification . . . . . . . . . . . . . . . . . . . . 733-31 Thrust pulses reconstructed using different identification routines . . . . . . . . . 733-32 Choice of order for least squares identification . . . . . . . . . . . . . . . . . . . 763-33 Beat in measurement signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763-34 Compensated thrust signal for filter of order 2 and 8 . . . . . . . . . . . . . . . 773-35 Frequency response for Inverse Transfer Function filter of order 2 and 8 . . . . . 773-36 Time response of Bessel and Butterworth filter. . . . . . . . . . . . . . . . . . . 783-37 Thrust stand sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793-38 Frequency and phase response of the Inverse Transfer Function method. . . . . . 803-39 Forward, reverse and bidirectionally filtered thrust signal . . . . . . . . . . . . . 803-40 Phase jump when frequency passes through 0 . . . . . . . . . . . . . . . . . . . 81


List of Figures xi

3-41 One and bidirectionally filtered thrust pulse . . . . . . . . . . . . . . . . . . . . 813-42 Pole-zero plot of the Inverse Transfer Function compensator . . . . . . . . . . . 823-43 Frequency response of Inverse Transfer Function Filter. . . . . . . . . . . . . . . 833-44 Last pulse of measurement signal and transient response . . . . . . . . . . . . . 843-45 Error sources considered in the uncertainty quantification for Airbus DS. . . . . . 853-46 Electronic noise in a thrust measurement signal. . . . . . . . . . . . . . . . . . . 853-47 Histogram of electronic noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863-48 Frequency spectrum of electronic and ideal white Gaussian noise. . . . . . . . . . 863-49 Left: Identification error, Right: Histogram of modeling error. . . . . . . . . . . 873-50 Frequency spectrum of modelling error. . . . . . . . . . . . . . . . . . . . . . . 873-51 Measured (black) and predicted (red) output using an Autoregressive-Moving Av-

erage (ARMA) model of order 27. . . . . . . . . . . . . . . . . . . . . . . . . . 883-52 Example of firing noise caused by inconsistent combustion. . . . . . . . . . . . . 893-53 Histogram of firing noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893-54 Frequency spectrum of firing noise. . . . . . . . . . . . . . . . . . . . . . . . . . 903-55 Example of one input and output signal including noise . . . . . . . . . . . . . . 903-56 Uncertainty and 95% credibility intervals for a thrust signal . . . . . . . . . . . . 913-57 Uncertainty and 95% credibility intervals for a thrust signal . . . . . . . . . . . . 91

4-2 Deconvolution of a random input signal using Bora’s method [9] . . . . . . . . . 954-3 Convergence of Kalman gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964-4 Deconvolution of random input signal with initial guess for P determined in a pre-run. 964-5 Q/R-ratio for the Deconvolution Kalman Filter . . . . . . . . . . . . . . . . . . 974-6 Relative error for different Q/R ratios. . . . . . . . . . . . . . . . . . . . . . . . 984-7 Different effects of Butterworth low-pass filter. . . . . . . . . . . . . . . . . . . . 994-8 Thrust inputs deconvoluted with the Kalman Filter using Butterworth filters with

different order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004-9 Steady-state error caused by the Inverse Transfer Function and Kalman Filter . . 1014-10 Steady-state system error for different types of feedback systems [14]. . . . . . . 1024-11 Thrust signal recovered by Deconvolution Kalman Filter for different values of Q 1044-12 Absolute error x− x for different Q/R ratios. . . . . . . . . . . . . . . . . . . . 1044-13 Four pulses used to determine the value of Q . . . . . . . . . . . . . . . . . . . 1054-14 Error metrics J1 and J2 for the Deconvolution Kalman Filter . . . . . . . . . . . 1064-15 Input Signal and Output Signal used for Autocovariance Determination. . . . . . 1084-16 95% credible intervals reconstructed with the Deconvolution Kalman Filter . . . 1104-17 95% credible intervals calculated with a sequential Monte Carlo method . . . . . 1114-18 Input and output signal used for the sequential Monte Carlo method . . . . . . . 1124-19 Input and output signal used for the sequential Monte Carlo method for PMF test. 1124-20 Input and output signal for a PMF test . . . . . . . . . . . . . . . . . . . . . . 113


xii List of Figures

4-21 Uncertainty for a single thrust pulse, for which the rise of the thrust has been triple 113

5-1 Flowchart illustrating how synthetic data are generated to compare different de-convolution methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5-2 Measurement signal generated with the (n,n) or (n,n-1) transfer function. . . . . 1165-3 Frequency response of the two test models constructed to have the same frequency

response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175-4 Delay of one sample caused when identifying a (n,n) model with a (n,n-1) transfer

function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185-5 Bias in the TDMS, Inverse Transfer Function and Deconvolution Kalman Filter

method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195-6 Recovery of prescribed input signals with on and off times of the same duration:

10ms and 50ms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205-7 Effect of random noise with standard deviation of σ = 0.15 on deconvolution

methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225-8 Inverse Transfer Function method without and with Butterworth low-pass filter. . 1245-9 Effect of white noise with standard deviation 0.15N on deconvolution methods. . 1245-10 Synthetic input and output data for realistic test scenario . . . . . . . . . . . . . 1265-11 Reconstruction of input signal including firing noise. . . . . . . . . . . . . . . . . 127

6-1 The theoretical model built by Ren with a nonlinear modal analysis is accurateenough to replace the actual frequency response [15]. . . . . . . . . . . . . . . . 133

6-2 Effect of errors in the noise covariances of Kalman Filter and OUFIR filter estimates[10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134


Acknowledgements

I would like to thank my supervisors Johann Keppeler (Airbus Defence and Space) and Fer-dinand Schrijer (Delft University of Technology) for their advice and assistance during thewriting of this thesis. A special thanks goes to Robert Wagner (Airbus Defence and Space) forhis critical comments and Sascha Eichstädt (Physikalisch Technische Bundesanstalt Berlin)for his helpful insights and impulses.

Delft, University of Technology Richard B.H. AhlfeldAugust 19, 2014


xiv Acknowledgements


Chapter 1

Introduction

The International Space Station (ISS) remains on its nominal orbit by firing its thrusterson a regular basis to compensate for the effect of gravitational forces [6]. Likewise, everysatellite, space shuttle or spacecraft carries thrusters for orbit and gesture controlling, dockingrendezvous or landing operations [8].

Figure 1-1: ATV Edoardo Amaldi and the International Space Station firing their thrusters toboost their orbit, captured by ESA astronaut André Kuipers [1].

During an attitude adjustment of the ISS or a satellite, an imprecise firing of the thrusterscan cause an imbalance. By contrast, a more accurate firing can save fuel, extend the servicelife of the spacecraft and consequently increase the payload or reduce the cost of a mission[8]. Because precise thrust levels can be better obtained by a pulsed firing than by continuousfiring of the thrusters [15], the design of highly accurate measurement evaluation methodsfor pulsed liquid rocket engines has received more attention in the last decade [8, 15, 16, 17],even though most of the concepts have been known since the 1960s [4, 18].


2 Introduction

To determine the exact performance of a thruster in space, an experimental determinationof the thrust can be conducted in a vacuum chamber beforehand on earth. Although thethrust of a liquid rocket engine can by determined by analytical or computational methods,a reliable experimental validation of every thruster is unavoidable. It has to be ensured thatthe thruster will function flawlessly in space for many years, where there is no opportunity ofmaintenance. Even if a good computational or mathematical model is available, each thrusterstill has to be tested extensively on a thrust stand. A thrust stand is a mechanical structureequipped with force transducers onto which the thruster is mounted. An example of a thruststand operated by Airbus DS and the German Aerospace Center (DLR) in Lampoldshausenis shown in Figure 1-2.

Figure 1-2: Hot firing test of an AHRES engine by the German Aerospace Center DLR inLampoldshausen [2].

The thrust of chemical rocket engines is usually determined using a displacement type thruststand. For steady-state thrust measurements, during which the engine is fired continually,displacement-type thrust stands can be very precise [16]. However, most displacement-typethrust stands found in literature are not suitable for pulsed firing frequencies of higher than50Hz [16]. The reason is that during pulsed thrust measurements the transient response ofthe mechanical structure interferes with the thrust signal. The amplitude of the interferingthrust stand oscillation can become larger than the thrust value, so that the true thrustvalue cannot be determined without first compensating the oscillations of the thrust stand.Since the mathematical relation between the thrust signal and the thrust stand response isbest described by a convolution-type integral, the inverse process of separating the thrustfrom the transient system response is called deconvolution. The flexible bearing betweenthe thruster and the load cell is mainly responsible for the high amplitudes in the transientthrust stand response. Therefore, pulsed thrust can be measured more accurately when astiffer bearing with higher natural frequencies is used. Novel designs of thrust stands withfavourable frequency spectra were proposed by [15, 16, 17, 7]. While such thrust stands willimprove the precision of pulsed thrust measurements in the future, at the moment industrialrocket engine providers mostly use displacement type thrust stands. For such thrust standsit is a challenge to reduce the error below 20% for pulsed tests, where the on and off times ofthe thruster firing are twenty times shorter than sampling rate. The main aim of this thesisis to develop a digital deconvolution filter, which is able to reduce the measurement accuracyof pulsed thrust measurements below 5%, if a displacement type thrust stand is used.


3

This thesis was initiated by Airbus Defence and Space (Airbus DS)1 and carried out in thedepartment for data reduction and statics (TP24) in Ottobrunn in cooperation with the testdepartment of liquid rocket engines in Lampoldshausen. The data made available by AirbusDS are proprietary and cannot be published in this work. However, surrogate data containingthe relevant aspects of the problem were generated to perform tests. It is generally difficultto refer the reader to freely available data in the literature. There are, however, a few papersindicating the state-of-the-art [8, 16], which describe the typical model for dynamic thrustmeasurements. Chen [8] outlines three methods to compensate the transient thrust standresponse (see Section 2-3-3) and Xing [16] outlines the calibration and design of a new thruststand system for liquid pulsed rocket engines.

Figure 1-3: Airbus DS Bipro-pellant 10N Hydrazine Thruster[3]

For Airbus DS the primary goal of this thesis was the trans-fer of their digital thrust compensation method from a SciL-ab/Java environment in the local Technical Data ManagementSystem (TDMS) in Lampoldshausen to a Matlab/C++ envi-ronment in Airbus’ global Measurement Data Base (MDB) inOttobrunn. Furthermore, it should be investigated whether themethod could be improved considering recent advances in dy-namic measurement filtering. Any improvement or alternativemethod should take into account that Airbus DS has a wholerange of thrust stands and that the evaluation method shouldbe easily adaptable to all of them. The thruster fired on theAirbus DS 10N thrust stands is depicted in Figure 1-3.

Conventionally, the thrust stand response is filtered by a dig-ital IIR or FIR filter and the operator tries to improve theaccuracy by shifting the zeros and poles of the digital filter.However, the designing effort increases significantly for increas-ing demands on accuracy, because the design of a deconvolutionfilter requires extensive manual testing. In addition, the systemorder has to be chosen carefully as a trade-off it is necessary tochoose the system order carefully as a trade off between accu-racy and stability. Above all, it is important that the operatorcan discern whether a signal component is characteristic for thethrust signal or caused by the evaluation method.

Using data provided by Airbus DS, an entirely new methodfor pulsed rocket thrust measurements for displacement typethrust stands was developed: an augmented version of the op-timal linear Kalman Filter. As this filter essentially performsthe deconvolution of the thrust signal from the transient thruststand response the method is called a Deconvolution KalmanFilter. To apply it, only an Autoregressive-Moving Average (ARMA) model describing thetransient response has to be provided. Unlike for other filters, the Deconvolution KalmanFilter needs no stability analysis, even if an unstable model is used. It requires no other

1 The former Astrium Space Transportation.


4 Introduction

designing choices of the operator than a sensible choice of its covariance matrices and, if anadditional smoothing is wanted, a cut-off frequency for its low-pass filter component. Thecommonly known weak-spot of the Kalman Filter is that its measurement noise and processnoise covariance matrices Q and R are unknown and need to be determined. Guidelines onhow to choose the covariance matrices for dynamic thrust measurements were developed andshown to be effective. The Deconvolution Kalman Filter is simple to implement. Due toits special set-up, it is possible to conveniently integrate an evaluation of the measurementuncertainty into the deconvolution process. Tests on synthetic data have shown its great ca-pacity to extract even the shortest firing pulses with an error of less than 5%, for tests wherethe error of a comparable IIR Filter was never less than 20%.

Summary of Contents

The literature study in Chapter 2 introduces the basics concepts of rocket thrust measure-ments and the dynamic rocket thrust measurement problem. Three different methods arepresented to solve it: the analytical calculation of the dynamic system response, the con-struction of a distortionless measurement system and the experimental determination of thedynamic response with a subsequent removal by a digital filter. The basic properties of digitaldeconvolution filters are explained. The ill-posedness of the deconvolution problem requiresthat any thrust correction filter is complemented by a low-pass filter. Low-pass filters suitablefor thrust measurements are elaborated. The focus lies on IIR filters for deconvolution, as usedby Airbus DS. It is described how ARMA models can be identified using linear least squaressystem identification and how the modelling error can be quantified using error statistics. Asinnovative solution to most problems faced in dynamic thrust measurement evaluations, theuse of a Kalman Filter for deconvolution is proposed. An adaptation of the Kalman Filterfor deconvolution has already been attempted by Bora [9]. The choice of the Kalman Filtertuning parameters is different than for the linear optimal Kalman Filter and has not yet beentreated in the literature. Only two methods, one by Saha and the other by Rajamani, arefound to be adaptable for the Deconvolution Kalman Filter. The derivation of the KalmanFilter algorithm is used to illustrate how the Kalman Filter can be used for uncertainty quan-tification. The basics of uncertainty quantification are explained and the Monte Carlo methodis suggested to evaluate the uncertainty in dynamic thrust measurements. It results in a highaccuracy and has low memory requirements. The study is concluded by the industrial andacademic objectives of this thesis.

In Chapter 3, the Airbus DS main digital filter and thrust stand are described. First, anoverview of the entire measurement evaluation procedure is given and specific properties of theused measurement equipment are explained. Then, the design of the digital filter is describedin detail. The implementation of the described method in a Matlab/Octave environmentrequired a few changes, which are explained. It is attempted to improve the method bymaking several adaptations. The performance of the improved method is analysed in theTest Chapter 5. Finally, the measurement uncertainty is evaluated using a sequential MonteCarlo method provided by the Physikalisch Technische Bundesanstalt Berlin and conclusionsare drawn from the results. This chapter is not included in the public version of this thesis,but the accuracy of the Airbus DS filters is a point of reference for the new methods developed.It partly contains real test data in scaled form, while synthetic data are used throughout theother chapters of this thesis. This is done not only because real data are proprietary, but also


5

because it is only for synthetic test data that the true input signal is available and can beused to compare the accuracy of different deconvolution methods.

The Deconvolution Kalman Filter is developed and adapted to the dynamic thrust measure-ment problem in Chapter 4. The concept of the Deconvolution Kalman Filter is adaptedfrom a design made by Bora [9]. His method is tested and adapted for the dynamic thrustmeasurement problem. As the correct choice of the measurement and process noise covariancematrices is crucial for the accuracy of the method, three ways to determine correct values aredescribed: a heuristic approach, a rational method suggested by Saha [19] and an automaticmethod developed by Rajamani [20]. Due to the special set-up of the augmented KalmanFilter, an estimate of the uncertainty in the thrust signal can be extracted from the measure-ment state covariance matrix. It is found that the Deconvolution Kalman Filter produces astable system, robust against numerical errors, and provides the optimal recursive solution forlinear-time invariant systems when the noise is white Gaussian and its statistics are exactlyknown. Finally, the model uncertainty is evaluated once more using codes provided by thePhysikalische Technische Bundesanstalt Berlin [21] for a model of order two. This is neces-sary, because the determination of the model uncertainty for dynamic force measurements isnot yet standard practice in the industry [21].

In Chapter 5, the performance of the Deconvolution Kalman Filter, the Airbus DS IIRFilter and the alternative IIR Filter improving the Airbus DS method is evaluated in aseries of synthetic test cases with different on and off times. The evaluation is divided intodeconvolution filter, noise and low-pass filter effects to identify error sources separately. Thecomputational effort is compared and the results are summarised.

The results of this thesis are summarised in Chapter 6 and possibilities for further researchare pointed out.


6 Introduction


Chapter 2

Literature Review

This chapter introduces the basic concepts of rocket thrust measurements in Section 2-1. Thedynamic thrust measurement problem is introduced and three different methods of solvingit are presented: the analytical calculation of the dynamic system response in Section 2-2-1,the construction of a distortionless measurement system in Section 2-2-2 and the experimen-tal determination of the dynamic response with a subsequent removal by a digital filter inSection 2-2-3. The concept and basic properties of digital deconvolution filters are developedin Section 2-3. A digital deconvolution filter always consists of a compensating and a noiseattenuating part. This is described in Section 2-3-3 and Section 2-3-4, respectively. IIR filtersfor deconvolution, as used by Airbus DS, are explained in Section 2-4. It is described howAutoregressive-Moving Average (ARMA) models can be identified using linear least squaressystem identification and how the modelling error can be quantified. As innovative solutionto most problems faced in dynamic thrust measurement evaluations, the Kalman Filter isintroduced in Section 2-5. Its algorithm is derived in a way which illustrates how the KalmanFilter can be used for uncertainty quantification in Section 2-5-3. The basics of uncertaintyquantification are explained in Section 2-6. A Monte Carlo method suitable to evaluate theuncertainty in dynamic thrust measurements with high accuracy and low memory require-ments is explained in Section 2-6-2. The chapter is concluded by the industrial and academicobjectives of this work.

2-1 Rocket Thrust Measurements

2-1-1 Thrust Stands

The thrust force of a rocket engine can be determined by mounting the thruster onto amechanical measurement system equipped with force transducers: the thrust stand. Althoughvarious different thrust stand types exist, their measurement principles are the same. Therocket engine is mounted onto a movable platform. The platform is attached to a fixedbasement through one or several flexures. The flexures allow the thrust force to move the


8 Literature Review

platform, so that the displacement of the movable platform is proportional to the thrust to bemeasured. A sensing element or force transducer is fixed to the movable platform to detectthe displacement [16]. The thrust of chemical rocket engines is usually determined using aload cell as sensing element. Load cells can handle large amounts of thrust, but are also verysensitive [22]. They are the conventional choice for measuring thrust forces in the order ofseveral Newton [23]. A load cell measures force through a set of sensitive strain gages. Thestrain gages measure the deformation of a material with known elasticity by recording thechange in electrical resistance accompanying a deformation [24]. Three typical and simplifieddisplacement-type thrust stands are displayed in Figure 2-1. In all of them, the thruster (3)is attached to a load cell (2), which is firmly connected to a thrust butt and to the ground.In the graph on the far left-hand side, the thruster is not directly fixed to the load cell butto flexures (4) which bend when the thrust force is exerted. This intermediate step has theadvantage that the measurable strain of the bent beam can be described by a time-invariantand linear model of order 2. Moreover, the sensitivity of the thrust stand can be altered bychanging the stiffness of its flexures.

Figure 2-1: Typical configurations of thrust stands. One horizontal (left) and two vertical (middleand right) configurations [4].

2-1-2 Steady-State and Pulsed Thrust Measurements

Thrusters can be fired in two different modes: steady-state and pulsed. If the thruster isfired once, continuously and long enough for the relevant parameters like the thrust and thepressure in the combustion chamber to reach quasi-stationary conditions, the firing of thethruster is referred to as Steady-State Firing (SSF). If the thruster is fired repeatedly inan either periodic or non-periodic sequence of on and off times, the firing is referred to asPulse Mode Firing (PMF) and the measurement is known as dynamic thrust measurement.The particular case that the thruster is fired only once, but not long enough to reach anequilibrium can be counted as SSF.

Precise thrust levels can be better obtained by PMF than by SSF [15]. An exact knowledgeof the thrust saves fuel, extends the service life of the spacecraft, increases the payload andreduces the cost of a mission [8]. Therefore, the design of thrust stands with high naturalfrequencies and adequate sensitivity for pulsed liquid rocket engines has received more at-tention during the last decade. For SSF measurements displacement-type thrust stands canreach high precision.


2-2 The Dynamic Thrust Measurement Problem 9

Figure 2-2: Illustration of SSF and PMF, using an input signal of amplitude 10N and a transferfunction of order 2. One can see that the measurement signal reaches a new steady state at 10Nafter about 300ms in the graph on the left.

2-2 The Dynamic Thrust Measurement Problem

The classic reference work for dynamic rocket thrust measurement techniques was written byCrosswy and Kalb in 1967 at the Arnold Engineering Development Center (AEDC) in theUnited States [4]. The authors were the first to investigate the relevant components of theproblem and suggested different solutions. In a nutshell, the dynamic thrust measurementproblem can be formulated as follows: a thrust measurement is defined as dynamic if theinput thrust varies faster in time than the response of the measurement system. If the inputsignal varies more slowly than the rate at which the system responds, the measurement isdefined as static. For dynamic measurements, the fast variations of the input signal have agreat effect on the accurate estimation of the measurand1 and the measurement uncertaintyconnected with it. In addition, there is usually a time delay between the thrust input andmeasured thrust stand response and the amplitude scaling is wrong [26].

That the mechanical thrust stand system is dynamic can be seen when observing the oscilla-tions in the measurement signal following an abrupt change in the input thrust, as depictedin Figure 2-3. If the thruster is turned on or off (green curve), the thrust force (red curve)changes abruptly and the thrust stand responds with oscillations (blue curve). Since rocketengines do not produce exact sinusoidal waves [16], the sinusoidal oscillation can only haveone cause: the harmonics of a thrust stand frequency. If the thruster is fired with a frequencyclose to a natural frequency of the mechanical structure, the firing can even excite resonanceeffects.

1 Expression used in metrology to refer to the quantity sought in a measurement.


10 Literature Review

Figure 2-3: Example of a thrust measurement signal distorted by the thrust stand’s systemresponse (blue curve). The firing signal (green curve) indicates when the thruster was turned onor off (1 or 0) and the reconstructed thrust (red curve) is the thrust input estimated by a digitalfiltering method.

To solve the dynamic thrust measurement problem, a technique has to be devised, which canreconstruct an accurately scaled and undistorted replica of the thrust force from a measure-ment signal corrupted by thrust stand dynamics. The existing techniques can be classifiedinto three categories:

1. Calculating the system response analytically:

- distributed parameter methods- lumped parameter methods

2. Building a distortionless measurement system.3. Determining the system response experimentally:

- analogue filter- digital filter

2-2-1 Analytical Calculation of the System Response

The mechanical system of the thrust stand is a composition of continuous mass members,whose motion response is a function of both space and time. In the past, a treatment of thethrust stand as a distributed parameter system2 could only be approximated with methodslike Rayleigh-Ritz. Nowadays, the availability of high performance computers make it possi-ble (but still time-consuming) to set up a three-dimensional distributed model of the standand simulate its dynamic motions with finite element software using Galerkin or Collocation

2 For an introduction to the notion of a distributed parameter systems see Chapter 2 in [27].



methods [28]. In 2013, the thrust stand of the historic Russian Vostock rocket engines wastechnically overhauled with an ANSYS model, as this was much cheaper than designing anew thrust stand [29]. Nevertheless, an analytical approach characterising all objects as rigidand only connected by joints, springs or dampers, as illustrated in Figure 2-4, provides insightinto the parameters that govern the thrust stand and is less time intensive to perform. It canbe used as a guideline, especially when designing a new thrust stand.

Figure 2-4: Example for an analytical multi-degree of freedom thrust stand model suitable forall thrust stands.

For a lumped parameter model, the mass, elasticity and energy dissipation of all rigid objectsis assumed to be located at discrete points in space. In this way, only ordinary linear differ-ential equations with constant coefficients have to be solved instead of multi-variable partialdifferential equations. The system can be built analytically and solved with little computa-tional effort, as the equation of motion is derivable by application of D’Alembert’s principleor the Lagrangian formalism:

[M(t)] q + [B(t)] q + [K(t)] q = {F} (2-1)

where [M(t)] is the mass and moment of inertia matrix, [B(t)] is the damping matrix, and[K(t)] the stiffness matrix. If all coefficients [M(t)], [B(t)] and [K(t)] are constants, thedynamic system (2-1) is called time-invariant or stationary. All three matrices are squarematrices of size n× n, where n is the number of degrees of freedom; q and q and q are singlecolumn matrices with n elements.

Figure 2-5: Normed step response of a thrust stand behaving like a system of order 2.



The transient response of a thrust stand is mostly characterised by a single predominantfrequency and looks like the response of a stationary single-degree-of-freedom system of order2 as shown in Figure 2-5. To describe the system with a classical Mass-Spring-Damper (MSD)system of order 2, which can be illustrated as in Figure 2-6, all components (the rocket engine,the propellant lines and the mounting structure) have to be regarded as one lumped massto be contained in a single mass parameter M . The dynamic and static deflections need tobe small to assume linear damping with parameter B. The spring rate should be a linearand time invariant function of spring deflection to apply Hooke’s Law with stiffness K. Inaddition, it is assumed that the energy dissipation mechanism is linearly dependent uponvelocity and time invariant and that the thrust butt is rigid [4]. That an ordinary MSDsystem of order 2 with these assumptions is a good approximation for a limited range wasvalidated for example by [6, 8, 17]. However, the complex mechanical thrust stand structuredoes not respond like a single-degree-of-freedom system for an unlimited range of excitationfrequencies. It will be discussed in Chapter 3 why efficient dynamic response compensationusing a system of order 2 is difficult to realise [8].

Figure 2-6: Analytical single-degree-of-freedom thrust stand model [4].

For a single-degree-of-freedom system, the external force is equivalent to the sum of thesystem’s reaction forces. Summing them up to determine the thrust is called the ReactionForce Summation Technique and was extensively adopted to solve the dynamic thrust problem[8], for example in [30].

The adequacy of any model formulation still has to be verified by correlation with an experimentally-determined system response. To obtain a transfer function for the system of order 2, thecontinuous Laplace transform is performed under the assumption that the coefficients areconstant and the initial conditions are all set to 0:

Ms2Y (s) + CsY (s) + kY (s) = X (s) . (2-2)

Rearranging shows that the transfer function of the system is

H (s) = Y (s)X (s) = 1

Ms2 + Cs+ k. (2-3)

Using a regression method, a system with two poles and no 0 can be fitted to this model.



2-2-2 Building a Distortionless Measurement System

The perfect thrust measurement system would be a device that returned an exact replica of thethrust force produced by a thruster without time delay, noise or modulation. However, this ispractically impossible to accomplish. First of all, there is an unavoidable time delay td betweenthe action of the thruster and the reaction of the thrust stand structure. Furthermore, themeasurand of the thrust stand system is not the thrust force per se, but an electrical signal.Because the electrical signal has a different scaling than the thrust force, the ideal thrustmeasurement system could at best be a system h(t) with a different scaling but constant gainh and only a short known time delay td:

Fthrust (t)h (t− td) = y (t) , with h (t) = h h = scalefactor (2-4)

When building a thrust stand, the designer should try to conform as closely as practicallypossible to the ideal requirements described in Equation (2-4). However, even systems withnearly ideal specifications are hard to construct. The reason why a measurement system asdescribed in Equation (2-4) is virtually impossible lies in the mechanical structure of a thruststand. Its mechanical components require the system h(t) to be frequency dependent. Inaddition, stochastic processes in the force transducer and electronics overlay the measurementsignal y(t). Hence, realistically, the thrust stand dynamics can be only described with arelation

Fthrust (ω)H (ω) = Y (ω) + ∆. (2-5)

where ∆ is a vector containing the measurement errors. Such a system can be achievedby Laplace transforming the time dependent measurement signals. The procedure will beexplained in detail in Section 2-3-1.

Systems with a similar behaviour to the ideal system described by Equation (2-4) can beachieved, if the lowest natural frequency of the thrust stand is about five times higher thanthe ‘highest appreciable amplitude Fourier frequency component of the input thrust’ [4]. Asa consequence, the design of a PMF measurement thrust stand strongly depends on the firingfrequency of the thruster.

Building a Distortionless Systems for PMF Measurements

Most displacement-type thrust stands in the literature are not suitable for PMF frequencieshigher than 50Hz [16]. To avoid resonance effects in dynamic thrust measurements, the naturalfrequency of the thrust stand should be about five times higher than the PMF frequency ofthe thruster [16]. Since PMF measurements reach frequencies up to 50Hz, a PMF thruststand should have no natural frequency below 250Hz. While it is generally difficult to builda thrust stand with a higher natural frequency than 200Hz, it is virtually impossible to builda displacement-type thrust stand with such a high first natural frequency [16]. Buchholz[31] describes a time-resolved thrust measurement system using a strain-gage-based forcetransducer, which had a resonant frequency of 62Hz. Haag [32] built two-torsional thruststands for pulsed plasma thrusters but as the natural frequencies of both stands were too low,the stands could only register the average level of the PMF thrust.

A thrust stand capable of registering time-resolved thrust and with a natural frequency ofabout 1KHz was built by Cubbin [17]. The stand measured the dynamic response of a



swinging arm using two-sensor laser interferometers. Most systems developed for PMF thrustmeasurements were realised using a piezoelectric load cell. Ren [15] designed a piezoelectricthrust stand with a natural frequency of higher than 1000Hz. Xing [16] described the designof a piezoelectric thrust stand with nearly ideal properties. The lowest natural frequencyof the thrust stand is 1245Hz and the relation between thrust input and output is almostlinear with a linearity error of less than 0.24% and a repeatability error of less than 0.25%.This thrust stand also shows excellent properties regarding the reconstruction of short pulses.For 3ms rising to maximum thrust time, 4ms on time and 10ms off time, the error is lessthan 5% [16]. Gao [7] specifically investigated the possibilities of piezoelectric thrust standdesigns and was able to design a system, where the lowest resonant frequency along the axialdirection is larger than 2000 Hz, far exceeding their original design requirement of 1000Hz.Figure 2-7 shows the frequency response of the PMF measurement platform designed by Gao.Piezoelectric sensors are ideal for measuring small, volatile forces, but cannot pick up steadyforces. Their quartz crystals only register an electrostatic charge when the force applied tothem changes. During a SSF measurement, the measurable charge will eventually leak to 0,no matter how good the electrical insulation is [33]. While displacement-type thrust standscan be used to measure both SSF and PMF thrust, a piezoelectric thrust stand can only beused for the measurement of PMF thrust.3

Figure 2-7: Frequency response of the PMF measurement platform designed by Gao [7].

2-2-3 Experimental Determination of the System Response

In an experimental solution of the dynamic thrust measurement problem, the thrust force ismeasured using a thrust stand with not ideal properties. The effect of the thrust stand isfiltered from the measurement signal. For this purpose, the thrust stand is regarded as a block-box system as shown in Figure 2-8. The input to the system is the rocket engine thrust forceand the output is the analogue electric signal produced by the load cell. The electric signalcan be converted to a digital signal with an Analogue-Digital Converter. Both the electricand digital signal contain the thrust signal as well as the system response of the thrust standand measurement errors. Hence, to reconstruct the thrust force either an analogue or a digitalfilter has to be designed and joined up in circuit. Since the principal purpose of this filter is tocompensate the transient response, one can call it a thrust compensation filter. The process

3 More information on the comparison of strain gages and piezoelectric crystal can be found in [34, 35].



of removing the transient response from the measurement signal of liquid rocket engines issometimes called ‘thrust compensation’ in the literature, see for example [8, 18], and it is alsothe way it is referred to at Airbus DS.

Figure 2-8: Thrust stand system with electronic noise εy.

To design a digital filter which is able to compensate the dynamic system response in a thrustmeasurement, a more mathematical definition of the behaviour of the thrust stand is needed.Using system analysis, the thrust stand is regarded as a black box system with unknowninput and known output signal. Then, it is established that the thrust stand behaves like aLinear Time-Invariant (LTI) system. A system is classified as linear if its input and outputsignal are related by a linear mapping. The necessary requirements for a linear mapping areadditivity and homogeneity. Both requirements hold if one can show that

ay1 (t) + by2 (t) = h (ax1 (t) + bx2 (t)) . (2-6)

A system is time-invariant if only input and output but not the system change over time.Therefore an input x(t) will result in an output y(t), irrespective of the point in time, that isx(t+ δ), results in y(t+ δ) [26]:

y (t+ δ) = h (x (t+ δ)) . (2-7)

Whether a system behaves like a linear system can empirically be established: for examplewhen the amplitude of the input signal is doubled, the amplitude of the output doubles aswell. The system is time-invariant if the impulse response is the same irrespective of thetime. As the output signal is obtained from the input by convolution, the input signal canbe calculated from the output signal by performing the inverse operation of convolution. Theprocess of estimating the input signal from the output signal by correcting the effects of theinstrumentation and sensing system is called deconvolution [36].



2-3 Digital Deconvolution Filters

For an LTI system the relation between output y (t) and input signal x (t) is defined byconvoluting the input signal with the transfer function h (t):

y (t) =∫ +∞

−∞h (t− τ)x (τ) dτ =

∫ +∞

−∞h (t)x (t− τ) dτ. (2-8)

If, like in case of the thrust measurements, the data are only discretely available, the relationbetween input and output signal is described by a discrete convolution:

y [n] =∞∑

k=−∞h [n− k] · x [k], (2-9)

where n stands for the sample at time T = ndt, x[n] is the input thrust, y[n] is the measure-ment signal, and h [n] is a system model of the thrust stand at time n. The system modelh [n] and input thrust x[n] are unknown, while the measurement signal y[n] is known, butspoiled by measurement errors.

2-3-1 Deconvolution

Deconvolution is defined as the process of finding a solution x(t) to the convolution equation

y (t) =∫ +∞

−∞H (t) ∗ x (t− τ) dτ, (2-10)

whether in continuous form like in Equation (2-8) or in discrete form as in Equation (2-9) [36].The convolution equation can mathematically be classified as an inhomogeneous Fredholmintegral equation of the first kind. Solving an inhomogeneous Fredholm integral equation ofthe first kind is not trivial and commonly avoided in Digital Signal Processing (DSP) due tothe rather complicated mathematics involved. In addition, it has not yet been shown thatpurely numerical approaches are more accurate than commonly used filtering approaches [37].One possible advantage is that recently much progress has been made in the developmentof parallel algorithms for integral equations, so that a direct numerical method becomespreferable in terms of computation time. A fast and parallelised solver for integral equationswith convolution-type Kernel was proposed by Ye and Zhang [38].Instead of solving Equation 2-10 numerically, the typical approach in DSP is to transformthe convolution equation into the frequency domain using a Laplace transform. The Laplacetransform is a mostly bijective integral transform of a function. For a continuous functionh(t), it can be defined as

L {h (t)} =∫ ∞

0h (t) e−stdt = H(s), (2-11)

where t is the continuous time variable and the function space of h(t) is called the timedomain; s is the complex parameter s = σ + iω and the function space of H(s) is calledthe frequency domain. For the Fourier transform, s is evaluated at the imaginary arguments = iω. The Fourier transform can be applied to determine the frequency spectrum of adynamical system.


2-3 Digital Deconvolution Filters 17

The z-transform is the discrete equivalent of the Laplace transform. For a discrete functionh[n] it can be defined as

Z {h} =∞∑

n=−∞h [n] z−n = H(z), (2-12)

where the square brackets indicate the discreteness of the variable n [39].

Using the property of the Laplace and z-transform that, in the time domain, convolutionbecomes multiplication in the frequency domain, instead of solving

y (t) =∫ t

0h (t− τ)x (τ) dτ (2-13)

for x(t), one can Laplace transform Equation (2-13) and obtain

Y (s) = H (s) ·X (s) , (2-14)

so that X(s) can be obtained by a division in the Laplace domain:

X (s) = Y (s)H (s) , (2-15)

if Y (s) and H(s) are fully and accurately known. Unfortunately, for thrust measurements,Y (s) is affected by measurement noise and H(s) is unknown and can only be determinedusing the measurement signal.

2-3-2 Ill-Posedness of the Deconvolution Problem

To demonstrate the effect of errors on the solution of the deconvolution problem, let Ym(z)and Hm(z) denote the measurement signal and the identified transfer function, respectively.4As the signals are discrete, the z-transform is used. Using the flawed signals, the estimateXm(z) of the thrust input is determined:

Xm (z) = Ym (z)Hm (z) (2-16)

Since the true signal is hidden underneath the coat of errors, the measured signal is be splitinto a correct and an error part:

Ym (z) = Y (z) + Yε (z)Hm (z) = H (z) +Hε (z) , (2-17)

where H (z) and Y (z) are accurate and Hε (z) and Yε (z) contain the errors.

The accurate input X(z) can be calculated once the errors are known by rearranging Equation(2-17) as

X (z) = Ym (z)− Yε (z)Hm (z)−Hε (z) . (2-18)

4Note that while h[n] is a discrete function its z-transform H(z) is continuous.



In regions where Hm is as small as the errors contained in the signal, the denominator isclose to 0 and the input becomes very large. A small change in the data can trigger a largechange in the results. This is the textbook definition of an ill-conditioned or ill-posed prob-lem. ill-conditioned or ill-posed problems can be stabilized by using regularization methods[36]. The ill-posedness is also the reason why purely mathematical approaches are difficult.Nevertheless, Maleknejad [40], for instance, overcomes the ill-posedness of the deconvolutionproblem by using wavelets and the collocation method to solve the Fredholm integral equationand even obtains accurate error bounds for the solution.

In practical situations, the bandwidth of the measured signal is always finite and has regionsin which H(z) is fairly small. Practical deconvolution is thus inherently unstable [41]. Even ifthe errors are small enough to avoid the solution blowing up, the reconstructed signal is likelyto contain unrealistically high frequency oscillations. For this reason, every deconvolution hasto be combined with a suitable low-pass filter. When adding the low-pass filter it has to bedecided which range of the signal is relevant and which merely amplifies noise. It is importantto realise at this point that deconvolution is an estimation problem, and not a problem towhich an exact solution exists. This means that it is not possible to perform deconvolutionwithout subjective judgement. The filter designer has to decide, whether the found estimateis acceptable.

Summed up, a digital deconvolution filter hc corrects the output signal of a sensor for dynamiceffects by employing deconvolution. The input signal x can be estimated by convoluting thefilter h with the output signal y: _

x [n] =(hc ∗ _

y)

[n] where _y [n] is the sensor output signal at

time instant nTs after an A/D conversion with error ε [n] = _y [n]−y [nTs] where n ∈ N specifies

the sample and Ts = 1/fs denotes the used sampling rate [26]. Practical deconvolution isinherently unstable so that the compensator has to be complemented with a low-pass filter.

Figure 2-9: Every deconvolution filter is a combination of compensation filter and low-pass filter.

2-3-3 Thrust Compensation Filters

The documentation of digital filters specifically designed for the purpose of thrust compen-sation available in freely-available publications is rather limited, because research in thisparticular direction is mostly done by governmental research institutions or companies pro-tecting their intellectual property. The easiest approach to find a thrust compensation filter isto assume that the errors are small compared to the thrust force. In case ε� FT , also Yε ≈ 0and Hε ≈ 0 so that the measurement signal and the transfer function, which is identified fromit, can be assumed to be fairly consistent with their unknown correct values:

hm [n] ≈ h [n] ym [n] ≈ y [n] . (2-19)



In this case, an acceptable estimation of the solution is provided by:

X (z) ≈ Ym (z)Hm (z) . (2-20)

If this is not the case, a stable filter imitating the behaviour of the inverse transfer functionH−1 as closely as possible has to be found. The common procedure is to try to find an inversecompensation filter HC(z), such that the product of filter and compensation filter becomesone:

H (z)HC (z) = 1. (2-21)

A design according to Equation (2-21) is only feasible, if there is no time delay between inputand output signal. If there is a delay n, then a compensation filter satisfying

H (z)HC (z) = z−n (2-22)

has to be found. If the requirements to treat the thrust stand as a system of order 2 (describedin Section 2-2-1) apply, a compensation filter HC (z) based on the ideal transfer function oforder 2 can be designed:

HC (z) = C

H (z) = C(z2 + 2ζ0ω0z + ω2

0

), (2-23)

where C is a constant. The resulting filter is unstable, if its zeros lie outside the unit circle.Due to the ill-posedness of the problem, the filter is likely to amplify high frequency noise.One should therefore try to choose a denominator polynomial that creates a system witha high natural frequency. Placed in the denominator it will then have the reverse effect ofattenuating high frequencies.

Deadbeat Compensator Developed by Chen

The approach of Equation (2-23) is called the ‘construction of an ideal transfer function’ byChen [8]. He criticises this method for its limitation in bandwidth and suggests three differ-ent filters with a broader bandwidth and better noise attenuation instead: the non-deadbeat,deadbeat and two-deadbeat compensator. Chen shows that the accuracy of a thrust compen-sation filter can be influenced by changing the time delay caused by the compensator HC(z).If a filter causes a time delay of one sample they call it a deadbeat compensator, of two samplesa two-beat-delay or two-deadbeat and a combination of the two a non-deadbeat digital com-pensator. Comparing deadbeat filters with inverse filters without time delay, Chen [8] showsthat by foregoing a small amount of response speed, white noise is better attenuated by thenon-deadbeat digital compensator (of which the two-beat-delay and deadbeat compensatorsare the specific cases) Figure 2-10. So far only the design of filters specifically designed forthe purpose of dynamic thrust compensation have been discussed. In the following section,the concept is extended to the general design of digital filters.



Figure 2-10: White noise cancellation performance of deadbeat (top) and non-deadbeat (bottom)digital compensator [8].

2-3-4 Low-Pass Filter

The ideal low-pass filter to complement a thrust deconvolution filter should be designed so thatnone of the appreciable amplitude frequency components of the input thrust are amplitudeor phase-distorted [4]. To that end, the passband should be as flat as possible, the transitionband should be short to provide a clean cut-off, and the stop band should guarantee no noiseamplification, as depicted in Figure 2-11.

Figure 2-11: Ideal frequency response for a thrust compensation low-pass filter according to [4].

The phase response should be linear to prevent phase distortions. If ω is the cut-off fre-quency separating the relevant from the noisy frequency components, the ideal filter can be



characterised asHlow (ω) = cons ω ≤ ωHlow (ω) = 0 ω > ω

. (2-24)

The characteristics of the passband, transition and stop band are determined by the type oflow-pass filter used. The cut-off frequency has to been chosen dependent on the width of thetransition band and frequency spectrum of the thrust signal.

Common Low-Pass Filter Types

A low-pass filter is designed to extract a part of the entire frequency band of a signal andattenuate all other frequencies. As its name suggests, it extracts lower frequencies and at-tenuates higher frequencies. The ideal low-pass filter for thrust compensation should have afrequency response similar to the one characterised in Figure 2-11. The most commonly usedlow-pass filters in digital signal processing are Elliptic, Chebyshev and Butterworth filters.Their effect on a signal can be visualised by regarding their frequency response. An exemplaryfrequency response for the Elliptic filter is shown in Figure 2-12. It can be seen that Ellipticfilters have ripples in both their stop and pass band. If the number of ripples in the stopband nears 0, the filter becomes a Chebyshev type I filter. If the number of ripples in thepass band nears 0, the filter becomes a Chebyshev type II filter. If the number of ripples inboth bands approach 0, the filter becomes a Butterworth filter.

Figure 2-12: Schematic frequency response of an elliptic low-pass filter.

The ideal low-pass filter for thrust compensation can under no circumstances have ripples inthe passband, because a flat passband is essential to preserve the correct thrust amplitude.Hence, the Elliptic and Chebyshev type I filter can be ruled out. Only Butterworth andChebyshev type II filters have a flat passband. Butterworth filters have a slower roll-off thanChebyshev filters, but in the pass band, they have a more linear phase response. On thedownside, Butterworth filters need a higher order to achieve the same stop band attenuationobtained with a Chebychev filter. Both filters produce overshoot at the signal edges, as canbe seen in Figure 2-13, but the overshoot of the Chebyshev type II filter is larger for the sameorder. A high stop band attenuation and fast roll off are not as important as a maximallyflat passband, smaller overshoot and a more linear phase response in the pass band. Thus, ofthe common low-pass filter types, the Butterworth filter is the most suitable one for thrustcompensation.

Crosswy suggests an analogue Bessel filter for thrust measurements, because it does notproduce overshoot [4]. However, it produces blunter edges and slower rise times. Therefore,Crosswy recommends the use of a Butterworth-Thomson filter to combine the amplitude



characteristics of the Butterworth filter with the linear phase of the Thomson filter, which hasmaximally flat time delay, but a poor rise time and amplitude characteristics [42]. However,he has an analogue filter in mind, since the Bessel filter is not intended for digital use.Furthermore, it is very difficult to get access to the paper referred to by Crosswy.

Figure 2-13: Comparison of the amplitude characteristics of Butterworth of order 3, Chebychevand digitalised Bessel filter.

Nevertheless, he mentions another important property the low-pass filter should have forthrust compensation: fast rise and fall times. If the edges of the thrust pulse are to berepresented accurately, the filter should not smooth them too much. This is another argumentin favour of the Butterworth filter and is demonstrated in Section 3-5-4.

Cut-off Frequency

The low pass filter cut-off frequency should chosen such that most of the signal is kept butstill the noise is attenuated. A high cut-off frequency will minimise the error caused by amodulation of the relevant frequencies, but increases the error caused by noise amplification.A low cut-off frequency reduces the signal noise but distorts the signal and increases theestimation error. This trade-off is mostly done by a trial-and-error method in engineeringapproaches. There have been attempts to find automatic methods, but they are not whollyreliable and limited in their applicability to different cases. One approach is suggested byEichstädt [26]. They suggest the formulation of an optimisation problem by minimising the


2-4 IIR Deconvolution Filters 23

root-mean-squared (RMS) error r[n] to determine the optimal cut-off frequency:

r [n] =

√√√√ 1K

K∑k=1

(xk [n]− x [n])2 for k = 1, ..,K, (2-25)

where xk [n] is 1 of K estimates obtained for the input signal x using different cut-off fre-quencies. This approach can only be used if the true input signal is known. For the dynamicthrust measurement problem, the input signal is unknown, so that the approach can only beapplied on test cases with known solution.

2-4 Infinite Impulse Response (IIR) Deconvolution Filters

Airbus DS only uses IIR filters. Therefore, the focus of this thesis is also on IIR filters.An IIR filter is essentially a rational polynomial model which is applied to correct a signal.The determination of the polynomial coefficients for a system like the thrust stand fromits input and output signal is referred to as system identification. It can be done with anautomated iterative procedure if an appropriate model structure (like Autoregressive (AR),Moving Average (MA) or ARMA defined in Section 2-4-2) and the polynomial order of themodel are specified. The usual procedure to find a suitable model for a problem is to fitmany different models to the data. The standard method to find an optimal model for a LTIsystem is linear least squares regression and is explained in Section 2-4-3. To determine thebest model a goodness of fit statistic like Normalised Root Mean Square (NRMS), which isdefined in Section 2-4-5, can be used. Once the system has successfully been identified, themodel can be used to predict, analyse or filter the behaviour of the system.

2-4-1 Finite Impulse Response (FIR) and IIR Filters

Analogue and digital filters are classified according to how long they respond to a unit impulse:finite or infinite. For digital filters, this classification is only theoretical, because infiniteresponses also become finite when they are truncated by the digital floating point arithmetic.Digital FIR and IIR Filters are better distinguished by how they compute an output. FIRfilters express each output sample as a weighted sum of the last N input samples. IIR filtersexpress each output sample as a linear combination of the previous N input and M outputsamples, meaning that they use feedback. Any LTI system can be approximated arbitrarilywell with a FIR filter when an adequate time delay is introduced [43] and any FIR filter canbe approximated by a stable IIR filter [44].



Figure 2-14: Illustration of the working principle of FIR and IIR filters. IIR Filters take feedbackinto account and can therefore be unstable.

IIR filters are the digital equivalent of analogue filters and can be analysed using traditionalfilter design techniques. When an analogue filter is to be replaced with a digital filter, an IIRfilter can be designed based on the previous analogue filter design. The usage of a feedbackloop allows IIR filters to solve a problem with a significantly lower order than a FIR filterwould require. For IIR filters, the use of higher orders implies the possibility of instability andarithmetic overflow. Arithmetic overflow can be avoided if the order 10 is not exceeded [37].Instability can also occur for lower orders. The phase of IIR filters is a nonlinear function offrequency and can cause distortions.

FIR filters do not employ feedback, so they are naturally stable. If their coefficients arechosen symmetrically, they possess a linear phase which means that signals of all frequenciesequally delayed by them and the resulting signal is not distorted. Their main drawback isthat they require considerably higher orders and longer processing time than IIR Filters forthe same performance. IIR filters are harder to design than FIR filters. Table 2-1 shows thatthe pros and cons of FIR and IIR filters are rather well balanced. It mostly depends on theapplication, which filter is preferable.

FIR Filters IIR Filters+ unconditionally stable analogue equivalents

linear phase response more efficient– no analogue equivalents possibly unstable

less efficient nonlinear phase response

Table 2-1: Comparison of the key properties of FIR and IIR Filters.

2-4-2 Autoregressive-Moving-Average Models

ARMA models combine a MA model with an AR model. They are called autoregressive,because a regression is performed on the time series itself (auto). An ARMA model is appro-priate when a system is a function of a series of random shocks, described by the MA part,as well as its own behaviour, described by AR part [45].



Conventionally, ARMA models are applied to model the physical persistence of a randomtime series, where the random process is more dominant than the physical. When an ARMAmodel is used to model a thrust stand, the physical process is more dominant. The transientresponse is modelled by the AR part and measurement errors by the MA part. Essentially, anARMA model is based on the same equations as an IIR filter. The close connection betweenARMA models and IIR transfer function, can be demonstrated by considering a z-transformedtransfer function H (z) of an IIR filter:

H (z) = Y (z)X (z) =

z−dNa∑j=0

ajz−j

Nb∑i=0

biz−i=z−d

(a0 + a1z

−1 + ...+ aNaz−Na

)1 + b1z−1 + ...+ bNb

z−Nb, (2-26)

where z−d includes the possibility of a time delay. Rearranging Equation (2-26) to includethe input and output signal, gives:

Y (z)Nb∑i=0

biz−i = X (z) z−d

Na∑j=0

ajz−j . (2-27)

Taking the inverse z-transform of Equation (2-27) results in the discrete finite differenceequation used for describing ARMA processes:

Nb∑i=1

biy (k − i) =︸︷︷︸AR

Na∑j=0

ajx (k − d− j)

︸︷︷︸MA

, (2-28)

where the left-hand side stands for the AR, the right-hand side for the MA part and drepresents the delay between input and output signal. Taking the term for i = 0 out of theleft sum, results in the difference equation (2-29). This illustrates that in discrete form, thecurrent output value of y(k) depends on the n past values of the output y(k − i) and m+ 1values of the input x(k − d− j):

y (k) = −Nb∑i=0

biy (k − i) +Na∑j=0

ajx (k − d− j). (2-29)

The information contained in a discrete rational transfer function can be described by adifference equation and an ARMA model, just like the information contained in a continuoustransfer function can be described by a differential equation, as shown in Section 2-2-1.

2-4-3 Discrete-Time Linear Least Squares System Identification for IIR Filters

The time domain identification approach described in this section is one of the most elemen-tary approaches to start an IIR filter design. The coefficients, obtained with this approach,can be used directly for the design of a stable IIR filter or as basis for the design of an FIRfilter. In the following, it is assumed that a time delay between input and output signal hasalready been eliminated by an appropriate shift of input signal. A method to design an IIR



filter, which can automatically determine the existing time delay can be found in Vuerinckx[46]. Applying a linear least squares regression to an IIR transfer function means that thecoefficients of a rational polynomial, relating input to output signal, are determined in sucha way that the error between model and measurement is minimised. The coefficients of anIIR transfer function H(z) can be described by the discrete finite difference equation

y (k) = −Nb∑i=0

biy (k − i) +Na∑j=0

aju (k − d− j), (2-30)

where k = 1, ...,K andK is the number of samples, n andm are the orders of the denominatorand enumerator polynomials, respectively. To find the closed form solution to the linear leastsquares problem in the time domain, a system of equations is set up. For this purpose,Equation (2-30) is at first considered only for sample k. The input values uk and outputvalues yk are summarised in one vector xk:

x (k − 1) = [y (k − 1) y (k − 2) y (k −Na) u (k − 1) u (k − 2) ... u (k −Nb)]T (2-31)

and their corresponding coefficients are summarised in a vector θ

θ = [−a0 − a1 ... − aNa b0 b1 ... bNb]T . (2-32)

Regarding the problem for k = 1, ...,K where K is the total number of samples considered,Equation (2-30) can be reformulated in matrix form:

y (1)y (2)...

y (k)

︸︷︷︸

Yk

=

y (0) · · · y (1−Na) u (0) · · ·y (1) · · · y (2−Na) u (1) · · ·...

y (k − 1) y (k −Na) u (k − 1)

u (1−Nb)u (2−Nb)

u (k −Nb)

︸︷︷︸

XTk

a1 (k)...

aNa (k)b1 (k)

...bNb

(k)︸︷︷︸θk

.

(2-33)Summarising all vectors xk for k = 1, ...,K in the matrix Xk, Equation (2-33) can be abridgedto

Yk = XTk θk. (2-34)

where the hat indicates that θ is an estimate. Equation (2-34) has no unique solution. Thenumber of measurement points K usually exceeds the sum of numerator Nb and denominatororder Na, so that Equation (2-34) is an overdetermined system of equations. Hence, the prob-lem must be reformulated as an optimisation problem. For this purpose, the approximationerror produced by the overdetermined systems is added to Yk in Equation (2-34):

Yk + εk = XTk θk. (2-35)

This estimation error can be expressed as

εk = XTk θk − Yk. (2-36)



Because there is no unique solution to Equation (2-33), the problem is transformed into aminimisation problem by taking the square of the errors:

εTk εk =k∑l=i

[yi −XT

i−1θk]2. (2-37)

Since this is a quadratic equation in θ, the error function is convex and its minimum can befound where the derivative is 0:

∂εTk εk

∂θk= 0. (2-38)

The first derivative of Equation (2-37) is

− 2XkYk + 2XTk Xkθk = 0 (2-39)

and the second derivative is∂2εTk εk

∂θ2k

= 2XTk Xk > 0. (2-40)

That the second derivative is larger than 0 means that there is a unique minimum of the errorsquares function at

θk =(XTk Xk

)−1XTk Yk. (2-41)

If Xk is a square invertible matrix, the pseudo-inverse equals the inverse, so that an estimatefor the coefficients can be found with

θk = XTk Yk. (2-42)

In order for the closed form solution of Equation (2-41) to exist, all input and output valuesYk and Xk should be known. The optimal solution to the minimisation problem in Equation(2-41) or Equation (2-42) is not necessarily the best solution to the problem at hand. Thesystem may not be perfectly linear. In this case, a nonlinear least squares approach has to beused instead. In addition, the measurement signal is affected by noise. The presence of noiseis not so problematic if it is zero mean white Gaussian, but non-Gaussian noise will reducethe quality of the obtained estimate. Gaussian white noise can be defined as a sequence ofrandom uncorrelated variables αt with finite variance and zero mean:

E (αt) = µavar (αt) = σ2

a

cov (αt, αt+k) = 0 fork 6= 0. (2-43)

Gaussian white noise approximates many real-world situations, like measurement errors, fairlywell and generates mathematically tractable models [39].

For the deconvolution problem, the input signal Xk is unknown. This complication can besolved by generating an idealised input, such as an impulse, pulse or trapezium, for the inputsignal and inserting it as an approximation for Xk in Equation (2-41).



2-4-4 Stabilisation of IIR Filters

Unlike FIR filters, IIR filters are not inherently stable and their stability has to be ensured[47]. For a stable IIR transfer function, all poles have to be either in the left-half plane of thecomplex plane if the function is continuous, or within the unit circle if the function is discrete.The stability of an IIR Filter can be ensured using either of the four following methods:

- cancelling unstable poles with zeros [4, 8]- reflecting unstable poles into the unit circle see Section 2-4-4 or [46]- asynchronous time reversal filtering, see [48, 49]- decomposition into an all-pass and minimum-phase system, see [46, 47]

In this thesis, the reflection of poles outside the unit circle into the unit circle will be usedfor stabilisation of IIR filters.

Reflection of Poles Outside the Unit Circle

A discrete digital filter is stable when all poles of its z-domain transfer function lie within theunit circle. The unit circle serves as stability boundary. It is possible to stabilise a discretetransfer function by reflecting the zeros of the denominator polynomial, that is to say, thepoles, into the unit circle by inverting their radius [50]:

rp exp (±jθp)→1rp

exp (±jθp) , (2-44)

where rp is the radius and θp is the angle of the unstable pole. While this procedure preservesthe amplitude, it modifies the phase characteristics of the signal:

Astab (ωTs) = 1A(ω) φstab (ωTs) 6= −φ (ω) , (2-45)

where Astab is the amplitude and φstab is the phase of the stabilised transfer function Hstab[z].If the phase becomes strongly nonlinear, the signal becomes distorted [50]. The nonlinearphase can be corrected either by linearising it [51] or with an all-pass filterHall(z) of amplitude1 [52]:

Hcomp (z) = Hstab (z)Hall (z) . (2-46)

2-4-5 Approximation Error and Error Statistics

An effective least squares system identification requires the choice of a suitable model order.The frequency response is a useful tool to find the order. If one can restrict the identificationfrequency region to the bandwidth of the input signal, a lower filter order can be chosen.For thrust stands, it was explained in Section 2-2-1 that a rational polynomial of order 2provides a good analytical approximation. One can thus safely assume that a IIR filter oforder 2 will perform comparatively well. Nevertheless, a higher identification order can stilllower the approximation error at the cost of higher computational effort. This works only upto the point where a further increase of the identification order only integrates signal noise



into the model; or, the increase in numerical errors becomes larger than the improvementin estimation. The quality of the least squares fit can be determined either with an errorstatistic or visually, by looking at the reconstructed input signal. A reliable error statisticis the reprojection error. The measurement signal is re-calculated using the model and theapproximation error is determined as difference between model and true value. The reprojec-tion error is especially suitable for a visual comparison of the error. For a one-dimensionalthrust signal, the reprojection error can be defined as

εrp = ‖y − y‖ with y =∫ t

0H (t− τ) x (τ) dτ, (2-47)

where H (t− τ) is a transfer function and x is the deconvolved input thrust. Another wayto visually evaluate the performance of a deconvolution filter in the time domain is to applythe filter to a problem with known solution. To create such a synthetic test case, one takesa known input signal, for example a thrust pulse of uniform amplitude, convolutes it witha transfer function resembling the thrust stand dynamics and adds white Gaussian randomnoise. A visual evaluation in the time domain has the advantage that the designer can seehow the filter affects the reconstructed input signal, so that an incorrect time delay, a bias inthe thrust amplitude, or an inappropriate cut-off frequency (too much random noise in theend product) can all easily be seen. If a test scenario is used, the NRMS error values providesa good error statistic. The NRMS error is a normalised version of the commonly most usedaccuracy measure: the Root Mean Square (RMS) error.

εNRMS =

1− ‖x− xref‖∥∥∥∥x− 1n

n∑i=1

xref (i)∥∥∥∥ · 100 (2-48)

where ‖.‖ indicates the 2-norm, x is the deconvoluted estimate of the input signal and xref isthe prescribed thrust input. For the NRMS statistic, a 100% fit means that the new functionfits the original perfectly, 0% means that the new function is not better at approximatingthe original function than a straight line, and −∞ means a very bad fit. In this thesis, thereprojection error of reconstructed thrust input signals is used for visual comparison and theNRMS error for statistical comparison.

An upper bound for the time domain estimation error of a deconvolution filter was reportedby Eichstädt [21, 26] and is

|∆ [n]| ≤ 12π

∫ πfs

−πfs

∣∣∣eiωnd/fsHC

(eiω/fs

)H (iω)− 1

∣∣∣ · X (ω) dω, (2-49)

where ∆ [n] is the estimation error at sampling nTs, and X (ω) is an known upper boundfor the magnitude spectrum of the input signal. This error bound is especially suitable fordeconvolution filters, as it takes the low-pass filter into account: if the low-pass filter has abroad pass-band, the error is primarily influenced by the compensation filter. While Equation(2-49) provides an upper boundary for the compensation error, it does not evaluate to whatextent estimation and measurement errors degrade the accuracy of the deconvoluted inputsignal. This can be done by performing an uncertainty evaluation.



2-5 The Kalman Filter

2-5-1 Advantages of the Kalman Filter for the Dynamic Thrust MeasurementProblem

It was shown in Section 2-3-1 that the deconvolution problem is ill-posed and that this leadsto an amplification of the signal noise. The conventional way to deal with this problem isto design an IIR or FIR filter and to complement it with a low-pass filter to prevent noiseamplification, as described in Section 2-4 and 2-3-4. Both filter types are time-consuming todesign and require a secure foundation in DSP, such as knowledge of pole-zero plots, frequencyresponse and frequency spectra. IIR filters are especially difficult to design, because they canbe unstable and stabilisation is difficult to perform without destroying the phase, see Section2-4-4. The Kalman Filter is known for its simple design and intuitive algorithm. It doesnot require stabilisation. It improves the signal-to-noise ratio by taking the statistics of themeasurement and process noise into account. It provides the optimal solution for a LTIsystem if the noise is white Gaussian. Its only disadvantage is that the covariance matricesof the measurement and process noise have to be provided by the operator. Otherwise, itoffers an innovative solution to all the problems commonly encountered when designing afilter for dynamic thrust compensation. Thus, the Kalman Filter offers an ideal way to solvethe dynamic thrust measurement problem, if it can be adapted to deconvolute the thrustsignal from the transient response of the thrust stand.

IIR Filters Kalman Filter+ no choice of covariance matrices required optimal for white Gaussian noise

improves signal-to-noise ratiostable, even with unstable IIR coefficients

simple designsimple implementation

integrated uncertainty analysis– not optimal requires choice of covariance matrices

worsens signal-to-noise ratiounstable

complicated designcomplicated implementation

additional uncertainty analysis

Table 2-2: Comparison of the properties of IIR and Kalman Filters.


2-5 The Kalman Filter 31

The Kalman Filter was published by Rudolf E. Kalman as a recursive solution to the discretetime linear estimation problem [53]. Its first major application was in the on-board computerthat guided the descent of the Apollo 11 lunar module to the moon. The idea behind theKalman Filter is that of a two step predictor-corrector method: a state is predicted using amodel for the system dynamics based on previous steps only. This estimate for the next step isthen corrected, once the measurement outcome is known, by forming a probabilistic weightedaverage between the measurement outcome and the prediction, as illustrated in Figure 2-15.

Figure 2-15: Illustration of the working principle of the Kalman Filter. The next step of thesignal αk+1,k+1 is estimated by combining the probability of the prediction of the next step gainedby a model αk+1,k and the actual measurement yk+1.

The Kalman Filter’s typical role within the realm of dynamic force measurements is that ofa noise cancellation filter, and for good reason: if the noise is white Gaussian, as it usuallyis for measurement signals, the Kalman Filter is optimally suited for the task. Haddab [54],for instance, applies the Kalman Filter to smooth a micro-force measurement signal obtainedby a strain gage, before applying a deconvolution filter. In the following, a more unusualuse of the Kalman Filter is suggested: a modification to perform the deconvolution itself.A few simple adjustments are sufficient to make the Kalman Filter suitable for this task.Successful attempts to use the Kalman filter for deconvolution were already made in otherfields than thrust measurement. Sayman [55] uses a Deconvolution Kalman filter to recoverseismic traces, Rotunno [56] for ion quantisation and Rousseaux [57] describes the retrievalof the time-varying active states in a muscle with a Deconvolution Kalman Filter. Bora [9]reports that the signal-to-noise ratio could be improved significantly using a DeconvolutionKalman Filter, although deconvolution filters in general have the tendency to reduce thesignal-to-noise ratio. Furthermore, Bora was even able to reconstruct an input signal ofcomplete white Gaussian random noise after a convolution with a fifth order transfer functionusing a Deconvolution Kalman Filter. Especially this last result makes the Kalman Filtera promising candidate to extract even the sharper edges of short firing pulses overlaid withGaussian firing noise.

2-5-2 State-Space Representation

The state-space representation of a LTI system is a time-domain approach, in which a differ-ence equation of order N is rewritten as a vector difference equation of order 1. The statevariables constitute the smallest possible subset of variables by which the entire system can



be described. For a system of m inputs, n states and r outputs, the state-space representationcan be defined as:

α (t) = Aα (t) +Bx (t)y (t) = Cα (t) +Dx (t) , (2-50)

where α(t) is the state, x(t) is the input and y(t) the output signal. The matrix A (sizen× n) is the system matrix, B (size n×m) is the input matrix, C (size r× n ) is the outputmatrix and D (size r ×m) is the called the feedthrough matrix, because it represents directconnections between input and output signal [58] .The connection between the state-space model and the differential equation can be illustratedusing the analytical model of order 2, used to describe the thrust stand In Section 2-2-1, thedifferential equation (2-3) was transformed into a transfer function:

H (s) = Y (s)X (s) = 1

Ms2 + Cs+ k⇔ My (t) + Cy (t) +Ky (t) = x (t) . (2-51)

To transform the differential equation (2-51) into a state-space representation, the derivativesof y are substituted with components of a vector. In this case, two equations of order 1 replacea differential equation of order 2:

α1 (t) = y (t)α2 (t) = dy(t)

dt

. (2-52)

The derivatives of this state vector areα1 (t) = dy(t)

dt

α2 (t) = − CM y2 (t)− K

M y1 (t) + x (t). (2-53)

The two equations (2-53) can be written in matrix form as[α1 (t)α2 (t)

]=[

0 1−KM − C

M

]︸︷︷︸

A

[α1 (t)α2 (t)

]+[

01M

]︸︷︷︸

B

x (t) , (2-54)

which, introducing the matrix A and B can be summarised as

α (t) = Aα (t) +Bx (t) . (2-55)

The output equation is given byy (t) = α1 (t) , (2-56)

which can also be expressed in matrix form as

y (t) =[

1 0]

︸︷︷︸C

[α1 (t)α2 (t)

]+ [0]︸︷︷︸

D

x (t) , (2-57)

which is in shorty (t) = Cα (t) +Dx (t) . (2-58)

The matrix D represents direct connections between the output and the input feedthrough.Equation (2-55) and (2-58) together represent the commonly used short form of a linearstate-space model:

α (t) = Aα (t) +Bx (t)y (t) = Cα (t) +Dx (t) . (2-59)

The Kalman Filter is a particular algorithm that is used to solve state-space models in thelinear case. Occasionally, state-space models are even referred to as Kalman Filter models.



2-5-3 Kalman Filter Algorithm

The Kalman Filter is famous for its straightforwardness. Kalman himself declared in later lifethat the simplicity and intuitiveness of the final equations (2-91) had surprised him. For theKalman Filter, a set of simple equations is the result of a series of simplifying assumptions. Inthe following, the assumptions made during the derivation of the Kalman filter are identified.The general problem addressed by the Kalman Filter is that of estimating the state α ofa discrete-time controlled process that can be described by the linear stochastic differenceequations (2-60) and (2-61):

αk+1 = Aαk +Bxk + wk, (2-60)

yk = Ckαk + vk, (2-61)

where the subscript k denotes the sampling of the values at time tk = k∆t. The matrix A indifference Equation (2-60) relates the state at time step k to the state at step k + 1. MatrixB relates the control input x to the state α and matrix C in Equation (2-61) relates the stateα to the measurement signal y. For further reference, Equation (2-60) will be referred to asstate equation and Equation (2-61) as measurement equation. Essentially, the state equation(2-60) contains the model which is used to predict the behaviour of the next measurementvalue. The states and the measurements are corrupted by additive noise: the process noisewk and measurement noise vk, respectively. Both are usually assumed to be mutually uncor-related (E [wkvk] = 0), independent, and white Gaussian, because the Kalman Filter yieldsoptimal results for this case.

For the assumption of white Gaussian noise with zero mean, the probability distribution ofthe process and measurement noise can be defined as

p (w) ∼ N (0, Q)p (v) ∼ N (0, R) . (2-62)

where QA and R are the covariance matrices of the process and measurement noise, re-spectively. The prediction and update state essentially describe the same value of xk, manyauthors mark the prediction with a minus superscript to set it apart from the correction.In the following, however, a two index system will be used. The first index indicates theiteration step, while the second designates the availability of measurement data. This indexsystem is more conform with the typical notation for random processes and is more suitablefor probabilistic proofs and derivations. In this notation, the predicted (a priori) and thecorrected (a posteriori) estimates can be written as:

- αk,k−1 denotes the predicted a priori state estimate of step k given measurement dataup to yk−1

- αk,k denotes the updated/corrected a posteriori state estimate at step k+ 1 given mea-surement yk. Occasionally also αk is written for αk,k.

There are many ways to derive the Kalman Filter. The following derivation is based onthe idea that the Kalman filter is an efficient recursive weighted least-squares solution to anestimation problem considered from a probabilistic point of view [59]. This way of derivation



is chosen, because it will be used later on to extract an estimate of the uncertainty attachedto the measurement states for thrust measurements by using the Kalman Filter. Kalmanhimself insisted on regarding the estimation problem probabilistically, because he agreedwith Norbert Wiener [60] that the natural setting of the estimation problem belonged to therealm of probability theory and statistics [53]. Any estimation problem starts with defining amodel. The estimation aim is to fit a model f(α, θ) (where α are the states and θ the modelparameters) as closely as possible to the measurement data y by minimising the error betweenmodel and data:

ε = y − f (α, θ) . (2-63)To penalise modelling errors, a cost function J = J (ε) is defined, which fulfils the additionalrequirements

J (0) = 0J (ε2) ≥ J (ε1) ≥ 0 for ε2 ≥ ε1 ≥ 0J (ε) = J (−ε) .

(2-64)

The model coefficients are determined, such that the cost function becomes minimal

θ = arg min J (x, θ) , (2-65)

where θ again denotes an estimate for θ.5 For the Kalman Filter, a weighted least squarescost function is defined to equally penalise state and measurement prediction errors [61].

J = 12(αk+1,k+1 − αk+1,k)TP−1

k+1,k (αk+1,k+1 − αk+1,k)︸︷︷︸penalisestateprediction error

12 (yk+1 − Ck+1αk+1,k+1)TR−1

k+1 (yk+1 − Ck+1αk+1,k+1)︸︷︷︸penalisemeasurement prediction error

, (2-66)

where R is the covariance matrix of the measurement errors and P is the state estimationerror covariance matrix. The cost function in Equation (2-66) illustrates what is minimisedby the Kalman Filter. The next state αk+1,k+1 has to be predicted in such a way that theprediction error αk+1,k+1 − αk+1,k is small with respect to the variance of the state measure-ment errors represented by P . It also has to be chosen such that the measurement erroryk+1 − Ck+1αk+1,k+1 is minimised with respect to the measurement noise represented by R.

Optimal State Correction

To find a relation for the optimal estimate αk+1,k+1 of the next step αk+1, the cost functionis minimised with respect to the next step αk+1 by finding the location where ∂J

∂αk+1,k+1= 0:

∂J

∂αk+1,k+1= (αk+1,k+1 − αk+1,k)TP−1

k+1,k + (yk+1 − Ck+1αk+1,k+1)TR−1k+1Ck+1 = 0. (2-67)

Adding and subtracting CTk+1R−1k+1Ck+1αk+1,k to the right-hand side of Equation (2-67) and

rearranging it in terms of αk+1,k+1 and αk+1,k leads to(P−1k+1,k + CTk+1R

−1k+1Ck+1

)αk+1,k+1 =

5 argmin stands for the argument of the minimum, for which the given function attains its minimum value.



(P−1k+1,k + CTk+1R

−1k+1Ck+1

)αk+1,k + CTk+1R

−1k+1 [yk+1 − Ck+1αk+1,k] . (2-68)

Transposing the equation for αk+1,k+1 gives

αk+1,k+1 = αk+1,k + CTk+1R−1k+1 [yk+1 − Ck+1αk+1,k]

(P−1k+1,k + CTk+1R

−1k+1Ck+1

)−1. (2-69)

Below in Equation (2-87) is shown that the terms(P−1k+1,k + CTk+1R

−1k+1Ck+1

)−1CTk+1R

−1k+1 can

be summarised into one vector called the Kalman gain K, so that the formula for the statecorrection is

αk+1,k+1 = αk+1,k +Kk+1 [yk+1 − Ck+1αk+1,k] . (2-70)

Equation (2-70) demonstrates the working principle of the Kalman Filter. The next stateαk+1,k+1 is calculated as a weighted average between the measured and the predicted state.

State Covariance Prediction

The Kalman gain is found by minimising the state error covariance matrix Pk. The use of aquadratic cost function in Equation (2-66) allows to define the expectation of the states asthe estimate of the sought quantity [62]. Therefore, the estimates for αk are the expectedvalue of αk:

E [αk] = αk. (2-71)The covariance matrix of the state prediction error εk+1,k = αk+1,k −αk+1 is called Pk and isdefined as

E[(αk − αk) (αk − αk)T

]= E

[εkε

Tk

]= Pk. (2-72)

Hence, the probability density function of the states p (αk) is

N (αk, Pk) = N(E [αk] , E

[(αk − αk) (αk − αk)T

]). (2-73)

if the current measurement value yk is known.Using Equation (2-60), the next state is predicted as

αk+1,k = Aαk,k +Bxk (2-74)without consideration of the process noise. The real measurement value αk+1 is assumed todeviate from the predicted value by a random noise constant wk:

αk+1 = Aαk +Bxk + wk. (2-75)

The difference between Equation (2-74) and Equation (2-75) constitutes the state predictionerror εk+1,k = αk+1,k − αk+1, which has the covariance

E[εk+1,kε

Tk+1,k

]= E

[(Aεk,k + wk) (Aεk,k + wk)T

]. (2-76)

Multiplying Equation (2-76) out, results in

AE[εkε

Tk

]AT + E

[wkw

Tk

]T +AE

[εkw

Tk

]T + E

[wkε

Tk

]AT , (2-77)

The cross-covariances in Equation (2-77) are 0, because the process noise and the state pre-diction error are uncorrelated. The process and the state prediction variance were defined asE[wkw

Tk

]=Qk and E

[εkε

Tk

]= Pk, respectively. Therefore, the a priori error covariance can

be predicted withPk+1,k = APkA

T +Qk. (2-78)



State Covariance Correction

The a posteriori state error covariance Pk+1,k+1 is calculated as:

Pk+1,k+1 = cov (αk+1 − αk+1,k+1) . (2-79)

If αk+1,k+1 is replaced by the formula for the state correction of Equation (2-70):

= cov

αk+1 − (αk+1,k +Kk+1 (yk+1 − Ck+1αk+1,k))︸︷︷︸αk+1,k+1

(2-80)

and yk by the measurement relation in Equation (2-61):

Pk+1,k+1 = cov

αk+1 −

αk+1,k +Kk+1

Ck+1αk+1 + vk+1︸︷︷︸yk+1

−Ck+1αk+1,k

. (2-81)

The measurement error αk − αk,k−1 can be extracted as

Pk+1,k+1 = cov ((αk+1 − αk+1,k) (I −Kk+1Ck+1) +Kk+1vk+1) (2-82)

and the covariance can be applied on the individual parts of the sum:6

(I −Kk+1Ck+1) cov (αk+1 − αk+1,k) (I −Kk+1Ck+1) +Kk+1cov(vk+1)KTk+1 (2-83)

= (I −Kk+1Ck+1)Pk+1,k (I −Kk+1Ck+1) +Kk+1Rk+1KTk+1 (2-84)

Multiplying out and rearranging results in an extended formula for the a posteriori errorcovariance, gives:

Pk+1,k+1 =

Pk+1,k−Kk+1Ck+1Pk+1,k−Pk+1,kCTk+1K

Tk+1+Kk+1

(Ck+1Pk+1,kC

Tk+1 +Rk+1

)KTk+1. (2-85)

The best estimator is the one minimising this variance.

Kalman Gain

The Kalman gain is determined as the value for Kk resulting in the optimal estimator forPk. For this purpose, the trace of the matrix in Equation (2-85) is minimised by setting thematrix derivative with respect to Kk to 0:

∂trace (Pk+1,k+1)∂K

= −2Pk+1,kCTk+1 + 2Kk+1

(Ck+1Pk+1,kC

Tk+1 +Rk+1

)= 0. (2-86)

Rearranging yields the minimum mean square error gain, also called the optimal Kalmangain:

Kk+1 = Pk+1,kCTk+1

(Ck+1Pk+1,kC

Tk+1 +Rk+1

)−1. (2-87)

6It holds that cov(AX) = Acov (X) A.



When the Kalman gain equals the optimal value, the a posteriori error covariance can becomputed in a simplified manner. The formula for the optimal Kalman gain can then berearranged:

Kk+1(Ck+1Pk+1,kC

Tk+1 +Rk+1

)KTk+1 = Pk+1,kC

Tk+1K

Tk+1 (2-88)

and inserted into Equation (2-85), so that the last two terms cancel each other out and thea posteriori error covariance can be computed with

Pk+1,k+1 = Pk+1,k −Kk+1Ck+1Pk+1,k. (2-89)

Each measurement has an associated standard deviation σy and each input σx has an associ-ated standard deviation. Their noise covariance matrices can be defined as follows:

R =

σy1 0 0

0 . . . 00 0 σyn

Q =

σx1 0 0

0 . . . 00 0 σxn

(2-90)

The Kalman Filter algorithm can be summarised with the five equations:

αk+1,k = Aαk,k +Bxk state predictionPk+1,k = APk,kA

T +Qk covariance prediction

Kk+1 = Pk+1,kCTk+1

(Rk + Ck+1Pk+1,kC

Tk+1

)−1Kalman gain

Pk+1,k+1 = Pk+1,k (I −Kk+1Ck) covariance correctionαk+1,k+1 = αk+1,k +Kk+1 (yk+1 − Ck · αk+1,k) state correction

(2-91)

Measurement State Uncertainty

The derivation of the state covariance prediction and correction performed above have shownthat the Kalman Filter predicts and optimally corrects the variance Pk = E

[εkε

Tk

]of the

error contained in the state vector α in each iteration. The use of a quadratic cost functionin Equation (2-66) allows the definition of the mean of the posterior distribution of the statesas the estimate of the sought quantity [62], as in Equation (2-71). This assumption, namelythat E [αk] = αk, means that the covariance matrix P summarises the dispersion of theprobability distribution around the expected value. In other words, P contains the entireinformation of the uncertainty associated with the state error. Normally, the calculationof the variance for the estimates of a dynamic measurement evaluation method requires anextensive computational method, like the Monte Carlo method. For the Kalman Filter, suchan estimation of the uncertainty can be obtained simply by retrieving the information duringeach iteration from the measurement state error covariance matrix. This approach will bedeveloped in Chapter 4-4-1 for the Deconvolution Kalman Filter. To the knowledge of theauthor, this has not been attempted so far. However, to validate the results a Monte Carlomethod will be needed.



2-5-4 IIR Deconvolution from Noisy Observations Using the Kalman Filter

Bora [9] developed a Deconvolution Kalman Filter for the general scenario of extracting anARMA process after convolution with an IIR Filter and corruption by white Gaussian noise.The innovation of the method is to combine the Kalman Filter with a low-pass filter in state-space representation to form a single augmented state-space system. The resulting augmentedsystem can perform noise removal and deconvolution simultaneously. The set-up of Bora’smethod is depicted in Figure 2-16.

Figure 2-16: Kalman filtering to recover a noise corrupted input signal convoluted with an IIRtransfer function developed by Bora [9].

An unknown input signal x(k) is convoluted with an IIR transfer function H(z) and whiteGaussian random noise v(k) with probability distribution v ∼ N

(0, σ2) is added. The mea-

surement signal is theoretically generated with

y[n] =K∑k=1

h [n− k] · x [k] + σv[n] (2-92)

and the Deconvolution Kalman Filter is applied to find an estimate x of x. The z-transformedIIR transfer function H(z) can be described by

H (z) = a0 + a1z−1 + ...+ amz

−m

1 + b1z−1 + ...+ bnz−n=

m∑j=0

ajz−j

1 +n∑i=0

biz−i(2-93)

and is brought into state-space form by inserting the denominator coefficients bi and nu-merator coefficients ai into the matrices AH in Equation (2-94) and BH in Equation (2-95),respectively.

AH =

0 1 · · · 0... . . . . . . ...0 · · · 0 1−bN −bN−1 · · · −b1

(2-94)

BH =

00...1

CH =[aN aN−1 · · · a1

]DH = a0 (2-95)



The resulting state-space model is defined as

αk+1 = AHαk +BHxky′k = CHαk +DHxk

, (2-96)

where y′k is the output of the system H(z) without noise and α is the state-space variable. Ina similar way, the coefficients of a Butterworth low-pass filter can be expressed in state-spaceform as

βk+1 = AGβk +BGwkxk = CGβk +DGwk

, (2-97)

where β is the state-space variable for the low-pass filter. Both state-space systems, Equation(2-97) and Equation (2-96), are combined into a single augmented state-space system withmatrices A, B and C defined in Equation (2-98) and (2-99):

A =

AH BH 0 00 0 CG DG

0 0 AG BG

(2-98)

B =[

0 0 0 I]

C =[CH DH 0 0

](2-99)

and corresponding state vector ζ

ζk =[αTk xTk βTk wTk

]T. (2-100)

A closer look at matrix A reveals that the augmentation of the state-space system with thelow-pass filter is not very complicated. Following the way a matrix-vector multiplication iscarried out, αk is multiplied with AH , xK with BH , and the noise components of the signalare affected by the low-pass filter matrices only. The Deconvolution Kalman Filter for theaugmented state-space system is almost identical to the ordinary Kalman Filter:

ζk+1,k = Aζk,kPk+1,k = APk,kA

T +BQBT

Kk+1 = Pk+1,kCT(R+ CPk+1,kC

T)−1

Pk+1,k+1 = Pk+1,k (I −Kk+1C)ζk+1,k+1 = ζk+1,k +Kk+1 (yk+1 − C · ζk+1,k)

(2-101)

Compared to the conventional Kalman Filter code in Equation (2-91), the augmented codeshows only a few alterations:

1. In the normal Kalman Filter, the first line also contains the matrix B: αk+1 = Aαk +Bxk. In the augmented system, Bxk is contained in the matrix A, so that the nextstate can be predicted with ζk+1,k = Aζk,k.

2. In Pk+1 = AkPkATk + BQkB

T , BQkB appears to be different, but for the case that Qis a number, BBT = 1.

The algorithm of the Deconvolution Kalman Filter is not different to the algorithm of theordinary Kalman Filter. Its working principle can be understood by looking at the algorithm



in Equation (2-101). Instead of applying an inverse model, the Kalman Filter computes theoutput value of the model with ζk+1,k = CAζk,k and subtracts the calculated value fromthe measurement value at step k. Hence, the transient response described by the model isremoved from the measurement signal by subtracting it. The Kalman Filter corrects theamplitude of the calculated input by multiplying it with the optimal Kalman gain. As themodel is applied without inverting the transfer function, no stabilisation or regularisation isneeded. The problem is not ill-posed.

The drawback of the Deconvolution Kalman Filter becomes apparent from its way of working.The Kalman gain may be calculated incorrectly, if inappropriate values are chosen for thecovariance matrices Q and R. In this case, the Kalman Filter returns a qualitatively correctsignal, meaning that the shape of the signal is correct, but the amplitude is wrong. Since theamplitude of the thrust is the value which needs to be found, this kind of error is fatal andhas to prevented.

2-5-5 Determination of Q and R

The Kalman Filter provides a simple yet effective method for state estimation by accountingfor not modelled dynamics and measurement noise with the use of the system covariancematrices Q and R [19]. These filter tuning parameters have to be supplied by the designerto guarantee a convergence to the correct solution. Shmaliy [10] depicts the dependency ofthe Kalman Filter’s accuracy on the noise covariance as a parabola, shown in Figure 2-17, toillustrate that the Kalman Filter’s optimality is strictly dependent on the correct choice of Qand R.

Figure 2-17: Performance of the Kalman Filter for different noise covariance matrices [10].

Figure 2-17 shows that the Kalman Filter is only optimal if the noise covariance matricesare prescribed correctly. However, Q and R are still chosen heuristically most of the time inengineering applications [63]. If the Kalman Filter is applied as a low-pass filter the heuristictuning is based on the effect of the tuning parameters [64]:

- small Q: Dynamics trusted more- small R: Measurements trusted more


2-6 Uncertainty Evaluation for Deconvolution Filters 41

As the Deconvolution Kalman Filter’s main objective is not noise removal, but the removalof a transient system response, the same logic cannot be transferred to the DeconvolutionKalman Filter. To the knowledge of the author, no method exists which describes how tochoose the tuning parameters, if the Kalman Filter is used for deconvolution.

Various researchers have developed methods to rationalise and automatise the choice of thetuning parameters, see for example [65, 66, 67]. From the available methods only two methodscould be identified, which show the potential to be applied to the Deconvolution KalmanFilter. One was developed by Saha [19]. Saha suggest the use of two metrics, which measurethe effect of Q and R. The designer can use the two metrics to find suitable choices of Qand R for the deconvolution problem. The other is the Autocovariance Least Squares (ALS)method developed by Rajamani [20]. The method is available in form of an open-sourceOctave toolbox. The software routines can be adapted to any form of ARMA model. Hence,it should be possible to adapt them to the Deconvolution Kalman Filter.

2-6 Uncertainty Evaluation for Deconvolution Filters

The deconvoluted input signal is of little use unless its accuracy is known. The accuracy isdifficult to ascertain, because it is diminished by environmental factors, like ground vibra-tions, and mistakes made by the operator or the measuring equipment. The measured valueis merely an indication of the quantity that is really sought: the measurand. In an erroranalysis, systematic and random errors are quantified to give the best possible estimate ofthe measurand. In an uncertainty analysis, the best estimate of the measurand is given alongwith an associated measurement uncertainty and confidence intervals. An uncertainty analy-sis is based on the premise that, because it is impossible to state how well the true value of ameasurand is known, it is better to quantify the result of a measurement in terms of probabil-ity. Therefore, all quantities influencing the accuracy of the measurand are mathematicallydefined as random variables with a probability distribution assigned to them. It is possiblethat the assigned random variables are correlated and have joint distributions. Measurementerrors can often by described well with a normal distribution. If no estimate of a variable isknown, but only that the value is to be found within a certain interval, a uniform distributioncan be used. Relevant data to define distributions can be found by analysing the measurementdata in reference books and manufacturer specifications. Concerning the uncertainty causedby a deconvolution filter, exact formulas to calculate the mean and variance are available forFIR filters, but not for IIR filters [68]. The formulation of closed mathematical expressionsfor IIR filters generally requires linearisation. For this reason, the uncertainty attached toIIR filters is generally evaluated by performing a Monte Carlo simulation.

2-6-1 Monte Carlo Uncertainty Quantification for Thrust Deconvolution Meth-ods

The term Monte Carlo method is used to encompass stochastic computational methods, whichobtain numerical results by repeated random sampling for cases in which deterministic resultsare difficult to obtain. The random numbers are obtained using random number generators[69]. To apply the Monte Carlo method to the Deconvolution problem, the thrust input X is



regarded to be unknown and depending on one or more other quantities, denoted by Yn:

X = f (Y1, Y2, ..., Yn) . (2-102)

As before, the thrust input can be obtained by a convolution of the measurement signal Ywith an inverse filter hc, so that the input thrust becomes known up to a constant ∆[n],representing a local bias, introduced by the application of the deconvolution filter:

X [n] = (Y ∗ hc) [n] + ∆ [n] . (2-103)

To characterise the uncertainty, which propagates from the measurement signal through tothe model and the deconvoluted thrust input, a probability distribution needs to be assignedto the model coefficients ai and bi and to the error in the measurement signal y. Then, randomdraws y(r)b(r)a(r) can be made from the distributions and the input signal can be calculatedfor each draw, transferring the effect of the model and measurement uncertainty to the inputthrust:

x(r) = y(r) ∗ hc(r). (2-104)

The number of random draws made is referred to as the number of Monte Carlo trials M .The uncertainty attached to each vector of lengthM , calculated for each sample of the thrustinput by the Monte Carlo method, can be interpreted as the uncertainty in the input signal.Hence, for each sample M values have to be saved. This shows that while the procedure ofa Monte Carlo simulation is not complicated, the achievable accuracy depends strongly onthe computational power and storage capacity. The higher the number of trials M , the morereliable are the estimates as shown in Figure 2-18.

Figure 2-18: Number of Monte Carlo trials needed to obtain two and three digits accuracy foruncertainty evaluations in dynamic measurements [11]. While an accuracy of two digits is stillachievable with capacities of desktop computers today, the usage of a computer server system isneeded for three digits.


2-6 Uncertainty Evaluation for Deconvolution Filters 43

To achieve an accuracy of three digits, more than M = 106 Monte Carlo trials have to beperformed. That this is currently difficult to perform on a conventional desktop PC, as usedfor this thesis, can be shown by the following example: Eichstädt [21] reported an ‘Out ofmemory warning’ in Matlab for a signal length of N = 3000 with 106 Monte Carlo trials ona workstation computer with 24 cores, 96GB of RAM running on a 64-bit Linux operatingsystem. Hence, for this thesis, a memory efficient alternative to a direct Monte Carlo methodis needed to evaluate the uncertainty of a deconvolution filter: the sequential Monte Carlomethod. The signal of N = 3000 and 106 Monte Carlo trials of the example above runswithout difficulty on a desktop PC if a sequential Monte Carlo method is applied.

2-6-2 Sequential Monte Carlo Method

The sequential Monte Carlo method differs from the direct Monte Carlo method mainly inthe amount of computer memory that it requires. The Monte Carlo simulation is performedsequentially for each measurement point in the signal, column by column in Equation (2-105),instead of traversing and saving the entire measurement signal:

XMC =

x(1) [1] · · · x(1) [K]x(2) [1] · · · x(2) [K]

......

...x(M) [1] · · · x(M) [K]

Sequential Monte Carlo

−→

, (2-105)

where M is the number of Monte Carlo trials and K is the length of the measurementsignal. This procedure requires considerably less memory, because only the M Monte Carlorealizations of the Nθ = Na +Nb + 1 filter coefficients have to be stored:

X [n] =∑

bjY [n− j]︸︷︷︸2·(Nb+1)·M

−∑

ajX [n− j]︸︷︷︸2·Na·M

, (2-106)

in contrast to the direct Monte Carlo method where N filter input values and MK outputquantity values have to be stored. The storage requirements for the sequential Monte Carlomethod compared with the direct Monte Carlo method can be summarised to:

directMCsequentialMC

∣∣∣∣∣ (M + 2)K +Nθ

M (2Nθ + 2) , (2-107)

where the most striking difference is the absence of K for the sequential Monte Carlo method[68]. Storing the output quantity values for only a small number of time samples does notimpede the results obtainable with the sequential Monte Carlo method, which are identicalto those obtained from a direct implementation. Several efficient implementations of theMonte Carlo method to evaluate the measurement uncertainty of deconvolution filters wereprogrammed by Eichstädt [68] and can be downloaded online.7 In order to apply the toolbox,not only the model coefficients of an IIR or FIR filter are needed, but also either a proba-bility distribution connected to them or an uncertainty covariance matrix Uba for the modelcoefficients bi and ai has to be provided.

7 The software can be downloaded at http://www.ptb.de/cms/en/fachabteilungen/abt8/fb-84/ag-842/dynamischemessungen-842/download.html.


http://www.ptb.de/cms/en/fachabteilungen/abt8/fb-84/ag-842/dynamischemessungen-842/download.html



2-6-3 Setting up the Filter Covariance Matrix Uba

The filter covariance matrix Uba for an IIR filter with coefficients bi and ai can be foundby using another Monte Carlo simulation. During M Monte Carlo trials, random draws aremade from probability distributions assigned to the input and output signal to identify thefilter coefficients of the model under the influence of changing noise. For the case of randomzero mean Gaussian white noise, the noise distributions in the input and output identificationtime series are characterised as

εin ∼ N(0, σ2

inp

)εoutp ∼ N

(0, σ2

outp

), (2-108)

and the input and output identification time series become

Xε = X + εin Yε = Y + εoutp. (2-109)

The model parameters can be identified M times by solving the linear least squares problem:

θm,k =(XTε,kXε,k

)−1XTε,kYε,k for m = 1, ï£·,M. (2-110)

The set of M randomly flawed filter coefficients θm,k is divided into numerator and denom-inator coefficients, bm,i and am,i respectively. The numerator coefficients am,i are saved tomatrix A in Equation (2-111), and the denominator coefficient bm,i are saved to a matrix Bin Equation (2-112):

A =

a(1)1 a

(1)2 · · · a

(1)N−1 a

(1)Na

a(2)1

. . . a(2)Na......

a(M−1)1

. . . a(M−1)Na

a(M)1 a

(M)2 · · · a

(M)N−1 a

(M)Na

(2-111)

B =

b(1)1 b

(1)2 · · · b

(1)N−1 b

(1)Nb

b(2)1

. . . b(2)Nb......

b(M−1)1

. . . b(M−1)Nb

b(M)1 b

(M)2 · · · b

(M)N−1 b

(M)Nb

, (2-112)

where A is a M × Na and B is a M × Nb matrix, so that every column corresponds to anobservation and every row to a variable. The covariance matrix of the model Uba needed forthe toolbox can be found as covariance of the matrix Uba = [A, B] where the first column ofB is left out, because it is constantly 1:

B =

b(1)2 · · · b

(1)Nb...

......

b(M)2 · · · b

(M)Nb

. (2-113)


2-7 Conclusion 45

The covariance matrix Uba becomes an Na ×Nb matrix and can be calculated with

Uba = Cov ([A,B]) = 1M − 1

M∑i=1

(Ui − U

) (Ui − U

)T, (2-114)

where Ui for i = 1, ...,M represents a column of Uba, and U is the mean of the correspondingcolumn.8 The best estimate for the filter coefficients ai or bi can be determined by calculatingthe mean of each column in matrix A or B, respectively:

bk = 1M

M∑i=1

bi , for k = 1, ..., Nb

ak = 1M

M∑i=1

ai , for k = 1, ï£·, Na

. (2-115)

All the filter approaches described so far have focused on the characterisation of the transientresponse of the thrust stand and only suppressed measurement errors by using a low-passfilter. However, there are methods, which use the occurrence of random measurement errorsto their advantage by incorporating them into the optimisation process. One of them is theKalman Filter.

2-7 Conclusion

As private companies have ventured into the space sector during the last decade increasingthe competition, the development of more cost efficient rocket engines has attracted renewedinterest. The fuel efficiency of thrusters can be improved by pulsed firing. Therefore, themeasurement methodologies and evaluation strategies for pulsed rocket engines have beenrevisited. To improve their measurement accuracy, several scientific institutions have designednovel thrust stands based on piezoelectric force sensors for PMF measurements. However, thecurrently prevalent type of thrust stand is based on load cells containing strain gages. Thistype is not well suited to pulsed rocket thrust measurements as it produces large oscillations.To avoid costly new designs for the majority of industrial suppliers, a new evaluation methodis needed to adapt displacement type thrust stands to the demands of PMF measurements.Many different FIR and IIR filter methods are available and can be designed to fulfil thistask. Yet, trimming them to the high precision needed to increase the measurement accuracyin PMF measurements, is a difficult task. Therefore, in this thesis, an amended version of thelinear optimal Kalman Filter is proposed as a simple, highly accurate and easy-to-implementsolution to the dynamic force measurement problem. The Kalman Filter provides the optimalsolution, as long as the noise is white Gaussian (which it usually is for thrust measurements).To apply the Deconvolution Kalman Filter, only an ARMA model describing the transientresponse of the thrust stand has to be provided. As the Deconvolution Kalman Filter appliesthe model without inversion, the ill-posedness of the deconvolution problem does not affectit. Thus, it needs no stability analysis, even if an unstable model is used. It requires noother designing choices of the operator than a sensible choice of the noise covariance matricesand the low-pass filter’s cut-off frequency. It conveniently integrates the evaluation of themeasurement uncertainty into the deconvolution process, as it estimates states by minimising

8 In Matlab , the integrated cov function can simply be used.



the error variance. Results obtained by Bora [9] have shown that the Deconvolution KalmanFilter can reconstruct random noise which changes at the frequency of the sampling rate.This suggests that the Deconvolution Kalman Filter can also reconstruct short firing pulsesof the length of five to ten samples during a PMF measurement.

2-8 Objectives

Objectives by Airbus

This thesis was carried out for Airbus DS, which is in possession of a range of thrust stands anda huge data base of validated test results to correlate new methods with. For Airbus DS, theprimary goal of this thesis was the transfer of their digital thrust compensation method fromits SciLab/Java environment within the local Technical Data Management System (TDMS)in Lampoldshausen to a Matlab/C++ environment in the Airbus DS global MeasurementData Base (MDB). During this transfer, the method should be documented carefully. Itshould be investigated, whether the method could be improved or replaced by an alternativemethod considering recent advances in dynamic measurement filtering. Any improvementor alternative method should take into account that Airbus DS has a whole range of thruststands and that the evaluation method should be adaptable to all of them. While suggestionson how to improve the measurement efficiency by changes in hardware were encouraged, thefocus of this work should be the documentation and improvement of the software and not thehardware of the thrust stand.

It is suggested to proceed as follows: first of all, the characteristics of the Airbus DS rocketengine, thrust stand and measurement equipment should be specified. Then, the methodologyused by Airbus DS should be extracted from the Java and SciLab codes used in Lampold-shausen and implemented in Matlab and C++. After that, it could be investigated how themethod works and whether it is possible to improve it. Because only an error analysis, but nouncertainty analysis has been performed for the 10N rocket thrust measurements, an uncer-tainty analysis should be performed for the Airbus DS evaluation method. This uncertaintyquantification should try to include the most important sources of uncertainty caused by theAirbus DS thrust measurement equipment.

Academic Objectives

Deconvolution methods can only be validated using an analytical model or with synthetic testcases. Analytical models with sufficiently high accuracy to validate deconvolution filters aredifficult and time-consuming to develop, as described in Section 2-2-1. Therefore, only theeffectiveness of the simplest analytical model, a MSD system should be investigated. Severalauthors have mentioned that a system of order 2 can approximate the thrust stand dynamicswell, but have not discussed the degree to which this proposition is valid.

To compare the performance of deconvolution filters, including the Airbus DS thrust mea-surement method, a suitable test scenario has to be developed to generate realistic surrogatethrust measurement signals, for which the thrust input is known.


2-8 Objectives 47

It was shown in this literature study that the Kalman Filter offers an attractive solution tomost of the problems encountered when designing a deconvolution filter, if it can be adaptedto the purpose of deconvolution. Furthermore, it was found that Bora [9] already devel-oped a Kalman Filter for deconvolution. Therefore, the method developed by Bora shouldbe implemented and adapted to deconvolute the input thrust from dynamic rocket thrustmeasurements. As the only disadvantage of the Kalman Filter is that the noise covariancematrices Q and R have to be specified, a way has to be found to choose the matrices for PMFthrust measurements. From the methods available in the literature, only the two metrics de-veloped by Saha [19] and the ALS method developed by Rajamani [20] seem to be applicableto the Deconvolution Kalman Filter. Their applicability for deconvolution in dynamic thrustmeasurements should be tested. In addition, it was observed that the Deconvolution KalmanFilter calculates an estimate of the uncertainty associated to the input thrust during each it-eration. If this uncertainty could be extracted, an evaluation of the measurement uncertaintycould be directly incorporated into the deconvolution filter. That the uncertainty estimatecontained in this way is realistic could be validated with a sequential Monte Carlo method.Finally, the performance of the Deconvolution Kalman Filter should be compared with themethod used by Airbus DS to see whether it improves the measurement accuracy for PMFrocket thrust measurements.


Chapter 3

Airbus DS Thrust MeasurementEquipment and Evaluation Method

This chapter describes the software evaluation, which is used by Airbus DS to analyse thrustmeasurements performed with their 10N thrusters. To distinguish this method from alterna-tives developed throughout this thesis, it will be referred to as TDMS method, deduced fromthe Technical Data Management System (TDMS) in Lampoldshausen. First, an overview ofall required evaluation steps is given in Section 3-1, illustrating the evolution of data from thetest bench to the performance parameters of the rocket engine. The measurement equipmentand its characteristics are described in Section 3-2. The thrust compensation algorithm ofthe TDMS method is explained in detail in Section 3-3. One major aim of this thesis was toimplement the TDMS method, which was programmed in SciLab and Java, in Matlab. Toperform this task a few alterations were needed. They are described in Section 3-4. Results ofthe new Matlab edition’s performance are presented in Section 3-4-4. Further possibilitiesof improving the TDMS method are suggested in Section 3-5. They are implemented in analternative method referred to as Inverse Transfer Function method. Tests comparing theperformance of the two approaches can be found in Chapter 5.

3-1 Measurement Evaluation Procedure

The test conditions on ground are different to the operational conditions of the thrusterin space. Since an accurate prediction of the engine performance in space is needed, themeasurement data obtained on ground needs to be corrected. An overview of these correctionsis given in Figure 3-1: before a measurement, the thruster has to be mounted onto the thruststand, where it is fired continually in Steady-State Firing (SSF) or repeatedly in Pulse ModeFiring (PMF). The created thrust force bends the measurement bridge of the thrust stand likea beam. The deformation is registered by a load cell underneath the stand and transformedinto an electric signal by a quarter bridge. The electrical signal is interpreted and digitalisedby an Analogue-Digital (A/D) converter and stored on a server. The unaltered measurementsignal is referred to as raw thrust.

Master of Science Thesis CONFIDENTIAL Richard B.H. Ahlfeld

50 Airbus DS Thrust Measurement Equipment and Evaluation Method

Figure 3-1: Flowchart illustrating the steps of a thrust measurement evaluation.

Thrust compensation is the first correction taking place, because correct thrust values areneeded for all consecutive steps. The term thrust compensation is used as an umbrella termfor the removal of the dynamic response of the thrust stand and random measurement errors.While a simple low-pass filter is sufficient to perform this task for steady-state measurements,a more intricate filter design is needed for transient measurements. The main purpose ofthrust compensation is to compensate the dynamic response of the thrust stand. However,random errors, caused by ground fluctuations, the electric circuitry and inconsistent com-bustion, complicate this compensation because they interfere with the well-posedness of thedeconvolution problem and transform it into an ill-posed problem, as described in Section2-3-1. Furthermore, deconvolution filters have the unavoidable drawback of strongly ampli-fying high frequency noise. Hence, every deconvolution filter has to be complemented with alow-pass filter.The design of a deconvolution filter with the purpose of removing the thrust stand’s dynamicresponse from the measurement signal is the main topic of this thesis. The reason for this isthat the dynamic response causes by far the largest error for the series of test stands consideredhere. This is exemplified in Figure 3-2, where the red graph represents the deflection of theThrust Measurement Bridge (TMB) caused by rocket thrust. The true thrust value lieshidden underneath the oscillations. The green graph indicates the thrust value that onewould expect if there were no oscillations. This is called the firing signal. The amplitude ofthe firing signal is chosen to display the expected thrust amplitude (for example 10N for aAirbus DS Hydrazine thruster). In the following it is chosen as either 1 or 0, equivalent tothe thruster being either on or off.

Richard B.H. Ahlfeld CONFIDENTIAL Master of Science Thesis

3-1 Measurement Evaluation Procedure 51

Figure 3-2: Measurement signal of thruster showing constructive interference. The amplitude ofthe oscillation doubles from the first to the third firing.

The cause of the large oscillations of the TMB (the red graph in Figure 3-2) is the softbearing of the 10N thrust stands: it leads to a low damping, a high amplitude and a low firstnatural frequency. For strain gage type thrust stands, as used by Airbus DS, the first naturalfrequency is so low that during a PMF test the firing frequency of the thruster can approachto the first natural frequency and cause resonance effects. In Figure 3-2, one can see how theoscillation builds up after each firing of the thruster due the constructive interference.

Offset correction refers to correcting the drift in measurement values seen in Figure 3-3. Thehigh temperature of the thruster inevitably heats up the entire thrust stand so that the zeropoint of the measurement signal increases continuously. The offset can be corrected easily,because it is approximately linear.

Figure 3-3: Offset of zero point during longer SSF measurement



Vacuum correction is still embedded into the evaluation software, although it is not neededany more. Nowadays all data are obtained under almost vacuum-like conditions in a vacuumchamber. The vacuum correction routine was used to transfer data obtained at sea levelconditions - that is, at atmospheric pressure and room temperature - to vacuum conditions.

After these corrections, the performance parameters of the rocket engine (such as Isp, Ibitand thrust integral) are calculated.

3-2 Measurement Equipment

3-2-1 Thrust Stand

The thrust that a rocket engine generates is the most fundamental indicator of its perfor-mance. It can be determined by several basic methods, such as force measurement, internalmomentum balance and calculations using a nozzle coefficient [70]. Airbus DS mainly uses aforce measurement approach. The thrust force is measured using a strain gage thrust standand a digital correction filter. In this thesis the term thrust stand is used for a test standspecifically designed for the purpose of measuring rocket thrust. Another commonly-usedexpression is thrust bench. The 3D sketch in Figure 3-4 was designed to depict the mostimportant features of the Airbus DS thrust stand.

Figure 3-4: Left: Thrust stand sketch, side view with load cell and fuel lines, Right: Thruststand sketch displaying the Thrust Measurement Bridge (TMB).

The thruster (1) is vertically mounted into the arch-shaped part at the front (2) and it isrefuelled by the fuel lines (3). The basic construction of the stand can be divided into twomain components: a fixating structure (7), which is firmly attached to the ground, and thethrust measurement bridge (TMB) (8). The bridge is connected to the fixating structure bytwo elements: a soft spring (4) with variable stiffness and a thin steel plate at the end ofthe bridge (6). The spring is soft for the same reason the plate at the end of the bridge isthin: to ensure that the bending moment caused by the thruster results in a large deflection.


3-2 Measurement Equipment 53

The high flexibility is needed to make the TMB more sensitive to small changes in the thrustforce, which is required to evaluate SSF tests with an error of less than 5%. On the downside,the soft bearing responds dynamically to rapid changes in the thrust signal similar to a Mass-Spring-Damper (MSD) of order 2 described in Section 2-2-1. The load cell (5) sits below theTMB and records its deflection. The deflection can be mathematically converted into thethrust force, but first the transient response of the TMB has to be removed.

3-2-2 Thrust Force Sensor

The deflection of the TMB is transformed into an interpretable electrical signal by the loadcell underneath it. Force transducers are sensors used to convert force into other mea-surable entities. For that purpose many different devices like magnets, pressure probes,

Figure 3-5: Characteristics of SM S-TypeLoad Cell [12].

lasers, strain gages or piezoelectric crystals can beused. The force exerted on the stand by the thrusteris measured using a SM S-Type Load Cell. This isa general purpose beam strain gage cell produced byInterface Inc. Its main characteristics are describedin Figure 3-5. Forces can be measured with straingages as they change their electrical resistance pro-portionately to the undergone deformation. This lin-ear relationship is the reason that strain gages arethe most commonly used force measurement sensors.A conditioning circuit converts the electrical signal to a change in voltage, which is convertedto a binary signal by an A/D converter, so that the signal can be stored and edited by acomputer. As not only force but also temperature has an effect on elastic deformation andthe thrust chamber can reach temperatures up to 1500◦C, a highly temperature resistant typeof load cell is used. The SM S-Type Load Cell has less than 0.0015% temperature effect onoutput.



Figure 3-6: External influences on the TMB.

3-2-3 Thruster

The Airbus DS 10N Bi-Propellant Thruster is a small rocket engine for attitude, trajectoryand orbit control of large satellites and deep space probes. Its history in space goes backmore than 40 years with over 3000 units having steered numerous international scientificand commercial spacecraft. The thruster uses the storable propellants MonomethylhydrazineMMH as fuel and pure Di-Nitrogen-Tetroxide N2O4, or Mixed Oxides of Nitrogen (MON-1,MON-3) as oxidizer. Its operating range goes from 6 to 12N at regulated pressure, where thepressure is kept constant by an additional pressurant vessel, or in system blow down mode,where it decreases continually. Two important characteristics of the thruster performance areSpecific Impulse and Impulse Bit. The Specific Impulse (Isp) is a measure of how efficientlya thruster delivers a certain force. The Impulse Bit (Ibit) is used to describe the change inmomentum per pulse.

Figure 3-7: Thruster characteristics of the Airbus DS 10N Hydrazine Thruster [13].

For the Airbus DS 10N range, the Isp depends on several parameters but is generally higherfor higher thrust values, as seen in the graph on the left in Figure 3-8. If a manoeuvrerequires a lower thrust than 10N, the thrust can be throttled in one of the two followingways: either the thruster can be fired continually with lower chamber pressure or it can befired in pulses at its nominal operation point. As mentioned above, the first mode is calledSteady-State Firing (SSF), the latter Pulse Mode Firing (PMF). PMF is an efficient way toreduce the required average thrust level over time. The thruster is fired in pulses long enoughto guarantee the highest possible Isp (on times), but with breaks in between (off times) toreduce the average thrust level.

Figure 3-8: Left: Isp for SSF, Right: Isp for PMF [13].


3-2 Measurement Equipment 55

In pulse mode, full thruster efficiency is reached for pulse durations over 500 milliseconds, ascan be seen in the graph on the right in Figure 3-8. In general, efficiency is lower for shorteron times, since the share of start-up and turn-off times increasingly approach the total pulseduration. Another important performance indicator is the Ibit repeatability shown in thegraph on the left in Figure 3-9. The Ibit shows the change of momentum per pulse. A high Ibitrepeatability means that the thruster is capable of delivering pulses with similar momentumwhen fired in pulses. For ratios with on/off times higher than 1/100, the repeatability of theIbit of the thruster approaches the nominal operation point of 100mNs. To guarantee efficientand reliable pulse mode operation, Airbus DS tests its thrusters extensively for numerouscombinations of different on and off times. An overview of the test range is given in the graphon the right in Figure 3-9. The evaluation software used to quantify the performance of thethruster should be as accurate as possible for the entire range of on and off time tests.

Figure 3-9: Left: Ibit repeatability of the Airbus DS 10N thruster, Right: Range of on and offtime tests performed with 10N thrusters [13].



3-3 The Airbus DS Thrust Compensation Method

The Airbus DS thrust compensation method is a digital IIR deconvolution filter with manuallydesigned low pass filter. The finer details of the methodology depend on the involved SciLabroutines and their functionality. The flow chart in Figure 3-10 gives an overview of thealgorithms used. All steps displayed in the chart are elaborated in detail in the next sections.

Figure 3-10: The algorithm of the TDMS method.

3-3-1 The Digital Filter and its History

Until 1998 Airbus DS filtered raw thrust data with an analogue band-stop filter. A band-stopfilter simply blocks a band of unwanted frequencies as illustrated in Figure 3-11. Althoughthis was effective, it also resulted in phase shifts. In addition, the filtering was performedelectronically so that the raw thrust data were lost.


3-3 The Airbus DS Thrust Compensation Method 57

Figure 3-11: Band-stop filter.

Due to their higher accuracy and simply because digital filters allow for the possibility ofevaluating measurements repeatedly with different methodologies, the analogue filter wasreplaced with a digital filter in the late 1990s.

SSF tests are sampled at a rate of 125Hz. For PMF tests, the sampling rate is 1000Hz. ForSSF measurements, the compensation is performed by filtering the signal with a 40Hz low-passfilter. This removes all oscillations except the thrust and its amplitude. In principle, the samemethod can be applied to PMF tests, depending on the on and off time of the valve. However,the results obtained in this way are not very accurate for shorter on/off times. To remove theentire dynamic response of the thrust stand, the low-pass filter’s cut-off frequency has to bein the region of 30Hz. This removes almost the entire measurement signal and causes a largeapproximation error. Moreover, short pulses are smoothed out almost entirely by a stronglow-pass filter. Hence, a more refined method is needed for dynamic firing. For dynamicmeasurements, the signal is filtered with a low-pass filter of 500Hz to avoid aliasing effects.Afterwards, it is filtered with the compensation filter described later on in this chapter. Thefollowing sections are entirely devoted to the evaluation of PMF tests.

3-3-2 Linearity Check

Most of the analysis available for system models is performed for Linear Time-Invariant (LTI)systems. Thus, the assumption that a thrust stand can be described by a LTI system hasseveral advantages: the theory is better developed and easier than for nonlinear systems.Linearity is assumed for the TDMS method. A linearity check is performed before everymeasurement by attaching a range of different weights to the thrust stand and measuringthe force. This is to guarantee that the measured signal is still linearly dependent on theacting force and the load cell is operating flawlessly. Figure 3-12 shows the output of such acalibration run. The maximum discrepancy from a linear relationship is smaller than 0.0005for any of the measured weights. The static system is almost ideally linear.

Figure 3-12: Check of linearity.

Linearity is more difficult to determine for a dynamic force input. For this case, the internrule of thumb is that a linear relation between signal and acting force can only be assumed,



if the force does not change faster than a sine wave of less than 20% of the thrust stand’slowest natural frequency. For the 10N series, this means that if the thruster needs less than1/80s, or 12.5ms to increase its thrust from 0s to 10N, the relation cannot be linear. The10N thruster only needs about 2ms. Therefore, the measured signal cannot be considered tobehave linearly during the 2ms long start-up and turn-off phase of the thruster.

3-3-3 High-Pass Filter

If excited by the right firing frequency, the tubing system of the fuel lines can cause strong lowfrequency oscillations as shown in Figure 3-13. To remove these oscillations, the measurementsignal is filtered with a Butterworth high-pass filter of of order 6 and a cut-off frequency of10Hz.

Figure 3-13: Frequency spectrum of PMF test showing low frequency noise.

3-3-4 IIR System Identification

When the first filter was designed it had still been intended to determine the transfer func-tion coefficients by a calibration test. Calibration has to be performed for any thrust standconfiguration individually and again after any alteration. The transfer function can be as-sumed to remain unchanged, if all elements remain fixed and neither the spring stiffness norfuel lines are changed. Calibration means that before any measurement, the stand is excitedwith a known impulse, which triggers a clean impulse response. The known impulse can begenerated by an articulated hit with an impulse hammer, as demonstrated in Figure 3-14.By analysing the decaying vibrations immediately after the hit, the system parameters canbe determined. In theory, this is the cleanest way of obtaining the IIR coefficients.

In practice, the hit with the hammer results in a different impulse response depending onthe point where the hammer hits the stand. An impulse response identical to the one causedby the thruster is difficult to attain. The best approximation of the coefficients can beobtained by using an extract from the actual measurement signal as output for the systemidentification. The real input signal - the true rocket thrust - is unknown. Thus, an input forthe identification is missing and has to be created manually.



Figure 3-14: Calibration with impulse hammer.

In the TMDS method, the identification input is chosen as impulse realised as a zero vectorwith x(1) = 1. For the identification output, a time series, which can be seen as response tothis impulse, must be extracted from the measurement signal. Here, 1 second after the lastpulse is chosen. The last pulse is chosen to avoid interference of a next pulse. The startingpoint of the identification time series is chosen as the point in time when the thruster isturned off for the last rime during a test. First, a local maximum is sought within the vicinityof this point. When the maximum has been found, the beginning point of the identificationtime series is shifted forward in time to the first point after the signal falls below 40 % ofthe average thrust amplitude. This is done in order to ensure that the gain of the filter isnot adversely affected by the height of the thrust amplitude. To make absolutely certain thatonly the decaying curve goes into the identification series another 10 samples are added tothe starting point. How the identification output is chosen is illustrated in Figure 3-15.

Figure 3-15: Time series used for identification. Only the decaying oscillation is used for systemidentification.

A SciLab routine ‘time_id’ is used to determine a discrete transfer function H[z] with 60 polesand 59 zeros from input and output data employing a linear least squares system identificationapproach:

H [z] = Y [z]X [z] = a0 + a1z + ...+ a59z

59

1 + b1z + ...+ b60z60 . (3-1)



How the real unknown thrust signal is idealised in order to obtain a model with least squaressystem identification is depicted in Figure 3-16.

Figure 3-16: Flow chart illustrating the filter design through system identification.

3-3-5 Filter Order

According to the documentation of the method, the identification order is set to 60, becauseit should not be too small in comparison to the lowest eigenfrequency of the system. Whilethis reasoning could not be confirmed during testing (there was no difference between order60 and order 10), another reasoning could: more poles result in a better resolution of thefrequency spectrum of the thrust stand. From a physical point of view, the thrust stand canbe approximated well with a system of order 2 as described in Section 2-2-1. That this is truefor the Airbus DS 10N thrust stand is shown by the two frequency spectra in Figure 3-17.It can be seen that the main mode at 150Hz remains the same and only the other frequencycomponents vary from test to test, especially below about 100Hz. The single mode at 150Hzcan be described by a physical system of order 2.



Figure 3-17: Frequency spectra of two tests.

The cause of the varying components is the complex layout of the thrust stand and a changingexcitation through different firing frequencies. If the varying frequencies are integrated intothe compensation filter, the results become more inconsistent. Since robustness is the maindesign criterion for the TDMS method, the varying effect of the additional frequencies isremoved from the compensation filter. To that end, the transfer function is first identifiedwith a high order, usually 60. The high order guarantees that there are many poles availableto separate all occurring frequencies. To find the main mode, the high order system is splitinto sub-systems of order 2. From these sub-systems the system causing the highest amplitudeis extracted. Only the system causing the highest amplitude is used for the compensationfilter. The extraction is mathematically accomplished with a partial fraction decomposition:

Let m be the multiplicity of a pole, then the transfer function can be expressed as

H (z) = r1(z − p1)m + ... + rk

(z − pk)m, (3-2)

where pi for i = 1, .., k are the poles and rj for j = 1, ..., k the residues of the partial fractions.Numerically, partial fraction decomposition of a ratio of polynomials represents an ill-posedproblem.

From a control-theoretical point of view, partial fraction decomposition is equivalent to paral-lelising the transfer function of order 60 into a series of systems of order 1 and 2. The SciLabfunction computing the partial fraction decomposition does this automatically and returnsall systems of order 1 and 2. For the obtained systems, the frequency, damping, amplitudeand phase are calculated using equations (3-5) to (3-9). The results are plotted as shown inFigure 3-18.



Figure 3-18: Example for frequency, damping, amplitude and phase of a transfer function oforder 60 computed with a partial fraction decomposition for all residues.

Since the physical solution for a mode is a pair of conjugate complex poles, all systems oforder 60 can be neglected. Only the amplitudes belonging to the systems of order 2 have tobe computed and sorted in decreasing order. Let the system of order 2 responsible for thestrongest mode be denoted by HR and defined by

HR (z) = NR (z)DR (z) = a1 + a2z + a3z

2

1 + b1z + b2z2 . (3-3)

The poles of this transfer function are computed by solving

1 + b1z + b2z2 = 0. (3-4)

The solution of Equation (3-4) ought to be a conjugate complex pair of poles. Their real partshave the same value, and their imaginary parts have the same absolute value but differentsigns. Hence, it is sufficient to continue with one pole, say p1. In the Airbus DS method, thispole is first divided by the sampling rate and its logarithm is taken:

p1 = log (z1)dt

. (3-5)

The frequency f1 is computed with

f1 = imag (p1)2π . (3-6)

The damping d byd = real (p1)

imag (p1) . (3-7)

The amplitude A is withA =

∣∣∣HR

(e2iπdtf1

)∣∣∣ (3-8)

and the phase p by

p = imag

log

HR

(e2iπdtf1

)|HR (e2iπdtf1 )|

. (3-9)

The mode at 117Hz in Figure 3-18 is more than 50 times stronger than all the others. Thismode is chosen for the compensation filter and all the others are disregarded.



3-3-6 Gain Change Caused by Order Reduction

Reducing the order of the transfer function inevitably changes the gain, which in turn altersthe amplitude of the thrust. Hence, the reduced-order transfer function cannot recover thecorrect thrust amplitude, unless the gain is corrected. This can be done by making sure thatat least for t = 0, the gain of both functions is identical:

H(eiωt

)= HR

(eiωt

). (3-10)

This means practically H (1) = HR (1) for t = 0. The gain can be corrected with

HR = HR

HR (1)H (1) . (3-11)

That this way of correction is very efficient can be seen in Figure 3-19. The graph on the leftshows the frequency response and thrust signal without gain correction. The graph on theright was corrected using Equation (3-11).

Figure 3-19: Top Left: Frequency response of inverted transfer function, Top Right: Frequencyresponse of inverted transfer function without gain correction, Bottom Left: Thrust input re-constructed with inverted transfer function, Bottom Right: Thrust input reconstructed with gaincorrected inverted transfer function.

3-3-7 Second Order System Low-Pass Filter

Figure 3-20 shows the frequency response of the inverted transfer function H−1R of the main

mode and the frequency spectrum of a representative measurement signal. The spectrumshows that from 0.4fNyquist upwards the signal consists mainly of noise. Hence the signalamplification visible in the frequency response affects only noise from 250Hz upwards.



Figure 3-20: Top: Frequency response of inverted, reduced order transfer function, Bottom:Frequency spectrum of a typical thrust measurement.

In the TDMS method such amplification is avoided by constructing an artificial pole at 400Hz.This pole pushes the right part of the frequency response in the top graph in Figure 3-20 downto the frequency response seen in Figure 3-22. The value is chosen manually as 400Hz to keepthe noise attenuating properties of the pole away from the important frequency range ataround 150Hz. It is designed with a Second Order System (SOS). The SOS is created ascontinuous system by inserting the circular frequency

ω = 2dt

tan(

2πfSOS ·dt

2

)(3-12)

into

Hsos = ω2

(s− ω) (α− i) · (s− ω) (α+ i) . (3-13)

Equation (3-13) is discretised using a bilinear transform and frequency warping is counteractedby inserting Tustin’s formula

s = 2dt

(z − 1z + 1

). (3-14)

The denominator of Equation (3-13) is called DSOS and is used as low-pass filter componentfor the compensator.



Figure 3-21: Frequency response of low-pass filter SOS.

3-3-8 Design of the Compensation Filter

The numerator of the compensator is constructed by convolution of the polynomials DR (z)from Equation (3-3) and DSOS from Equation (3-13):

Hinv = DR (z) ∗Dsos (z)(z − 0.1)2 . (3-15)

For the compensator denominator a double pole at 0.1 is chosen. The value is set to 0.1,because for this value the arch between the two poles reaches exactly up to 0. Inserting adifferent value will move the arch either up or down. In this way, most of the signal canbe kept without amplifying the noise. For 10N thrusters the default value is 0.1. For otherthrusters the low-pass filter’s noise attenuating characteristics can be tuned by inserting adifferent value.

Figure 3-22: Frequency response of the Airbus DS thrust compensation filter.

3-3-9 Error Analysis for PMF Measurements

The error analysis performed in this section follows the error analysis performed by Dr. Garusin Lampoldshausen for 10N thrusters, as documented in the intern technical report ‘IP43-TN-02-015’. It assumes that the reconstructed input thrust FT (i) can be regarded as the sumof a physical process Fp plus stochastic random noise FS :

FT (i) = Fp (i) + Fs (i) . (3-16)

The random noise is white Gaussian with an expected value of E [Fs (i)] = 0 and standarddeviation σ. An important quantity to characterise the rocket engine in pulse mode is the



thrust integral, because the movement of the spacecraft is determined by the entire thrustdelivered and not a temporary fluctuation. In the following, formulas are derived to approx-imate the error of the physical and stochastic thrust integrals in PMF. The integral of thephysical part is given by

IFT=∫ t2

t1Fp (t) dt+

∫ t2

t1Fs (t) dt. (3-17)

It is assumed that the thruster is fired periodically, with period T = 2πω defined as

T = Ton (i+ 1)− Ton (i) . (3-18)

where i counts the number of times the thruster is fired. Let F (t) denote the time-resolvedthrust. F (t) can be approximated as a periodic function using a Fourier Series Expansion as

F (t) = a02 +

∞∑k=1

ak sin (kωt) +∞∑k=1

bk cos (kωt) (3-19)

with k ∈ N. The integral of this function over one period is always a02 T , where a0/2 is the

amplitude of F (t) in its Fourier spectrum at ω = 0. For a LTI system, the frequency responseconverges to 1 for ω → 0 and the value of the frequency response at ω = 0 is invariant if onlylinear operations are performed like convolution. Hence, if only linear transforms are used,the integral of the physical component of the reconstructed thrust can be estimated with∫ t2

t1Fp (t) dt ≈ a0

2 T = FnomT, (3-20)

where t1 = Ton (i) and t2 = Ton (i+ 1) and Fnom is the nominal thrust value.

The integral of the stochastic part of the reconstructed thrust is a stochastic quantity itself.Using the trapezoidal rule of integration, the integral can be defined as

IFs =i2∑i1

Fs (i)− Fs (i1)− Fs (i2)2 · dt, (3-21)

where dt is the time step. For a sampling rate of 1Khz dt is 0.001. Since E [Fs (i)] = 0, IFs

has mean 0:

IFs =i2∑i1

E [Fs (i)]− E [Fs (i1)]− E [Fs (i2)]2 · dt = 0. (3-22)

The variance can be calculated as1

var (IFs) = var

i2∑i1

Fs (i)− Fs (i1)− Fs (i2)2 dt

= (n− 1)σ2(dt)2, (3-23)

if n = i2 − i1 + 1. The standard deviation is σ√n− 1dt.

1 The result is obtained using the fact that if the normal random variable Fs(i) has variance σ2 then Fs(i)dthas variance σ2dt2.


3-4 The MATLAB Edition of the TDMS Method 67

The entire thrust integral error can be approximated by

εI =√var (IFs)∫Fp

∼ σ√n− 1dt

Fnommdt= σrel

√n− 1m

, (3-24)

where σrel is the standard deviation in relation to the nominal thrust value σrel = σFnom

.The factor m is introduced as the number of samples during which the thruster is turned on,and n is the entire number of samples used for integration. From Equation (3-24) it can beconcluded that the integral thrust error increases if the on time decreases. This is confirmedby the test results in Section 5-1-1.

3-4 The MATLAB Edition of the TDMS Method

SciLab and Matlab share a similar programming syntax. Nonetheless, most of the tools androutines in SciLab have a different functioning than their counterparts in Matlab. Overall,the implementation was performed by finding the counterparts and adapting the programme’ssyntax to yield the same results as SciLab. This worked well in general. Only SciLab’s linearleast squares identification and partial fraction decomposition had to be adapted to havethe exact same functionality available in Matlab. Moreover, the transfer function classwas omitted in Matlab and replaced with vectors of polynomial coefficients. This wasdone because the class of transfer function is different in Matlab , and its freely-availableclone Octave, and the method should be executable from both platforms. Otherwise, themethodology could be kept, except for the three points described below.

3-4-1 Lower Identification Order

The high identification order of 60 sometimes causes the order reduction to go wrong. Fig-ure 3-23 and Figure 3-24 show an example of an amplitude spectrum found for a transferfunction of order 60 obtained for a step input, which cannot be correct. There are two sim-ilarly strong frequencies in the decomposition in Figure 3-23 marked in green and red. 1 isthe correct main amplitude found at 151Hz, the other is found at 96Hz.

Figure 3-23: Frequency, damping, amplitude and phase obtained by applying a partial fractiondecomposition to a transfer function of order 60 for test 0M15_010.

The frequency spectrum in Figure 3-24 shows that there is no similarly strong frequencyat 96Hz in the measurement signal. Numerical errors have distorted the partial fractiondecomposition.



Figure 3-24: Frequency spectrum of the entire measurement signal for test 0M15_010.

To avoid such errors, the identification order was lowered to 10. No drawbacks could be foundfor this lower order. Up to an order of 10, numerical errors do not become disproportion-ately large. If it is discovered at a later point that the higher order is required for certaincases, a numerically more stable alternative to the partial fraction decomposition should beimplemented.

3-4-2 Step-Function as Identification Input

The SciLab method uses an impulse as input for the identification: a zero vector with x(1) = 1as illustrated in the graph on the left in Figure 3-25. This input cannot be used in Matlab.An input vector consisting almost entirely of zeros means that the right half of the systemidentification matrix in Equation (2-33) is filled with zeros, so that the matrix has an insuffi-cient rank. SciLab ignores this fact when computing the coefficients, but the insufficient rankresults in extremely small numerator coefficients:

As the numerator coefficients are disregarded (only the denominator coefficients are used forthe compensator, see Section 3-3-8), this error is of no importance for the TDMS method.Unlike SciLab, the Matlab equation solver aborts the iteration when the numerator coeffi-cients fall below a certain tolerance level. This results in a bad estimate for the denominatorcoefficients in Matlab:

The matrix’s insufficient rank cannot be disregarded in Matlab. Either an alternative equa-tion solver or a different input has to be chosen. As a different solver has no effect on the



matrix’s rank, the latter approach is preferable. Instead of an impulse, a step-function inputis suggested, as shown in the graph on the right in Figure 3-25. This has the advantagethat the matrix is filled with ones instead of zeros, so that the rank becomes sufficient. Thedecaying signal can be transformed into a step-response with

outpID = outpID − outpID (1)outpID (1) , (3-25)

where outpID denotes the extract of the measurement signal used for identification in thetime domain.

Figure 3-25: Impulse and step-function input used for system identification. They behavesimilarly, except that the step response oscillates around 1 and the impulse response around 0.

3-4-3 Zero-Point Correction

The comparison of results obtained by the Matlab edition of the TDMS method with TDMSdata base values revealed a small bias correction, shown in Figure 3-26. This bias correctionis not corrected in the SciLab code, which is used as foundation of this chapter. It must becorrected in a later post-processing step, which is currently not included in the documentationof the TDMS method.



Figure 3-26: Zero point correction of a small signal bias caused by the deconvolution filter inthe TDMS data base and the Matlab edition of the TDMS method. For all tests contained inthe TDMS data base, a zero point correction is performed after 4s. In the new Matlab edition,it is performed from the beginning of a measurement.

The blue graph in Figure 3-26 is shifted to 0 after 4s. This correction is performed for alltests found in the TDMS data base, always shortly before the first ignition of the thruster.Hence, this zero point correction was also included into the Matlab edition. In contrast tothe SciLab version, however, the signals are shifted to 0 from the beginning, as seen in thered graph. The zero point correction is achieved in Matlab by determining the mean of thenoise before the first pulse and shifting the entire measurement signal y by this mean:

y0 = y − µn, (3-26)

where y0 is the nulled measurement signal. The mean of the noise is computed in the usualway:

µn = 1N

N∑1y (i), (3-27)

where N is the index of a sample 10ms before the first pulse and y is the entire measurementsignal.

3-4-4 Results Obtained by the Matlab Edition

Figure 3-27 compares results computed with the Matlab and SciLab edition of the TDMSmethod to data saved to the TDMS data base for the same test.



Figure 3-27: Thrust signals obtained using Airbus DS’s own SciLab codes (red), the new Mat-lab edition (blue) and the result saved to the TDMS data base (dark green). The red and theblue curve are qualitatively identical apart from a small bias. The green curve extracted from thedata base in Lampoldshausen has a different thrust amplitude.

The Matlab and SciLab codes yield the same result, apart from a small bias of about 0.01.The bias is constant for all tests, so it can be corrected. However, neither of the methodsconfirms the graph saved to the TDMS data base. The deviation of both methods suggeststhat something else, not contained in the SciLab codes used as foundation for this chapter, isdone to the signals either before or after the SciLab codes are applied. One possibility is thatthe SciLab and Matlab version used here determine the transfer function coefficients for eachtest individually, while the TDMS method might use one test to determine the coefficientsfor an entire test series.

The two graphs in Figure 3-28 confirm this. In both, the dynamic response is not entirelyremoved from the thrust pulse. The reason for this could be that coefficients from another testwere used, which had a slightly different frequency spectrum. To differentiate between thevalue saved to the TDMS data base and the one computed with Matlab for the MeasurementData Base MDB in Ottobrunn, the TDMS thrust is denoted by Fcomp TDMS and the Matlabcomputed thrust by Fcomp MDB.

Figure 3-28: Two tests for which the Matlab method (FcompMDB) yields smoother resultsthan TDMS (FcompTDMS).



3-5 Improvement Possibilities

This section investigates further possibilities with the potential to improve the TDMS method.The most promising alterations are implemented in a method, which is named Inverse TransferFunction method in Chapter 5 to distinguish it from the Matlab edition of the TDMSmethod. The filter is called Inverse, because it is essentially obtained by inverting the transferfunction after reducing the order.

3-5-1 Inverse Transfer Function Method

The methodology of the Inverse Transfer Function method depicted in Figure 3-29 has itsorigin in the insufficient rank of the matrix in the original SciLab TDMS method. The systemidentification with an impulse as input signal created a system identification matrix withinsufficient rank in the original TDMS method. Due to the insufficient rank, the numeratorcoefficients obtained by the order reduction were meaningless. The problem can be solvedby replacing the impulse input with a step function input for the system identification. Thisincreases the rank of the matrix so that the correct numerator can be found and the transferfunction can be inverted. This was already explained in Section 3-4-2. However, due tothe new way of identification, the reduced order transfer function can be inverted instead ofchoosing the compensator poles manually, as in Section 3-3-8. This is the basic idea behindthe Inverse Transfer Function method discussed in the following sections.

Figure 3-29: Algorithm of the Inverse Transfer Function method.

The Inverse Transfer Function method was developed during the process of validating thesingle design parts of the TDMS method. It was investigated why certain parts of the TDMS


3-5 Improvement Possibilities 73

method were chosen in the way they are and what effect a change in the methodology has.The changes mostly concern the system identification and the low-pass filter. As can be seenfrom Figure 3-29, these were also the only parts which were eventually changed in the newmethod.

3-5-2 System Identification with Firing Signal

Another feature that was added to the codes is a special identification mode for tests withshort on times. The option can be enabled in the code by passing the string ‘dynamic’ tothe function ‘pulseextraction()’. The idea behind the alternative identification mode is thatfor short pulses, the decaying measurement signal used for system identification is influencedby the rising and falling flanks of the pulse. The effect of the falling flank on the transientresponse can be seen clearly in Figure 3-30. Hence, whenever the on time ton of a pulse is notlong enough for the dynamic response to subside (which is usually the case for any on timeshorter than 300ms), a trapezoidal pulse should be used instead of a step-input function foridentification in order to integrate the effect of the rising and the falling flank into the systemidentification.

Figure 3-30: Left: Firing signal pulse used as input signal for identification with padding, timedelay, on time and effect of falling flank labelled, Right: Thrust pulse reconstructed using staticand dynamic identification routines.



Figure 3-31: Left: Thrust pulse reconstructed using static and dynamic identification routines,Right: Synthetic thrust pulse reconstructed dynamic identification routines.

According to a warning in Matlab , large changes should not occur too early in the measure-ment signal. To avoid this, the signal should be padded with zeros at the beginning. However,the number of zeros in the beginning can change the outcome of the system identification.The correct padding order also depends on the order of the system. The best results wereobtained by a padding with 2n zeros for a transfer function of order n.

The system identification fails completely, if the timing of the pulse used as input signal doesnot correspond to the timing of the output signal. The firing signal records the theoreticthrust level. It was tried to use the firing signal as input signal for identification. However,there is an unknown time delay between the firing and the measurement signal and it is notconstant for different tests. It sometimes varies by ± one sample. The variation is enough toadversely affect the system identification. In an early prototype of the TDMS method, thiswas addressed by pausing the code while a pop-up window showed the firing and measurementsignal for the last pulse. The operator could then read out the time delay by counting thesamples between the edges of the firing and measurement signal. While this is reliable, it isvery time-consuming. To reduce the effort, a short routine was programmed, which determinesthe time delay automatically. It computes the difference between two vector elements andshifts the firing and the measurement signal so that the maxima of the difference vectorscoincide.

An alternative identification routine was programmed, which extracts a pulse from the firingsignal as input signal for identification, ensures that enough zeros are padded to the beginningof the identification signals, compensates the time delay automatically and performs a linearleast squares system identification. This identification mode can be enabled by passing thestring ‘dynamic’ to the main identification method. The identification with a step-functionis used with the string ‘static’. In Figure 3-30, the dynamic identification mode was enabledto evaluate the real measurement test 0M15012. The method yields similar results to anidentification with a step function input, but the ripples in the signal caused by firing noiseof the combustion are at different positions. This effect becomes more important for shorterpulses. While the deviation looks small for 0M15012, which has an on time of 30ms, it lookslarger for test 0M15011 in the graph on the left in Figure 3-31. To validate which identificationmode is more reliable, both identification modes were used on synthetic test cases. While theresults coincide most of the time, there are cases in which the dynamic identification modefails. The test case shown in the graph on the right in Figure 3-31, gives an example forthe reconstruction of a wrong amplitude with the dynamic identification mode, whereas thestatic mode is correct. No reason has been found so far why the dynamic mode is unreliablesometimes. Nevertheless, the default option for system identification is set to ‘static’.



3-5-3 Filter Order

It is well known that a physical Autoregressive-Moving Average (ARMA) model of order 2can approximate the transient response of a thrust stand [4, 8, 17]. In this section, it isinvestigated to what degree this proposition is valid for the Airbus DS thrust stand by usingthe MatlabSystem Identification Toolbox. The toolbox is used to identify a model from thedecaying signal after a long pulse and to obtain a goodness of fit value in percent.

IIR Ident. Order Zeros 1 2 3 4Poles1 9.22% 9.51% 10.40% 11.35%2 89.38% 89.43% 90.61% 91.08%3 93.76% 93.76% 93.76% 93.75%4 89.67% 93.76% 93.76% 94.62%5 94.30% 94.65% 94.60% 94.55%

Table 3-1: Nonlinear least squares fit to measurement data with Matlab SI Toolbox.

A (n, n − 1) transfer function (denominator degree n, enumerator degree n − 1) is used inthe TDMS method. In Table 3-1, the goodness of fit for different combinations of numeratorand denominator orders is shown. Lower order transfer functions of degree (n, n − 1) asused by Airbus are highlighted in red. Bi-proper transfer functions (n,n) are highlighted inblue. In general, the goodness of fit increases for an increasing number of poles. The fitalso becomes better for an increasing number of zeros, but the effect is much weaker. Fourpoles increase the accuracy from 10% to 95% for one zero, while four zeros only increase theaccuracy from 9.22% to 11.35% for one pole. Comparing the result obtained for four and fivepoles, it becomes apparent that there is an upper boundary to the achievable accuracy. Inconclusion, from the results in Table 3-1, the reason why an identification order of n, n − 1is advantageous does not become apparent. On the contrary, bi-proper transfer functions aremore accurate and easier to invert.

The identification order of the TDMS method is chosen as 60. As mentioned before, this highorder can result in the divergence of the method. Tests have shown that a maximum possibleidentification order exists, up to which the identification is not affected by noise. For any orderhigher than this maximum order, numerical errors will adversely affect the identification. Inthe following, it is attempted to find the optimal linear system identification order. Figure 3-32 shows the NRMS error in percent for linear least squares identifications of different order.The upper boundary to the achievable accuracy is approximately 95% and is already reachedfrom an order of 4 upwards. Comparing the results of the nonlinear optimisation performedby Matlab in Table 3-1 with the linear identification of a (n, n) transfer function Figure 3-32,no big difference in accuracy can be seen.



Figure 3-32: Left: Measurement signal and simulated response of a least squares identifiedtransfer function of order 8, Right: Goodness of Fit for increasing linear least squares (n,n)transfer function identification orders.

Figure 3-33: Left: The amplitude of the transient system response is not accurately representedby the model of order 2, Right: The amplitude of the transient system response is accuratelyrepresented by a model of order 8, but the beat in the signal is not captured.

Linear least squares system identification is efficient enough for the dynamic thrust measure-ment problem. It is only for a transfer function of order 2 that the linear fit is considerablyworse than a nonlinear one, in this case 78% for the linear fit and 95% for the nonlinear one.The main reason for the worse fit is that the damping is not estimated correctly. The linearlyidentified model curve decays too rapidly as illustrated in the left graph of Figure 3-33. Inaddition, it is difficult to obtain a NRMS fit better than 80% with a linear least squaresapproach if the off time between pulses is short enough for a beat to develop, as shown inFigure 3-33. A beat is a periodic variation in amplitude and occurs if two sounds with similarfrequencies interfere. As shown in Figure 3-33, a higher polynomial order, in this case 8,improves the model fit to the measurement data. As described in Section 3-3-5, this problemis solved in the TDMS method with a combination of high order linear system identificationand partial fraction decomposition.



Figure 3-34: Left: Filter of order 2, inverted and stabilised, Right: Filter of order 8, invertedand stabilised.

The two graphs in Figure 3-34 exemplify, however, that a better fit to the decaying curveis no guarantee for a better deconvolution filter. The graph on the left was reconstructedwith the model of order 2 from Figure 3-33, the graph on the right with the model of order8. Although the model of order 2 only had a NRMS fit of 78% and not 96% as the modelof order 8, the model of order 2 reconstructs the thrust for the real thrust test 0M15010significantly better. The overshoot at the beginning and at the end of a pulse, seen in thegraph on the right, becomes more extreme when the order of the Inverse Transfer Functionfilter increases. A good explanation for this particular phenomenon has not yet been found.It appears to be a type of overshoot, however. A reason why the lower system is better suitedfor compensation can be found by looking at the frequency response of both filters, as shownin Figure 3-35. The high order creates a filter with a nonlinear phase. This nonlinear phasedistorts the signal when the filter is applied to it.

Figure 3-35: Left: Inverse Frequency Response for a transfer function of order 2, Right: InverseFrequency Response for a transfer function of order 8.



One can conclude that for a thrust stand filter, the correct identification of the entire transientsystem response is less important than the correct identification of the main thrust standmode. The order reduction approach used by Airbus is an excellent way to find a goodcompensation filter. The additional poles of the first higher identification step absorb weakerfrequencies and allow the construction of a stable and robust filter of order 2.

3-5-4 Low-Pass Filter Design

A digital low-pass filter is suitable to complement a thrust compensator if it does not alterthe amplitude characteristics or amplify the noise. That these requirements are fulfilled bythe Butterworth, the Chebyshev type II and the Bessel filter was introduced in Section 2-3-4.

The Bessel filter suggested for thrust compensation by Crosswy [4] produces no overshoot,but it is an analogue filter not intended to be used digitally. In analogue signal processing,the main reason for choosing the Bessel Filter is its approximately equal group delay. Abilinear transform destroys this group delay. Hence, the main advantage of the Bessel filter isdestroyed by digitalisation. For thrust compensation the Bessel Filter is not attractive becauseof its equal group delay, but because it does not cause overshoot. Thus, the analogue Besselfilter contained in the Matlab Digital Signal Processing (DSP) toolbox can be digitalisedusing an impulse invariant transform. An impulse invariant transform converts an analogueto a digital transfer functions filter in such a way that the impulse response of the analoguefilter is not changed [71].2

Figure 3-36: Time response of Bessel and Butterworth filter. As Butterworth filters overshootfor higher orders, they are not suitable for thrust compensation.

In Section 3-3-9, it was established that one of the main disadvantage of the TDMS methodis that it is less accurate for shorter pulses. In Chapter 5 the main cause of this is going to beidentified as the wrong characterisation of the rising and falling rate in the pulse flanks. But-terworth filters can produce overshoot, but have superior rising rate characteristics than themanually designed low-pass filter of the TDMS method. Moreover, their overshoot becomesnegligibly small, if a filter of order 1 or 2 is used. Figure 3-36 shows how the rectangularpulse is best preserved by the Butterworth filter of order 1. Hence, a Butterworth filter oforder 1 is chosen as low-pass filter. The Butterworth filter of order 1 takes only two samplesinto account and, in consequence, can only have a small averaging effect on the flanks of the

2 More Information on impulse invariant methods can be found at https://ccrma.stanford.edu/~jos/pasp/Impulse_Invariant_Method.html.


https://ccrma.stanford.edu/~jos/pasp/Impulse_Invariant_Method.html

https://ccrma.stanford.edu/~jos/pasp/Impulse_Invariant_Method.html


signal. In general, the higher the order of the filter, the more samples are considered and thesmoother the flanks of a pulse become. The time output of two Butterworth and one Besselfilter is depicted in Figure 3-36.

The cut-off frequency of the low-pass filter can be selected manually by convoluting the inversefilter with a low-pass filter in the frequency domain and plotting the frequency response ofthe resulting combination. The picture on the left in Figure 3-37 shows the inverse frequencyresponse of a reduced order transfer function. One can see how the inverse filter amplifies allfrequencies higher than 0.35fNyquist. An application of the inverse filter without additionallow-pass filter will amplify all signal components higher than 0.35fNyquist. For the AirbusDS thrust measurements, the signal only consists of noise above 0.4 as can be seen in thefrequency spectrum in Figure 3-20. If the inverse filter is applied without adding a low-passfilter, the reconstructed thrust will look like the signal in the picture on the right in Figure ??.To avoid such noise amplification, the frequency response in picture on the left in of Figure 3-37 has to be changed to look like the frequency response in Figure 3-38. To achieve this,the cut-off frequency and the noise attenuation properties of the low-pass filter have to beadapted. Since the signal above 0.4fNyquist only consists of noise, the low-pass filter needs acut-off frequency in this region. The order of the Butterworth filter determines the behaviourof its transition band and stop band. The transition band becomes shorter and steeper andthe noise attenuation stronger for higher order Butterworth filters. On the downside, theyproduce overshoot. A Butterworth filter of order 1 does not cause overshoot, but its transitionband is larger and flatter and the noise cancellation is weaker.

Figure 3-37: Left: Frequency response of the inverse of the IIR model describing the thrust stand,Right: Example of noise amplification taking place through deconvolution without additional low-pass filter.

In order to avoid a reduction of the relevant thrust frequencies, the cut-off frequency is chosenhigher than the required 0.3fNyquist: it is set to 0.5fNyquist. This keeps many of the highfrequency dynamics, but still has the desired effect of suppressing the noise amplificationproperties of the inverse reduced transfer function from 0.4fNyquisty upwards as can be seenin Figure 3-38.



Figure 3-38: Frequency and phase response of the Inverse Transfer Function method.

Unlike the manually designed low-pass filter of the TDMS method, the Butterworth filterallows the designer to increase the noise attenuation when a test has higher background noisethan usual. Should the operator not be content with the remaining noise in the final result, hecan easily increase the order of the Butterworth filter by 1 or 2. As long as he does not lowerthe cut-off frequency below 0.4fNyquist, the important frequency components of the signalwill not be harmed. Only the signal flanks will become smoother when using a higher order.

3-5-5 Phase Correction

The graph on the left in figure Figure 3-39 shows a thrust pulse reconstructed with theTDMS method and shows that the rising and falling edges of the reconstructed thrust pulseslook different. The thrust curve was created by applying the TDMS method to a syntheticmeasurement signal generated as explained in Chapter 5. The asymmetry in Figure 3-39 isnot typical for the thrust signal, but caused by the nonlinear phase of the filter. If a signalshows a left-right symmetry after the application of a filter, it indicates that the filter’s phaseis either 0 or linear. If the falling and rising edges look different, it may be due to the nonlinearphase of the filter.



Figure 3-39: 1. Left-right applied, 2. Right-left applied, 3. Bidirectionally applied TDMS filteron surrogate signal without noise.

At first glance, the TDMS filter’s phase response shown in Figure 3-40 seems mostly linear.Its most striking nonlinearity is a jump of π-radians when the frequency passes through the0 belonging to the main mode. This jump results in a phase inversion (a sign inversion inthe signal) of any oscillation occurring above 0.3fNyquist. As this frequency range is mostlycharacterised by noise, a sign reversal is of no importance. Less obvious, but still recognisableis that the phase is not actually linear from 0 to 0.3fNyquist, in Figure 3-40. This is whatcauses the different edges on the left and the right.

Figure 3-40: The phase jumps when the frequency passes through the 0 belonging to the mainmode.

The conventional way to correct the nonlinear phase of a recursive filter is to filter the signalbidirectionally, which means applying the filter on the signal and subsequently on the reversedfilter signal [47]. Bidirectional filtering is implemented in Matlab with the function filtfilt().It is usually employed for low-pass, high-pass or band-pass filters with a nonlinear phase,which are ideally 1 in their passband and 0 in their stop band. For such cases, a reductionof the amplitude can be excluded. This is, however, not the case for the inverse filter createdhere. Its passband changes the signal.

Figure 3-41: One and bidirectionally filtered thrust pulse



On the other hand, applying bidirectional filtering, as illustrated in graph 3 of Figure 3-39and in Figure 3-41, to either a synthetic or a real measurement signal does not have an effecton the amplitude of the thrust signal. The real measurement signal in Figure 3-41 even showsanother possible advantage: the curve is much smoother than the one-directionally filteredone. One disadvantage for bidirectional filtering is a duplication of computational effort.Another is that the causality of the signal is impaired. The thrust may start rising before therocket has been turned on, as is the case in graph number 2 of Figure 3-39.

3-5-6 Stability

To ensure the stability of the inverted reduced order system, all poles must be within the unitcircle and not too close to the circle boundary to avoid poles with a very small damping. In alltests, the system’s poles lie well inside the unit circle (red crosses in Figure 3-42). Therefore,the reduced order system is a minimum-phase system, which means that the transfer functionand its inverse are stable [26]. Therefore, it is not necessary to check the stability of thetransfer function of reduced order.

Figure 3-42: Pole-zero plot of the Inverse Transfer Function compensator (zeros are circles, polesare x-es).

3-5-7 Design of Inverse Compensation Filter

Similar to the Matlab TDMS method, the system is first identified with an IIR transferfunction with order 8 using a step function input as described in Section 3-3-4. Then, thesystem of order 2 describing the main mode is extracted using a partial fraction decompo-sition as described in Section 3-3-5. The resulting compensator of order 2 is inverted andconvoluted with a Butterworth low-pass filter of order 1 in the frequency domain to suppressthe compensator’s tendency to amplify high-frequency noise:

HR = (z − p1) (z − p1)(z − r1) (z − r2)︸︷︷︸

Compensator

(z − b1)(z − b2)︸︷︷︸Low−Pass

. (3-28)



The frequency response of the system in Figure 3-43 shows that all frequency componentshigher than 0.3fNyquist are attenuated. As the Butterworth filter is easily adaptable, thecut-off frequency and noise attenuation can simply be changed if a stronger attenuation ofthe noise should be required.

Figure 3-43: Frequency response of Inverse Transfer Function Filter.



3-6 Uncertainty Analysis for the Airbus DS Thrust Measurements

The measurement uncertainty for the thrust measurements performed by Airbus DS canbe determined by means of a Monte Carlo simulation as described in Section 3-3-4. Inthe subsequent analysis, the uncertainty attached to results obtained by the TDMS filteris quantified using a representative test case: 0M15_011 (TDMS data base test number).The test 0M15_011 is used, because it is a PMF test with 10ms on time and 1s off timeas can be seen in the graph on the left in Figure 3-44. Consequently, there is virtually nointerference between the pulses and, due to the short on time, the response can approximatelybe considered as impulse response. For these reasons, a transfer function of order 2 describedin Equation (3-29) can be identified directly using the linear least squares system identificationapproach described in Section 2-4-3:

H11 = 0.5786z2 + 0.1772z + 0.06977z2 − 1.152z + 0.9849 . (3-29)

No higher identification with subsequent order reduction is necessary for the transient responsein Figure 3-44. The response simulated with transfer function H11 fits the measured transientresponse very well, as can be seen in the graph on the right.

Figure 3-44: Left: Last pulse in the measurement signal of test 0M15_011, Right: Transientresponse simulated using model H11.

To determine the measurement uncertainty, a probability distribution has to be assignedto each quantity that influences the deconvolution process. The error sources which wereincluded in the uncertainty analysis are depicted in Figure 3-45. The electronic noise iscaused by the A/D conversion and the entire electric circuitry. It adds random noise tothe entire measurement signal. The identification error is caused by a loss of informationwhen the real transient response of the thrust stand is approximated with the model H11 oforder 2. The repeatability error of the load cell is ±0.01 of the measured signal amplitudeaccording to specifications supplied by the manufacturer. The firing noise occurs due toinconsistent combustions in the rocket combustion chamber and introduces randomness tothe measurement signal during the on times of the thruster. These four sources of uncertaintyare investigated below and a probability distribution is assigned where possible.


3-6 Uncertainty Analysis for the Airbus DS Thrust Measurements 85

Figure 3-45: Error sources considered in the uncertainty quantification for the deconvolutionmethod used by Airbus DS to evaluate thrust measurements.

3-6-1 Electronic Noise

An easily identifiable random component of the measurement signal is the noise caused bythe electronic circuitry and the Analogue-to-Digital conversion. The electronic noise becomesapparent during measurement intervals with no thrust input and no thrust stand oscilla-tions, for example before the start of a measurement. Typical electronic noise for a thrustmeasurement is depicted in Figure 3-46.

Figure 3-46: Electronic noise in a thrust measurement signal.



The histogram in Figure 3-47 shows that the probability distribution of the electronic noiseis approximately Gaussian with a mean of -0.05. A histogram can only be fed by a limitednumber of samples. Hence, it is often difficult to ascertain in practice to what degree a realprocess is Gaussian [53]. While the distribution of the electronic noise is not a perfect bellcurve, the approximation as Gaussian noise is still legitimate, because the noise is evenlydistributed around a mean which occurs most often.

Figure 3-47: Histogram of electronic noise obtained from a thrust measurement before the firstignition of the thruster.

Figure 3-48: Top: Frequency spectrum of electronic measurement noise with a standard deviationof σ = 0.025, Bottom: Frequency spectrum of ideal white Gaussian noise also with a standarddeviation of σ = 0.025.



That the electronic noise is also white, can be seen when computing its frequency spectrum.Figure 3-54 compares the frequency spectrum of the electronic noise with ideal Gaussian whitenoise generated in Matlabwith the same variance. The electronic noise is mostly distributedevenly over the entire frequency spectrum. There are only two more dominant frequencies,one at about 50Hz and the other at 130Hz. In general, the electronic noise has a similaramplitude as the ideal noise, so that the it can be approximated well with white Gaussiannoise generated by Matlab. For the uncertainty analysis, electronic noise is generated usingthe random number generator from Matlabwith a standard deviation of σ = 0.008. This isthe average standard deviation of the electronic noise, determined using several thrust tests.

3-6-2 Identification Error

The identification error can be determined by subtracting the simulated from the actualsystem response of a real measurement. The transient response is obtained as before bylooking at the decaying signal after the last firing pulse. The resulting identification erroris shown in Figure 3-49 and looks stochastic. Its histogram shows that the series can beapproximated as zero-mean Gaussian. However, a closer inspection reveals that there aredeterministic elements contained in the signal, like a periodicity of approximately 0.2s.

Figure 3-49: Left: Identification error, Right: Histogram of modeling error.

The deterministic part can clearly be revealed by analyzing the frequency spectrum of theidentification error, which is shown in Figure 3-50. Several peaks are observed: the mostdominant ones occur at about 90Hz and 150Hz.

Figure 3-50: Frequency spectrum of modelling error.



Therefore, a simple modelling of the identification error with white Gaussian random noiseis not possible. The best representation of the physical and random components of thesignal can be obtained by simulating the noise itself with another Autoregressive-MovingAverage (ARMA) process (see Section 2-4-2). Due to the strong random character of thenoise, it is better to identify it with the Matlab System Identification Toolbox instead ofusing a linear least squares approach. Due to the complexity of the noise, only higher orderprocesses are capable of describing it. A model of order 4 can only achieve a fit of 25%. Thebest fit is achieved by using a model of order 27, which results in a goodness of fit of 86%.The simulated model response is shown in Figure 3-51.

Figure 3-51: Measured (black) and predicted (red) output using an ARMA model of order 27.

The coefficients of the identified ARMA process are integrated into the uncertainty evaluationby passing the coefficient vectors φ and θ to the function. As noted in Equation (2-28), thevectors φ and θ describe the random process with

φXk = θYk, (3-30)

where φ represents the coefficients of the autoregressive part and θ of the moving averagepart.

3-6-3 Non-Repeatability Force Transducer

According to the manufacturer Interface Advanced Force Measurement, the SM S-TypeLoad Cell (described in Section 3-2-2) used in the Airbus DS 10N thrust stands has a non-repeatability error of ± 0.01% of the measured output. The repeatability error is defined asthe maximum difference in output when the same load is measured repeatedly under identicalconditions. It is expressed as a percentage of the rated output.3

No probability distribution for the repeatability error is specified by Interface Inc. Thissuggests the use of a uniform distribution. A normal distribution cannot be used, because its

3 See https://www.tml.jp/e/product/transducers/catalog_pdf/terminology.pdf.


https://www.tml.jp/e/product/transducers/catalog_pdf/terminology.pdf


standard deviation would have to be guessed. In the following, the assignment of a probabilitydistribution is entirely avoided by making use of a feature of the measurement uncertaintyevaluation toolbox programmed by Eichstädt [68]. The software provides a tool to evaluateuncertainty caused by a local bias ∆k at each sample influencing the measurement outcome:

Yk = Xk ∗H + ∆k. (3-31)

The repeatability error is incorporated into the uncertainty analysis as a local bias ∆k = 0.01at each sample for a thrust amplitude of 1N.

3-6-4 Firing Noise

The term firing noise is used to describe random fluctuations occurring in the combustionchamber during the firing of the thruster. An exemplary extract of firing noise recordedduring a long firing session is depicted in Figure 3-52.

Figure 3-52: Example of firing noise caused by inconsistent combustion.

Using data from a very long firing test to feed the histogram, it can be shown that the firingnoise is almost ideally Gaussian distributed, as can be seen in Figure 3-53.

Figure 3-53: Histogram of firing noise in a thrust measurement signal recorded during the ontime of a long SSF test.



The frequency spectrum of the firing noise shows that it is evenly distributed, but not withthe same magnitude. The magnitude is larger in the region from 0 to 100Hz. Consequently,firing noise could be simulated using noise of one amplitude from 0 to 100Hz and anotherfrom 100Hz to 500Hz. However, the noise up to 100Hz is not changed by the low-pass filterwhich follows after the deconvolution filter. The noise above 100Hz, on the other hand, isfiltered. Hence, for simulation purposes, it is sufficient to approximate the firing noise withwhite noise of only one amplitude, corresponding to the stronger noise from 0 to 100Hz. Thisnoise is added to the input signal before convolution with the system model.

Figure 3-54: Frequency spectrum of firing noise.

3-6-5 Monte Carlo Simulation

In this section, a sequential Monte Carlo simulation is performed using Matlab codes pro-vided by Eichstädt [72]. The simulation input is generated manually as a thrust pulse with800ms on time and an amplitude of 1N to which random white Gaussian firing noise with astandard deviation of 0.02 is added. The measurement output is calculated by convolutionwith model H11 and electronic noise of standard deviation 0.0008 is added to the syntheticmeasurement signal. The non-repeatability error is included as ∆k = 0.0001 corresponding to0.01% of the nominal thrust amplitude of 1N. The generated measurement signal is displayedin Figure 3-55.

Figure 3-55: Example of one input and output signal including noise used for the sequentialMonte Carlo simulation.



The uncertainty is calculated with a sequential Monte Carlo method with M = 106 trials. Itcan be seen in graph on the left in Figure 3-56 that the calculated uncertainty is constantlyabout 0.035N. The 95% credibility intervals in the graph on the right suggest that the resultingdeconvoluted thrust can be trusted up to a degree of ±0.1N , that is, 10%. This is quite ahigh degree of uncertainty. It is mostly due to the system identification error described withthe ARMA model, as testing the different uncertainty components separately revealed.

Figure 3-56: Uncertainty and 95% credibility intervals for a thrust signal, calculated with asequential Monte Carlo method with 106 trials.

If system identification errors are left out of the simulation, the uncertainty becomes almostten times smaller and the uncertainty margins are reduced to 0.03N as shown in the graphon the right in Figure 3-57. Hence, the uncertainty would only be ± 0.015% of the thrustamplitude. The uncertainty without consideration of identification errors is shown in Figure 3-57. There are two significant peaks. One when the thruster is turned on, and one when thethruster is turned off. Therefore, the uncertainty is dependent on the amplitude of thetransient response, but the effect is overshadowed by the larger error caused by the systemidentification.

Figure 3-57: Uncertainty and 95% credibility intervals for a thrust signal without identificationerror, calculated with a sequential Monte Carlo method with 106 trials.



3-7 Conclusion

The strain gage type thrust stand used by Airbus DS is highly accurate for SSF tests, forwhich it achieves accuracies of 0.3%. However, the strain gage type thrust stand is not suitablefor PMF measurements with on times lower than 100ms, corresponding to firing frequenciesof higher than 10Hz (depending on the chosen on and off times).

The TDMS compensation algorithm used by Airbus DS to evaluate PMF measurements isvery robust. Its performance data were largely confirmed by other departments. The orderreduction is a very efficient and innovative way to extract the main mode of the thruststand. For certain tests, like the ones in Figure 3-28, it still has to be checked why themethod fails. Moreover, it also has to be investigated why it is not possible to recreate thesame measurement results saved to the TDMS data base with the original Airbus DS SciLabmethod. The answer to both questions might be that the coefficients of one test are used toevaluate several others.

For PMF measurements, the NRMS error is always higher than 4%. Depending on thecombination of on and off times, it can rise up to 40%. For on times shorter than 100ms,the error is always higher than 10%. The error decreases for longer on or off times. Thethruster becomes more efficient for longer on times and is most efficient for on times longerthan 500ms. For such cases, the accuracy is about 5%. Thus, the current evaluation methodis sufficiently accurate for cases in which the thruster is fired most efficiently. Nevertheless,the Normalised Root Mean Square (NRMS) error for on times shorter than 100ms is too high.There is still room for improving the accuracy. The test series in Chapter 5 shows that theerror is largest at the flanks of the signal pulses. They are not hit accurately, because thesmoothing effect of the method is too great.

Of the improvement possibilities described, it is recommended to change the input signal forthe system identification to a step input and to reduce the identification order to achieve morenumerical stability. The other adaptation suggested in Section 3-5 will inevitably change thedesign of the filter so that new results can no longer be compared to old results obtainedby the TDMS method. Nevertheless, the use of the numerator coefficients of the reducedorder system results in more accurate results than the manual choice of a double pole at 0.1.Furthermore, a Butterworth low-pass filter allows significantly more flexibility in terms offilter tuning.

As result of the performed Monte Carlo simulation, it should be noted that the uncertaintyis tripled immediately after the thruster is turned on or off. Therefore, the results obtainedfor PMF tests with high frequencies are more uncertain. For the future, it is recommendedto perform more detailed Monte Carlo simulations with a higher number of trials and toincorporate the knowledge into the designing process of new thrusters.

In the short term, it is recommended to increase the sampling rate for PMF measurementsfrom 1000Hz to 5000Hz. Independent of the evaluation method used, higher accuracy can beobtained at the pulse flanks, without changing the algorithm. In addition, the implementationof the Kalman Filter described in Section 4 is suggested to improve the accuracy for shorttests. In the long term, the construction of a piezoelectric thrust stand with a higher firstnatural frequency is recommended for the evaluation of PMF measurements, as described inChapter 2 .


Chapter 4

Kalman Filter for Deconvolution

This chapter discusses the theory that is required to apply the linear optimal Kalman Filter forthe purpose of deconvolution to the dynamic thrust measurement problem. The requirementsneeded to apply the Kalman Filter are checked in Section 4-1. The design of the DeconvolutionKalman Filter suggested by Bora [9] is described in Section 4-1. The adaptations needed toapply Bora’s method to the dynamic thrust measurement problem are presented in Section 4-3. The most important adaptation concerns the choice of the measurement and process noisecovariance matrices for the case of Pulse Mode Firing (PMF) thrust measurements. How tochoose them is discussed in Sections 4-3-2 to 4-3-4. In Section 4-4-1, it is explained how themeasurement state error covariance matrix of the Deconvolution Kalman Filter can be usedto obtain an estimate of the uncertainty in the input thrust. A time-varying analysis of theuncertainty in the input thrust using a Monte Carlo simulation is performed in Section 4-4-2.The analysis is similar to the one performed in Section 3-6, but is repeated here because theuncertainty analysis in Chapter 3 is tailored to Airbus DS data and could not be included inthe public version of this thesis.

4-1 Preliminary Conditions for the Use of the Kalman Filter

To apply the Kalman Filter it is assumed that the measurement signal can be modelledwith an Autoregressive-Moving Average (ARMA) process, introduced in Section 2-4-2. Thisassumptions implies that

- The model effectively describes the persistence in the signal.- The model residuals are random, uncorrelated in time and approximately normallydistributed.

If the model residuals are random, uncorrelated in time and approximately normally dis-tributed, it is likely that the ARMA model also successfully captures the persistence in thesignal. If this random series is white Gaussian, the Kalman Filter is optimal.


94 Kalman Filter for Deconvolution

To check whether the Kalman Filter is optimal to evaluate thrust measurements, the outputof the model simulating the transient system response has to be subtracted from the realtransient system response. This was done in Section ??, using measurement data provided byAirbus DS. It was found that the frequency spectrum of the thrust stand is too complex tobe modelled by a transfer function. The model residues are not white Gaussian. Figure 3-50shows that there are still physical components. However, they are distributed approximatelynormal, as can be seen in Figure 3-49. In Section ??, the measurement noise could be shownto be white Gaussian.Hence, the Kalman Filter’s optimality for the dynamic thrust measurement problem is ques-tionable. The measurement noise is white Gaussian, and a physical model can describe themain persistence in the signal. However, the model residues are only Gaussian, not whiteGaussian. The Infinite Impulse Response (IIR) filters described in Chapter 3 are capable ofreconstructing the unknown input signal, even if the model residuals contain certain deter-ministic elements. Since the same model coefficients are inserted into the Kalman Filter, thefilter should be equally able to perform the deconvolution. However, there is no guaranteethat the result is optimal.

4-2 Test of Bora’s Deconvolution Kalman Filter

To check if the Deconvolution Kalman Filter suggested by Bora is implemented correctly,it is applied to the same test scenario as described in their publication. That the methodis implemented correctly is demonstrated in Section 4-2-1 The sought input signals can bereconstructed with the Deconvolution Kalman Filter. However, two difficulties are encoun-tered. First, the initial conditions prescribed by Bora result in a larger initial error as shownin Bora’s publication. Second, the accuracy of the method depends strongly on the valueschosen for the process noise covariance matrices Q and R. However, no recommendation onhow to choose these values is given by Bora [9]. In the following section, the set-up of thetest scenario is described. Then, the two difficulties are investigated: the dependency of thesolution on the initial values and the covariance matrices Q and R.

4-2-1 Set-Up of Test Scenario

The Deconvolution Kalman Filter is applied to reconstruct an ARMA process x(k), which haspassed through an IIR filter F (z) and is corrupted by measurement noise v(k). The ARMAprocess x(k) is generated by exciting a system G(z) with white noise w(k). The systemG(z) is designed as a low-pass filter using the Yule-Walker equations with cut-off frequencywc = 0.5π and order 10. The measurement noise v(k) is chosen as white Gaussian randomnoise of unit variance. This procedure is illustrated in Figure 4-1.

Figure 4-1: Flowchart showing how Bora tests the Kalman Filter for Deconvolution [9].


4-2 Test of Bora’s Deconvolution Kalman Filter 95

The IIR transfer function F (z), which is used by Bora to add a transient response to theinput signal x(k), is defined as

F (z) = 0.4652− 0.1254z−1−0.3151z−2+0.0975z−3−0.0259z−4

1− 0.6855z−1+0.3297z−2−0.0309z−3+0.0032z−4 . (4-1)

The state vector ζ is initialised as ζ (0, 0) = 0 and the state error covariance matrix asP (0, 0) = I. Bora gives no indication how to choose the covariance matrices Q and R. Forthe reconstruction of his results depicted in Figure 4-2, they are chosen as Q = 1000000 andR = 1.

Figure 4-2: Deconvolution of a random input signal from a noisy signal using Bora’s method.Left: Reconstruction of Bora’s test results, Right: Benchmark test results supplied by Bora [9].

Although the instructions given by Bora are closely followed, it can be seen in Figure 4-2 thatthe imitated results differ from the original results in two ways:

- The absolute error xr − x, which subtracts the input signal reconstructed with theDeconvolution Kalman Filer xr from the original input signal x, is significantly largerfor the initial 20 iterations than claimed by Bora.

- The scale in the left graph is 2x1011 and not only 2, as in Bora’s graph on the right. Thisis because the input signal could only be reconstructed, if Q was chosen significantlyhigher than R. Bora prescribes random noise v of unit variance. Hence R should bechosen as 1. For this case, it was only possible to reconstruct the input signal when Qwas set to values at least 10000 times larger.

4-2-2 Removal of Initialisation Error

It can be seen in Figure 4-2 that the largest error occurs during the initialisation phase of theDeconvolution Kalman Filter. This error is caused by using 0 as initial condition for the stateerror covariance matrix P . For Linear Time-Invariant (LTI) systems, the entries of the matrixP and the Kalman gain converge to constant values after a few iterations. In Figure 4-3, itis illustrated how the Kalman gain K converges to the constant value 1.381 within the first10 iterations. Similarly, all entries of the 16 × 16 matrix P converge to constant values.



Figure 4-3: The Kalman gain converges after 6 iterations and then remains constant.

The large initialisation error can be prevented if the matrix P is obtained in a short pre-run.Usually, the solution for P is obtained by solving the Riccati Equation. The Riccati Equationis an ordinary differential equation of order 1 of the state error covariance matrix P . It canbe derived by combining the update and prediction steps of the Kalman Filter in such a waythat a quadratic equation in P is obtained. However, because the equation is quadratic, itis simpler to find P using a different method instead of the standard Riccati approach: theDeconvolution Kalman Filter is applied to the problem until P has converged to its finalvalue. If this value is then used as as initial guess for P , the initial error is reduced, as canbe seen in Figure 4-4.

Figure 4-4: Deconvolution of random input signal, for which the initial guess for P was deter-mined in a pre-run.

4-2-3 Determination of Q and R

For Bora’s test scenario the value of R is chosen as 1, because the measurement noise v issupposed to be of unit variance. However, no value for Q is suggested by Bora. Because R


4-2 Test of Bora’s Deconvolution Kalman Filter 97

is constant, different Q/R ratios, rather than single values of Q, are used for a series of teststo find a suitable value for Q. The code is run in a loop for Q/R ratios from Q/R = 10−5 toQ/R = 200000 and the absolute error ε = x− x is computed.

In all cases, in which the Q/R ratio is chosen smaller than or close to 1, the DeconvolutionKalman Filter greatly reduces the amplitude of the input signal, so that the reconstructionerror becomes extremely large. The graphs in Figure 4-5 illustrate this. The input signalswere reconstructed using a Q/R ratio of 1 for the graph on the left and 0.1 for the graph onthe right. One can see that the amplitude for the Q/R ratio of 0.1 is smaller than that of 1.When using more values, it can be observed that the amplitude of the input signal convergesto 0, if the Q/R ratio converges to zero. Hence, the Q/R ratio has to be chosen larger than1 for a correct reconstruction of the amplitude.

Figure 4-5: Left: Result obtained by the Deconvolution Kalman Filter for Q/R = 1, withQ = 1000 and R = 1000, Right: Q/R = 0.1, with Q = 100 and R = 1000

The effect of Q/R ratios from 1 to 200000 on the estimation error is shown in Figure 4-6. Thetwo graphs show the relative error ‖x−x‖‖x‖ , because the absolute error varies greatly for differentQ/R ratios. The graph on the left is plotted to scale and shows dominant random fluctuationsfor the relative error for different Q/R ratios. The graph on the right is logarithmic and showsthat the error decreases linearly for Q/R from 1 to 1000. Evaluating the information obtainedin Figure 4-6, it can be attempted to find a general rule how to choose the Q/R ratio. Therandom fluctuations of the error in both graphs show local minima. For example, for a Q/Rratio of 1.59e5 the error is only 10−4 compared to the error of 10−2, which occurs mostoften. This suggests that an optimal Q/R ratio could be obtained using a gradient descentalgorithm, which minimises the relative error. This is only possible when using syntheticdata. Because, for real test cases, the input signal is unknown. The chance that a Q/R ratio,which is optimal for a test case, is also optimal for a corresponding real test is very small.For example, in Figure 4-6, the error to the left and right of the minimum error found at1.59e5 is a hundred times larger. Hence, finding a global minimum for a real measurementby employing a gradient descent method on a corresponding test case is unlikely.



Figure 4-6: Relative error ‖x−x‖‖x‖ for different Q/R ratios. Left: Unscaled plot, Right: Double

logarithmic plot showing a linear behaviour for Q/R ratios between 1 and 1000.

For a practical application of the method, it is more helpful to realise that from a certainQ/R ratio upwards, an upper boundary for the estimation error can be found. In Figure 4-6,there is a boundary for the estimation error of 0.01 for Q/R ratios from 10000 to 200000.Due to the wide range of ratios for which this rule applies, it is probable that a similar ratiois applicable to a real test. In a similar way, a heuristic choice will be developed for thrustmeasurements using synthetic data in Section 4-3-2.


4-3 Deconvolution Kalman Filter for Dynamic Thrust Measurements 99

4-3 Deconvolution Kalman Filter for Dynamic Thrust Measure-ments

In this section, the Deconvolution Kalman Filter is adapted for the application on the dynamicthrust measurements problem. Two problems were encountered. One was already elaboratedin the Section 4-2-3 and is widely known to be the main drawback of the Kalman Filter: thenecessity to choose the covariance matrices Q and R correctly. The second problem is that thelow-pass filter integrated into the augmented state-space system sometimes causes oscillationsin the reconstructed input signal. A case where this has happened is shown in Figure 4-7and Figure 4-8. Both effects are investigated in the following sections for synthetic thrustmeasurement data, which was generated exactly as described in Chapter 5. The transferfunction used to model the system dynamics of the thrust stand is

H(z) = 0.5887z2 + 0.2072z + 0.02314z2 − 1.15 + 0.9771 . (4-2)

The input signal is a sequence of four pulses with an on time of 30ms and an off time of 30msand a rise and fall time of 2ms. In the visual comparisons, always the third pulse is lookedat.

4-3-1 Choosing the Integrated Low-Pass Filter

When applying the Deconvolution Kalman Filter on thrust measurements, it is found thatthe Butterworth low-pass filter integrated into the state-space equations of the DeconvolutionKalman Filter behaves differently than it does in combination with an inverse filter. This isexemplified in Figure 4-7 where a thrust signal reconstructed with the Deconvolution KalmanFilter is compared with a thrust signal recreated with the Inverse Transfer Function method,which was developed in Section 3-5. For both graphs, a Butterworth filter of order 6 witha cut-off frequency 0.1fNyquist has been used. This filter is capable of removing the entiretransient system response, but causes overshoot. This effect is visible in the graph on the rightin Figure 4-7. However, integrated into the state-space system of the Deconvolution KalmanFilter, this low-pass filter has a completely different effect. The original transient response isonly slightly reduced in amplitude, but remains almost intact, as can be seen in the graphon the left in Figure 4-7. In order to be able to predict the behaviour of the Butterworthlow-pass filter integrated into the Kalman Filter, different filters are used and the result aredocumented.



Figure 4-7: Different effects of Butterworth low-pass filter. Left: Integrated into the augmentedstate-space system of the Deconvolution Kalman Filter, Right: Applied in combination with aninverse filter.

It is found that the effect of the Butterworth low-pass filter depends on the cut-off frequencyand on the order. Figure 4-8 shows four thrust signals reconstructed with four differentlow-pass filters integrated into the augmented state-space system. The two graphs on thetop illustrate that the overshoot is dependent on the cut-off frequency. Both graphs wereobtained with Butterworth filters of order 3. For the graph on the left, the cut-off frequencyis 0.3fNyquist and for the graph on the right 0.1fNyquist. The amplitude of the oscillation inthe graph on the right is higher. The graph in the bottom left-hand corner shows that theovershoot depends on the order. The cut-off frequency is 0.3fNyquist, as in the graph above.However, the ringing has a higher amplitude due to the higher order of 6. The graph in thebottom right-hand corner is obtained with a Butterworth filter of order 2 and has a highcut-off frequency of 0.8fNyquist. This low-pass filter has an effect on the highest 20% of thefrequency range and no overshoot occurs.

Figure 4-8: Thrust inputs deconvoluted with the Kalman Filter using Butterworth filters withdifferent order and cut-off frequency. In the following abbreviation the first number describesthe order, the second number the cut-off frequency. Top left: Butterworth(3,0.3), Top right:Butterworth(3,0.1), Bottom left: Butterworth(6,0.3), Bottom right: Butterworth(2,0.8).

From the results in Figure 4-8, it can be concluded that the ideal low-pass filter for theaugmented state-space system of the Deconvolution Kalman Filter is a filter which has only



little effect on the entire bandwidth of the signal and is of low order. This insight led tothe idea to include an integrator instead of a Butterworth filter into the state-space system.In control theory, an integrator is usually described in the frequency domain as I = 1

z . Itis used to remove the steady-state error caused by a filter. The steady-state error is thedifference between input and output signal of a system as time goes to infinity. It can be seenin Figure 4-9 that the Inverse Transfer Function method has a steady-state error while theDeconvolution Kalman Filter with integrator has none.

Figure 4-9: Left: Steady-state error caused by the Inverse Transfer Function method, Right: Nosteady-state error occurs for the Deconvolution Kalman Filter with integrator.

The effect, which the integrator has on the Deconvolution Kalman Filter, can be explainedmathematically by adapting results obtained by Chen [8]. The steady-state error in thereconstructed input thrust can be defined as

X (z)− X (z) with X (z) = X (z)H (z)HC (z) , (4-3)

where X is an estimate for the thrust signal, H(z) is the thrust stand transfer function andHC(z) is the compensator. The steady-state error of a system can be calculated using thefinal value theorem:

e (∞) = limz→0

zH (z) , (4-4)

where e is the steady-state error. To obtain an equation how the steady-state error can beremoved with the compensator HC in Equation (4-3), the final value theorem can be adaptedto

e (∞) = limz→1

(z − 1)(X (z)− X (z)

), (4-5)

because H(z)HC(z) ≈ 1, the steady-state error vanishes for the limit to 1. The multiplication

with (z − 1) ensures that the steady state-error still goes to 0. Inserting Equation (4-3) intoEquation (4-5) results in

limz→1

(z − 1)

X (z)− X (z)︸︷︷︸error

= limz→1

(z − 1)

X (z) (1−H (z)HC (z))︸︷︷︸X(z)=X(z)H(z)HC(z)

= 0, (4-6)



which is a general equation for the steady-state error for a compensator HC dependent onthe input signal X(z).Suitable input signals X(z) to simulate a thrust pulse are the unit step function and the unitramp function. The general z-transform of a unit step and unit ramp input is:

X (z) = A (z)(1− z−1)m (4-7)

with m = 1 and A(z) = 1 for the unit step and m = 2 and A(z) = Tz−1 for the unit ramp.Inserting, for example, the unit step into the steady-state error equation (4-6) results in

limz→1

(z − 1) 1(1− z−1) (1−H (z)HC (z)) = 0. (4-8)

From Equation (4-8), it can be seen that the limit z → 1 can only be 0 if

1−H (z)HC (z) =(1− z−1

), (4-9)

otherwise it becomes infinity. Rearranging Equation (4-9) results in

H (z)HC (z) = z−1. (4-10)

Therefore, the compensator must fulfil

HC (z) = z−1

H (z) = 1zH (z) (4-11)

in order to avoid a steady-state error. One can see that the term z−1 is needed to have asteady-state error of 0 for a unit step input. This term causes a time delay of 1 sample. Ina similar way, it can be found that higher order polynomials of z in the denominator arerequired to remove the steady-state error for a different input signal. A polynomial of order2 can be used for a ramp input.Although the derivation is different, the obtained result is common knowledge in DigitalSignal Processing (DSP). The steady-state error for an open loop system is known to bedependent on the number of times 0 occurs in the denominator. This number is also calledthe type of the system. The type of a feedback system is defined as the power n of z whichcan be factored from the denominator:

H (z) = N (z)znD (z) . (4-12)

How many additional poles at 0 is required to avoid a steady-state error for different inputsignals is shown in Figure 4-10.

Figure 4-10: Steady-state system error for different types of feedback systems [14].



The thrust input signal can either be approximated with a step input or a ramp input. Theramp input is more realistic, because it considers the rise and fall time of the thrust. Thetransfer function of the Deconvolution Kalman Filter is always bi-proper and its poles occurin complex conjugate pairs for every mode of the thrust stand. Hence, there are no 0 polesand the Deconvolution Kalman Filter transfer function is of type 0. As figure Figure 4-10shows, for type 0 systems, a finite steady-state error can occur for a step input [73]. Adding anintegrator to the augmented state-space system increases the system type by 1 and removesthe possibility of a steady-state error. To include an integrator 1

z into the state-space system,which removes the steady-state error for a step input, the matrices AG, BG, CG and DG

described in Equation (2-97) have to be chosen as

AG= 0 BG = 1 CG= 1 DG= 0. (4-13)

To include a double integrator 1z2 into the state-space system, which removes the steady-state

error for a ramp input, the matrices AG, BG, CG and DG have to be chosen as

AG =[

0 01 0

]BG =

[10

]CG =

[0 1

]DG = 0

. (4-14)

In the following, the second, double integrator is chosen to remove the steady-state errorcaused by the ramp-like thrust input.

4-3-2 Heuristic Tuning of Q and R

In this section, a heuristic approach is developed to choose the filter tuning parameters Q andR for the Deconvolution Kalman Filter for the application in dynamic thrust measurements.A good guess for the measurement noise covariance matrix R is easily obtained from thesensor noise. The statistics of the electronic sensor noise can be determined during periodsin which the sensing apparatus is turned on, but no thrust force is exerted, see Section 3-6-1.This is the case at the beginning and end of each thrust measurement. The variance and meanof the electronic noise can be considered to be time-invariant. Hence, the measurement noisecovariance matrix R is kept constant at a value typical for electronic measurement noise, forexample σR = 0.005N, in the following test series.

The tuning of the process noise covariance matrix Q is more difficult. All the model uncer-tainties and inaccuracies as well as the noise affecting the process should quantitatively beincorporated into Q [19]. Therefore, Q is determined in the same way as in Section 4-2-3,by reconstructing an input signal using different Q/R ratios. Instead of using an exampleby Bora [9], synthetic thrust measurements with known input are used. How the synthetictest case is generated is explained Chapter 5. Because R is constant, the Q/R ratio can beused as a single indicator for the chosen values of Q and R. In Figure 4-11, input signalsreconstructed for the Q/R ratios 0.1, 1, 10 and 100 are depicted. As before, it can be seenthat for Q/R ratios smaller than or equal to 1 the resulting amplitude is so inaccurate that itcannot be used. The accuracy increases for Q/R ratios larger than 1, but it is only for ratiosabove a 100 that the thrust pulse amplitude reaches an acceptable level of accuracy.



Figure 4-11: Thrust signal recovered by Deconvolution Kalman Filter for different values of Qand constant value of R.

In Figure 4-12 is depicted how the error decreases for ratios higher than 1, where the Nor-malised Root Mean Square (NRMS) error, defined in Equation (2-48), is plotted for Q/Rratios from 1 to 1000. It can be seen in the graph on the left-hand side that the error de-creases almost exponentially for ratios until 100 and then levels off. To find the limit of theaccuracy, the relative error ‖x− x‖ / ‖x‖ between the input signal x and the reconstructedsignal x was calculated for Q/R ratios from 1 to 12000 in the graph on the right-hand side.



Figure 4-12: Left: NRMS error, Right: Relative error ‖x−x‖‖x‖ for different values of Q and

constant value of R = 0.005.

It can be seen that the error decreases up to Q/R ratios of 10000, where it becomes constantwith a residual error of 0.006. The value reflects the standard deviation of the electronic noiseadded to the signal of σR = 0.005. This suggests that the noise level of the electronic noise isthe limit of the accuracy.

A Q/R ratio of approximately 10000 can be obtained by using the standard deviation of themeasurement signal as an estimate for Q. This is illustrated in Figure 4-13, where a testwith an on time of 30ms and an off time of 30ms was plotted. The standard deviation ofthe test from the first ignition of the thrust till it is turned off is Q = 54, as can be seen inthe graph on the left. The electronic noise contained in the signal has a standard deviationof R = 0.005, as can be seen in the graph on the right. Hence, the Q/R ratio is about 104.This choice for Q falls directly within the region of the lowest possible error in Figure 4-12as indicated by the red line. Hence, the standard deviation of the measurement signal can beused as an indication for the correct value of Q. However, it is more reliable to use only onepulse than an entire series. Otherwise, long off times between the firing pulses can lower thestandard deviation.

Figure 4-13: Left: First four pulses used to determine the value of Q, Right: White Gaussianrandom noise simulating the electronic noise, used to determine the value of R.

Using this knowledge, the following heuristic choice for the tuning parameters Q and R indynamic thrust measurements can be recommended:

Q = σ2Q R = σ2

R, (4-15)

where σQ =√

var (y∆t) is the standard deviation of the measurement signal from the firstignition of the thruster till it is turned off for the last time, if the off time is shorter thanthe time it takes for the transient response of the thrust stand to subside. If the off timeis longer, σQ =

√var (y∆t) is the standard deviation of a decaying pulse. σR =

√var (yt0)

is the standard deviation of the system noise measured before the first ignition of the thruster.



4-3-3 Rational Tuning of Q and R

A completely different approach to use the covariance matrices Q and R for calibration issuggested by Saha [19]. Instead of estimating the tuning parameters, he suggests the usage oftwo metrics, J1 and J2, by which the designer can manually tune the linear optimal KalmanFilter. A higher value of the metric J1 improves the sensitivity of the filter performance whilean increase in the metric J2 improves the filter robustness. Thus, the designer can eitherchoose the property which fits him best or find a compromise by balancing the two factors:

J1k = trace{

(Ak +Bk +Rk)−1Rk}

J2k = trace{

(Ak +Bk)−1Bk}, (4-16)

where Ak and Bk are defined as Ak = CkAkPkATkC

Tk and Bk = CkQkC

Tk . The two metrics

can be derived by rearranging the basic equations of the linear optimal Kalman Filter. Thefiner details of the derivation can be found in Equations (10-15) in Saha’s publication [19].In essence, the metric J1 measures the effect of the measurement noise covariance matrix Rk,and the metric J2 the effect of the process noise covariance matrix Qk.

In Figure 4-3, it was shown that the Kalman gain and measurement state error covariancematrix P converge to constant values after a few iterations because the model is a LTI system.Therefore, the metrics J1 and J2 similarly converge to constant values for one set of valuesof Q and R. In order to plot a graph, such as the ones in Figure 4-14, from which suitablevalues for Q and R can be obtained by use of the two metrics, Saha suggests a constant valueof R as initial choice Q0 = R and then the calculation of both metrics for Q = 10p · Q0,where p ∈ N is contained in the interval [−10, 10] and the x-axis values are computed withnq = log (trace {Bk}).

The values of both metrics for the p values of Q and R for the Deconvolution Kalman Filterapplied on a thrust test are plotted in the graph on the left in Figure 4-14.1

Figure 4-14: Left: Error metrics J1 and J2 for the Deconvolution Kalman Filter applied to athrust test, Right: Benchmark results provided by Saha [19] for the Kalman Filter (green) andthe Extended Kalman Filter (red).

Figure 4-14 suggests a choice of Q of the same or similar magnitude as R for the best trade-off between robustness and sensitivity. From Section 4-3-2 it is known, however, that such a

1 The value for R = Q = 0.005 was an outlier in J2 and had to be removed.



choice does not result in the correct amplitude when using the Deconvolution Kalman Filter.Hence, Saha’s method cannot be used for the Deconvolution Kalman Filter to find a bettertrade-off between Q and R, but it shows that for high choices of Q, the Deconvolution KalmanFilter is more robust. Another advantage of Saha’s method is that not only single values canbe chosen for Q and R but also full matrices, as can be seen in Equation (4-16). Their effectcan still be evaluated with the same metrics. Normally, it is very difficult to evaluate theeffect of a full covariance matrix.

4-3-4 Automatic Tuning of Q and R

The Autocovariance Least Squares (ALS) method was developed by Rajamani in his PhDthesis [74] and can be used to determine the values for Q and R [20]. It is considered by manyresearchers as the most promising approach to determine the covariance matrices [75]. Themethod developed by Rajamani [20] was adapted for the Kalman Filter by Aakesson [75], butthe details are beyond the scope of this thesis. The main idea behind the ALS method is toaddress the covariance determination problem by focussing on the innovations of the KalmanFilter. The innovation is defined as qk = yk −Cxk,k−1 and originates from the linear optimalKalman Filter equation:

αk+1,k+1 = αk+1,k +Kk+1 (yk+1 − Ck · αk+1,k)︸︷︷︸innovation

, (4-17)

The autocovariance, which is minimised by the method, is defined as the variance of thisinnovation qk:

var(qk) = E[qkq

Tk

]. (4-18)

Application of GNU Octave ALS Toolbox

To apply the version of the ALS method available as a GNU Octave Toolbox2 for the Decon-volution Kalman Filter, a few things have to be noted. First of all, for the matrices A, B andC, the model matrices AH , BH and CH from Equation (2-94) and Equation (2-95) have tobe used and not the extended matrices in Equation (2-98):

AF =[

0 1.00−0.982880 1.149515

]CF =

[−0.529724 0.841263

]. (4-19)

BF is an optional choice in the toolbox and is left out.3 The toolbox requires a coefficientvector G for the random noise vt, which is not considered for the Deconvoluted Kalman Filtermodel here, so G was set to 1, G = [1, 1]T . As initial guesses Q0 = 10 and R0 = 0.005 areused. Then, the method requires an extract from the assumed input and output signal usedto identify the model. The best results are obtained by using a step input as input function,and the decaying curve after the last pulse in the output signal as output signal. Both signalsare depicted in Figure 4-15.

2 The GNU Octave package of the Autocovariance Least Squares Toolbox can be downloaded from http://jbrwww.che.wisc.edu/software/als/download.html.

3 The entries of CF are computed with ai=ai−a0bi, where ai and bi are the coefficients of the numeratorand denominator polynomial, respectively.


http://jbrwww.che.wisc.edu/software/als/download.html

http://jbrwww.che.wisc.edu/software/als/download.html


Figure 4-15: Input Signal and Output Signal used for Autocovariance Determination.

Under these conditions, the ALS method estimates the covariance matrices as

Qest = 3.081590Rest = 1.659573e−4,

(4-20)

which gives a Q/R ratio of about 18000. Inserting Qest and Rest into the DeconvolutionKalman Filter results in a relative error of 0.00624, which is slightly smaller than the smallesterror obtained by the heuristic choice, which is 0.00674.

The ALS method is integrated into the code of the Deconvolution Kalman Filter right afterthe identification step when the input signal, output signal and the corresponding modelcoefficients are determined. To increase the probability of reliable results, the heuristic choicefor Q and R suggested in Section 4-3-2 is used as initial guess for Q and R. Even so, duringtesting, the ALS method produces inconsistent results. Occasionally, it returns a largerestimate for R than for Q, which results in far too an low amplitude. The reason for thisestimation error could not yet be identified. Therefore, the autocovariance matrix is alwaysdisabled as it is not fit for routinely performed thrust evaluation.

4-4 Uncertainty Analysis for the Deconvolution Kalman Filter

4-4-1 Uncertainty Analysis Using the Deconvolution Kalman Filter

During the application of the Deconvolution Kalman Filter, the variance of the measurementstate error is updated in each iteration in form of the state error covariance matrix P :

Pk,k = E

[(ζk,k − ζk

) (ζk,k − ζk

)T ]. (4-21)

The variance describes the dispersion of a probability density function around the expecta-tion. It can be interpreted to be the uncertainty associated with the expected value. In


4-4 Uncertainty Analysis for the Deconvolution Kalman Filter 109

the Deconvolution Kalman Filter, the state-space system is augmented to contain the thrustinput signal. The state prediction is performed with

ζk+1,k = Aζk,k with ζk =[αTk xTk βTk wTk

]T, (4-22)

where A and ζ are defined as in Section 2-5-4. The corresponding state error covariancematrix P contains the entries

P =

σ2α1 σ2

α1,α2 · · · σ2α1,x

σ2α2,α1

. . . · · · σ2α2,x

...... σ2

αn

...σ2x,α1 σ2

x,α2 · · · σ2x

, (4-23)

where σ2x is the variance calculated for the input signal, and σ2

x,αiare the covariances between

the input signals and the other states. To retrieve an estimate for the uncertainty attachedto the thrust input, the covariances have to be evaluated as well. This can be done byusing the conditional covariance theorem [76]: if a random vector Z with multivariate normaldistribution can be partitioned in two components X and Y , then the mean and variance ofZ can also be partitioned into a mean vector µ and a variance-covariance matrix Σ:

µ =(µXµY

)Σ =

(ΣX ΣX,Y

ΣY,X ΣY

). (4-24)

The conditional variance-covariance matrix of Y given that X = x is equal to the variance-covariance matrix of Y minus the terms involving the covariances between X and Y :

var (Y |X = x) = ΣY − ΣY XΣ−1X ΣXY . (4-25)

The theorem can be applied directly to the matrix P by abbreviating the state vector as

ζ =[α1 · · · αn x

]=[

Λ x], (4-26)

where the state variables αi were replaced with the vector Λ. In the same way, the matrix Pcan be summarised as

P =[

ΣΛ ΣΛ,xΣx,Λ Σx

], (4-27)

where the sub-matrices of P are defined as

ΣΛ =

σ2α1 · · · σ2

α1,αn

... . . . ...σ2αn,α1 · · · σ2

αn

Σx = σ2x (4-28)

and

Σx,Λ =[σ2x,α1 · · · σ2

x,αn

]ΣΛ,x =

σ2α1,x...

σ2αn,x

. (4-29)



Using Equation (4-28) and (4-29), the variance of the input state x can be calculated with

var (x|Λ = Λk) = Σx − ΣxΛΣ−1Λ ΣΛx, (4-30)

given a realisation of the coefficient vector Λ = Λk. Using Equation (4-30), the DeconvolutionKalman Filter can automatically calculate the uncertainty attached to the each reconstructedthrust sample during every iteration. If the Deconvolution Kalman Filter is applied to a LTIsystem, the measurement state error covariance matrix P converges to a constant value aftera few iterations. Hence, only one estimate for the uncertainty in the input thrust signal canbe obtained with Equation (4-30).To test the developed method, the 95% credible intervals were computed for a thrust signalof amplitude 10N containing white Gaussian firing noise of 0.3N and electronic measure-ment noise of 0.008N. The Deconvolution Kalman Filter estimates a variance of the error inthe input signal of 0.1106N, which corresponds to 95% credible intervals of ±0.1836. Thisuncertainty margin is displayed in Figure 4-16.

Figure 4-16: Input thrust reconstructed with the Deconvolution Kalman Filter and 95% credibleintervals extracted from the state error covariance matrix P .

To validate the obtained estimate, a Monte Carlo simulation was performed for the sametest conditions. The simulation returned 95% credible intervals of width ±0.1N, as shownin Figure 4-19. Hence, the credible intervals calculated by the Deconvolution Kalman Filterare validated. The Monte Carlo simulation calculates the uncertainty for each sample and isconsequently more accurate than the Deconvolution Kalman Filter, which calculates only oneuncertainty margin for one entire test. However, though less accurate, the credible intervalscalculated by the Deconvolution Kalman Filter still provide a realistic upper boundary for theuncertainty contained in the input thrust. Moreover, they can be obtained with significantlyless computational effort than with the Monte Carlo method. The performed Monte Carlosimulation is described in the next section.



Figure 4-17: 95% credible intervals attached to the deconvoluted input signal calculated with asequential Monte Carlo method with 106 trials.

4-4-2 Uncertainty Analysis Using an Efficient Implementation of the MonteCarlo Method

The Deconvolution Kalman Filter provides the variance of the reconstructed input signal,but only in a time-invariant form. To obtain a time-resolved description of the uncertainty, aMonte Carlo simulation has to be performed. For this, a toolbox developed by Eichstädt [68]is used. The toolbox is based on the same assumptions as used for the Kalman Filter:

- Knowledge of the system outputs y[n] is available in form of discrete-time observations.- The measurement system is linear and time-invariant.- The observational noise process ε[n] can be modelled as a stationary zero-mean ARMAprocess. (That this last requirement is fulfilled, was shown in Section 3-6-2)

The covariance matrix Uba, required for the toolbox, is calculated as described in Section2-6-3. As input signal, a step function is used without additional noise. The output signalis calculated using the transfer function described by Equation (4-31). Noise of varianceσoutp = 0.008 is added, representative for the electronic measurement noise. The used transferfunction is the same as used for all Kalman Filter tests so far:

H(z) = 0.5887z2 + 0.2072z + 0.02314z2 − 1.15 + 0.9771 . (4-31)

The filter covariance matrix Uba obtained in this way is

Uba = 10−3 ·

0.081738 −0.139598 0.0579391 −0.000881 0.000919−0.139598 0.281667 −0.142762 0.002297 −0.0029220.057939 −0.142762 0.085481 −0.001416 0.002030−0.000881 0.002297 −0.001416 0.000066 −0.0000660.000919 −0.002922 0.002030 −0.000066 0.000093

. (4-32)

The mean of the coefficient vectors resulting from the used Monte Carlo simulation are:

a =[

0.595895 0.156678 0.072552]

b =[

1.000000 −1.148859 0.981404]. (4-33)



The uncertainty is evaluated using Equation (4-32) and (4-33) for a test case with an on timeof 800ms and a total signal length of 2000 samples, as displayed in Figure 4-18. Firing noiseof amplitude 0.3N is added to the input signal during the firing of the thruster and electronicmeasurement noise of amplitude 0.008N is added to the entire resulting measurement signal.

Figure 4-18: Input and output signal used for the sequential Monte Carlo simulation in state-space form (SMC-SS).

Using a sequential Monte Carlo method in state-space form (SMC-SS) with 106 trials, theuncertainty associated with the reconstructed input signal was calculated and depicted inFigure 4-19.

Figure 4-19: Uncertainty associated with the deconvoluted input signal calculated with a se-quential Monte Carlo method in state-space form (SMC-SS) with 106 trials.

When comparing the measurement signal inFigure 4-18 and the associated uncertainty inFigure 4-19, it can be seen that the uncertainty is dependent on transients in the thrust inputsignal. It is highest after big changes in amplitude, which occur shortly after the thruster isturned on at 0.2s and off at 1.0s.

Therefore, the overall measurement uncertainty is dependent on the PMF frequency of thethruster and can be assumed to be higher for higher PMF frequencies. To illustrate this, a



PMF test case with an on and off time of 100ms is shown in Figure 4-20. The uncertainty,shown in the graph on the right-hand side, takes about 20ms to decay after the thruster hasbeen turned on or off. Hence the uncertainty is larger for PMF tests with shorter on or offtimes.

Figure 4-20: Left: Input and output signal for a PMF test with an on and off time of 100ms,Right: Associated uncertainty.

Figure 4-19 and Figure 4-20 show that the uncertainty is higher after an abrupt change inthe thrust signal. These abrupt changes are caused by the ignition of the thruster. Hence,the accuracy of a PMF engine could be improved if the firing of the engine was less abrupt.Figure 4-21 show the uncertainty for one thrust pulse, for which the rise time of the thrustis tripled: the thrust needs 6ms instead of 2ms to reach 10N. The high uncertainty peaksobserved in Figure 4-19 and Figure 4-20 have almost vanished.

Figure 4-21: Uncertainty for a single thrust pulse, for which the rise of the thrust has beentripled from 2ms to 6ms.

One can conclude that a fast pulsed firing of the thruster results in a significantly higheruncertainty. The uncertainty can be reduced by increasing the turn on or turn off time of thethruster. However, this requires a change in the valves of the rocket engine, which is difficult



to achieve. Nevertheless, the results obtained by the uncertainty analysis can be taken intoaccount to improve the PMF accuracy of thrusters. For example, when defining how a thrustlevel is established, combinations which result in high uncertainty can be avoided.


Chapter 5

Test Results

In this chapter the Deconvolution Kalman Filter and the Inverse Transfer Function method(an improved version of the TDMS method) are compared to the Matlab edition of theTDMS method, which is used by Airbus DS in Lampoldshausen to evaluate thrust mea-surement of 10N Hydrazine thrusters. The methods are tested for surrogate test scenarioscovering the entire range of on and off times of the Airbus DS thrusters.1 To identify theweight and source of different errors, tests with different configurations are performed. First,only the parts of the deconvolution filters compensating the transient thrust stand responseare compared in Section 5-1-1. Second, the effect of random noise on the methods is in-vestigated in Section 5-1-2. Third, the influence of the low-pass filters on the measurementaccuracy is tested in Section 5-1-3. The ability of the three methods to cope with firing noiseis tested in the most realistic test scenario in Section 5-2. Firing noise is left out in fromtests in Section 5-1-1 to 5-1-3 as including a strong random component into the measurementsignal makes it more difficult to evaluate the effect of low-pass filter and deconvolution filter.Finally, the computational effort of all methods is compared in Section 5-3 and the test resultsare summarised in Section 5-4.

5-1 Performance of Deconvolution Filters in Test Scenarios

The test scenario by which the digital deconvolution filters are validated and compared isshown in Figure 5-1. An input signal (1) is created by manually prescribing a periodic seriesof at least four rectangular or trapezoidal pulses shown in Figure 5-2. The rise time and falltime of a thrust pulse is chosen realistically as 2ms. This is done to investigate how well themethods can capture short rise times. Moreover, the rise time of many thrusters is shorterthan 5ms. A series of four pulses is chosen to evaluate the effect of different on and off timeson the total error. A periodic series with the same on and off times is selected, because ananalytical error analysis using Fourier series is not possible for aperiodic cases (as describedin Section 3-3-9). White Gaussian firing noise (2) with a standard deviation of approximately

1 The entire performance range of Airbus DS thrusters is publicly available in [13].


116 Test Results

3% of the nominal thrust value can be added optionally during the on times of the thruster. Itis optional, because it is easier to compare most features of the methods without the presenceof firing noise. Firing noise is included into the evaluation in the most realistic test scenariodescribed in Section 5-2.

Figure 5-1: Flowchart illustrating how synthetic data are generated to compare different decon-volution methods.

Figure 5-2: Left: Self-generated input pulses with 10ms on and 100ms off time, Right: Measure-ment signal generated with the (n,n) or (n,n-1) transfer function H(z). The oscillation resultingfrom both models is almost identical.

Assuming that the thrust stand can be modelled as an Linear Time-Invariant (LTI) system,a transient thrust stand response can be integrated into the surrogate measurement signalby convolution with a transfer function (3). The following two discrete 2nd order transferfunctions are chosen to imitate the behaviour of a thrust stand:

(n, n) : 0.5887z2 + 0.2072z + 0.02314z2 − 1.15 + 0.9771 (n, n− 1) : 0.5887z + 0.2295

z2 − 1.15z + 0.9761 . (5-1)

Both transfer functions are designed to have a single main mode at 0.3fNyquist. For a samplingrate of 1000Hz this means approximately 150Hz. The main mode at 0.3fNyquist is constructed


5-1 Performance of Deconvolution Filters in Test Scenarios 117

with a high gain of approximately 30 and strong damping for all higher frequencies. Thefrequency response of both transfer functions is shown in Figure 5-3.

Figure 5-3: Frequency response of the two test models. Left: (2,2) model, Right: (2,1) model.Both models are constructed to have the same frequency response.

After convolution with the transfer function, white Gaussian random noise (4) with a stan-dard deviation of 0.05% of the nominal thrust value is added, so that an artificial thrustmeasurement signal (5) of the thrust stand is created. The deconvolution filters (6) can beapplied to this signal to obtain an estimate x (7) of the input thrust x.

The two transfer functions described in Equation (5-1) are needed, because the TDMS andInverse Transfer Function Filter are programmed for a transfer function of denominator degreen and numerator degree n − 1 (from now on abridged with the tuple (n,n-1))2, while theDeconvolution Kalman Filter needs a (n,n) transfer function for the proper set-up of itsstate-space system. Hence, either a (n,n-1) transfer function has to be re-identified using a(n,n) model for the Deconvolution Kalman Filter, or a (n,n) model has to be re-identifiedby an (n,n-1) model for the two IIR filter methods. If a (n,n-1) transfer function is re-identified using a (n,n) model, the numerator coefficient of z2 is identified as a very smallconstant (10e-10) which can adversely affect the calculations. If a (n,n) transfer function isre-identified using a (n,n-1) model, the transfer function can be only identified correctly forthe Deconvolution Kalman Filter. For the two IIR filters, the model can at best be identifiedup to a time delay of one sample. This is due to the fact that if a transfer function withone zero has to describe a transfer function with two zeros, the missing zero is interpreted asa time delay, see Chapter 2-3-3 and Figure 5-4. Therefore, two different transfer functionshave to be used to ensure that the results are not affected by system identification errors.The measurement signal for the IIR filter methods is created with a (n,n-1) transfer functionand for the Deconvolution Kalman Filter with a (n,n) transfer function. In this way, bothdeconvolution filters can possess a full and accurate knowledge of the system dynamics andno time delay occurs.

2 This is not a conventional notation, but merely introduced in this thesis to differentiate between the twotransfer functions.


118 Test Results

The deconvolution methods are evaluated by comparing the NRMS error statistic describedin Equation (2-48) of x and x. To allow a better comparison of the methods, the deviationof the perfect fit in percent is used, instead of the goodness of fit. This changes the NRMSstatic such that 0% means a perfect fit and 4% means a deviation of 4 % from a perfect fit.The changed Normalised Root Mean Square (NRMS) statistic can be defined as

εfit = 100− 100

1− ‖x− x‖∥∥∥∥x− 1n

n∑i=1

x (i)∥∥∥∥ , (5-2)

where x is the test input and x is the estimate of thrust input obtained by the deconvo-lution method. The NRMS errors are always computed for the full length of the createdmeasurement signals. For a signal consisting of four consecutive pulses and their off times,the measurement signal length is ∆t = 4 ·ton+4 ·toff . In this way, the calculated error is morerepresentative of the total error occurring during Pulse Mode Firing (PMF) measurements.In visual comparisons, like in Figure 5-6, only the third pulse is looked at. The system istested for a set of input pulses with an amplitude of exactly 10N and varying on and off times:5ms, 10ms, 50ms, 100ms and 1s. This set is chosen to cover the whole range of PMF tests, inwhich the thrusters are tested by Airbus DS (see Figure 3-9) and to investigate the influenceof the pulse length on the measurement accuracy.

5-1-1 Comparison of Deconvolution Filters

At first, only the component of the deconvolution methods compensating the transient thruststand response is evaluated by excluding as many external influences as possible. No mea-surement noise is added. Therefore, no low-pass filters are needed to complement the com-pensation filters. Although the compensation filters still amplify frequencies higher thanapproximately 0.4fNyquist, there is no noise contained in the signal which can be amplified.Hence, the low-pass filters can be left out to avoid their effect on the signals.

Figure 5-4: Delay of one sample caused when identifying a (n,n) model with a (n,n-1) transferfunction.



While the low-pass filters can easily be disabled for the Inverse Transfer Function methodand Deconvolution Kalman Filter, disabling the low-pass component of the original TDMSmethod poses a problem. In the TDMS method, noise amplification is avoided by manuallyadding a pair of zeros. By omitting these additional two zeros, the design of the TDMS filteris somewhat crippled. However, if they remain, the TDMS method applies its low-pass filterwhile the other two methods do not. To solve this, the additional low-pass filter zeros areleft out of the design of the TDMS method. Furthermore, the gain of the filter is correctedas follows:

HTDMS_without_lp (iω) = HTDMS_without_lp (iω)HTDMS (1) HTDMS (1) . (5-3)

In this way, the gain of the filter is not changed when the additional zeros are omitted.Zooming into a longer test (with 1000ms on time) as shown in Figure 5-5, reveals that theTDMS method and Inverse Transfer Function method have a constant bias of approximately0.1N when reconstructing a 10N pulse. For a thrust amplitude of 10N, this is a 1% error, whilethe Deconvolution Kalman Filter only has a bias of 0.00002N, that is an error of 0.0002%.

Figure 5-5: Bias in the TDMS, Inverse Transfer Function and Deconvolution Kalman Filtermethod.

For all methods the largest part of the error is caused at the rising and falling edges of thereconstructed thrust pulses, although none of the IIR filters causes overshoot without theuse of low-pass filters. A test with an on time of 10ms is shown in Figure 5-6. One can seethat the TDMS method needs 5ms to reach the true thrust amplitude, the Inverse Transfer


120 Test Results

Function method needs 2ms and the Deconvolution Kalman Filter needs 1ms. Hence, theDeconvolution Kalman Filter is the most accurate for short pulses. The TDMS method’ssmoothing effect on the edges of a pulse is constant. Consequently, the error caused by theflanks has a stronger weight, when the pulse length is shorter. For example, while for an ontime of 10ms in Figure 5-6 only 5ms in the middle of the pulse are hit correctly, 45ms are hitcorrectly for an on time of 50ms.

Figure 5-6: Recovery of prescribed input signals with on and off times of the same duration:10ms and 50ms.



The dependence of the error on different on and off times can be seen in Table 5-1. Toquantify the error, the NRMS error statistic introduced in Equation (5-2) is used. The erroris calculated for the entire measurement signal. The true error of the TDMS method isconcealed to protect the intellectual property of Airbus DS. For the TDMS method, the tableonly shows, whether the computed error is larger or smaller than 5%. It can only be revealedthat the error of the TDMS method increases for shorter on times as predicted in Section3-3-9. The overall NRMS error is below 5% only for an on and off time of 1000ms. The InverseTransfer Function method behaves similarly to the TDMS method, only that the errors arealways smaller. The adaptations to the design of the filter increase the accuracy.

TMDS Method on|off time 5 ms 10 ms 50 ms 100ms 1000ms

5 ms >5 >5 >5 >5 >510 ms >5 >5 >5 >5 >550 ms >5 >5 >5 >5 >5100 ms >5 >5 >5 >5 >51000 ms >5 >5 >5 >5 <5

Inverse Filter

5 ms 0.9465 0.9464 0.9457 0.9441 0.939310 ms 0.9556 0.9553 0.9540 0.9507 0.941150 ms 0.9910 0.9900 0.9863 0.9768 0.9486100 ms 1.1060 1.1030 1.0920 1.0629 0.97421000 ms 2.0796 2.0636 2.0043 1.8410 1.2577

Kalman Filter

5 ms 0.00002 0.00002 0.00002 0.00003 0.0000310 ms 0.00002 0.00002 0.00002 0.00002 0.0000350 ms 0.00002 0.00002 0.00002 0.00002 0.00003100 ms 0.00003 0.00003 0.00002 0.00002 0.000041000 ms 0.00005 0.00005 0.00005 0.00004 0.00003

Table 5-1: Goodness of Fit for different on and off times for TDMS, Inverse Transfer Functionand Deconvolution Kalman Filter method without noise and low-pass filter (rows contain thesame on time, columns the same off time).

The average error of the Inverse Transfer Function method is about 1%. The minimum erroris approximately 0.94% and occurs for on times of 5ms and all off times tested. The maximumerror occurs for an on time of 1000ms and off time of 5ms. This large error can be explainedwith the bias of the method observed in Figure 5-6. Because of the bias for the InverseTransfer Function method holds: the longer the on time, the higher is the error caused bythe bias. For the same on time, the error decreases slightly for increasing off time. This effectcan be best observed for an on time of 1000ms, where the error is 2.07% for an off time of2ms and 1.25% for an off time of 1000ms. This can be explained by the increasing weight ofthe off times. As the NRMS error is computed for the entire signal and the thrust oscillationtakes only 400ms to decay, each off time adds 600 samples of comparing 0 with 0 to the error.


122 Test Results

5-1-2 White Gaussian Random Noise

The effect of random errors on the performance of the deconvolution filters is evaluated byadding white Gaussian random noise with standard deviation 0.005N (corresponding to 0.05%of the nominal thrust force for 10N) to the created measurement signals. The NRMS error fordifferent combinations of on and off times is listed in Table 5-2. The occurrence of randomnoise decreases the accuracy for all methods. The average error for the Inverse TransferFunction method increases from 1% to about 3% with noise. Similarly, the average error ofthe Deconvolution Kalman Filter is increased from 0.002% to about 1%. Nonetheless, theDeconvolution Kalman Filter is still the most accurate overall.

The only value which stands out, is the high error of 12% for the Inverse Transfer Functionmethod for an on time of 5ms and an off time of 1000ms. It can be explained by consideringthat about 99.5% of this signal is 0. In the 0 regions, random noise is compared to 0, addingto the overall error. This is also true for the Deconvolution Kalman Filter. However, for theDeconvolution Kalman Filter the error is only 3.4%, because it improves the signal-to-noiseratio while the Inverse Transfer Function method does not. The NRMS statistic is not agood error indicator for tests with long off times and short on times. For such cases a visualcomparison of the true and deconvoluted input thrust is preferable.

Figure 5-7: Effect of random noise with standard deviation of σ = 0.15 and NRMS error for allthree deconvolution methods.

To visualise how sensitive the different deconvolution methods are to an increase in the noise,



TDMS Method on|off time 5 ms 10 ms 50 ms 100ms 1000ms

5 ms >5 >5 >5 >5 >510 ms >5 >5 >5 >5 >550 ms >5 >5 >5 >5 >5100 ms >5 >5 >5 >5 >51000 ms >5 >5 >5 >5 >5

Inverse Filter

5 ms 5.7754 5.8817 6.1530 6.8278 12.775510 ms 4.1244 4.1165 4.5193 5.3928 8.949850 ms 2.8686 2.8803 2.8648 3.2034 5.3235100 ms 2.0274 2.0291 2.1416 2.1671 3.21831000 ms 2.8984 2.8311 2.7906 2.5780 2.0192

Kalman Filter

5 ms 2.3214 2.2983 2.3391 2.3649 3.365810 ms 1.7400 1.7348 1.7400 1.8333 2.450550 ms 1.1062 1.1086 1.1073 1.0816 1.3327100 ms 1.0529 1.0088 1.0197 1.0165 1.07111000 ms 1.6272 1.6023 1.5187 1.4366 1.0057

Table 5-2: Goodness of Fit for different on and off times for TDMS, Inverse Transfer Functionand Deconvolution Kalman Filter method with white Gaussian noise with a standard deviation of0.005N, but without low-pass filter.

measurement noise with a standard deviation of 0.15N is imposed on the synthetic measure-ment signal. This noise is about 30 times larger than the commonly encountered electronicnoise. The reconstructed inputs thrusts are depicted in Figure 5-7. The noise is smoothed outmost evenly by the TDMS method. Figure 5-7 explains why the error is generally higher forthe Inverse Transfer Function method than for the Deconvolution Kalman Filter. Withouta low-pass filter, the Inverse Transfer Function method amplifies the imposed measurementnoise. The Deconvolution Kalman Filter’s output still shows a certain randomness. However,the standard deviation of the noise in the reconstructed input signal is only 0.8244N whileit had a standard deviation 0.15N in the measurement signal. The Deconvolution KalmanFilter does not amplify the noise: it decreases its amplitude and improves the signal-to-noiseratio. The effect can be explained with results obtained by Chen in [8]. He discovered thatadditional poles at 0 can improve the cancellation properties of a filter for white Gaussianrandom noise. The integrator has the same effect, only that it does not cause a time delay.

5-1-3 Low-Pass Filter Effects

To avoid noise amplification, the low-pass filter of the Inverse Transfer Function method isactivated and a Butterworth filter of order 1 with a cut-off frequency of 0.5fNyquist is used.A comparison of the result obtained with and without low-pass filter for electronic noise with


124 Test Results

standard deviation 0.005N is shown in Figure 5-8. It shows that the effect of the low-passfilter on the signal flanks outweighs the effect of the noise cancellation considerably in theNRMS error.

Figure 5-8: Input thrust of 10ms on time deconvoluted with Inverse Transfer Function methodwithout (left) and with (right) Butterworth low-pass filter of order 1.

Figure 5-9: Effect of white noise with standard deviation 0.15N on the TMDS, Inverse TransferFunction and Deconvolution Kalman Filter method for a 10ms long pulse.



To visualise the effect of the low-pass filter for stronger noise, noise with a standard deviation0.15N is imposed and a Butterworth filter of order 3 with cut-off frequency 0.3fNyquist is in-cluded into the Inverse Transfer Function method and into the augmented state-space systemof the Deconvolution Kalman Filter. The results are displayed in Figure 5-9. The TDMSmethod shows regular ripples where the random noise has been smoothed. The applicationof a stronger low-pass filter in the Inverse Transfer Function method becomes clearly visiblein the overshoots caused at the beginning and end of a firing pulse. For the DeconvolutionKalman Filter, the noise cancellation is not as effective as for the other two methods, but noovershoots occur and the signal flanks are both accurately represented.

TDMS Filter on|off time 5 ms 10 ms 50 ms 100ms 1000ms

5 ms >5 >5 >5 >5 >510 ms >5 >5 >5 >5 >550 ms >5 >5 >5 >5 >5100 ms >5 >5 >5 >5 >51000 ms >5 >5 >5 >5 >5

Inverse Filter

5 ms 22.8822 22.8360 22.7947 22.7060 22.688910 ms 95.7037 95.7305 95.8219 95.2246 91.040550 ms 13.7061 13.7422 13.6629 13.5271 13.1509100 ms 8.4377 8.4058 8.3264 8.1260 7.49341000 ms 5.4076 5.3672 5.2244 4.7988 3.2978

Kalman Filter

5 ms 1.3512 1.3577 1.5509 1.5799 3.039710 ms 1.0778 1.0172 1.1165 1.2222 2.197450 ms 0.6048 0.5929 0.6157 0.6343 1.0527100 ms 0.5090 0.5144 0.5303 0.5305 0.78641000 ms 0.6906 0.6869 0.6707 0.6176 0.5027

Table 5-3: Goodness of Fit for different on and off times for TDMS, Inverse Transfer Functionand Deconvolution Kalman Filter method with Gaussian white noise and Butterworth low-passfilter of order 1 with 0.5fNyquist cut-off frequency.

In Table 5-3, it can be seen that the low-pass filter causes larger errors than the randomnoise. Therefore, it can be concluded that for the design of a low-pass filter for short PMFtests, more attention should be paid to the flanks than to the noise cancellation. If a separateButterworth low-pass filter is applied as a follow-up after the Deconvolution Kalman Filter,almost the same NRMS errors are produced as for the Inverse Transfer Function method.Hence, the Deconvolution Kalman Filter works better without a subsequent low-pass filter.Tests with higher noise levels, as well as tests with real data have revealed that a Butterworthfilter included into the state-space system has an improving effect on the thrust signal, if thenoise in the measurement signal is strong enough. For noise of standard deviation 0.005, itis sufficient to replace the Deconvolution Kalman Filter with the same Butterworth filter of


126 Test Results

order 1 as used by the Inverse Transfer Function method to improve the accuracy as seenin Table 5-3. Hence, the included low-pass filter in the state-space system has a smallereffect on the measurement signal’s flanks than the separate execution after application of theDeconvolution Kalman Filter.From the test results above, it can be concluded that the integration of the low-pass filter intothe state-space system of the Deconvolution Kalman Filter can become useful for strongerrandom noise in the output signal. For weak random noise in the output signal, the applicationof an integrator produces in the best results. Similarly, the random firing noise in the inputsignal can be reconstructed most accurately if an integrator is used. This is demonstrated inthe next section.

5-2 Realistic Test Scenario

The following test scenario tries to reconstruct realistic test conditions for small liquid rocketengines. The surrogate input signal is generated as a series of trapezoidal pulses with anamplitude of exactly 10N, a rise and fall time of 2ms, an on and off time of 50ms. The ontime is chosen as 50ms to make the effects of firing noise visible. Electronic noise with standarddeviation 0.008N is added to the output signal and firing noise with standard deviation 0.3Nis added during the on times of the thruster as depicted in Figure 5-10. The TDMS, InverseTransfer Function and Deconvolution Kalman Filter method all use their integrated low-passfilter components.

Figure 5-10: Synthetic input and output data with firing noise in the input and electronic noisein the output signal.

The deconvoluted input results are depicted in Figure 5-11. The smoothest results are ob-tained by the TDMS method. The Inverse Transfer Function method follows the randomnoise little better, but not consistently. The Deconvolution Kalman Filter is significantlybetter suited to the task of extracting the random firing noise from the system response thanthe other two methods. It is the only method capable of extracting the time-resolved firingsignal.It is difficult to evaluate whether a better fit to the firing noise makes the DeconvolutionKalman Filter more or less accurate. If the error is computed with respect to a smooth and


5-2 Realistic Test Scenario 127

noise-free input signal despite the presence of firing noise, the following NRMS errors areobtained: 13.5145 for the TDMS method, 14.8277 for the Inverse Transfer Function method,and only 3.574 for the Deconvolution Kalman Filter. The error of the TDMS method becomessmaller compared with a smooth input signal, but it is still higher than the other two values.Again, the reason is the bad fit at the signal flanks. The error of the Inverse Transfer Functionmethod also decreases slightly, because it smooths the firing noise. Still, even for a smoothinput signal the NRMS error is considerably smaller for the Deconvolution Kalman Filter thanfor the other IIR filter methods. This can be explained by the Deconvolution Kalman Filter’sability to hit the amplitude most accurately. In addition, for the Deconvolution KalmanFilter, the firing noise is zero-mean Gaussian around the correct amplitude value of 10N.Once again, its mean gives the correct amplitude. Hence, even if a smooth input is desired,the Deconvolution Kalman Filter is still the most accurate.

Figure 5-11: TDMS, Inverse Transfer Function and Deconvolution Kalman Filter method decon-volution of input signal including firing noise with standard deviation 0.3N and electronic noisewith standard deviation 0.008N.

It still has to be tested whether the smooth reconstruction of the input signal or a recon-struction of the input signal with firing noise is eventually more accurate. This is done bycomparing real thrust integrals and centroid times. However, following the indications of thetest series, it is most likely that the Deconvolution Kalman Filter creates the most accu-rate thrust integrals and centroid times, because it has the smallest bias and hits the thrustamplitude and signal edges most accurately.


128 Test Results

To obtain results comparable to those of the TDMS method for real test cases, a stronger low-pass filter has to be integrated in the Inverse Transfer Function method and DeconvolutionKalman Filter. The results most similar to the ones achieved with the TDMS method wereobtained by a Butterworth filter of order 3 with a cut-off frequency of 0.4fNyquist.

5-3 Computational Effort

The computation time needed by the three methods for measurement signals of increasingsignal length is depicted in Figure 5-12. The Inverse Transfer Function method is the fastest,immediately followed by the TMDS method. The Deconvolution Kalman Filter needs sig-nificantly longer. While all three methods show linear computation time, the DeconvolutionKalman Filter’s gradient is more than ten times higher than that of the other two methods.The reason for this is that the Deconvolution Kalman Filter has to solve a system of equa-tions in each iteration to find the measurement state error covariance matrix P . The transferfunction filter methods only have to find their model once and then traverse the output signalwith it.3

Figure 5-12: Comparison of computational effort of the Deconvolution Kalman Filter and thetwo transfer function approaches.

Long measurement tests with signal lengths of more than 300000 samples can take about 5to 6 seconds when evaluated with the Deconvolution Kalman Filter. This is no significantdisadvantage for two reasons. First, a measurement signal of 300000 samples is 30 secondslong. Hence, the Deconvolution Kalman Filter could already evaluate the measurement whileit is taken and finish in time without causing a delay. Second, even if the measurements areevaluated a posteriori, the required evaluation time is not unreasonable. Even if 300 testswere performed in one day, they could all be evaluated in 25 minutes. One can conclude thatthe increase in computation time can be justified by the improvement in accuracy.

3 The measurement time was computed using 4 cores of an Intel i7 processor with 8GB RAM.


5-4 Summary of Test Results 129

5-4 Summary of Test Results

The accuracy of all methods shows a dependency on the length of on and off times duringPMF. In general, shorter on and off times disturb the oscillation of the system and increasethe error. The inaccurate representation of the true signal flanks in the reconstructed thrustsignal has been identified as the main reason behind the increasing error for shorter pulses.

The TDMS method has a strong smoothing effect on the measurement noise. White Gaussianrandom noise has no great effect up to a certain magnitude and the firing noise occurringduring real tests is smoothed. Moreover, the TDMS method requires only little computationaleffort. However, it has a constant bias of about 1% of the thrust level and produces a largeerror at the signal flanks.

The Inverse Transfer Function method hits the signal flanks more accurately, but it, too,has a bias of 1%. It needs an additional low-pass filter to avoid noise amplification due tothe ill-posedness of the inversion problem. How much the firing noise occurring during realtests is smoothed depends on the order of this low-pass filter. The Inverse Transfer Functionmethod requires a little more computational effort than the TDMS method.

The Deconvolution Kalman Filter represents the signal flanks most accurately and with asteady-state error so low, that it is in the order of the electronic noise. The DeconvolutionKalman Filter does not invert the transfer function and, therefore, does not amplify the noise.Its integrator has no effect on the system flanks and also reduces the noise amplitude. It ismore suitable for an accurate extraction of short pulses than the tested IIR filters. It canreconstruct the firing noise as accurately as a smooth pulse. This accurate reconstruction ofthe firing noise could be beneficial for the computation of the correct centroid times and thrustintegrals. This will have to be investigated further using real test data. The DeconvolutionKalman Filter requires more computational effort than the other methods, but the additionaleffort can be justified by the increase in accuracy.


130 Test Results


Chapter 6

Conclusion

In an age of fierce competition, as more private companies venture into the space sector,the fuel and cost efficiency of satellites and their thrusters is of primary importance. Aprecise firing of the thrusters during a position adjustment of a satellite can save preciousfuel and increases the service life [8]. Such precise thrust levels can be better obtainedby pulsed than by steady-state firing [15]. For this reason, the design of highly accuratemeasurement evaluation methods for pulsed liquid rocket engines has received more attentionin the last decade [8, 15, 16, 17]. To avoid costly new designs for the majority of industrialsuppliers, a new evaluation method is needed to adapt commonly used strain gage type thruststands to the demands of pulsed thrust measurements. In this thesis, a new method wasdeveloped to increase the measurement accuracy for such Pulse Mode Firing (PMF) rocketthrust measurements. The method is intended as a cost-efficient alternative for industrialrocket manufacturers who wish to increase their measurement accuracy without building anew thrust stand specifically designed for PMF measurements.

Using a design made by Bora [9], the Kalman Filter was adapted to deconvolute the inputthrust from a noise-corrupted transient thrust stand response. This Deconvolution KalmanFilter distinguishes itself from conventional, industrial IIR filtering approaches by its algorith-mic simplicity and the few designing choices it requires. The designer only needs to supplyan Autoregressive-Moving Average (ARMA) model describing the thrust stand behaviour.The model can be identified with, for example, a linear least squares method. The modelcoefficients can be obtained most reliably by assuming a step function input and extractinga corresponding decaying oscillation from the measurement signal. For Infinite Impulse Re-sponse (IIR) filters, a stability analysis has to be conducted for the identified coefficients incase they are unstable. This is not necessary for the Deconvolution Kalman Filter, becauseit does not invert the model.

During testing, it was discovered that including an integrator into the augmented state-spacesystem of the Deconvolution Kalman Filter offered a way to remove the steady-state errorand, at the same time, attenuate white Gaussian measurement noise. A similar observationwas already made by Chen [8]. They noticed that a thrust compensator can be improvedby inserting an additional pole at 0 into the transfer function at the cost of causing a time


132 Conclusion

delay. An integrator has the same effect, only that no time delay is caused. For dynamicthrust measurements, the use of an integrator has two more advantages. First, white Gaussianrandom components in the input signal can be reconstructed accurately even if it changesat the rate of the sampling rate. For thrust measurements, this means that the combustionnoise of the rocket engine can be recovered accurately. Second, with the help of an integrator,thrust pulses can be reconstructed in rectangular form with a reconstruction error in the orderof the standard deviation of the electronic measurement noise.

The ability to evaluate rectangular pulses makes the Deconvolution Kalman Filter especiallysuitable for the evaluation of PMF rocket thrust measurements with short firing times. Thiswas demonstrated by comparing the Deconvolution Kalman Filter with two IIR filtering meth-ods. It was found that a low-pass filter, incorporated into the augmented state-space systemof the Deconvolution Kalman Filter results in a more accurate reconstruction of rectangularpulses than when applied in combination with an IIR filter.

The commonly known weak-spot of the Kalman Filter is the determination of its measurementand process noise covariance matrices Q and R. It was found that for PMF thrust measure-ments, Q can be chosen as the standard deviation of the oscillations following a sharp thrustpulse and R can be chosen as the standard deviation of electronic noise picked up at the be-ginning of a measurement. If these heuristic choices do not lead to good results, the rationalmetrics developed by Saha [19] or the Autocovariance Least Squares (ALS) method developedby Rajamani [20, 75] can be used.

In the special set-up of the augmented state-space Kalman Filter, the input signal is addedas an additional state. The state error covariance matrix P contains the variance of theerror for this state. The variance of the error in the input signal can be interpreted as anestimate of the uncertainty in the reconstructed thrust signal and can be extracted using theconditional covariance theorem. In this way, an evaluation of the measurement uncertaintycan be conveniently integrated into the deconvolution process for the Kalman Filter. Bycontrast, the deterministic models used by FIR and IIR Filter approaches do not include anevaluation of the measurement uncertainty, but need an additional Monte Carlo simulation.

A Monte Carlo simulation was carried out for the model used by the Deconvolution KalmanFilter to obtain a time-resolved estimate of the uncertainty at each sampling point. ThisMonte Carlo simulation showed that the uncertainty in dynamic thrust measurements isgreatly influenced by transients in the thrust input signal. To the knowledge of the author,the uncertainty caused by the transients in dynamic thrust measurements has so far not beenconsidered in the design of pulsed rocket engines. The analysis performed in this thesis showsthat the uncertainty is tripled immediately after the thrust is turned on or off. Hence, afast PMF results in a significantly higher uncertainty level overall. The uncertainty couldbe reduced by increasing the turn on and turn off times of the thruster. As this requires achange in the valves of the rocket engine, it is difficult to achieve. Nonetheless, the resultsobtained by the uncertainty analysis can be taken into account to improve the PMF accuracyof thrusters. When defining how a thrust level is established, combinations which result inhigh uncertainty can be avoided.


6-1 Further Research Possibilities 133

6-1 Further Research Possibilities

Improvement of System Identification: Nonlinear Modal Analysis

The entire analysis performed in this thesis was based on the assumption that the thruststand can be modelled as a Linear Time-Invariant system. While it could be shown that thisassumption is legitimate, further research could investigate to what extent the accuracy canbe improved by including dynamic nonlinearities into the model. Ren [15] developed a verypromising recursive nonlinear method, which can accurately describe the frequency responseof system modes up to 2500Hz, as shown in Figure 6-1.

Figure 6-1: The theoretical model built by Ren with a nonlinear modal analysis is accurateenough to replace the actual frequency response [15].

A similar approach could be used to reduce the system identification error and the uncer-tainty caused by it. However, a better fit to the frequency response is no guarantee that thedeconvolution filter will perform more accurately, as shown in Section 3-5-3 of this thesis.This problem would need to be addressed before implementing Ren’s method.

Improvement of Low-Pass Filter: Gaussian Filter

A low-pass filter designed especially for the purpose of complementing deconvolution filtersfor dynamic thrust measurements still has to be found. Not only the Deconvolution KalmanFilter, but all FIR and IIR filtering methods could be improved with a low-pass filter withuniform time delay, abrupt cut-off, no pass-band ripples, no overshoot, no phase distortionand, most importantly, short rise and fall times. One promising method is the Gaussian filter.It produces no overshoot for a step function input and minimises the rise and fall time.

Improvement of Deconvolution Kalman Filter

The Kalman Filter can only be optimal, if the process and measurement noise covariancematrices Q and R are predicted absolutely correctly. Although the heuristic and rational


134 Conclusion

choices suggested in this thesis were shown to produce only small errors, the exact values forQ and R could not be determined. Among the automatic methods to estimate the covariancematrices Q and R, the ALS method is the most promising approach. Nevertheless, manyresearchers have made similar experiences as in this thesis: the obtained estimates are notalways reliable. Further research could focus on improving the reliability of the ALS method.Due to the popularity of the Kalman Filter, research in this direction is likely to find greatappreciation by the Digital Signal Processing (DSP) community.

Alternative Methods: Bayesian Deconvolution, OUFIR Filters, Numerical Approach

Two other slight disadvantages of the Deconvolution Kalman Filter are its restricted appli-cability, as it can only be used for cases with white Gaussian noise, and its slightly highercomputation time in comparison to IIR filters. These problems could be avoided by switchingto different methods, which, however, might have disadvantages of their own.It could be seen throughout this thesis that it is difficult to determine to what extent noise iswhite Gaussian. This is a disadvantage, because the optimality of the Deconvolution KalmanFilter depends on the assumption that the noise is white Gaussian. Different probabilitydistributions could be used by exchanging the Deconvolution Kalman Filter for the moregeneral concept of Bayesian Deconvolution, which operates similarly to the DeconvolutionKalman Filter, but without the assumption of white Gaussian noise. The concept is notnew and a considerable amount of research in this direction has already been done [77, 78].However, Bayesian Deconvolution has not yet been applied to the dynamic rocket thrustmeasurement problem.Less ambitious, but similarly fruitful would be the application of an approach, which ignoresthe noise statics altogether. Attempts in this direction have been made by several researchersand were recently summarised by Shmaliy [10]. According to Shmaliy, Optimal Unbiased FIR(OUFIR) filters are better and more robust under real-world conditions with uncertaintiesand non-Gaussian noise than the Kalman Filter. While the Kalman Filter is more accuratein cases where the real noise covariance matrices are known exactly, OUFIR filters produceonly slightly less accurate estimates and are significantly more robust, because their accuracyis not reduced by errors in the noise covariances, as illustrated by Figure 6-2.


6-1 Further Research Possibilities 135

Figure 6-2: Effect of errors in the noise covariances of Kalman Filter and OUFIR filter estimates[10].

A simple way to improve the computation time of the Deconvolution Kalman filter is theparallelisation of the method. If the initial conditions are determined either with the Riccatiequation or in a short pre-run, the measurement signal can be divided into several sections, sothat the Deconvolution Kalman Filter algorithm can be parallelised to some extent. Alterna-tively, it might be interesting to look at the problem from a completely different perspectiveusing a purely numerical approach. For instance, a fast and parallelised solver for integralequations with convolution-type Kernel was proposed by Ye and Zhang [38]. Such approachescould provide a distinctive advantage in terms of computation time, but have not yet beenrigorously compared to conventional deconvolution approaches applied in DSP.


136 Conclusion


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[6] T. Esward, C. Matthews, B. Hughes, J. Perez-Luna, C. Edwards, and D. Robinson,“System identification and uncertainty evaluation for dynamic force measurements atthe micronewton level for space satellite applications,” in 7th International Workshop onAnalysis of Dynamic Measurement, October 2012.

[7] C. Gao and B. Sun, “Investigation of the piezoelectric thrust test stand for rocket engine,”Science, Measurement Technology, IET, vol. 3, no. 6, pp. 395–402, 2009.

[8] L. Chen, M. Wang, J. Li, and H. Dai, “Digital compensators for dynamic thrust mea-surement,” in 8th IEEE International Conference on Control and Automation (ICCA),pp. 842–847, 2010. cited By (since 1996) 0.

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Glossary

List of Acronyms

AEDC Arnold Engineering Development Center

ALS Autocovariance Least Squares

AR Autoregressive

ARMA Autoregressive-Moving Average

ATV Automated Transfer Vehicle

DSP Digital Signal Processing

FIR Finite Impulse Response

IIR Infinite Impulse Response

LTI Linear Time-Invariant

MA Moving Average

MSD Mass-Spring-Damper

NRMS Normalised Root Mean Square

PMF Pulse Mode Firing

RMS Root Mean Square

SOS Second Order System

SSF Steady-State Firing

TDMS Technical Data Management System

TMB Thrust Measurement Bridge


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