Nonlinear Robust Control Synthesis Methods for … · Nonlinear Robust Control Synthesis Methods...

Nonlinear Robust Control Synthesis Methods for SpacecraftApplications

by

Stefan LeBel

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Aerospace Science and EngineeringUniversity of Toronto

c© Copyright 2014 by Stefan LeBel

Abstract

Nonlinear Robust Control Synthesis Methods for Spacecraft Applications

Stefan LeBelDoctor of Philosophy

Graduate Department of Aerospace Science and EngineeringUniversity of Toronto

2014

This thesis focuses on practical methods for constructing robust nonlinear control systems. In general,the development of such control systems is characterized by the solution to one or more Hamilton-Jacobipartial differential equations (HJE). However, no general analytical solution has yet been obtained tosolve this optimization problem. Solutions have thus far only been obtained under certain conditions.Therefore, the first significant contribution of this thesis is a method for obtaining analytical expressionsfor approximate solutions to a common form of HJE (under certain assumptions regarding the class ofnonlinear systems used).

Additionally, modern state space controller synthesis techniques typically result in state estimatorsof equal or greater dimension than the plant model. However, it is often desirable, or even necessary,to approximate these controllers by models of lower state dimension. Presently, methods for developingnonlinear state balancing transformations are not very well understood. Therefore, the second significantcontribution of this thesis is a proper algorithm for the application of state balancing techniques tononlinear control systems and the subsequent reduction of the number of control states. The method tobe developed for state balancing is based on the above framework for constructing analytical solutionsto the HJE.

In this thesis we will make use of three existing robust nonlinear control methods from the literature.These three methods have the advantage that they can all be constructed from solutions to a singleform of HJE. Thus, by developing a method for obtaining analytical expressions for the solution to asingle form of HJE, we are able to develop explicit polynomial solutions for each of these three controlmethods.

Due to the difficulties associated with quantifying robustness and performance properties for nonlin-ear systems, the effectiveness of the three control methods considered shall be demonstrated via numericalsimulations. The particular applications of interest to us here are space systems. First, we will considerthe attitude control of a single spacecraft. Second, we examine the problem of formation flying controlfor a pair of spacecraft. The third and final problem we consider is the control of a nonlinear mass-springchain.

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Acknowledgements

To Chris:For your brilliant advice and guidance, as well

as your patience and encouragement throughout.

To Karen:For encouraging and believing in me through thegood and bad, especially across a great distance.

To UTIAS:For providing me with an interesting, enjoyableand educational graduate student experience.

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Contents

Abstract ii

Acknowledgements iii

Table of Contents iv

List of Tables viii

List of Figures ix

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Proposed Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Summary of Objectives and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Background Theory 7

2.1 Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 Vector Signal Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.2 Vector Signal Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Properties of Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 System Models, Feedback Configurations, and Properties . . . . . . . . . . . . . . . . . . 102.2.1 Class of Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 System Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 Graph Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.4 Projection Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Dissipative Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.1 Dissipativity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.2 Disturbance Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.3 L2-Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Hamilton-Jacobi Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Robustness in the Gap Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.5.1 Small Gain Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

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2.5.2 The Gap Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5.3 Simultaneous Perturbations in Plant and Controller . . . . . . . . . . . . . . . . . 202.5.4 Calculation of the Gap Metric for Linear Systems . . . . . . . . . . . . . . . . . . . 21

2.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Full-State-Order Controller Synthesis 23

3.1 Control Synthesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.1 Control Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.2 Control Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Analytical Solutions to HJE Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.1 Stabilizing HJE Solution Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2.2 Antistabilizing HJE Solution Gradient . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.3 Combined HJE Solution Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 H∞ Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.1 Linear H∞ Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.3.2 Polynomial H∞ Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 H∞ Loop Shaping Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.4.1 Linear Filter Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4.2 Linear H∞ Loop Shaping Controller . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4.3 Polynomial H∞ Loop Shaping Controller . . . . . . . . . . . . . . . . . . . . . . . 43

3.5 Weighted Mixed Sensitivity Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . 443.5.1 Linear Weighting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.5.2 Linear Weighted Mixed Sensitivity Controller . . . . . . . . . . . . . . . . . . . . . 493.5.3 Polynomial Weighted Mixed Sensitivity Controller . . . . . . . . . . . . . . . . . . 49

3.6 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Reduced-State-Order Controller Synthesis 52

4.1 Balanced Nonlinear Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1.1 Balanced Nonlinear H∞ Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.1.2 Balanced Nonlinear H∞ Loop Shaping Controller . . . . . . . . . . . . . . . . . . . 554.1.3 Balanced Nonlinear Weighted Mixed Sensitivity Controller . . . . . . . . . . . . . 56

4.2 Controller State Balancing Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3 Nonlinear Balancing Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.1 First-Order Balancing Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 604.3.2 Second-Order Balancing Transformation . . . . . . . . . . . . . . . . . . . . . . . . 614.3.3 Third-Order Balancing Transformation . . . . . . . . . . . . . . . . . . . . . . . . 614.3.4 Fourth-Order Balancing Transformation . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 Nonlinear Balancing Transformation Gradient . . . . . . . . . . . . . . . . . . . . . . . . . 654.4.1 First-Order Balancing Transformation Gradient . . . . . . . . . . . . . . . . . . . . 654.4.2 Second-Order Balancing Transformation Gradient . . . . . . . . . . . . . . . . . . 664.4.3 Third-Order Balancing Transformation Gradient . . . . . . . . . . . . . . . . . . . 664.4.4 Fourth-Order Balancing Transformation Gradient . . . . . . . . . . . . . . . . . . 66

4.5 Inverse of Nonlinear Balancing Transformation Gradient . . . . . . . . . . . . . . . . . . . 674.5.1 Zeroth-Order Balancing Transformation Gradient Inverse . . . . . . . . . . . . . . 67

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4.5.2 First-Order Balancing Transformation Gradient Inverse . . . . . . . . . . . . . . . 674.5.3 Second-Order Balancing Transformation Gradient Inverse . . . . . . . . . . . . . . 684.5.4 Third-Order Balancing Transformation Gradient Inverse . . . . . . . . . . . . . . . 69

4.6 Balanced Nonlinear Dynamics Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.6.1 First-Order Balanced Dynamics Matrix . . . . . . . . . . . . . . . . . . . . . . . . 714.6.2 Second-Order Balanced Dynamics Matrix . . . . . . . . . . . . . . . . . . . . . . . 724.6.3 Third-Order Balanced Dynamics Matrix . . . . . . . . . . . . . . . . . . . . . . . . 724.6.4 Fourth-Order Balanced Dynamics Matrix . . . . . . . . . . . . . . . . . . . . . . . 73

4.7 Balanced Nonlinear Input Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.7.1 Zeroth-Order Balanced Input Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 754.7.2 First-Order Balanced Input Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.7.3 Second-Order Balanced Input Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 754.7.4 Third-Order Balanced Input Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.8 Balanced Nonlinear Output Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.8.1 First-Order Balanced Output Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 764.8.2 Second-Order Balanced Output Matrix . . . . . . . . . . . . . . . . . . . . . . . . 774.8.3 Third-Order Balanced Output Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 774.8.4 Fourth-Order Balanced Output Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.9 Balanced HJE Solution Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.9.1 First-Order Balanced HJE Solution Gradient . . . . . . . . . . . . . . . . . . . . . 784.9.2 Second-Order Balanced HJE Solution Gradient . . . . . . . . . . . . . . . . . . . . 794.9.3 Third-Order Balanced HJE Solution Gradient . . . . . . . . . . . . . . . . . . . . . 794.9.4 Fourth-Order Balanced HJE Solution Gradient . . . . . . . . . . . . . . . . . . . . 80

4.10 Balanced Output Injection Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.11 Reduced-State-Order Nonlinear Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.11.1 Reduced-State-Order Nonlinear H∞ Controller . . . . . . . . . . . . . . . . . . . . 854.11.2 Reduced-State-Order Nonlinear H∞ Loop Shaping Controller . . . . . . . . . . . . 874.11.3 Reduced-State-Order Nonlinear Weighted Mixed Sensitivity Controller . . . . . . . 87

4.12 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5 Simulation Results 89

5.1 Attitude Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.1.1 Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.1.2 Existing Control Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.1.3 Simulation Results and Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.2 Formation Flying Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.2.1 Formation Flying Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.2.2 Simulation Results and Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.3 Nonlinear Mass-Spring Chain Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.3.1 Nonlinear Mass-Spring Chain Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 1105.3.2 Simulation Results and Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4 Computational Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.5 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

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6 Conclusions and Future Work 116

6.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Bibliography 119

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List of Tables

5.1 Disturbance rejection results for existing controllers . . . . . . . . . . . . . . . . . . . . . . 975.2 Disturbance rejection results for H∞ controllers . . . . . . . . . . . . . . . . . . . . . . . . 975.3 Robustness to actuation time-delay for existing controllers . . . . . . . . . . . . . . . . . . 1025.4 Robustness to actuation time-delay for H∞ controllers . . . . . . . . . . . . . . . . . . . . 1025.5 State-order-reduction comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.6 Computational Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

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List of Figures

2.1 Feedback configuration [P ,K] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Generalized feedback configuration [P?,K] . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 Feedback configuration [P ,K] (repeated) . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2 Generalized feedback configuration [P?,K] (repeated) . . . . . . . . . . . . . . . . . . . . 243.3 Construction of modified plant P?

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Feedback configuration [P ,K] with loop shaping filter on P . . . . . . . . . . . . . . . . . 413.5 Feedback configuration [P ,K] with loop shaping filter on K . . . . . . . . . . . . . . . . . 423.6 Feedback configuration [P ,K] with mixed sensitivity weights W1 and W2 . . . . . . . . 463.7 Magnitude plot of tracking error weighting function . . . . . . . . . . . . . . . . . . . . . 473.8 Magnitude plot of control effort weighting function . . . . . . . . . . . . . . . . . . . . . . 48

5.1 MRP trajectories for I = diag10, 20, 30 kg·m2 with 4th-order H∞ control system . . . 945.2 Control torques for I = diag10, 20, 30 kg·m2 with 4th-order H∞ control system . . . . 945.3 RMS tracking error with respect to 1st-order actuator bandwidth . . . . . . . . . . . . . . 985.4 RMS control effort with respect to 1st-order actuator bandwidth . . . . . . . . . . . . . . 995.5 RMS tracking error with respect to 2nd-order actuator bandwidth . . . . . . . . . . . . . 1005.6 RMS control effort with respect to 2nd-order actuator bandwidth . . . . . . . . . . . . . . 1005.7 Gap metric with respect to 1st-order actuator bandwidth . . . . . . . . . . . . . . . . . . 1015.8 Gap metric with respect to 2nd-order actuator bandwidth (with damping ratio ζ = 0.5) . 1015.9 Formation flying dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.10 RMS tracking error vs. loop shaping parameter . . . . . . . . . . . . . . . . . . . . . . . . 1075.11 RMS control effort vs. loop shaping parameter . . . . . . . . . . . . . . . . . . . . . . . . 1075.12 Tracking errors for D = 10−4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.13 Tracking errors for 1st-order H∞ controller with nc = 6 . . . . . . . . . . . . . . . . . . . 1095.14 Tracking errors for 1st-order H∞ controller with nc = 3 . . . . . . . . . . . . . . . . . . . 1095.15 Mass-spring chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.16 RMS tracking error vs. number of controller states . . . . . . . . . . . . . . . . . . . . . . 1135.17 RMS control effort vs. number of controller states . . . . . . . . . . . . . . . . . . . . . . 113

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Chapter 1

Introduction

The primary purpose of feedback control is to enhance performance, especially in the presence of un-certainty. However, control systems are generally developed based on mathematical models that are, atbest, a close approximation of real-world phenomena. Thus, for control methods to have any practicalvalue, they must be robust with regards to uncertainties and disturbances that may affect a real system.The study of robust control is therefore an essential part of the application of control theory to physicalsystems. The overall goal of this thesis is to develop nonlinear robust control synthesis techniques.

1.1 Motivation

The modern world is a technology-rich environment where control systems are prevalent. They touchevery aspect of our lives, whether we are aware of it or not. Control systems have become completelyengrained in our world, so much so that we often take them for granted. Applications can range fromcommon home appliances and automobiles, to rare and complex systems such as spacecraft for landingon other planetary bodies. Such a broad range of applications means that the level of complexity incontrol systems also varies tremendously.

For many simple applications, linear system theory is more than adequate. More complex systems,on the other hand, often require advanced control methods. In general, though, control systems aredeveloped based on mathematical models that make various assumptions. The overall result of theseassumptions is a discrepancy between the model and reality, leading to uncertainties. Uncertainties in amodel can arise from various sources internal to the system, such as parametric uncertainty, nonlinearitiesthat are too complex to understand properly or model, time delays that may be related to controllercomputation time, unknown model structure at high frequency, or unmodeled dynamics such as sensorsand actuators, which may be intentionally neglected for tractability purposes. Additionally, real systemsare often subject to disturbances that are a result of the external environment. These are typicallydue to phenomena that are too complex or chaotic to predict. Depending on the complexity of thephysical system to be controlled, these uncertainties and disturbances can have a large influence on theeffectiveness of a control system.

Typically, the more complex a system is, the more nonlinear and complex the associated mathematicalmodel. Unfortunately, these complex systems are generally the ones that require the most accuratecontrol. Thus, it is our goal in this thesis to examine methods for synthesizing control systems that are

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Chapter 1. Introduction 2

robust with regards to various uncertainties and disturbances. Moreover, we shall also be interested innonlinear control systems, since nearly all physical systems are in fact nonlinear.

1.2 Background Literature

Robust Control Synthesis Methods

In general, the development of an optimal nonlinear state feedback control law is characterized bythe solution to a Hamilton-Jacobi partial differential equation (HJE) [55], while a robust nonlinearcontroller is obtained from the solution of one or more HJEs [6, 67, 69]. It is noted that the concept ofdissipativity, which is closely related to optimal and robust control, is characterized by a Hamilton-Jacobiinequality [28, 73, 74]. However, no exact analytical solution method has yet been obtained to solve thisoptimization problem in general. Solutions have thus far only been obtained under certain conditions.In the case of linear systems with a quadratic performance index, the HJE reduces to the well-knownalgebraic Riccati equation (ARE). However, linear robust control methods have limited applicability innonlinear systems.

As a result, extensive work has been carried out to approximate the solution of Hamilton-Jacobiequations or inequalities through a Taylor series expansion [1, 18, 19, 30, 41]. Although such a seriesexpansion results in an infinite-order polynomial, finite-order approximations can be used to obtainsub-optimal solutions to an HJE. It has been shown that a local solution to an HJE can be obtainedby solving the ARE for the linear approximation of the system [67, 68, 69]. Methods that have beendeveloped over the past decades to attempt to solve this problem include the Zubov procedure [27, 44],the state-dependent Riccati equation [13, 14], the Galerkin method for the equivalent sequence of first-order partial differential equations [8, 9], and the use of symplectic geometry to examine the associatedHamiltonian system [56]. However, all of these approaches rely on numerical solution methods. Oneaspect that is lacking in the literature is an analytical solution to the approximate equations. Therefore,the first significant contribution of this thesis is a method for obtaining analytical expressions for theapproximate solutions to Hamilton-Jacobi equations.

State-Order-Reduction Methods

Modern state space controller synthesis techniques typically result in state estimators of equal or greaterdimension than the plant model. However, it is often desirable, or even necessary, to approximate thesecontrollers by models of lower state dimension. Several different approaches have been proposed in theliterature to address the model reduction problem. However, due to its prominence, we shall only considerthe balanced truncation method, which was first introduced by Moore in 1981 [48]. Moore’s methodwas restricted to stable open-loop systems, and suffered from several drawbacks. In particular, it wasshown that a stable system could become unstable when a controller was designed for a reduced-orderplant if the plant reduction was based on an open-loop technique [33]. Thus, the LQG-balancing methodwas developed by Jonckheere and Silverman [34] to address the issue of balancing closed-loop feedbacksystems with Linear Quadratic Gaussian (LQG) controllers. Similarly, H∞-balancing was introducedby Mustafa [49] and further analyzed by Mustafa and Glover [50] to make use of H∞ controllers. Asan alternative approach, Liu and Anderson [40] extended the method of Moore to unstable open-loopsystems using coprime factorization theory [72]. Anderson and Liu [3] then extend this to include


weighted coprime factor reduction, while Meyer [47] made use of normalized coprime factors. Thesecoprime factorization methods have been shown to be equivalent to the closed-loop balancing methods[51].

As a natural extension of these linear balancing techniques, Scherpen introduced balanced truncationfor stable open-loop nonlinear systems in 1993 [59]. Scherpen then extended this method to closed-loop H∞-balancing for nonlinear systems [60]. Nonlinear versions of the open-loop technique have alsobeen developed using coprime factorizations [2, 61]. Pavel and Fairman [53] studied stability-preservingconditions for a nonlinear extension to the H∞ method of Mustafa and Glover [50]. Krener has alsoworked on nonlinear balancing methods for polynomial systems [38]. However, while Scherpen [59, 60]and Krener [38] discuss the theoretical aspects of nonlinear balancing, they do not provide any actualmeans of obtaining the balancing transformation. More recently, Fujimoto and Tsubakino [17] introduceda method for calculating balancing transformations based on a Taylor series approach. However, thereare certain issues that are not adequately addressed in their work. Therefore, the second significantcontribution of this thesis is a proper algorithm for the application of state balancing techniques toclosed-loop nonlinear systems.

Robustness in the Gap Metric

The small gain theorem was introduced by Zames in 1966 [75, 76] to define an open-loop criterion forclosed-loop stability. In fact, the small gain theorem lies at the heart of classical (linear) H∞ controltheory. However, the use of norms as a metric limited the applicability of this theory. The gap metricwas thus developed by Zames and El-Sakkary [16, 77] specifically for the purpose of studying uncertaintyin feedback systems. The gap metric was originally conceived to provide a suitable topology in whichsmall errors in the gap between open-loop systems would correspond to small errors in the norm of thestable closed-loop configuration [32, 77]. The gap provides a metric on (possibly) unstable systems interms of the distance between their graphs. Vidyasagar introduced an equivalent metric, called the graphmetric, using normalized coprime factorizations [71]. It was shown that these two metrics induce thesame topology, which is the weakest topology in which feedback stability is a robust property [23, 71].

The gap metric can be used to define a stability criterion similar to the small gain theorem. Moreover,the gap metric provides a convenient framework for the study of robustness with regards to simultaneousperturbations in the plant and the controller. We will make use of the gap metric as a robustness criterionthroughout this thesis.

1.3 Proposed Methodology

Based on the above background discussion, we are now in a position to describe the methodology tobe followed in this thesis. Although linear control systems are well understood, they have limitedapplicability in nonlinear systems. Nonlinear systems, on the other hand, are much more difficult towork with: proofs are rarely constructive and it is often impossible to develop general methods to solvethe associated nonlinear equations. Such methods are nearly useless from a practical standpoint becausethey cannot be used to construct a control system that can actually be implemented in a real system.

The overall approach taken in this thesis is to make use of the existing theory on nonlinear robustcontrol systems to develop practical algorithms. The three nonlinear control systems we will focus on canall be constructed from solutions to a single form of HJE. However, one of the primary issues outstanding


in the literature is a closed-form method for obtaining solutions to the nonlinear HJE. We thus focuson developing a methodology for constructing analytical expressions to approximate solutions of theHamilton-Jacobi equation. Although our goal will not be to solve this problem directly, the proposedresearch will develop methods to solve finite-order approximations to this problem using Taylor series,resulting in (sub-)optimal robust nonlinear controllers. In developing these approximate HJE solutions,certain assumptions will be made regarding the class of nonlinear systems used so as to simplify theensuing derivations.

Developing balanced and reduced nonlinear control systems is also based on the solution of nonlinearequations. We thus develop a methodology for constructing polynomial state transformations usingthe same approach as above. In fact, we can construct this state transformation using the nonlinearsolution to the same HJE used for constructing the three nonlinear control systems. This is particularlyadvantageous because we already developed these solutions when constructing the nonlinear controllers.

Due to the difficulty of proving the robustness or performance of such nonlinear control systems,the effectiveness of the methods developed in this thesis are demonstrated via numerical simulations.In contrast with most of the methods used for comparisons, which were developed to address specificproblems, the methods developed in this thesis are more general and can be applied to a broad class ofsystems (although some restrictions do still apply, as we will see). Indeed, this is due to the fact thatour synthesis methodology is based on general nonlinear control systems.

1.4 Applications

The particular applications of interest to us here are space systems. We present three space applicationsfor the purpose of highlighting different aspects of our novel control synthesis method. First, we willconsider the attitude control of a single spacecraft. The attitude control problem is critical for mostsatellite applications and has thus attracted extensive interest. A prime example is the Hubble SpaceTelescope, which has precise pointing requirements for the observation of stellar phenomena. While manycontrol methods have been developed to address this problem, most are concerned primarily with theoptimality and speed of attitude maneuvers [24, 58, 65, 66]. In contrast, the nonlinear control methodsdeveloped in this thesis focus more on robustness.

Second, we examine the problem of controlling the relative motion of two spacecraft. Satelliteformation flying has become very popular in recent years. Certain applications can better be servedby replacing single-purpose, monolithic satellites with a larger number of smaller, cheaper satellites.Consequently, improved methods are needed to control these formations of satellites with a certaindegree of accuracy and robustness. Linear control systems are usually developed based on the relativeequations of motion obtained through linearization about one satellite’s orbit [12, 29]. Instead, weshall apply our robust nonlinear controllers to this problem using higher-order approximations of thetrue formation flying dynamics. We will also compare controller results using the state-order-reductiontechniques developed in this thesis.

The third and final problem we consider is the control of a nonlinear mass-spring chain. Althoughthis is not directly a space systems application, such a model could be considered an approximation of alarge flexible space structure. The objective for the mass-spring chain problem is to control the positionand velocity of the tip mass while simultaneously suppressing disturbance vibrations along the entirechain. The dynamics for this problem can be derived with an arbitrary number of masses and springs.


This can result in a very large number of plant and controller states. Hence, for this problem we willprimarily be interested in comparing results using the controller state-order-reduction techniques.

1.5 Summary of Objectives and Contributions

The following is a summary of the objectives and contributions of the proposed research:

1. Develop a framework for obtaining analytical solutions to a Hamilton-Jacobi equation (HJE):

(a) Approximate nonlinear system dynamics and HJE solution gradient via Taylor series expan-sions by using a new structure;

(b) Construct a sequential set of equations for solving successively higher orders of approximationto the HJE solution gradient;

(c) Develop analytical expressions for each order of approximation of the HJE solution gradient;

(d) Demonstrate this methodology for obtaining both the stabilizing and antistabilizing HJEsolution gradients.

2. Develop a synthesis method for nonlinear feedback control systems using analytical expressions forthe HJE solution gradient:

(a) Design explicit solutions to three different robust nonlinear control methods:

i. H∞ control problem

ii. H∞ loop shaping control problem

iii. weighted mixed sensitivity control problem

(b) All three control methods use a single form of HJE;

(c) All three control methods yield robust nonlinear dynamic output feedback control systems;

(d) All three control systems provide stability and disturbance attenuation over a neighbourhoodof the equilibrium that is larger than existing linear methods.

3. Develop a method for obtaining and applying a nonlinear controller state balancing transformation:

(a) Develop an explicit solution technique for obtaining a nonlinear state balancing transformationusing a Taylor series expansion of the HJE solution;

(b) Devise the first technique in the literature for completely balancing a nonlinear control system;

(c) Develop a single systematic methodology for balancing all matrices used to construct thethree nonlinear control systems;

(d) Develop a methodology for constructing reduced-state-order nonlinear control systems thatapplies to all three control methods;

(e) Demonstrate via numerical simulations that state-order-reduction induces a trade-off betweenperformance and robustness.

4. Demonstrate the novel methods developed in this thesis using space applications:

(a) Spacecraft attitude control;

(b) Spacecraft formation flying control;

(c) Nonlinear mass-spring chain.


1.6 Thesis Outline

We now present an outline of this thesis. Chapter 1 presents the introduction, the motivation behindthe work in this thesis, a survey of the relevant literature, a description of the proposed methodology,and a summary of the objectives and contributions. This chapter concludes with the present outline ofthe thesis.

Chapter 2 presents the background theory to be used throughout the remainder of this thesis. Itbegins with a review of signal norms and spaces. These are essential for the subsequent description offeedback configurations, state space models, and system properties. Finally, the chapter describes ourcontroller synthesis objectives, namely with regards to robustness.

In Chapter 3, we develop solutions to the nonlinear H∞, loop shaping, and weighted mixed sensitivitycontrol problems. Each of these controllers can be constructed from the solutions to Hamilton-Jacobiequations with a particular structure. We thus develop an approach to obtain analytical expressions forapproximate solutions to Hamilton-Jacobi equations of this form. We also show how to construct theexplicit polynomial versions of the three control methods discussed above.

Chapter 4 presents a method for reducing the number of controller state variables using Taylor seriesapproximations. In performing this state-order-reduction, we attempt to affect closed-loop stability aslittle as possible. The state balancing transformation is obtained such that the solutions to our Hamilton-Jacobi equations of the previous chapter are in a specific form. Then, we apply this state transformationto each of our three nonlinear control methods to obtain balanced controllers. And finally, using ourspecial structure for nonlinear terms in the Taylor series, we develop reduced-state-order nonlinearcontrollers.

Chapter 5 presents numerical simulation results demonstrating the effectiveness of the proposedmethods. We examine three control problems in this chapter: the spacecraft attitude problem, satelliteformation flying control, and a nonlinear mass-spring chain. With each problem, we shall emphasizedifferent aspects of the three control methods.

Finally, Chapter 6 states some conclusions resulting from the work presented in this thesis. Wediscuss the main achievements of the work presented in this thesis, as well as some of its limitations.Based on these conclusions, we also suggest areas for future work.

Chapter 2

Background Theory

The purpose of this chapter is to provide the background theory necessary to understand the problemswith which we are concerned in this thesis. We begin with some basic definitions regarding signal normsand spaces, as well as some properties of quadratic forms. Then, we describe in detail the system modelsand feedback configurations of interest to us. This is followed by a discussion of several important anddesirable system properties along with how they are related to the Hamilton-Jacobi equation, which willbe used extensively in the next chapters. The chapter concludes with a description of the small gaintheorem and its connection with the gap metric for robustness analysis and synthesis, which play animportant role in the controller synthesis methods we will use in the next chapter.

2.1 Mathematical Preliminaries

The purpose of this section is to introduce various mathematical concepts that will be required through-out this thesis. We begin this section by introducing signal norms. Then, we make use of these normsto define the signal spaces we will require. Finally, this section concludes by examining some importantproperties of quadratic forms.

2.1.1 Vector Signal Norms

We begin by defining a vector-valued function of time.

Definition 1 A time-varying vector function v : R+ → Rn is written

v(t) =

v1(t)v2(t)

...vn(t)

.

Thus, the value of v at time t ∈ R+ is v(t) ∈ Rn. It will be assumed throughout this thesis that allfunctions v(t) are piecewise continuous. We will often omit the time-dependence notation for the sakeof brevity.

7

Chapter 2. Background Theory 8

We now examine the properties of a norm before going on to define the types of norms we will require.

Definition 2 The norm of a vector v, denoted by ‖v‖, is a real number which satisfies the followingproperties:

• ‖v‖ ≥ 0

• ‖v‖ = 0 ⇔ v = 0

• ‖c v‖ = |c| · ‖v‖, c ∈ R

• ‖v1 + v2‖ ≤ ‖v1‖+ ‖v2‖

where | · | denotes the absolute value of a scalar. The last property is called the triangle inequality.

We now have the following general definition for the norm of a time-varying vector function.

Definition 3 Given some integer p > 0, the Lp-norm of a time-varying vector function v : R+ → Rn

is defined as

‖v(t)‖p =

[∫ ∞0

n∑i=1

|vi(t)|p dt]1/p

.

The most common specialization of this definition is the case where p = 2:

‖v(t)‖2 =[∫ ∞

0

vT (t)v(t) dt]1/2

. (2.1)

We will next use these definitions of signal norms to define the signal spaces we require in this thesis.

2.1.2 Vector Signal Spaces

Here we define the signal spaces we require for our treatment of systems and mappings. We begin witha definition of the Lp space [45]. We will use the notation Lnp to represent the space of Lp functions withdimension n.

Definition 4 Given some integer p > 0, the Lnp space consists of all piecewise continuous, time-varyingvector functions v(t) satisfying

Lnp =

v : R+ → Rn∣∣∣ ‖v(t)‖p <∞

. (2.2)

In other words, we say that a vector v(t) = [v1(t) . . . vn(t)]T is contained in the space Lnp if and onlyif vi(t) ∈ Lp for all i ∈ 1, . . . , n. Note that a function v(t) that satisfies Def. 4 is said to have finiteenergy. The most common specialization of Def. 4 is the case where p = 2.

For our purposes, however, the Lnp space of functions is somewhat restrictive. In order to circumventthis issue, we require the notion of a truncation operator.


Definition 5 For each T ∈ R+, the truncation of a time-varying vector function v : R+ → Rn to thetime interval [0, T ] is defined by

vT (t) =

v(t) , 0 ≤ t < T

0 , t ≥ T (2.3)

Note that‖v(t)‖p = lim

T→∞‖vT (t)‖p. (2.4)

The truncation operator now allows us to define a more general, and more useful, signal space.

Definition 6 The extended Lnp space, also referred to as the extension of Lnp , is defined by

Lnpe =

v : R+ → Rn∣∣∣ vT (t) ∈ Lnp , T ∈ R+

. (2.5)

The space Lnpe contains functions that may have infinite escape times (i.e., that may grow withoutbound as time approaches infinity) in addition to the ones in Lnp . Thus, Lnpe consists of functions whosetruncation lies in Lnp . It is therefore obvious that Lnp ⊂ Lnpe. Again, the most common specialization ofthis last definition is the case where p = 2, resulting in the signal space Ln2e.

We will next discuss some important properties of quadratic forms.

2.1.3 Properties of Quadratic Forms

Here we will define certain properties of symmetric matrices that will be required later in this thesis.We begin by introducing the notation λi(M), which refers to the ith eigenvalue of the matrix M.Additionally, we use the notation λmax(M) to refer to the maximum eigenvalue.

Definition 7 A symmetric matrix M = MT ∈ Rn×n is called positive definite if

xTMx > 0 ∀x ∈ Rn, x 6= 0.

Equivalently, the matrix M is positive definite if λi(M) > 0 for all i ∈ 1, . . . , n.

A somewhat weaker condition is the following.

Definition 8 A symmetric matrix M = MT ∈ Rn×n is called positive semidefinite if

xTMx ≥ 0 ∀x ∈ Rn, x 6= 0.

Equivalently, the matrix M is positive semidefinite if λi(M) ≥ 0 for all i ∈ 1, . . . , n.

Related to these two definitions, we have the following.

Definition 9 A symmetric matrix M = MT ∈ Rn×n is called negative definite if −M is positivedefinite.


Definition 10 A symmetric matrix M = MT ∈ Rn×n is called negative semidefinite if −M is positivesemidefinite.

These definitions will be useful throughout the thesis. We next discuss the types of systems andfeedback configurations that we require in the development of our approach to the synthesis of nonlinearcontrol systems.

2.2 System Models, Feedback Configurations, and Properties

This section describes the class of systems with which we are concerned in this thesis. We begin with adescription of the basic feedback configuration we require, along with the associated input-output (IO)and state space models. We then define several important characteristics of these systems. Finally, wediscuss an alternate and somewhat more general system representation that will also be required.

2.2.1 Class of Systems

We shall be interested in the feedback configuration shown in Fig. 2.1, denoted by [P ,K], for which thefollowing relations hold:

y1(t) = Pu1(t)u2(t) = Ky2(t)u1(t) = u0(t)− u2(t)y2(t) = y0(t)− y1(t)

(2.6)

In general, the plant P and controller K are assumed to be causal, time-invariant and (possibly) nonlinearmappings from their respective inputs and outputs, that is, P : U → Y and K : Y → U , which satisfyP0 = 0 and K0 = 0. The signals ui belong to U ⊆ Lmp

2e and the signals yi belong to Y ⊆ Lpp

2e

for i ∈ 0, 1, 2, where mp is the dimension of the plant input and pp is the dimension of its output.Additionally, it shall be assumed throughout this thesis that the feedback system described by Eq. (2.6)is always well-posed. We shall define the new terminology used here in the next subsection.

P

K

u1 y1

−+

+−u0

u2 y2 y0

Figure 2.1: Feedback configuration [P ,K]

In particular, the plant and controller just described can be represented by state space models. Thenonlinear plant shall be described by the input-affine system

P :

x(t) = a(x) + b(x)u1(t)y1(t) = c(x)

(2.7)


where a(x), b(x) and c(x) are assumed to be smooth (i.e., C∞) nonlinear functions of the plant statevector x(t) with a(0) = 0 and c(0) = 0 (i.e., x(t) = 0 is an equilibrium). Similarly, the nonlinearcontroller shall be described by

K :

xc(t) = ac(xc) + bc(xc)y2(t)u2(t) = cc(xc)

(2.8)

where ac(xc), bc(xc) and cc(xc) are smooth nonlinear functions of the controller state vector xc(t).For the purposes of our discussions throughout this thesis, we shall often require linear versions of

the plant and controller models. The linear analog to the nonlinear plant of Eq. (2.7) is described by

P :

x(t) = A1x + Bu1(t)y1(t) = Cx

(2.9)

and the linear analog to the nonlinear controller of Eq. (2.8) is described by

K :

xc(t) = Acxc + Bcy2(t)u2(t) = Ccxc

(2.10)

In the above linear and nonlinear systems, x(t) is the plant state vector with dimension np, xc(t) isthe controller state vector with dimension nc, u0(t) is the exogenous (disturbance) input, u1(t) is theactual plant input, u2(t) is the control signal, y0(t) is the reference signal (which may contain sensornoise), y1(t) is the actual plant output, and y2(t) is the tracking error. The signal x belongs to X ⊆ Lnp

2e

and the signal xc belongs to Xc ⊆ Lnc2e . Additionally, as above, the input signals ui belong to U ⊆ Lmp

2e

and the output signals yi belong to Y ⊆ Lpp

2e for i ∈ 0, 1, 2.As a final note here, we define the notation [46]

P =

[A1 B

C 0

](2.11)

as an alternate representation of the linear state space model given by Eq. (2.9). This short-handnotation will occasionally be useful in this thesis.

Now that we have defined the basic class of systems we are interested in, we will next define theterminology used above to describe these systems.

2.2.2 System Properties

In the previous subsection, it was assumed that the feedback configuration described by Eq. (2.6) is well-posed and that the plant and controller in this feedback configuration are both causal, time-invariantand (possibly) nonlinear. We shall now define these properties. We begin with the notion of causality.

Definition 11 A plant P defined by Eq. (2.7) is said to be causal if the output y1(t) and internal statex(t) depend only on the current and past inputs u1(t) and never on any future inputs.

It is obvious from this definition that all physically meaningful systems are, in fact, causal.The second definition we present concerns time-invariance.


Definition 12 A plant P defined by Eq. (2.7) is said to be time-invariant if the system matrices a(x),b(x) and c(x) do not depend explicitly on time.

We next define a nonlinear system.

Definition 13 Consider a plant P defined by Eq. (2.7). Given any two input signals u1(t) and u′1(t)with resulting output signals y1(t) = Pu1(t) and y′1(t) = Pu′1(t), along with any two scalars c1 and c2,the plant P is said to be nonlinear if it does not satisfy the following principle of superposition:

c1y1(t) + c2y′1(t) = P [c1u1(t) + c2u′1(t)] .

As mentioned in the previous subsection, we will assume that all systems in this thesis are possiblynonlinear, which means they may also include linear terms.

These last three definitions pertain to the plant and controller in the feedback configuration of Eq.(2.6). We now define the concept of a well-posed feedback system (based on [32]).

Definition 14 The feedback configuration [P ,K] defined by Eq. (2.6) is called well-posed if for anyw = (u0,y0) ∈ W, where W = U × Y, there exist unique u1,u2 ∈ U and y1,y2 ∈ Y such that Eq. (2.6)holds and the map

HP,K :W →W ×W

:

[u0

y0

]7→([

u1

y1

],

[u2

y2

])(2.12)

is causal.

Note that the basic problem of feedback stabilization is to ensure that the feedback configuration of Eq.(2.6) is indeed well-posed [23].

We next describe a system representation that will be required for the robustness analysis to bediscussed later in this chapter.

2.2.3 Graph Representations

It is often convenient to represent operators by their graphs. We now define the plant and controllergraphs.

Definition 15 Consider the feedback configuration [P ,K] defined by Eq. (2.6). The input-output rela-tions describing the plant and controller can be represented by their respective graphs:

GP :=

[u1

Pu1

]: u1 ∈ U , Pu1 ∈ Y

(2.13)

and

GK :=

[Ky2

y2

]: Ky2 ∈ U , y2 ∈ Y

, (2.14)

where GP ,GK ⊂ W = U × Y.


In other words, the graph of the operator P is the subspace of L2 × L2 consisting of all pairs (u1,y1)such that y1 = Pu1 [22]. A similar statement can be made with regards to the graph of the controllerK.

Note that we shall often denote the plant and controller graphs more compactly as

M = GP and N = GK. (2.15)

In the next subsection, we will describe another system representation that will be useful in ourtreatment of nonlinear controller synthesis.

2.2.4 Projection Operators

In the study of robust control systems in this and the next chapters, it will be useful to have an alternaterepresentation of the feedback system in Eq. (2.6). This alternate representation is based on the notionof projection operators. We begin with the defining parallel projection property [23].

Definition 16 An operator Π :W →W is called a (nonlinear) parallel projection if, for any w1,w2 ∈W, the following relationship holds:

Π (Πw1 + (I −Π)w2) = Πw1, (2.16)

where I denotes the identity operator.

We now define a particular pair of projection operators [15, 23].

Definition 17 Consider the feedback configuration [P ,K] defined by Eq. (2.6) with the input-outputmap HP,K given by Eq. (2.12). The parallel projection operators are defined by

ΠM‖N = Π1HP,K

ΠN‖M = Π2HP,K

(2.17)

where Πi : W × W → W denote the natural projections onto the ith component of W → W, withW = U × Y and i ∈ 1, 2.

In this definition, M and N are the plant and controller graphs, respectively, per Eq. (2.15). Theseparallel projection operators satisfy the following identities:

HP,K =(ΠM‖N ,ΠN‖M

)and ΠM‖N + ΠN‖M = I. (2.18)

These two parallel projection operators can be represented by the generalized feedback configuration[P?,K] shown in Fig. 2.2. A consequence of the above properties is that the stability of the feedbacksystem [P ,K] is equivalent to stability of the generalized feedback system [P?,K] as given by eitherparallel projection ΠM‖N or ΠN‖M.


K

w z

u yP?

Figure 2.2: Generalized feedback configuration [P?,K]

The generalized plant corresponding to either of the parallel projection operators in Def. 17 can berepresented via the following state space model:

P? :

x(t) = a(x) + b1(x)w(t) + b2(x)u(t)z(t) = c1(x) + d11(x)w(t) + d12(x)u(t)y(t) = c2(x) + d21(x)w(t)

(2.19)

The first equation here defines the plant dynamics with state variable x(t), control input u(t) andsubject to a set of exogenous inputs w(t), which includes disturbances (to be rejected) and/or references(to be tracked and, if noise is present, filtered). The second equation defines the penalty variable z(t)representing the outputs of interest, which may include a tracking error, a function of some of theexogenous variables w(t), as well as a cost of the input u(t) needed to achieve the prescribed controlgoal. The third equation defines the set of measured variables y(t), which are functions of the plantstate x(t) and the exogenous inputs w(t). The state space model for the controller K in the generalizedfeedback configuration of Fig. 2.2 is again given by Eq. (2.8).

From Def. 17, the parallel projection operators map the following sets of signals:

ΠM‖N :

[u0(t)y0(t)

]7→[u1(t)y1(t)

](2.20)

and

ΠN‖M :

[u0(t)y0(t)

]7→[u2(t)y2(t)

]. (2.21)

For our purposes, we shall focus on the parallel projection ΠN‖M. Therefore, in the state space modelof Eq. (2.19) we will have

w(t) =

[u0(t)y0(t)

], z(t) =

[u2(t)y2(t)

], u(t) = u2(t), y(t) = y2(t). (2.22)


Similar to the case of the feedback configuration [P ,K], we shall at times in this thesis require alinear analog to the plant of Eq. (2.19), which is described by

P? :

x(t) = A1x + B1w(t) + B2u(t)z(t) = C1x + D11w(t) + D12u(t)y(t) = C2x + D21w(t)

(2.23)

In the next section, we will present the theory of dissipative dynamical systems. This theory will beimportant in our treatment of robustness analysis and controller synthesis.

2.3 Dissipative Dynamical Systems

In the study of physical systems, the behaviour of a system can often be deduced using some concept ofstored energy. However, in many control problems, the model used is some mathematical abstraction ofreality. In such cases, it may still be possible to use energy methods, even though the “energy” storedin a system may have no physical meaning. Thus, in the following, when we talk about energy we referto this abstract concept of energy. Interestingly enough, Aleksandr Mikhailovich Lyapunov developedhis famous stability method in 1892 using a concept similar to the energy ideas used here [42, 43].

We begin this section by formally defining the theory encompassing systems that lose, or dissipate,energy. We then introduce the concepts of signal attenuation and finite L2-gain, and show how theseproperties relate to dissipativity.

2.3.1 Dissipativity Theory

We introduce the concept of dissipativity with the following definition.

Definition 18 [73, 74] A plant P defined by Eq. (2.7) is said to be dissipative with respect to a givensupply rate φ : U×Y → R, if there exists a storage function V : X → R that is nonnegative and vanishesat x = 0 satisfying

V (x(t1))− V (x(t0)) ≤∫ t1

t0

φ(u1(t),y1(t)) dt (2.24)

for every input u1(t) ∈ U , every initial state x(t0) ∈ X , and every t1 > t0, where x(t1) is the state of thesystem at time t1 resulting from initial condition x(t0) and the input function u1 acting over the timeinterval [t0, t1].

The storage function V (x(t?)) represents the energy stored by the system at a particular instant in timet?. The integral term on the right-hand side of Eq. (2.24) represents the energy externally supplied tothe system over the time interval [t0, t1]. Eq. (2.24) is called the dissipation inequality and says, roughlyspeaking, that a system must absorb more energy from the external environment than it supplies for itto be dissipative [28]. We note here that the storage function does not need to vanish at x = 0. However,enforcing this condition simplifies the ensuing mathematics by not requiring a constant bias term. Wewill show in the next subsection that the storage function V (x) can actually be considered a Lyapunovfunction.

If the storage function V (x) is at least once continuously differentiable (i.e., at least C1), then thedissipation inequality defined by Def. 18 can be replaced by its equivalent infinitesimal version, which


is also called the differential dissipation inequality [70]:

V (x) = ∇V (x)x(t) ≤ φ(u1(t),y1(t)), (2.25)

where ∇V (x) is the Jacobian matrix of the storage function V (x) and is defined as

∇V (x) =∂V

∂xT= rowi

∂V

∂xi

(2.26)

with rowi· denoting a row matrix in the index i ∈ 1, . . . , np. Note that the supply rate in Def. 18satisfies the following condition [45, 73]:

0 ≤∫ t1

t0

φ(u1(t),y1(t)) dt < ∞. (2.27)

A system is said to be lossless if this last integral is identically zero. It can be shown that the storagefunction for a lossless system is unique [73].

We now define a specific, although still quite general, form of supply rate [28, 45].

Definition 19 A plant P defined by Eq. (2.7) is said to be QSR-dissipative if it is dissipative withrespect to the supply rate

φ(u1,y1) = yT1 Qy1 + 2yT1 Su1 + uT1 Ru1, (2.28)

for given constant matrices Q = QT ∈ Rpp×pp , S ∈ Rpp×mp , and R = RT ∈ Rmp×mp .

Various types of passivity, a concept related to dissipativity, can be defined as special cases of the supplyrate given by Def. 19 [45]. However, we will not elaborate on these here. Instead, in this section we willfocus in the particular supply rate that arises when Q = − 1

21, S = 0 and R = 12γ

21, where γ > 0 is agiven scalar that will be discussed shortly. This particular form of QSR-dissipativity yields the supplyrate

φ(u1,y1) = 12γ

2uT1 u1 − 12yT1 y1. (2.29)

We now make use of this supply rate to define and relate two important perspectives of dissipativity.

2.3.2 Disturbance Attenuation

Substituting the supply rate of Eq. (2.29) into the integral condition of Eq. (2.27) yields the followingdefinition.

Definition 20 A plant P given by Eq. (2.7) is said to attenuate signals from the input to the outputby a factor of at least γ if it satisfies the inequality∫ T

0

yT1 (t)y1(t) dt ≤ γ2

∫ T

0

uT1 (t)u1(t) dt (2.30)

for a given scalar γ.

This relationship provides us with a performance perspective of dissipativity. In particular, this inequal-ity means that the system P mapping inputs u1(t) to outputs y1(t) attenuates signals by a factor of atleast γ. This definition will typically be used in the context of disturbance attenuation, where we will be


interested in the effects of the inputs w(t) on the outputs z(t) of the feedback system [P?,K] shown inFig. 2.2. We will explore the synthesis of controllers to achieve specific disturbance attenuation boundsin the next chapter.

2.3.3 L2-Gain

An alternative perspective of dissipativity relies on the following definition [70, 35].

Definition 21 Consider the plant P given by Eq. (2.7). The L2-gain of P is defined as

‖P‖∞ = sup0 6=u1∈ U

‖y1‖2‖u1‖2 = sup

0 6=u1∈ U

‖Pu1‖2‖u1‖2 , (2.31)

where all initial conditions of the system are quiescent.

The norm in this definition is also called an L2-induced norm because it is induced on a system by theL2-norm for signals.

Comparing the integral condition of Eq. (2.27) with this last definition, it is easy to show how theconcept of dissipativity relates to finite L2-gain.

Definition 22 A plant P given by Eq. (2.7) is said to have an L2-gain less than or equal to γ if it isdissipative with respect to the supply rate of Eq. (2.29) for a given scalar γ.

Thus, a system that is dissipative with respect to the supply rate of Eq. (2.29) for some given scalar γequivalently attenuates inputs by a factor of at least γ and has L2-gain less than or equal to γ.

Although we presented these concepts here with regards to open-loop systems, later in this chapterwe will see how they allow us to introduce the notion of robustness. Using this notion of robustness,in the next chapter we will be interested in the synthesis of controllers to achieve specific bounds onthe L2-gain from the inputs w(t) to the outputs z(t) of the feedback system [P?,K] shown in Fig. 2.2.Next, we will examine how dissipativity relates to Hamilton-Jacobi inequalities and equations.

2.4 Hamilton-Jacobi Equations

Hamilton-Jacobi inequalities (HJI) and equations (HJE) play an important role in the synthesis ofnonlinear optimal and robust control systems. These inequalities and equations can be derived from anonlinear optimization perspective using different cost functions. Here we will show that they can alsobe obtained from dissipativity theory. We will then show that these inequalities and equalities can beexpressed using a common framework.

Using the supply rate of Eq. (2.29), the infinitesimal version of the dissipation inequality given byEq. (2.25) can be written as

∇V (x)x ≤ 12γ

2uT1 u1 − 12yT1 y1. (2.32)

Substituting the plant dynamics of Eq. (2.7) into this inequality and rearranging terms, we obtain

∇V (x)(a(x) + b(x)u1

)− 1

2γ2uT1 u1 + 1

2cT (x)c(x) ≤ 0. (2.33)


The left-hand side of this inequality is a quadratic function in u1, which achieves a global maximumwhen

u?1 = γ−2bT (x)∇V T (x). (2.34)

Substituting this optimal robust state feedback control law back into the inequality of Eq. (2.33), weobtain the following HJI:

∇V (x)a(x) + 12γ−2∇V (x)b(x)bT (x)∇V T (x) + 1

2cT (x)c(x) ≤ 0, (2.35)

which must be satisfied for all x ∈ X . We shall be limiting ourselves in this thesis to the strict equalitycase. Therefore, we consider the Hamilton-Jacobi equation (HJE) corresponding to the HJI (2.35), whichyields

∇V (x)a(x) + 12γ−2∇V (x)b(x)bT (x)∇V T (x) + 1

2cT (x)c(x) = 0. (2.36)

Since the HJE (2.36) can be constructed from H∞ optimization principles and is typically used toconstruct H∞ control systems, we call it an H∞ form of HJE. However, its dependence on γ meansthat the disturbance attenuation factor must be fixed before the HJE can be solved. We will nextconsider another HJE, one that is independent of γ. Because this alternate HJE can be derived usingH2 optimization principles, we call it an H2 form of HJE. Here we shall derive it using the dissipativityframework.

Consider now the supply rate that results from QSR-dissipativity when Q = − 121, S = 0 and

R = 121. This particular form of QSR-dissipativity yields the supply rate

φ(u1,y1) = 12uT1 u1 − 1

2yT1 y1. (2.37)

Note that this supply rate is independent of the disturbance attenuation factor γ.Using the supply rate of Eq. (2.37), the infinitesimal version of the dissipation inequality given by

Eq. (2.25) can be written as∇V (x)x ≤ 1

2uT1 u1 − 12yT1 y1. (2.38)

Substituting the plant dynamics of Eq. (2.7) into this last inequality and rearranging terms, we obtain

∇V (x) (a(x) + b(x)u1)− 12uT1 u1 + 1

2cT (x)c(x) ≤ 0. (2.39)

The left-hand side of this inequality is a quadratic function in u1, which achieves a global maximumwhen

u?1 = bT (x)∇V T (x). (2.40)

Substituting this optimal state feedback control law back into the inequality of Eq. (2.39), we obtain thefollowing HJI:

∇V (x)a(x) + 12∇V (x)b(x)bT (x)∇V T (x) + 1

2cT (x)c(x) ≤ 0, (2.41)

which must be satisfied for all x ∈ X . Once again, we shall be limiting ourselves in this thesis to thestrict equality case. Therefore, we consider the Hamilton-Jacobi equation (HJE) corresponding to theHJI (2.41), which yields

∇V (x)a(x) + 12∇V (x)b(x)bT (x)∇V T (x) + 1

2cT (x)c(x) = 0. (2.42)


It can easily be seen that Eqs. (2.36) and (2.42) possess the same overall structure. We will takeadvantage of this fact to define a general framework for solving Hamilton-Jacobi equations. In particular,the general structure we shall use in the next chapters is represented as follows:

∇V (x)A(x)− 12∇V (x)R(x)∇V T (x) + 1

2Q(x) = 0, (2.43)

where A(x), R(x) and Q(x) are matrices related to the plant dynamics and whose construction dependson the particular problem we are addressing, as we will see later in this thesis. In the next chapter,we will introduce a novel technique for the development of approximate solutions to Hamilton-Jacobiequations with the structure given by Eq. (2.43).

Next, however, we make use of the concepts of system gain recently introduced to develop a criterionfor robust control synthesis.

2.5 Robustness in the Gap Metric

The small gain theorem was introduced by Zames [75, 76] to define an open-loop criterion for closed-loop stability. In fact, the small gain theorem lies at the heart of classical (linear) H∞ control theory.However, the use of norms as a metric limited the applicability of this theory. The gap metric was thendeveloped by Zames and El-Sakkary [16, 77] to provide a metric on (possibly) unstable systems in termsof the distance between their graphs.

This section begins by presenting the small gain theorem and the gap metric. This is followed by adiscussion of how the gap metric can be used to define a closed-loop stability criterion similar to the smallgain theorem and examine how simultaneous perturbations in the plant and controller affect closed-loopstability. Finally, we show how to calculate the gap metric between two (linear) systems.

2.5.1 Small Gain Theorem

The small gain theorem was first introduced by George Zames in 1966 [75, 76]. The main motivationwas to obtain an open-loop criterion that ensures closed-loop stability. The small gain theorem can bestated as follows (based on [75], with proof therein).

Theorem 1 The closed-loop feedback system [P ,K] defined by Eq. (2.6) is guaranteed to be stable ifthe following inequality holds:

‖P‖∞ ‖K‖∞ < 1. (2.44)

It should be emphasized that the small gain theorem only provides a sufficient condition for stability,not a necessary one.

The implications of the small gain theorem are immense and we immediately see how it can beinterpreted in two ways. The first is that any controller satisfying the norm bound would be guaranteedto stabilize the given plant. This has important consequences for controller synthesis. The secondperspective is that any (stable) plant can be controlled by a given control system so long as the normbound is satisfied. This relates to the problem of robust stability, which involves finding a controller tostabilize a set of plants.


The primary disadvantage of Thm. 1 is that it relies on the L2-gain to define the “size” of a system.However, the norm of an unstable system does not exist. In the next two subsections, we will define thegap metric and examine how it may be used to alleviate the deficiencies of the small gain theorem asdefined above.

2.5.2 The Gap Metric

The gap metric was developed by Zames and El-Sakkary [16, 77] specifically for the purpose of studyinguncertainty in feedback systems. The (linear) gap metric was originally conceived to provide a suitabletopology in which small errors in the gap between open-loop systems would correspond to small errors innorm of the stable closed-loop configuration [32, 77]. The gap provides a metric on (possibly) unstablesystems in terms of the distance between their graphs. Vidyasagar introduced an equivalent metric,called the graph metric, using normalized coprime factorizations [71]. It was shown that these twometrics induce the same topology, which is the weakest topology in which feedback stability is a robustproperty [23, 71]. We shall only be interested in the gap metric, which we now define (based on [16, 32],with proof therein).

Definition 23 Consider a (nominal) plant P and some (perturbed) plant P ′. The plants P and P ′must have the same number of inputs and the same number of outputs. The ρ-gap between these twoplants is defined as

ρg(P ,P ′) = max−→ρg(P ,P ′),−→ρg(P ′,P)

, (2.45)

where the directed gap from P to P ′ is defined by

−→ρg(P ,P ′) = sup06=v∈M

infv′∈M′

(‖v − v′‖2 + ‖Pv −P ′v′‖2)1/2(‖v‖2 + ‖Pv‖2)1/2

(2.46)

and −→ρg(P ′,P) is similarly defined.

It should be noted that the gap and norm metrics are equivalent for stable (linear) systems [77].As an extension of the above discussion on the small gain theorem, we will next use the gap metric

to define a condition for robust stability in the presence of simultaneous perturbations in the plant andcontroller.

2.5.3 Simultaneous Perturbations in Plant and Controller

We now examine the situation of simultaneous perturbations in the plant and controller. Previously,we discussed how the small gain theorem can be interpreted in a robust stability sense with regards toplant or controller perturbations. We shall now provide a small-gain-type of criterion for robust stabilitythat makes use of the gap metric to quantify tolerable uncertainty in both the plant and the controller(based on [21, 23], with proof therein).

Theorem 2 Consider a (nominal) plant P and a (nominal) controller K for which the feedback con-figuration [P ,K] is stable. Additionally, consider some (perturbed) plant P ′ with the same number ofinputs and outputs as P and some (perturbed) controller K′ with the same number of inputs and outputsas K. If

ρg(P ,P ′) ‖ΠN‖M‖∞ + ρg(K,K′) ‖ΠM‖N ‖∞ < 1, (2.47)


then the feedback configuration [P ′,K′] is also stable.

Typically, we will be interested in the maximum tolerable plant uncertainty for a given controlleruncertainty. Thus, rearranging the inequality of Thm. 2 yields the following bound:

ρg(P ,P ′) < ‖ΠN‖M‖−1∞[1− ρg(K,K′) ‖ΠM‖N ‖∞

]. (2.48)

The above definition of robustness in the gap metric can easily be reduced to the more common caseof purely plant perturbations (based on [23], with proof therein).

Theorem 3 Consider a (nominal) plant P and a (nominal) controller K for which the feedback con-figuration [P ,K] is stable. Additionally, consider some (perturbed) plant P ′ with the same number ofinputs and outputs as P. If

ρg(P ,P ′) ‖ΠN‖M‖∞ < 1, (2.49)

then the feedback configuration [P ′,K] is also stable.

Moreover, an upper bound on the L2-gain of the (perturbed) feedback configuration [P ′,K] is given by

‖ΠN‖M′‖∞ ≤ ‖ΠN‖M‖∞ 1 + ρg(P ,P ′)1− ρg(P ,P ′) ‖ΠN‖M‖∞

. (2.50)

Note that ‖ΠN‖M‖∞ is the L2-gain of the closed-loop mapping from the exogenous inputs w(t) tothe outputs z(t). Therefore, to minimize the effects of the disturbances w(t) on the outputs z(t), whichsimultaneously achieves optimal robustness, the objective is to design a controller K such that ‖ΠN‖M‖∞is minimized. However, we shall instead be concerned with designing K such that ‖ΠN‖M‖∞ ≤ γ forsome constant scalar γ. We will discuss this in more detail in the next chapter.

Next, we shall see how to calculate the gap metric for linear systems.

2.5.4 Calculation of the Gap Metric for Linear Systems

It was shown by Georgiou [20] that the calculation of the directed gap in Eq. (2.46) is equivalent to thesolution of the H∞-optimization problem

−→ρg(P ,P ′) = infQ∈H∞

‖M−M′Q‖∞, (2.51)

where Q is treated as a free parameter (see Green et al. [25] for more details). The calculation of−→ρg(P ′,P) is similarly defined.

Consider two linear systems given by

P =

[A1 B

C 0

]and P ′ =

[A′1 B′

C′ 0

](2.52)

where we use the superscript (·)′ in P ′ to distinguish the state space matrices associated with the twosystems. Recall that the number of inputs mp to the two systems must be the same, as must the numberof outputs pp of the two systems.


For the calculation of −→ρg(P ,P ′), we begin by defining the indefinite matrix

J(γ) =

[1(mp+pp) 0

0 −γ21(mp)

], (2.53)

where γ > 0 and 1(mp+pp) denotes an (mp + pp) × (mp + pp) identity matrix. Additionally, we definethe following state space system:

[A B

C D

]=

A1 + BBTP1 0 B 0

0 A′1 + B′(B′)TP′1 0 B′

BTP1 (B′)TP′1 1 1

C C′ 0 0

0 0 0 1

(2.54)

where all 0 and 1 matrices are assumed to be of the appropriate dimensions. The matrices P1 andP′1 are the stabilizing solutions to the algebraic Riccati equation (ARE) associated with the two linearsystems. We discuss the ARE further in the next chapter.

From model matching and J-spectral factorization theories [25], there exists a Q ∈ RH∞ such that‖M−M′Q‖∞ < γ if and only if the Hamiltonian

H =

[A 0

−CTJ(γ)C −AT

]−[

B

−CTJ(γ)D

](DTJ(γ)D)−1

[DTJ(γ)C BT

](2.55)

has no eigenvalues on the imaginary axis. The infimum γ can be calculated, to within a prescribedtolerance, using a γ-iteration procedure [25]. Note that this infimum γ is the value of the gap metric.Moreover, for the calculation of the gap, we do not actually need to find Q, but rather simply determinewhether or not it exists. Equivalently, we can simply verify whether or not the Hamiltonian in Eq. (2.55)has any purely imaginary eigenvalues.

Numerous attempts have been made in the literature to extend the above ideas to the computationof the gap metric for nonlinear systems [10, 64]. However, one of the major obstacles in these attemptsis the need to extend the concept of system gain to a nonlinear setting. Due to the difficulty associatedwith such an endeavour, there is still no concrete method for calculating the nonlinear gap metric. Weshall not tackle this problem in the present thesis.

2.6 Chapter Summary

This chapter provided some of the background theory necessary to understand the problems with whichwe are concerned in this thesis. In particular, we described various basic concepts regarding signal normsand spaces, as well as some properties of quadratic forms. Then, we described in detail the system modelsand feedback configurations we will work with. This was followed by a discussion of several importantand desirable system properties along with how they are related to the Hamilton-Jacobi inequality. Wewill use Hamilton-Jacobi equations extensively in the next chapters. The chapter concluded with adescription of the gap metric and its application to robustness analysis and synthesis, which will playan important role in the controller synthesis methods we will use in the next chapter.

Chapter 3

Full-State-Order Controller

Synthesis

The development of optimal nonlinear state feedback control systems is generally characterized by thesolution to a Hamilton-Jacobi partial differential equation (HJE) [55], while a robust nonlinear controlleris obtained from the solution of one or more HJEs [6, 67, 69]. However, no exact analytical solutionmethod has yet been obtained to solve this optimization problem in general. Solutions have thus faronly been obtained under certain conditions. In the case of linear systems with a quadratic performanceindex, the HJE reduces to the well-known algebraic Riccati equation (ARE). As a result, extensive workhas been carried out to approximate the solution of Hamilton-Jacobi equations or inequalities througha Taylor series expansion [1, 18, 19, 30, 41]. Although such a series expansion results in an infinite-order polynomial, finite-order approximations can be used to obtain sub-optimal solutions to an HJE.It has been shown that a local solution to an HJE can be obtained by solving the ARE for the linearapproximation of the system [67, 68, 69]. All the approaches that currently exist in the literature relyon numerical solution methods [8, 9, 13, 14, 27, 44, 56]. One aspect that is lacking in the literature is amethod for obtaining analytical expressions for the approximate solutions to Hamilton-Jacobi equations.

This chapter focuses on developing practical nonlinear controller synthesis techniques. We begin byreiterating our general control objectives, which can be viewed from either a robustness or a performanceperspective. Then, we shall develop a novel technique for solving a general form of Hamilton-Jacobiequation (HJE) using a Taylor series approximation. Our methodology results in analytical expressionsfor the approximate HJE solutions by building recursively from the linear solutions. This analyticalapproximation technique forms the core of this thesis. We then explore three important robust controlmethods from the literature, all of which rely on solutions to the same form of HJE. Finally, we willdevelop explicit polynomial expressions for these three control systems using our approximate solutiontechnique.

3.1 Control Synthesis Summary

We begin this chapter by briefly repeating our general control synthesis objectives. Consider again thefeedback configuration [P ,K] shown in Fig. 3.1, which consists of the nonlinear plant from Eq. (2.7),

23

Chapter 3. Full-State-Order Controller Synthesis 24

repeated here for convenience:

P :

x(t) = a(x) + b(x)u1(t)y1(t) = c(x)

(3.1)

along with the nonlinear controller from Eq. (2.8), also repeated here:

K :

xc(t) = ac(xc) + bc(xc)y2(t)u2(t) = cc(xc)

(3.2)

P

K

u1 y1

−+

+−u0

u2 y2 y0

Figure 3.1: Feedback configuration [P ,K] (repeated)

Additionally, this feedback configuration can be transformed into the generalized configuration [P?,K]shown in Fig. 3.2, which consists of the nonlinear plant from Eq. (2.19), repeated here:

P? :

x(t) = a(x) + b1(x)w(t) + b2(x)u(t)z(t) = c1(x) + d11(x)w(t) + d12(x)u(t)y(t) = c2(x) + d21(x)w(t)

(3.3)

along with the nonlinear controller of Eq. (3.2). We use the symbol ΠN‖M to represent the closed-loop mapping from inputs w(t) to outputs z(t), where M and N are the plant and controller graphs,respectively. For our purposes, we have

w(t) =

[u0(t)y0(t)

], z(t) =

[u2(t)y2(t)

], u(t) = u2(t), y(t) = y2(t). (3.4)

K

w z

u yP?

Figure 3.2: Generalized feedback configuration [P?,K] (repeated)


Our primary goal in this thesis is to develop nonlinear controllers of the general form given by Eq.(3.2). The control objective can be viewed from two equivalent perspectives: robustness and performance.We examine these two perspectives next.

3.1.1 Control Objectives

In the previous chapter, it was shown how to optimize robustness using the gap metric. In particular,for a stable feedback configuration [P ,K], if a system P ′ is such that

ρg(P ,P ′) < ‖ΠN‖M‖−1∞ (3.5)

then the feedback system [P ′,K] is also stable. Thus, for optimal robustness, we must find a controllerK that minimizes ‖ΠN‖M‖∞. However, we shall instead be concerned with the sub-optimal robustnessproblem of designing a controller K such that ‖ΠN‖M‖∞ ≤ γ for some constant scalar γ.

In addition to the robustness point of view, our control objective can be interpreted from a perfor-mance perspective. This means that, for a given controller K, the effects of disturbances in the inputsw(t) on the outputs z(t) are attenuated by a factor of at least γ if∫ T

0

zT (t)z(t) dt ≤ γ2

∫ T

0

wT (t)w(t) dt (3.6)

for all T ≥ 0 with initial state x(0) = 0. Thus, optimal robustness simultaneously achieves the objectiveof minimizing the effects of input disturbances w(t) on the outputs z(t). Similarly, sub-optimal robust-ness is equivalent to the synthesis of a controller K such that the inequality of Eq. (3.6) is satisfied forsome constant scalar γ.

3.1.2 Control Methods

The first control method we will examine in this thesis addresses the H∞ control problem. Our objectivewill be to synthesize controllers such that the L2-gain of the closed-loop mapping from the exogenousinputs y0(t) and u0(t) to the tracking error y2(t) and the control effort u2(t) is bounded by some givenscalar γ. This method differs from the traditional H∞ control problem, which is concerned with theL2-gain from the same exogenous inputs to the plant output y1(t) and the control effort u2(t). Althoughwe are concerned with robust (i.e., H∞) controllers, we will see that we only require the solution toa Hamilton-Jacobi equation (HJE) traditionally associated with the synthesis of H2 controllers. Theparticular advantage of this approach to H∞ control is that we are able to solve the HJE independentlyof γ. The controller is subsequently constructed using the solutions to the HJE and an arbitrary valueof γ, which can be chosen to obtain a particular dynamic response from the controller. One of the maindisadvantages of linear H∞ control methods in general is that they attempt to optimize equally over allfrequencies [46]. This motivates us to examine two control methods that take into consideration a prioriknowledge about the system and the types of exogenous inputs that will affect it.

The idea behind the H∞ loop shaping control problem arose in the linear system theory from a desireto manipulate, or shape, the open-loop gain of a system with regards to frequency. In particular, the H∞loop shaping technique can be used to allow for a trade-off between closed-loop system performance androbustness as a function of frequency. It should be noted, however, that the open-loop shaping technique


to be described does not guarantee that the closed-loop system will be internally stable [46]. We willformulate the nonlinear H∞ loop shaping problem in such a way that it follows the same synthesismethodology as the one developed for our H∞ problem.

While the H∞ control method mentioned above is concerned with minimizing the L2-gain from theexogenous inputs y0(t) and u0(t) to the tracking error y2(t) and the control effort u2(t), the weightedmixed sensitivity control problem seeks to minimize the effects of the input y0(t) (with u0(t) = 0,∀t ≥ 0) on some weighted versions of y2(t) and u2(t). The idea again comes from linear system theory,and provides an alternate method for including some a priori knowledge about the system and the typesof exogenous inputs that will affect it. Although the underlying objective is similar to the loop shapingmethod described above, we will use a different controller synthesis approach.

As we explore these three control methods more thoroughly in the next sections, we will see thatthey each require the solution to a Hamilton-Jacobi equation with a particular structure. Additionally,even though these three control methods satisfy H∞ objectives, we will see that the first two can beconstructed from the solutions to an H2 form of HJE. Of prime importance to us here is that, since theHJE used in all three cases has the exact same structure, we can develop a method of finding approximatesolutions to this form of Hamilton-Jacobi equation and apply it to all three control problems.

Therefore, in the next section we will focus on describing in detail our novel approach to solvinga general HJE with this particular structure. Then, in the sections that follow, we will examine thethree nonlinear control methods in detail. For each control method, we will discuss their linear analogs,explicitly show how the details of the HJE changes, and then describe how to use our approximate HJEsolutions to construct polynomial controllers.

3.2 Analytical Solutions to HJE Approximation

In the previous section, we briefly discussed the fact that all three control methods that interest ushere can be constructed from the solution to a Hamilton-Jacobi equation with a particular structure.In this section, we will examine in detail how to solve this HJE using a Taylor series expansion. Thenovel approach formulated here forms the basis for the methods to be developed in the remainder of thisthesis.

We make a small note here about the notation used throughout this section. We shall use the symbolx(t) to denote a generic state variable that can represent either plant or controller states and may includestates from the loop shaping filter or from the mixed sensitivity weighting functions. This will facilitateour development of a general method for solving the Hamilton-Jacobi equation. Then, in subsequentsections, we will use a more precise notation, which we will describe as needed.

Consider the general HJE (2.43) defined in the previous chapter, repeated here for convenience:

∇V (x)A(x)− 12∇V (x)R(x)∇V T (x) + 1

2Q(x) = 0. (3.7)

The matrices A(x), R(x) and Q(x) are related to the plant dynamics and their construction depends onthe particular problem we are addressing, as we shall see throughout this chapter. Additionally, ∇V (x)is the gradient of the storage function V (x) and is defined as

∇V (x) =∂V

∂xT= rowi

∂V

∂xi

(3.8)


with rowi· denoting a row matrix in the index i ∈ 1, . . . , np. We are generally interested in twoparticular solutions to this HJE. First, we define the stabilizing solution.

Definition 24 The unique smooth stabilizing solution to the HJE (3.7) is denoted by V+(x) and satisfiesV+(x) ≥ 0 for all x ∈ Rnp and V+(0) = 0, with(

A(x)−R(x)∇V T+ (x))

(3.9)

asymptotically stable.

Similarly, we have the following definition for the antistabilizing solution.

Definition 25 The unique smooth antistabilizing solution to the HJE (3.7) is denoted by V−(x) andsatisfies V−(x) ≤ 0 for all x ∈ Rnp and V−(0) = 0, with

−(A(x)−R(x)∇V T− (x)

)(3.10)

asymptotically stable.

Note that we make use of the subscripts (·)+ and (·)− to denote the stabilizing and antistabilizingsolutions, respectively.

Note that these two definitions assume global semidefinite solutions to the HJE. However, due tothe difficulty associated with verifying the sign-definiteness of a general multivariate polynomial, theglobal sign-definiteness of these solutions cannot be guaranteed. Although it has been shown that alocal solution to an HJE can be obtained by solving the ARE for the linear approximation of the system[67, 68, 69], due to the nature of the Taylor series approximation method, it is not possible to show thatthis holds for any arbitrary order of approximation. Therefore, it is important to note that the resultspresented in this thesis are only assumed to hold locally.

We shall now make two simplifying assumptions regarding the class of systems we will use throughoutthe remainder of this thesis. We emphasize, however, that these assumptions are not required in orderto proceed with the derivation of our synthesis method. Rather, they are introduced here solely tosimplify the ensuing derivations. It should be noted that many physical systems naturally satisfy thesetwo assumptions and they are in part motivated by the characteristics of the systems we will simulatelater in this thesis. In particular, we will assume that b(x) = B and c(x) = Cx, where B and C areconstant (real) matrices. The first simplification resulting from these assumptions is that R(x) in Eq.(3.7) will be a constant matrix, which we will denote simply by R. The second simplification is thatQ(x) in Eq. (3.7) will be a quadratic form, which we will simply write as xTQ2x. We will see later inthis chapter that these two simplifications hold for all three of the control problems we will examine asa result of the two assumptions stated here.

As a consequence of the nonlinear plant dynamics a(x), the term A(x) in the HJE (3.7) shall containnonlinearities. To simplify the following derivations, we will assume for now that A(x) = a(x); we willsee later in this chapter how to construct A(x) for each of the different control problems. For thepurpose of the results to be presented, the nonlinear function a(x) will be approximated to fourth-orderas follows:

a(x) = a1(x) + a2(x) + a3(x) + a4(x), (3.11)


wherea1(x) = A1x

a2(x) = colkxTA2kx

a3(x) = matmn

xTA3mnx

x

a4(x) = colkxTmatmn

xTA4kmnx

x (3.12)

In the above, colk· denotes a column matrix in the index k and matmn· denotes a matrix with rowindex m and column index n. The numbers in the subscripts here are used to indicate the order of aterm with respect to the variable x(t). Note that some of the terms in the approximation may be zero,depending on the particular system considered.

Due to the nonlinearities present in the term A(x), the solutions to the HJE will also be nonlinear.We will now discuss our approach to obtaining approximate solutions the HJE (3.7).

3.2.1 Stabilizing HJE Solution Gradient

The stabilizing solution to the Hamilton-Jacobi equation (3.7) can be approximated using a Taylor seriesexpansion to fifth-order as

V+(x) = V+2(x) + V+3(x) + V+4(x) + V+5(x). (3.13)

Recall that we make use of the subscript (·)+ to denote stabilizing terms and the numbers in thesubscripts are used to indicate the order of a term with respect to the generic state variable x(t). As weshall see, for the purposes of our three controller synthesis methods, we will only require the gradient ofthe HJE solutions and not the solutions themselves. Thus, we approximate the stabilizing HJE solutiongradient as follows:

∇V+(x) = ∇V+2(x) +∇V+3(x) +∇V+4(x) +∇V+5(x), (3.14)

where∇V+2(x) = xTP1

∇V+3(x) = rowkxTP2kx

∇V+4(x) = xTmatmn

xTP3mnx

∇V+5(x) = rowk

xTmatmn

xTP4kmnx

x (3.15)

Note that, because the approximation in Eq. (3.14) is a result of taking the gradient of the storagefunction in Eq. (3.13), the numbers in the subscripts no longer correspond to the order of a term in thevariable x(t).

We now substitute the approximate plant dynamics of Eq. (3.11) and the stabilizing solution gradientof Eq. (3.14) into the HJE (3.7). Grouping terms of the same order in x yields

O [‖x‖2] : ∇V+2(x) a1(x)− 12∇V+2(x) R ∇V T+2(x) + 1

2xTQ2x = 0 (3.16)

and the sequence

O [‖x‖k] :k∑i=2

(∇V+i(x) a(k−i+1)(x)− 1

2∇V+i(x)R∇V+(k−i+2)(x))

= 0 (3.17)

for k ≥ 3. At each order k we will only be interested in solving for ∇V+k(x). Therefore, extracting all


terms containing ∇V+k(x) from the summation and rearranging we obtain:

O [‖x‖k] : ∇V+k(x)(a1(x)−R∇V T+2(x)

)+∇V+2(x) a(k−1)(x)

+∑k−1i=3 ∇V+i(x)

(a(k−i+1)(x)− 1

2R∇V T+(k−i+2)(x))

= 0(3.18)

for k ≥ 3. Examining this last equation inspires us to define the following stabilizing closed-loop terms:

a+1(x) = a1(x)−R∇V T+2(x)a+j(x) = aj(x)− 1

2R∇V T+(j+1)(x) j ≥ 2(3.19)

Using this new notation to simplify the above equation yields

O [‖x‖k] : ∇V+k(x) a+1(x) +∇V+2(x) a(k−1)(x) +k−1∑i=3

∇V+i(x) a+(k−i+1)(x) = 0 (3.20)

for k ≥ 3. Note that the summation term in Eq. (3.20) is equal to zero for k = 3, since i > k − 1. Theexpressions for orders k = 2 to k = 5 are given explicitly as

O [‖x‖2] : ∇V+2(x)a1(x)− 12∇V+2(x)R∇V T+2(x) + 1

2xTQ2x = 0

O [‖x‖3] : ∇V+3(x)a+1(x) +∇V+2(x)a2(x) = 0

O [‖x‖4] : ∇V+4(x)a+1(x) +∇V+3(x)a+2(x) +∇V+2(x)a3(x) = 0

O [‖x‖5] : ∇V+5(x)a+1(x) +∇V+4(x)a+2(x) +∇V+3(x)a+3(x) +∇V+2(x)a4(x) = 0

(3.21)

Thus, for each order k, the objective is to solve for the unknown ∇V+k(x). Our approach to obtaininganalytical expressions for the unknowns at each order k is sequential: each order of the solution uses theknown terms in the plant dynamics and builds on the previous solutions. At each step of this iterativemethod, we will develop analytical expressions. This is one of the primary contributions of this thesis.

Second-Order Stabilizing HJE Solution Gradient

The solution sequence begins with the second-order term ∇V+2(x). In particular, solving Eq. (3.16) isequivalent to finding the stabilizing solution P1 to the control algebraic Riccati equation (ARE)

P1A1 + AT1 P1 −P1RP1 + Q2 = 0. (3.22)

This particular ARE is the linear analog to the HJE (3.7). Analogous to Def. 24, we characterize thestabilizing solution to the ARE (3.22) as follows.

Definition 26 The unique positive definite stabilizing solution to the ARE (3.22) is denoted by P1 =PT

1 ∈ Rnp×np and is such that the eigenvalues of the matrix

A+1 =(A1 −RP1

)(3.23)

all have strictly negative real parts.


Note that the stabilizing solution to the ARE (3.22) exists only if the system under consideration is bothstabilizable and detectable. The stabilizing solution to the ARE (3.22) can be determined by standardnumerical methods [4].

Third-Order Stabilizing HJE Solution Gradient

The third-order stabilizing HJE solution gradient is obtained from Eqs. (3.20) or (3.21) with k = 3:

∇V+3(x)a+1(x) +∇V+2(x)a2(x) = 0. (3.24)

This expression can be expanded to yield

rowk

∑i

∑j

xiP(i,j)2k xj

A+1x + rowk

∑i

∑j

xiA(i,j)2k xj

P1x = 0. (3.25)

Canceling all the xi, xj and x’s, this last expression becomes

rowkP

(i,j)2k

A+1 + rowk

A

(i,j)2k

P1 = 0. (3.26)

Rearranging terms, the third-order stabilizing HJE solution gradient is given by

rowkP

(i,j)2k

= −rowk

A

(i,j)2k

P1A−1

+1. (3.27)

Fourth-Order Stabilizing HJE Solution Gradient

The fourth-order stabilizing HJE solution gradient is obtained from Eqs. (3.20) or (3.21) with k = 4:

∇V+4(x)a+1(x) +∇V+3(x)a+2(x) +∇V+2(x)a3(x) = 0. (3.28)


xTmatmnxTP3mnx

A+1x + rows

xTP2sx

cols

xTA+2sx

+xTmatms

xTA3smx

matsn

P

(s,n)1

x = 0.

(3.29)

We now note that∇V+3(x) = rows

xTP2sx

= rows

∑m

∑i xmP

(m,i)2s xi

= rows

rowmxmcolm

∑i P

(m,i)2s xi

= rows

xT colm

∑i P

(m,i)2s xi

= xT rows

colm

∑i P

(m,i)2s xi

= xTmatms

∑i P

(m,i)2s xi

(3.30)


and thata+2(x) = cols

xTA+2sx

= cols

∑j

∑n xjA

(j,n)+2s xn

= cols

rown

∑j xjA

(j,n)+2s

colnxn

= cols

rown

∑j xjA

(j,n)+2s

x

= cols

rown∑

j xjA(j,n)+2s

x

= matsn∑

j xjA(j,n)+2s

x

(3.31)

Therefore, we obtain

xTmatmn∑

i

∑j xiP

(i,j)3mnxj

A+1x + xTmatmn

∑i

∑j xi

(∑s P

(m,i)2s A

(j,n)+2s

)xj

x

+xTmatmn∑

i

∑j xi

(∑sA

(i,j)3smP

(s,n)1

)xj

x = 0.

(3.32)

Canceling all the xi, xj and x’s, this last expression becomes

matmnP

(i,j)3mn

A+1 + matmn

∑s P

(m,i)2s A

(j,n)+2s

+ matmn

∑sA

(i,j)3smP

(s,n)1

= 0. (3.33)

Rearranging terms, the fourth-order stabilizing HJE solution gradient is given by

matmnP

(i,j)3mn

= −

[matmn

∑s P

(m,i)2s A

(j,n)+2s

+ matmn

∑sA

(i,j)3smP

(s,n)1

]A−1

+1, (3.34)

where A(j,n)+2s = A

(j,n)2s − 1

2

∑tR

(s,t)P(j,n)2t .

Fifth-Order Stabilizing HJE Solution Gradient

The fifth-order stabilizing HJE solution gradient is obtained from Eqs. (3.20) or (3.21) with k = 5:

∇V+5(x)a+1(x) +∇V+4(x)a+2(x) +∇V+3(x)a+3(x) +∇V+2(x)a4(x) = 0. (3.35)


rownxTmatmk

xTP4nmkx

x

A+1x + xTmatmsxTP3msx

cols

xTA+2sx

+xTmatms

xTA+3smx

cols

xTP2sx

+ rown

xTmatmk

xTA4nmkx

x

P1x = 0.(3.36)

We now note that

∇V+5(x) = rownxTmatmk

xTP4nmkx

x

= rown∑

m

∑k xm

(∑i

∑j xiP

(i,j)4nmkxj

)xk

= rown

xT colm

∑k

(∑i

∑j xiP

(i,j)4nmkxj

)xk

= xT rown

colm

∑k

(∑i

∑j xiP

(i,j)4nmkxj

)xk

= xTmatmn

∑k

(∑i

∑j xiP

(i,j)4nmkxj

)xk

= xTmatmn

∑i

∑j

∑k xiP

(i,j)4nmkxjxk

(3.37)


and thataT4 (x) = rown

xTmatmk

xTA4nmkx

x

= rown∑

m

∑k xm

(∑i

∑j xiA

(i,j)4nmkxj

)xk

= rown

xT colm

∑k

(∑i

∑j xiA

(i,j)4nmkxj

)xk

= xT rown

colm

∑k

(∑i

∑j xiA

(i,j)4nmkxj

)xk

= xTmatmn

∑k

(∑i

∑j xiA

(i,j)4nmkxj

)xk

= xTmatmn

∑i

∑j

∑k xiA

(i,j)4nmkxjxk

(3.38)

Therefore, we obtain

xTmatmn∑

i

∑j

∑k xiP

(i,j)4nmkxjxk

A+1x

+xTmatmn∑

i

∑j

∑k xi

(∑s P

(i,j)3msA

(k,n)+2s

)xjxk

x

+xTmatmn∑

i

∑j

∑k xi

(∑sA

(i,j)+3smP

(k,n)2s

)xjxk

x

+xTmatmn∑

i

∑j

∑k xiA

(i,j)4nmkxjxk

P1x = 0.

(3.39)

Canceling all the xi, xj , xk and x’s, this last expression becomes

matmnP

(i,j)4nmk

A+1 + matmn

∑s P

(i,j)3msA

(k,n)+2s

+matmn

∑sA

(i,j)+3smP

(k,n)2s

+ matmn

A

(i,j)4nmk

P1 = 0.

(3.40)

Rearranging terms, the fifth-order stabilizing HJE solution gradient is given by

matmnP

(i,j)4nmk

= −

[matmn

∑s P

(i,j)3msA

(k,n)+2s

+ matmn

∑sA

(i,j)+3smP

(k,n)2s

+matmn

A

(i,j)4nmk

P1

]A−1

+1,(3.41)

where A(i,j)+3sm = A

(i,j)3sm − 1

2

∑tRstP

(i,j)3mt .

3.2.2 Antistabilizing HJE Solution Gradient

Following the same approach used in the previous subsection, the antistabilizing solution to the Hamilton-Jacobi equation (3.7) can be approximated using a Taylor series expansion to fifth-order as

V−(x) = V−2(x) + V−3(x) + V−4(x) + V−5(x). (3.42)

We make use of the subscript (·)− to denote antistabilizing terms and the numbers in the subscripts areonce again used to indicate the order of a term with respect to the variable x(t). As stated previously,we will only require the gradient of the antistabilizing HJE solution and not the solution itself. Thus,we approximate the antistabilizing HJE solution gradient as follows:

∇V−(x) = ∇V−2(x) +∇V−3(x) +∇V−4(x) +∇V−5(x), (3.43)


where∇V−2(x) = xTQ1

∇V−3(x) = rowkxTQ2kx

∇V−4(x) = xTmatnm

xTQ3nmx

∇V−5(x) = rowk

xTmatnm

xTQ4knmx

x (3.44)

Additionally, we will define the following antistabilizing closed-loop terms:

a−1(x) = a1(x)−R∇V T−2(x)a−i(x) = ai(x)− 1

2R∇V T−(i+1)(x) i ≥ 2(3.45)

We now substitute the approximate plant dynamics of Eq. (3.11) and antistabilizing solution gradientof Eq. (3.43) into the HJE (3.7). Grouping terms of the same order in x yields

O [‖x‖2] : ∇V−2(x) a1(x)− 12∇V−2(x) R ∇V T−2(x) + 1

2xTQ2x = 0 (3.46)

and the sequence

O [‖x‖k] : ∇V−k(x) a−1(x) +∇V−2(x) a(k−1)(x) +k−1∑i=3

∇V−i(x) a−(k−i+1)(x) = 0 (3.47)

for k ≥ 3. Note that, in Eq. (3.47), the summation term is equal to zero for k = 3, since i > k− 2. Theexpressions for orders k = 2 to k = 5 can be given more explicitly as

O [‖x‖2] : ∇V−2(x)a1(x)− 12∇V−2(x)R∇V T−2(x) + 1

2xTQ2x = 0

O [‖x‖3] : ∇V−3(x)a−1(x) +∇V−2(x)a2(x) = 0

O [‖x‖4] : ∇V−4(x)a−1(x) +∇V−3(x)a−2(x) +∇V−2(x)a3(x) = 0

O [‖x‖5] : ∇V−5(x)a−1(x) +∇V−4(x)a−2(x) +∇V−3(x)a−3(x) +∇V−2(x)a4(x) = 0

(3.48)

Thus, for each order k, the objective is to solve for the unknown ∇V−k. We follow the same approachas in the previous subsection: each order of the solution uses the known terms in the plant dynamicsand builds on the previous solutions.

Second-Order Antistabilizing HJE Solution Gradient

The solution sequence begins with the second-order term ∇V−2(x). In particular, solving Eq. (3.46)is equivalent to finding the antistabilizing solution Q1 to the ARE (3.22). Analogous to Def. 25, wecharacterize the antistabilizing solution to the ARE (3.22) as follows.

Definition 27 The unique negative definite antistabilizing solution to the ARE (3.22) is denoted byQ1 = QT

1 ∈ Rnp×np and is such that the eigenvalues of the matrix

A−1 = +(A1 −RQ1

)(3.49)

all have strictly positive real parts.


Note that the antistabilizing solution to the ARE (3.22) exists only if the system under considerationis both stabilizable and detectable. As in the stabilizing case, the antistabilizing solution to the ARE(3.22) can be determined by standard numerical methods [4]. It can easily be shown that the stabilizingsolution to the filter algebraic Riccati equation, which is the dual to the control ARE (3.22), is equal to−Q−1

1 , although we do not delve into this here.

Third-Order Antistabilizing HJE Solution Gradient

The third-order antistabilizing HJE solution gradient is obtained from Eqs. (3.47) or (3.48) with k = 3in the same way the third-order stabilizing HJE solution gradient was obtained from Eqs. (3.20) or(3.21). The only differences here are the following substitutions:

P2k ⇒ Q2k

P1 ⇒ Q1

A+1 ⇒ A−1

(3.50)

The third-order antistabilizing HJE solution gradient is given by

rowkQ

(i,j)2k

= −rowk

A

(i,j)2k

Q1A−1

−1. (3.51)

Fourth-Order Antistabilizing HJE Solution Gradient

The fourth-order antistabilizing HJE solution gradient is obtained from Eqs. (3.47) or (3.48) with k = 4in the same way the fourth-order stabilizing HJE solution gradient was obtained from Eqs. (3.20) or(3.21). The only differences here are the following substitutions:

P3mn ⇒ Q3mn

P2s ⇒ Q2s

P1 ⇒ Q1

A+2s ⇒ A−2s

A+1 ⇒ A−1

(3.52)

The fourth-order antistabilizing HJE solution gradient is given by

matmnQ

(i,j)3mn

= −

[matmn

∑sQ

(m,i)2s A

(j,n)−2s

+ matmn

∑sA

(i,j)3smQ

(s,n)1

]A−1−1, (3.53)

where A(j,n)−2s = A

(j,n)2s − 1

2

∑tR

(s,t)Q(j,n)2t .

Fifth-Order Antistabilizing HJE Solution Gradient

The fifth-order antistabilizing HJE solution gradient is obtained from Eqs. (3.47) or (3.48) with k = 5 inthe same way the fifth-order stabilizing HJE solution gradient was obtained from Eqs. (3.20) or (3.21).


The only differences here are the following substitutions:

P4nmk ⇒ Q4nmk

P3mn ⇒ Q3mn

P2s ⇒ Q2s

P1 ⇒ Q1

A+3sm ⇒ A−3sm

A+2s ⇒ A−2s

A+1 ⇒ A−1

(3.54)

The fifth-order antistabilizing HJE solution gradient is given by

matmnQ

(i,j)4nmk

= −

[matmn

∑sQ

(i,j)3msA

(k,n)−2s

+ matmn

∑sA

(i,j)−3smQ

(k,n)2s

+matmn

A

(i,j)4nmk

Q1

]A−1−1,

(3.55)

where A(i,j)−3sm = A

(i,j)3sm − 1

2

∑tRstQ

(i,j)3mt .

3.2.3 Combined HJE Solution Gradient

The nonlinear H∞ control problem we will examine in the next section follows the approach of James,Smith and Vinnicombe [32]. In order to make use of their method, however, it is necessary to define aspecial HJE solution that combines the stabilizing and antistabilizing solutions described above. Thiscombined solution can actually be used to prove stability of the control system resulting from theirmethod, which we will explore in the next section. This combined HJE solution is given by [32]

r2e(x) = β−2V+(x) + γ2V−(x), (3.56)

where β =√

1− γ−2.Once again, we will be interested in the gradient of this combined HJE solution and not the solution

itself. Thus, we define the Taylor series approximation of the combined HJE solution gradient in exactlythe same way as the stabilizing and antistabilizing HJE solution gradients. This yields

∇r2e(x) = ∇r2e,2(x) +∇r2e,3(x) +∇r2e,4(x) +∇r2e,5(x), (3.57)

where∇r2e,2(x) = xTR1

∇r2e,3(x) = rowkxTR2kx

∇r2e,4(x) = xTmatnm

xTR3nmx

∇r2e,5(x) = rowk

xTmatnm

xTR4knmx

x (3.58)

We thus haveR

(i,j)1 = β−2P

(i,j)1 + γ2Q

(i,j)1

R(i,j)2k = β−2P

(i,j)2k + γ2Q

(i,j)2k

R(i,j)3nm = β−2P

(i,j)3nm + γ2Q

(i,j)3nm

R(i,j)4knm = β−2P

(i,j)4knm + γ2Q

(i,j)4knm

(3.59)

These expressions will be used shortly when we create the output injection gain for our explicit solution


to the nonlinear H∞ controller.Now that we have shown how to obtain the stabilizing and antistabilizing solutions to the HJE

(3.7), we will examine how to use these to construct three nonlinear controllers. Historically, the linearapproaches to the following methods were developed first. However, for our purposes, we will discussthe more general nonlinear versions and then show how to construct polynomial controllers using thelinear systems as a starting point.

3.3 H∞ Control Problem

As discussed previously, we are concerned with rendering ‖ΠN‖M‖∞ ≤ γ. In other words, we wish todesign a controller such that the L2-gain of the closed-loop mapping from the disturbance inputs w(t)to the outputs z(t) of the system in Eq. (3.3) is bounded by some constant scalar γ. In order to achievethis, we make use of the nonlinear H∞ controller synthesis method of James, Smith, and Vinnicombe[32]. The nonlinear method is based on the linear H∞ controller of McFarlane and Glover [46]. Thisparticular control synthesis approach has the distinct advantage that it relies only on the solutions toan H2 form of HJE. We begin this section by reviewing the nonlinear method of James, Smith, andVinnicombe [32]. We then describe the linear analog and our use of Taylor series approximations toextend this to a polynomial controller.

Consider once again the feedback configuration [P ,K] shown in Fig. 3.1. It was mentioned previouslythat this feedback configuration can be transformed into the generalized configuration [P?,K] shownin Fig. 3.2. This can be accomplished quite easily by substituting the relations of Eq. (3.4) into thestate space plant model of Eq. (3.1) and rearranging terms. The resulting state space model for thegeneralized plant is given by

P? :

x(t) = a(x) +

[b(x) 0

]w(t) +

[−b(x)

]u(t)[

z1(t)z2(t)

]=

[0

−c(x)

]+

[0 0

0 1

]w(t) +

[1

0

]u(t)

y(t) =[−c(x)

]+

[0 1

]w(t)

(3.60)

Without showing the details, we note that the HJE corresponding to the system model in Eq. (3.60)is complicated by the presence of a non-zero d11(x) term. However, this generalized system can betransformed into an equivalent system in which the d11(x) block is equal to zero [26, 32]. The resultingHJE for this transformed system is a technically simpler problem to solve. We now show how thistransformation is performed.

The fundamentals of the transformation are shown in Fig. 3.3. We begin by defining the constantmatrix

Θ =

[Θ11 Θ12

Θ21 Θ22

]= γ−1

[γ−1d11(x)

(1− γ−2d11(x)dT11(x)

)1/2− (1− γ−2dT11(x)d11(x)

)1/2γ−1dT11(x)

](3.61)

which has the property that ΘΘT = γ−21. The square root of some matrix M is defined such that(M1/2)T (M1/2) = 1. Note from Eq. (3.60) that d11(x) is constant.

The constant matrix Θ in Eq. (3.61) and the generalized plant in Eq. (3.60) are now combined usingthe composition operator, also known as the Redheffer Star-Product [26, 78]. This composition results


Θ

P

γ

u2 y2

γ

β y2

(u2,y2)(u0,y0)

(u0, y0)(u0, y0)(u2, y2)(u2, y2)

Figure 3.3: Construction of modified plant P?

in the following system:

P?= C` (Θ, P)

=

264 a + b1Θ22(1− d11Θ)−1c1 b1(1−Θ22d11)−1Θ21 b2 + b1Θ22(1− d11Θ22)−1d12

Θ12(1− d11Θ22)−1c1 0 Θ12(1− d11Θ22)−1d12

c2 + d21Θ22(1− d11Θ22)−1c1 d21(1−Θ22d11)−1Θ21 d22 + d21Θ22(1− d11Θ22)−1d12

375 (3.62)

Note that the d11(x) block is now equal to zero. The final plant model resulting from the set oftransformations depicted in Fig. 3.3 is described by

P?:

x(t) = a(x) +

[−b(x) 0

]w(t) +

[−b(x)

]u(t)[

z1(t)z2(t)

]=

[0

−β−1c(x)

]+

[1

0

]u(t)

y(t) =[−β−1c(x)

]+

[0 −1

]w(t)

(3.63)

where β =√

1− γ−2.The modified plant in Eq. (3.63) can now be used to synthesize a nonlinear H∞ control system using

the HJE (3.7) with the following parameters:

A(x) = a(x)R(x) = b(x)bT (x)Q(x) = cT (x)c(x)

(3.64)

Due to the above transformation, it should be noted that any controller that stabilizes the modifiedplant in Eq. (3.63) also stabilizes the original plant given by Eq. (3.60).

Following the approach of James, Smith, and Vinnicombe [32], a local solution to the nonlinear H∞control problem for the modified plant in Eq. (3.63) is obtained if the following conditions are satisfied:

1. There exists a C3 positive definite function V+(x) defined in a neighbourhood of the origin withV+(0) = 0 that satisfies the HJE (3.7) with parameters from Eq. (3.64).

2. There exists a C3 negative definite function V−(x) defined in a neighbourhood of the origin withV−(0) = 0 that satisfies the HJE (3.7) with parameters from Eq. (3.64), and additionally satisfies

∇2−∇r2e

(A + γ−2β−2R∇V T+)

+ 12γ−2∇r2eR∇rT2e

+ 12β−4∇V+R∇V T+ − 1

2γ2β−2Q

< 0,

(3.65)


where r2e(x) = β−2V+(x) + γ2V−(x).

3. There exists a C2 function G2(x) such that

∇r2e(x)G2(x) = γ2β−1cT (x) (3.66)

in a neighbourhood of the origin.

The nonlinear H∞ controller that stabilizes the system in Eq. (3.60) and results in a closed-loopL2-gain from disturbances w(t) to outputs z(t) less than or equal to γ is given by

K :

xc(t) = a(xc)− b(xc)bT (xc)∇V T+ (xc) + G2(xc)

[β−1c(xc) + βy2(t)

]u2(xc) = β−2bT (xc)∇V T+ (xc)

(3.67)

In the previous section, we saw how to obtain an approximate solution to the HJE (3.7). In thefollowing subsections, we present the linear version of the H∞ controller in Eq. (3.67) and then extendthis using our Taylor series approximation method.

3.3.1 Linear H∞ Controller

We shall now describe the linear version of the above H∞ controller and then extend it using the Taylorseries approach for solving the HJE in the next subsection to obtain a polynomial controller. Using thelinear analog to the nonlinear system of Eq. (3.1) along with the stabilizing solution gradient for theARE (3.22), the nonlinear H∞ controller of Eq. (3.67) simplifies as follows:

K :

xc(t) = A1xc(t)−BBTP1xc(t) + G2(xc)

[β−1Cxc(t) + βy2(t)

]u2(xc) = β−2BTP1xc(t)

(3.68)

Additionally, using the linear form of the combined HJE solution from Eq. (3.57), the output injectiongain condition from Eq. (3.66) becomes

xTR1G2(x) = γ2β−1xTCT . (3.69)

Canceling the leading xT on both sides of this equation and then inverting R1 yields an expression forthe output injection gain:

G2(x) = γ2β−1R−11 CT . (3.70)

Therefore, the linear H∞ controller corresponding to Eq. (3.67) is given by

K :

xc(t) = A1xc(t)−BBTP1xc(t) + γ2β−1R−1

1 CT[β−1Cxc(t) + βy2(t)

]u2(xc) = β−2BTP1xc(t)

(3.71)

As an aside, we note that the linear H∞ controller of Eq. (3.71) is effectively the same as the oneby McFarlane and Glover [46], although expressed in a somewhat different form. In fact, the method ofJames, Smith and Vinnicombe [32] is based primarily on the work of McFarlane and Glover.

We will next extend this linear controller to the nonlinear case using our methodology for the synthesisof polynomial controllers based on Taylor series expansions.


3.3.2 Polynomial H∞ Controller

We now form an explicit polynomial solution to the nonlinear H∞ control problem described above.Since the Taylor series approximation method presented earlier in this chapter is novel, the polynomialcontrol system to be presented here is also new. For the sake of generality, the above description ofthe controller considered all nonlinear terms. On the other hand, here we will include the Taylor seriesapproximation of the nonlinear plant dynamics a(x) given by Eq. (3.11) and the simplifying assumptionsb(x) = B and c(x) = Cx. This means solving the HJE (3.7) using the following parameters:

A(x) = a(x)R(x) = BBT = R

Q(x) = xTCTCx = xTQ2x

(3.72)

Note that R(x) is in fact constant and Q(x) is a quadratic form in x. Therefore, obtaining the stabilizingand antistabilizing solution gradients to the HJE (3.7) using these parameters results exactly in theexpressions obtained in Sec. 3.2.

It remains for us to describe our approach to constructing the nonlinear output injection gain G2(xc).For notational simplicity, we will derive this result in terms of the generic state variable x(t) insteadof xc(t). As discussed at the beginning of this section, the output injection gain G2(x) is obtained bysolving the equation

∇r2e(x)G2(x) = γ2β−1cT (x), (3.73)

where r2e(x) = β−2V+(x) + γ2V−(x). We solve for G2(x) in two steps. First, we extract and cancelthe leading xT from both sides of this equation. Then, we invert the resulting left-hand square matrixnumerically.

Extracting the leading xT from the combined HJE solution gradient of Eq. (3.57), we obtain

∇r2e(x) = xTΨ(x), (3.74)

where we can define the Taylor series approximation of the term Ψ(x) as follows:

Ψ(x) = Ψ0(x) + Ψ1(x) + Ψ2(x) + Ψ3(x), (3.75)

withΨ0(x) = Ψ0

Ψ1(x) = matstΨ1stxΨ2(x) = matstxTΨ2stxΨ3(x) = matstrowkxTΨ3stkxx

We now show how to extract the leading xT from each term in the combined HJE solution gradientand thus create the individual terms for Ψ(x) in Eq. (3.75). The second-order term in x becomes

∇r2e,2(x) = xTR1 = xTΨ0 (3.76)


where Ψ0 is a square matrix with Ψ(i,j)0 = R

(i,j)1 . The third-order term in x becomes

∇r2e,3(x) = rowkxTR2kx= rowk

∑m

∑i xmR

(m,i)2k xi

= rowkxT colm∑iR

(m,i)2k xi

= xTmatmk∑iR

(m,i)2k xi

= xTmatmk∑i Ψ1mk,ixi

= xTmatmkΨ1mkx

(3.77)

where Ψ1st = rowiΨ1st,i is a row matrix with Ψ1st,i = R(s,i)2t . The fourth-order term in x becomes

∇r2e,4(x) = xTmatstxTR3stx= xTmatstxTΨ2stx

(3.78)

where Ψ2st is a square matrix with Ψ(i,j)2st = R

(i,j)3st . And finally, the fifth-order term in x becomes

∇r2e,5(x) = rowkxTmatstxTR4kstxx= rowk

∑s

∑t xs

(∑i

∑j xiR

(i,j)4kstxj

)xt

= rowkxT cols∑t

(∑i

∑j xiR

(i,j)4kstxj

)xt

= xTmatsk∑t

(∑i

∑j xiR

(i,j)4kstxj

)xt

= xTmatskrowt∑i

∑j xiR

(i,j)4kstxjx

= xTmatskrowt∑i

∑j xiΨ

(i,j)3sktxjx

= xTmatskrowtxTΨ3sktxx

(3.79)

where Ψ3skt is a square matrix with Ψ(i,j)3skt = R

(i,j)4kst.

Note that, since the output matrix c(x) is linear, extracting the leading xT from the right-hand sideof Eq. (3.73) is trivial. Therefore, canceling the leading xT from both sides of Eq. (3.73) results in

Ψ(x)G2(x) = γ2β−1CT . (3.80)

Inverting the square matrix Ψ(x) numerically, the nonlinear output injection gain is given by

G2(x) = γ2β−1 [Ψ(x)]−1 CT . (3.81)

Therefore, an explicit solution to the nonlinear H∞ controller of Eq. (3.67) is given by

K :

xc(t) = a(xc)−BBT∇V T+ (xc) + γ2β−1 [Ψ(xc)]

−1 CT[β−1Cxc(t) + βy2(t)

]u2(xc) = β−2BT∇V T+ (xc)

(3.82)

where the approximate plant dynamics a(xc) are given by Eq. (3.11), the stabilizing HJE solutiongradient ∇V+(xc) is given by Eq. (3.14), and the output injection gain term Ψ(xc) is given by Eq.(3.75) and is inverted numerically.


3.4 H∞ Loop Shaping Control Problem

In this section, we examine the concept of nonlinearH∞ loop shaping and formulate a controller synthesismethod similar to the one developed in the previous section. The idea of loop shaping arose in the linearsystem theory from a desire to manipulate, or shape, the open-loop gain of a system with regards tofrequency. In particular, the H∞ loop shaping technique can be used to allow for a trade-off betweenperformance and robustness [46].

We begin with the plant in Eq. (3.1) and include a (possibly) nonlinear filter given by

F :

xF (t) = aF (xF ) + bF (xF )uF (t)yF (t) = cF (xF ) + dF (xF )uF (t)

(3.83)

where xF (t) is the filter state, uF (t) is the filter input, and yF (t) is the filter output. This configurationis shown in Fig. 3.4.

P

K

uF = u1 y1

−+

+−u0 = 0

u2 y2 y0

F yF

P

Figure 3.4: Feedback configuration [P ,K] with loop shaping filter on P

The plant and filter are combined to create the augmented system

P :

[x(t)

xF (t)

]︸︷︷︸

x

=

[a(x) + b(x)cF (xF )

aF (xF )

]︸︷︷︸

a(x)

+

[b(x)dF (xF )

bF (xF )

]︸︷︷︸

b(x)

u1(t)

y1(t) = c(x)︸︷︷︸c(x)

(3.84)

where the augmented state variable is defined as

x(t) =

[x(t)

xF (t)

](3.85)

and has dimensions np + nF .Using the augmented system in Eq. (3.84) in place of the plant model in Eq. (3.1), the method of

the previous section can be applied to design an H∞ controller. The resulting nonlinear controller isgiven by

K :

xc(t) = a(xc)− b(xc)b

T(xc)∇V T+ (xc) + G2(xc)

[β−1c(xc) + βy2(t)

]u2(xc) = β−2b

T(xc)∇V T+ (xc)

(3.86)


where the controller state xc(t) is an estimate of both the plant and filter states in x(t) as defined inEq. (3.85). Note, however, that this controller is designed for the augmented system. The filter F mustthen be combined with the controller, as shown in Fig. 3.5.

P

K

u1 y1

−+

+−u0 = 0

u2 = yF y2 y0F uF

KFigure 3.5: Feedback configuration [P ,K] with loop shaping filter on K

The resulting augmented nonlinear controller is given by

K :

xc(t) = a(xc)− b(xc)b

T(xc)∇V T+ (xc) + G2(xc)

[β−1c(xc) + βy2(t)

]uF (xc) = β−2b

T(xc)∇V T+ (xc)

xF (xc,xF ) = aF (xF ) + bF (xF )uF (xc)u2(xc,xF ) = cF (xF ) + dF (xF )uF (xc)

(3.87)

In the next subsection, we will discuss in more detail the motivation behind the linear filteringfunction that we will use. Then, we focus on developing the linear solution to this problem. We finallycombine this with our Taylor series method to form an approximate solution to the nonlinear H∞ loopshaping control problem.

3.4.1 Linear Filter Function

It is well known that “we cannot expect to achieve arbitrarily good performance and robustness overall frequencies” [46] as a result of the Bode integrals. Therefore, the motivation for our linear filtercomes from a desire to increase control authority in the low-frequency region where performance isimportant, while simultaneously limiting the allowable control effort at higher frequencies where noiseand uncertainty may be present. Traditionally, this concept translates into specifying bounds on thesingular values of the combined plant-filter open-loop system [46].

In order to simplify our lives, we shall consider a diagonal low-pass filter given by

F(s) = kF

(s+ 10ωFs+ ωF

)1, (3.88)

where ωF is the bandwidth, kF is the gain, and 1 is the identity matrix. The state space model of F inEq. (3.83) then takes the form

F :

xF (t) = AFxF (t) + BFuF (t)yF (t) = CFxF (t) + DFuF (t)

(3.89)


with constant matrices

AF = (−ωF ) 1, BF = (9ωF kF ) 1, CF = 1, DF = (kF ) 1. (3.90)

Using this linear filter in our synthesis methodology will considerably simplify matters. Althoughthis linear concept does not intuitively translate to the nonlinear case, we will make use of the generalidea. In the next subsection, we will focus on developing the linear solution to this problem. We thencombine this with our Taylor series method to form an approximate solution to the nonlinear H∞ loopshaping control problem.

3.4.2 Linear H∞ Loop Shaping Controller

We now describe the linear version of the H∞ loop shaping problem. It should be noted that this open-loop shaping technique does not guarantee that the closed-loop system will be internally stable. Thiscaveat applies equally to the nonlinear situation, although we do not address the issue in this thesis. It isfurther assumed that the combined plant-filter open-loop system does not contain any hidden unstablemodes [46]. Additionally, it is worth noting that even for linear systems “the appropriate selection ofweights over frequency and between objectives is not straightforward, and tends to be developed for eachspecific example” [46].

Combining the linear plant of Eq. (2.9) with the filter model in Eq. (3.89), we obtain the linearanalog to the augmented system in Eq. (3.84):

P :

[x(t)

xF (t)

]︸︷︷︸

x

=

[A1 BCF

0 AF

]︸︷︷︸

A

[x(t)

xF (t)

]︸︷︷︸

x

+

[BDF

BF

]︸︷︷︸

B

u1(t)

y1(t) =[C 0

]︸︷︷︸

C

[x(t)

xF (t)

] (3.91)

Using this augmented system, the linear method of the previous section can be applied to design anaugmented controller analogous to Eq. (3.87). Therefore, an explicit solution to the linear H∞ loopshaping control problem is given by

K :

xc(t) = Axc(t)−BB

TP1xc(t) + γ2β−1R−1

1 CT [β−1Cxc(t) + βy2(t)

]uF (xc) = β−2B

TP1xc(t)

xF (xc,xF ) = aFxF (t) + bFuF (xc)u2(xc,xF ) = cFxF (t) + dFuF (xc)

(3.92)

We will next extend this linear controller to the nonlinear case using our methodology for the synthesisof polynomial controllers based on Taylor series expansions.

3.4.3 Polynomial H∞ Loop Shaping Controller

We now form an explicit polynomial solution to the nonlinearH∞ loop shaping control problem describedabove. Once again, since the Taylor series approximation method presented earlier in this chapter isnovel, the polynomial control system to be presented here is also new. For the sake of generality, the


above description of the controller considered all nonlinear terms. On the other hand, here we willinclude the Taylor series approximation of the nonlinear plant dynamics a(x) given by Eq. (3.11) andthe simplifying assumptions b(x) = B and c(x) = Cx, in addition to the linear filtering function of Eq.(3.89).

The HJE associated with the construction of the nonlinear H∞ loop shaping controller will keepthe same general form as defined in Eq. (3.7). However, we replace our generic variable x(t) bythe augmented state variable x(t) defined in Eq. (3.85). The HJE solution gradients will thus havedimensions of np + nF instead of simply np. This means solving the HJE (3.7) using the followingparameters:

A(x) = a(x) =

[a(x) + BCFxF

AFxF

]

R(x) = b(x)bT

(x) =

[BDFDT

FBT BDFBTF

BFDTFBT BFBT

F

]= R

Q(x) = cT (x)c(x) =

[x

xF

]T [CTC 0

0 0

][x

xF

]= xTQ2x

(3.93)

Note that R(x) is in fact constant and Q(x) is a quadratic form in x. Moreover, to obtain the stabilizingand antistabilizing solution gradients to the HJE (3.7) using these parameters, we simply replace x(t)by x(t) in the corresponding expressions in Sec. 3.2.

Using the augmented system matrices B and C from Eq. (3.91) and replacing x(t) by x(t) whereappropriate, the polynomial method of the previous section can be applied to design an augmentedcontroller analogous to Eq. (3.87). Therefore, an explicit solution to the nonlinear H∞ loop shapingcontrol problem is given by

K :

xc(t) = a(xc)−BB

T∇V T+ (xc) + γ2β−1 [Ψ(xc)]−1 C

T [β−1Cxc(t) + βy2(t)

]uF (xc) = β−2B

T∇V T+ (xc)xF (xc,xF ) = aFxF (t) + bFuF (xc)u2(xc,xF ) = cFxF (t) + dFuF (xc)

(3.94)

In the next section, we will examine the weighted mixed sensitivity control problem.

3.5 Weighted Mixed Sensitivity Control Problem

The standard H∞ control method above was concerned with minimizing the L2-gain from the exogenousinputs y0(t) and u0(t) to the tracking error y2(t) and the control effort u2(t). We now consider theweighted mixed sensitivity problem, which seeks to minimize the effects of the exogenous input y0(t)(with u0(t) = 0 ∀t ≥ 0) on some weighted versions of y2(t) and u2(t), as shown in Fig. 3.6. The ideaagain comes from the linear case, where we may be interested in the output from some frequency-weightedversions of y2(t) and u2(t). This provides a method for including some a priori knowledge about thesystem and the types of exogenous inputs that will affect it. Although the underlying concept is similarto the loop shaping method described above, we will use a different controller synthesis approach.

Consider the configuration shown in Fig. 3.6, where a state space representation of the two weighting


functions can be written as

W1 :

x1(t) = aw1(x1) + bw1(x1)y2(t)z1(t) = cw1(x1)

(3.95)

and

W2 :

x2(t) = aw2(x2) + bw2(x2)u2(t)z2(t) = cw2(x2) + dw2(x2)u2(t)

(3.96)

Consider once again the feedback configuration [P ,K] shown in Fig. 3.1. It was shown in Sec. 3.3how this feedback configuration can be transformed into the generalized configuration [P?,K] shownin Fig. 3.2. Now, we construct the generalized system P? by combining the states of the plant in Eq.(3.1) with those of the two weighting functions in Eqs. (3.95) and (3.96). Performing these substitutionsyields

P?:

8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:

264 x(t)

x1(t)

x2(t)

375| z

x

=

264 a(x)

aw1(x1)− bw1(x1)c(x)

aw2(x2)

375| z

a(x)

+

264 0

bw1(x1)

0

375| z

b1(x)

y0(t) +

264 −b(x)

0

bw2(x2)

375| z

b2(x)

u(t)

"z1(t)

z2(t)

#| z

z

=

"cw1(x1)

cw2(x2)

#| z

c1(x)

+

"0

0

#|z

d11(x)

y0(t) +

"0

dw2(x2)

#| z

d12(x)

u(t)

y(t) =h−c(x)

i| z

c2(x)

+h1i

|zd21(x)

y0(t) +h0i

|zd22(x)

u(t)

(3.97)

where the augmented state variable is define as

x(t) =

x(t)x1(t)x2(t)

(3.98)

and has dimensions np + nw1 + nw2.Following the approach of Ball and van der Schaft [7], a solution to the nonlinear H∞ control problem

for the generalized plant in Eq. (3.97) is obtained if the following conditions are satisfied:

1. The d21(x) block in Eq. (3.97) is square and invertible for all x. The disturbance feedforward isthen given by

y0(x, t) = d−1

21 (x) [y(t)− c2(x)] . (3.99)

2. The vector field (a(x)− b1(x)d

−1

21 (x)c2(x))

(3.100)

is asymptotically stable.

3. There exists a positive definite function V+(x) with V+(0) = 0 that satisfies the HJE (3.7) with


P

K

u1 y1

−+

+−u0 = 0

u2 y2 y0

W2 W1

z2 z1

Figure 3.6: Feedback configuration [P ,K] with mixed sensitivity weights W1 and W2

parametersA(x) = a(x)− b2(x)e−1(x)d

T

12(x)c1(x)

R(x) = b2(x)e−1(x)bT

2 (x)− γ−2b1(x)bT

1 (x)

Q(x) = cT1 (x)[1− d12(x)e−1(x)d

T

12(x)]

c1(x)

(3.101)

and with (A(x)−R(x)∇V T+ (x)

)(3.102)

asymptotically stable, where e(x) = dT

12(x)d12(x) is positive definite for all x(t).

The resulting nonlinear controller is given by

K :

xc(t) = a(xc) + b1(xc)y0(xc) + b2(xc)u(xc)

y0(xc) = d−1

21 (xc) [y(t)− c2(xc)]

u(xc) = −e−1(xc)[dT

12(xc)c1(xc) + bT

2 (xc)∇V T+ (xc)] (3.103)

where the controller state xc(t) is an estimate of the combined plant and weighting states in x(t) asdefined in Eq. (3.98).

In the next subsection, we will discuss in more detail the motivation behind the linear weightingfunctions that we will use. Then, we focus on developing the linear solution to this problem. We finallycombine this with our Taylor series method to form an approximate solution to the nonlinear weightedmixed sensitivity control problem.

3.5.1 Linear Weighting Functions

As a particular scenario for this problem, we shall consider the case of first-order filters on each of y2(t)and u2(t). Furthermore, these filters will be made diagonal such that they weight each of the tracking


errors and control signals equally among themselves (i.e., all the tracking errors are equally weightedand all the control signals are equally weighted).

For the tracking error weighting function, we consider the low-pass filter

W1(s) =(

ωys+ ωyεy

)1, (3.104)

where ωy is the desired closed-loop bandwidth and εy is a low-frequency tracking error specification.This function sets an upper bound on the sensitivity function S(s) according to

|S(jω)| < |W−11 (jω)| (∀ ω ≥ 0). (3.105)

A rough sketch of the magnitude plot is shown in Fig. 3.7, where the notation | · | refers to the magnitude(Bode plot) of the quantity in question.

ω

εy

ωy

∣∣W−11 (jω)

∣∣

|S(jω)|

1

Figure 3.7: Magnitude plot of tracking error weighting function

The state space model of W1(s) in Eq. (3.95) then takes the form

W1 :

x1(t) = Aw1x1(t) + Bw1y2(t)z1(t) = Cw1x1(t)

(3.106)

with constant matricesAw1 = (−εy/ωy) 1, Bw1 = (ωy) 1, Cw1 = 1. (3.107)

For the control effort weighting function, we consider the high-pass filter

W2(s) =(s+ ωu/mu

εus+ ωu

)1, (3.108)

where mu is the actuator saturation limit, ωu is the actuator bandwidth and εu is a property influenced


by the sensor noise characteristics. This function sets an upper bound on K(s)S(s) according to

|K(jω)S(jω)| < |W−12 (jω)| (∀ ω ≥ 0). (3.109)

A rough sketch of the magnitude plot is shown in Fig. 3.8.

ω

εu

ωu

Mu∣∣W−1

2 (jω)∣∣

|K(jω)S(jω)|

1

Figure 3.8: Magnitude plot of control effort weighting function

The state space model of W2(s) in Eq. (3.96) then takes the form

W2 :

x2(t) = Aw2x2(t) + Bw2u2(t)z2(t) = Cw2x2(t) + Dw2u2(t)

(3.110)

with constant matrices

Aw2 = (−ωu/εu) 1, Bw2 = (εu/mu − 1)(ωu/ε2u) 1,

Cw2 = 1, Dw2 = (1/εu) 1.(3.111)

Thus, W1(s) is chosen to keep the tracking error small at low frequency while simultaneously allowingfor greater uncertainty due to sensor noise at high frequencies. For W2(s) this translates into allowinggreater, although still bounded, control authority in the low-frequency region which then ‘rolls off’ athigher frequencies.

Using these linear weighting functions in our synthesis methodology will simplify matters consider-ably. Like in the loop shaping problem, although this linear concept does not intuitively transfer to thenonlinear case, we will make use of the general idea. In the next subsection, we will focus on developingthe linear solution to this problem. We then combine this with our Taylor series method to form anapproximate solution to the nonlinear weighted mixed sensitivity control problem.


3.5.2 Linear Weighted Mixed Sensitivity Controller

We now describe the linear version of the weighted mixed sensitivity problem. Combining the linearplant of Eq. (2.9) with the weighting functions in Eqs. (3.106) and (3.110), we obtain the linear analogto the augmented generalized system in Eq. (3.97):

P?:

x(t)x1(t)x2(t)

︸︷︷︸

x

=

A1 0 0

−Bw1C Aw1 0

0 0 Aw2

︸︷︷︸

A

x(t)x1(t)x2(t)

︸︷︷︸

x

+

0

Bw1

0

︸︷︷︸

B1

y0(t) +

−B

0

Bw2

︸︷︷︸

B2

u(t)

[z1(t)z2(t)

]︸︷︷︸

z

=

[0 Cw1 0

0 0 Cw2

]︸︷︷︸

C1

x

x1

x2

+

[0

0

]︸︷︷︸D11

y0(t) +

[0

Dw2

]︸︷︷︸

D12

u(t)

y(t) =[−C 0 0

]︸︷︷︸

C2

x

x1

x2

+[1]

︸︷︷︸D21

y0(t) +[0]

︸︷︷︸D22

u(t)

(3.112)

Due to the invertibility of the D21 block in (3.112), the last equation of (3.112) can be rearranged toobtain an expression for the exogenous input: y0(t) = D

−1

21

[y(t)−C2x

]. Under the assumption that(

Ax−B1D−1

21 C2x)

is an asymptotically stable vector field, the controller is then given by

K :

xc(t) = Axc(t) + B1y0(xc) + B2u(xc)

y0(xc) = D−1

21

[y(t)−C2xc(t)

]u(xc) = −E

−1[DT

12C1xc(t) + BT

2 P1xc(t)] (3.113)

where E = DT

12D12.We will next extend this linear controller to the nonlinear case using our methodology for the synthesis

of polynomial controllers based on Taylor series expansions.

3.5.3 Polynomial Weighted Mixed Sensitivity Controller

In this subsection, we will form an explicit polynomial solution to the nonlinear weighted mixed sen-sitivity control problem described above. Once again, since the Taylor series approximation methodpresented earlier in this chapter is novel, the polynomial control system to be presented here is also new.For the sake of generality, the above description of the controller considered all nonlinear terms. On theother hand, here we will include the Taylor series approximation of the nonlinear plant dynamics a(x)given by Eq. (3.11) and the simplifying assumptions b(x) = B and c(x) = Cx, in addition to the linearweighting functions of Eqs. (3.106) and (3.110).

The HJE associated with the construction of the nonlinear weighted mixed sensitivity controller willkeep the same general form as defined in Eq. (3.7). However, we replace our generic variable x(t) bythe augmented state variable x(t) defined in Eq. (3.98). The HJE solution gradients will thus have


dimensions of np + nw1 + nw2. This means solving the HJE (3.7) using the parameters:

A(x) = a(x)−B2E−1

DT

12C1x

=

a(x)0

0

+

0 0 B(DTw2Dw2

)−1DTw2Cw2

−Bw1C Aw1 0

0 0 Aw2 −Bw2

(DTw2Dw2

)−1DTw2Cw2

x

x1

x2

(3.114)

and

R(x) = B2E−1

BT

2 − γ−2B1BT

1

=

B(DTw2Dw2

)−1BT 0 −B

(DTw2Dw2

)−1BTw2

0 −γ−2Bw1BTw1 0

−Bw2

(DTw2Dw2

)−1BT 0 Bw2

(DTw2Dw2

)−1BTw2

= R

(3.115)

and

Q(x) = xTCT

1

[1−D12E

−1DT

12

]C1x

=

x

x1

x2

T

0 0 0

0 CTw1Cw1 0

0 0 CTw2

[1−Dw2

(DTw2Dw2

)−1DTw2

]Cw2

x

x1

x2

= xTQ2x

(3.116)

where E = DT

12D12 and the constant matrices are explicitly given in Eq. (3.112). Note that R(x) is infact constant and Q(x) is a quadratic form in x. Moreover, to obtain the stabilizing and antistabilizingsolution gradients to the HJE (3.7) using these parameters, we simply replace x(t) by x(t) in thecorresponding expressions in Sec. 3.2. It is assumed here that

(a(x)−B1D

−1

21 C2x)

is an asymptoticallystable vector field.

Therefore, an explicit solution to the nonlinear weighted mixed sensitivity control problem is givenby

K :

xc(t) = a(xc) + B1y0(xc) + B2u(xc)

y0(xc) = D−1

21

[y(t)−C2xc(t)

]u(xc) = −E

−1[DT

12C1xc(t) + BT

2∇V T+ (xc)] (3.117)

where E = DT

12D12.

3.6 Chapter Summary

This chapter focused on developing practical nonlinear controller synthesis techniques. We began byreiterating our general control objectives, which can be viewed from either a robustness or a performanceperspective. Then, we developed a novel technique for solving a Hamilton-Jacobi equation (HJE) with aspecific structure using a Taylor series approximation. Our methodology results in analytical expressionsfor the approximate HJE solutions by building recursively from the linear solutions. We then exploredthree important robust control methods from the literature, all of which rely on solutions to a Hamilton-Jacobi equation with the same structure. We also developed expressions for these control systems using


our approximate solution technique.

Chapter 4

Reduced-State-Order Controller

Synthesis

Modern state-space controller synthesis techniques typically result in state estimators of equal or greaterdimension than the plant model. However, it is often desirable, or even necessary, to approximate thesecontrollers by models of lower state dimension. In doing this, though, we wish to avoid adversely affectingclosed-loop stability as much as possible. This is accomplished by first transforming the controller statevector into a special set of balanced coordinates. The state transformation is obtained in such a way thatthe balanced controller states are sorted in decreasing order of their relative importance with regards tothe closed-loop input-output (IO) map. Thus, eliminating the least important controller states impactsthe IO map the least. Throughout this thesis, we shall refer to this process of reducing the number ofstates in a system as “state-order-reduction.” It is not to be confused with the process of reducing thepolynomial order of a system, which we will not examine here.

The balanced truncation method was originally developed for linear open-loop plant models [48].However, it was found that the subsequent synthesis of controllers for these reduced-state-order plantscould have undesirable consequences [33]. Thus, the LQG-balancing method was developed by Jonck-heere and Silverman [34] to address the issue of balancing closed-loop feedback systems with LinearQuadratic Gaussian (LQG) controllers. Similarly, H∞-balancing was introduced by Mustafa [49] andfurther analyzed by Mustafa and Glover [50] to make use of H∞ controllers. These closed-loop methodsprovided a means of balancing the controller states in such a way that the closed-loop system char-acteristics were taken into consideration. As a natural extension of these linear balancing techniques,Scherpen introduced balanced truncation for stable open-loop nonlinear systems [59]. Scherpen thenextended this method to closed-loop H∞-balancing for nonlinear systems [60]. Fujimoto and Tsubakino[17] introduced a method for calculating the balancing transformation based on a Taylor series approach.However, there are certain issues that they do not adequately address, which we discuss later in thischapter.

Our approach to the synthesis of nonlinear reduced-state-order controllers will make use of the Taylorseries techniques discussed in Chapter 3. In particular, we saw in the previous chapter that our methodfor the synthesis of nonlinear control systems is based on the solutions to a Hamilton-Jacobi equation(HJE) with a particular structure. In this chapter, our approach to the development of a nonlinear statebalancing transformation shall be based on the exact same HJE solutions. Although we are balancing

52

Chapter 4. Reduced-State-Order Controller Synthesis 53

H∞ controller systems here, the approach we use is analogous to the LQG-balancing techniques in theliterature.

We shall begin this chapter by examining how a nonlinear state transformation can be applied tobalance our three nonlinear control systems. After that, we introduce the conditions that will be usedto derive such a state transformation. Then, we proceed to derive the nonlinear state transformation,its gradient, and an approximate inverse of its gradient. Once the transformation has been obtained,we show how it can be applied to the different terms that comprise the three nonlinear control systems.Finally, we discuss how to explicitly construct the three reduced-state-order control systems.

4.1 Balanced Nonlinear Controllers

In this thesis, we shall be interested in a general nonlinear state balancing transformation

x = T(z). (4.1)

As in the previous chapter, we shall use the symbol x(t) to denote a generic state variable that canrepresent either plant or controller states and may include states from the loop shaping filter or fromthe mixed sensitivity weighting functions. Additionally, we use the symbol z(t) to represent the corre-sponding balanced state vector. Note that the use of the symbol z(t) here should not be confused withthe outputs-of-interest in the generalized plant of Eq. (2.19). The meaning of this symbol shall be clearfrom the context.

In the next section we will present our novel method for actually obtaining such a transformation.First, however, we will demonstrate how such a nonlinear state transformation can be applied to each ofthe three nonlinear controllers that interest us. It was shown in the linear case by Mustafa and Glover[50] that the reduced-state-order controller is the “full-state-order” controller for the reduced-state-orderplant, provided the state transformation was derived from the appropriate closed-loop ARE solutions.We will see shortly that the same holds in the nonlinear case as well. A consequence of this is that areduced-state-order controller can be designed in one of two equivalent ways: either by first balancingand reducing the state-order of the plant and then designing the reduced-state-order controller, or bydesigning the full-order controller and then balancing and reducing the state-order of the controller. Weshall take the second approach in this thesis.

In each of the three controller cases, we shall see that the individual matrices that define the controllercan be balanced independently of each other. Additionally, the nonlinear terms that define each controllermust be completely balanced before any state-order-reduction can be performed. Obtaining analyticalexpressions for the balanced nonlinear expressions is an issue that does not appear to be addressed inthe existing literature. The recent work of Fujimoto and Tsubakino [17] presents a solution method forthe balancing of nonlinear systems, but does not properly address the question of obtaining a completelybalanced model before performing the state-order-reduction.


4.1.1 Balanced Nonlinear H∞ Controller

We now examine how a nonlinear state transformation can be applied to obtain a balanced nonlinearH∞ controller. Taking the time derivative of the state transformation in Eq. (4.1), we obtain

x(t) = ∇T(z)∣∣∣z=F(x)

z(t)∣∣∣z=F(x)

(4.2)

where ∇T(z) denotes the gradient of the transformation and z = F(x) is the inverse transformationfrom x(t) to z(t) such that T

(F(x)

)= x. The equation for the derivative can be rearranged to obtain

∇T(z)z(t) = x(t)∣∣∣x=T(z)

(4.3)

Additionally, we denote the matrix inverse of the transformation gradient as ∇T−1(z), such that ∇T(z) ·∇T−1(z) = 1.

Combining the transformation derivative of Eq. (4.3) with the nonlinear H∞ controller in Eq. (3.67)yields

K :

∇T(zc)zc(t) = a(T(zc))− b(T(zc))bT (T(zc))∇V T+ (T(zc))

+G2(T(zc))[β−1c(T(zc)) + βy2

]u2(zc) = β−2bT (T(zc))∇V T+ (T(zc))

(4.4)

which can be rearranged to obtain

K :

zc(t) = ∇T−1(zc)a(T(zc))−∇T−1(zc)b(T(zc))bT (T(zc))∇V T+ (T(zc))

+∇T−1(zc)G2(T(zc))[β−1c(T(zc)) + βy2

]u2(zc) = β−2bT (T(zc))∇V T+ (T(zc))

(4.5)

Recall that a general nonlinear control system should reduce to its linear analog. This reasoning mustalso apply to a balanced control system. Comparing some of the “balanced” terms in this last system withthose of a fully balanced linear system, we see that not all of the matrices here are completely balanced.Therefore, using concepts from linear balancing techniques, the completely balanced nonlinear controlleris given by

K :

zc(t) =

[∇T−1(zc)a(T(zc))]

− [∇T−1(zc)b(T(zc))] [

bT (T(zc))∇T−T (zc)] [∇TT (zc)∇V T+ (T(zc))

]+[∇T−1(zc)G2(T(zc))

] [β−1c(T(zc)) + βy2

]u2(zc) = β−2

[bT (T(zc))∇T−T (zc)

] [∇TT (zc)∇V T+ (T(zc))] (4.6)

which can be rewritten as

K :

zc(t) = a(zc)− b(zc)bT (zc)∇V T+ (zc) + G2(zc)

[β−1c(zc) + βy2

]u2(zc) = β−2bT (zc)∇V T+ (zc)

(4.7)


wherea(zc) = ∇T−1(zc) a

(T(zc)

)b(zc) = ∇T−1(zc) b

(T(zc)

)c(zc) = c

(T(zc)

)∇V+(zc) = ∇V+

(T(zc)

) ∇T(zc)G2(zc) = ∇T−1(zc) G2

(T(zc)

)(4.8)

Examining the relations in Eq. (4.8), we see that we will require expressions for the balancingtransformation, its gradient and the inverse of its gradient. We shall therefore derive these expressionsin the next three sections. Later in this chapter, we will present analytical expressions for the balancedsystem matrices a(zc), b(zc) and c(zc), the balanced stabilizing HJE solution gradient ∇V+(zc), and thebalanced output injection gain G2(zc). Once these matrices are obtained, we develop explicit expressionsfor the balanced nonlinearH∞ controller and then show how to construct the reduced-state-order version.Next, however, we show how to balance the other two nonlinear control systems of interest to us in thisthesis.

4.1.2 Balanced Nonlinear H∞ Loop Shaping Controller

The process of applying a nonlinear state balancing transformation to the H∞ loop shaping controller isnearly identical to the derivation of the previous subsection. However, the controller state vector zc(t)is now a balanced estimate of the augmented vector x(t) in Eq. (3.85), which includes both the plantand filter states. Thus, the state balancing transformation T(x) in Eq. (4.1) now has dimensions ofnp+nF , where nF is the state dimension of the loop shaping filter. Additionally, it is important to notethat the loop shaping filter F in Eq. (3.83) does not get transformed in any way. Rather, the balancingtransformation is applied to the states xc(t) in the controller K in Eq. (3.86) that was designed for theaugmented system in Eq. (3.84) resulting from the combined plant and filter.

Applying the approach derived in the previous subsection to the controller K in Eq. (3.86), thebalanced nonlinear controller is given by

K :


[β−1c(zc) + βy2

]u2(zc) = β−2bT (zc)∇V T+ (zc)

(4.9)


(T(zc)

)b(zc) = ∇T−1(zc) b

(T(zc)

)c(zc) = c

(T(zc)

)∇V+(zc) = ∇V+

(T(zc)

) ∇T(zc)G2(zc) = ∇T−1(zc) G2

(T(zc)

)(4.10)

Then, combining this balanced controller with the loop shaping filter, the balanced augmented nonlinearH∞ loop shaping controller is given by

K :


[β−1c(zc) + βy2

]uF (zc) = β−2bT (zc)∇V T+ (zc)xF (zc,xF ) = aF (xF ) + bF (xF )uF (zc)u2(zc,xF ) = cF (xF ) + dF (xF )uF (zc)

(4.11)


The methods for balancing the matrices in Eq. (4.10) are identical to those required for Eq. (4.8),except for the matrix dimensions. Once these matrices are obtained, we develop explicit expressions forthe balanced nonlinear H∞ loop shaping controller and then show how to construct the reduced-state-order version. Next, however, we show how to balance the last nonlinear control systems of interest tous in this thesis.

4.1.3 Balanced Nonlinear Weighted Mixed Sensitivity Controller

We now examine how to apply the nonlinear state balancing transformation to the weighted mixedsensitivity controller. This process is similar to the derivation of the previous two subsections. However,the controller state vector zc is now a balanced estimate of the augmented vector x(t) in Eq. (3.98),which includes the plant states and the states from both weighting functions. A result of this is thatthe state balancing transformation T(x) in Eq. (4.1) now has dimensions of np + nw1 + nw2, where nw1

is the state dimension of the tracking error weighting function and nw2 is the state dimension of thecontrol effort weighting function.

Combining the transformation derivative of Eq. (4.3) with the nonlinear controller in Eq. (3.103)yields

K :

∇T(zc)zc(t) = a(T(zc)) + b1(T(zc))y0(T(zc)) + b2(T(zc))u(T(zc))

y0(zc) = d−1

21 (T(zc)) [y(t)− c2(T(zc))]

u(zc) = −e−1(T(zc))[dT

12(T(zc))c1(T(zc)) + bT

2 (T(zc))∇V T (T(zc))] (4.12)

which can be rearranged to obtain

K :

zc(t) = ∇T−1(zc)a(T(zc)) +∇T−1(zc)b1(T(zc))y0(T(zc))

+∇T−1(zc)b2(T(zc))u(T(zc))

y0(zc) = d−1

21 (T(zc)) [y(t)− c2(T(zc))]


12(T(zc))c1(T(zc)) + bT

2 (T(zc))∇V T (T(zc))] (4.13)

Again, not all of the matrices in this last set of equations are completely balanced. Using concepts fromlinear balancing techniques, the completely balanced nonlinear controller is given by

K :

zc(t) = ∇T−1(zc)a(T(zc)) +∇T−1(zc)b1(T(zc))y0(T(zc))+∇T−1(zc)b2(T(zc))u(T(zc))

y0(zc) = d−1

21 (T(zc)) [y(t)− c2(T(zc))]


12(T(zc))c1(T(zc))

+

bT

2 (T(zc))∇T−T (zc)∇TT (zc)∇V T (T(zc))

](4.14)

which can be rewritten as

K :

zc(t) = a(zc) + b1(zc)y0(zc) + b2(zc)u(zc)

y0(zc) = d−1

21 (zc)[y(t)− c2(zc)

]u(zc) = −e

−1(zc)

[dT

12(zc)c1(zc) + bT

2 (zc)∇V T+ (zc)] (4.15)



(T(zc)

)b1(zc) = ∇T−1(zc) b1

(T(zc)

)b2(zc) = ∇T−1(zc) b2

(T(zc)

)c2(zc) = c2

(T(zc)

)d12(zc) = d12

(T(zc)

)d21(zc) = d21

(T(zc)

)e(zc) = e

(T(zc)

)∇V+(zc) = ∇V+

(T(zc)

) ∇T(zc)

(4.16)

Once again, the methods for balancing many of the matrices in Eq. (4.16) are identical to thoserequired for Eq. (4.8), except for the matrix dimensions. In particular, we note that the method forbalancing the input matrices b1(zc) and b2(zc) is identical to the method for b(zc). Similarly, balancingthe matrix c2(zc) is identical to the method for c(zc). Additionally, it should be noted that the matricesd12(zc), d21(zc) and e(zc) are in fact constant for our purposes, as seen in the previous chapter. Thus,they do not require any balancing. Later in this chapter, we develop explicit expressions for the balancednonlinear weighted mixed sensitivity controller and then show how to construct the reduced-state-orderversion.

In the next section, we introduce the conditions that will be used to derive a nonlinear state trans-formation of the form defined by Eq. (4.1). Then, in subsequent sections, we proceed to derive thenonlinear state transformation, its gradient, and an approximate inverse of its gradient. Each of thesederivations will make use of the same Taylor series approximation technique used in the previous chapter.

4.2 Controller State Balancing Objectives

Now that we understand how the nonlinear state transformation in Eq. (4.1) can be used to balancethe three nonlinear control systems we will use in this thesis, we will proceed to define conditions forobtaining such a state transformation. In general, the transformation of Eq. (4.1) can be derived indifferent ways to obtain any number of new coordinate systems [59]. Our objective here, however, isto obtain a transformation such that the controller states are in balanced coordinates. These balancedcoordinates are such that the controller state variables are sorted in decreasing order of their relativeimportance with regards to the closed-loop input-output map. In general, balancing transformationsare characterized by different system properties, depending on whether an open-loop or a closed-loopbalancing approach is desired. The original balancing method developed by Moore [48] for linear open-loop systems made use of the observability and controllability gramians. In the case of closed-loopbalancing techniques, the type of controller used also plays a role. Thus, the solutions to different formsof algebraic Riccati equations (AREs) were used, depending on whether LQG or H∞ controllers weredesired [34, 49, 50]. When these concepts were extended to nonlinear open-loop systems, nonlinearobservability and controllability functions were introduced to replace the gramians [59]. Then, variousforms of past and future energy functions were used for nonlinear closed-loop systems instead of AREsolutions [60].

The approach used here is based on the nonlinear closed-loop H∞ balancing method of Scherpen [60]and makes use of nonlinear past and future energy functions. Recall, however, that we made use of anH2 form of Hamilton-Jacobi equation for controller synthesis in the previous chapter. In consequence,


we now make use of H2 versions of the past and future energy functions. We define these two conceptsnow.

Definition 28 The H2 past energy function is defined as

E−(x0) = maxu∈L2(−∞,0]

12

∫ 0

−∞

(yTy + uTu

)dt (4.17)

with x(−∞) = 0 and x(0) = x0. This energy function is the smooth nonnegative solution to theHamilton-Jacobi equation

∇E−(x)a(x) + 12∇E−(x)b(x)bT (x)∇ET−(x) + 1

2cT (x)c(x) = 0 (4.18)

with E−(0) = 0, such that−(a(x) + b(x)bT (x)∇ET−(x)

)(4.19)


Definition 29 The H2 future energy function is defined as

E+(x0) = minu∈L2[0,∞)

12

∫ ∞0

(yTy + uTu

)dt (4.20)

with x(0) = x0 and x(∞) = 0. This energy function is the smooth nonnegative solution to the Hamilton-Jacobi equation

∇E+(x)a(x)− 12∇E+(x)b(x)bT (x)∇ET+(x) + 1

2cT (x)c(x) = 0 (4.21)

with E+(0) = 0, such that (a(x)− b(x)bT (x)∇ET+(x)

)(4.22)


It should be noted from these definitions that the energy functions are related to the stabilizing andantistabilizing solutions of the HJE (3.7) according to

V+(x) = +E+(x)V−(x) = −E−(x)

(4.23)

Although different types of balanced coordinates exist, they are all essentially equivalent [60]. Theapproach we take in this thesis is based on the so-called input-normal form [17, 60]. Making use ofthe relationship between the energy functions and the HJE solutions as defined in Eq. (4.23), the statebalancing transformation will be derived such that

V+(z) = V+

(T(z)

)= 1

2zT z

V−(z) = V−(T(z)

)= Σ(z)

(4.24)

where Σ(z) is an arbitrary nonlinear function. Although the choice of balanced coordinates is somewhatarbitrary, we opt to use this particular form because the state transformation T(z) can be uniquelydetermined from the stabilizing solution to the HJE, as seen by examining Eq. (4.24). We shall takeadvantage of this fact in deriving our nonlinear state balancing transformation.


In the next sections, we shall see how to use the above conditions to develop expressions for thebalancing transformation, its gradient and the inverse of its gradient. Following this, we shall derive thebalanced matrices required to subsequently construct our three balanced nonlinear control systems.

4.3 Nonlinear Balancing Transformation

We shall now approximate the nonlinear state transformation of Eq. (4.1) using a Taylor series expansionas follows:

x = T(z) = T1(z) + T2(z) + T3(z) + T4(z), (4.25)

whereT1(z) = T1z

T2(z) = colkzTT2kz

T3(z) = matmn

zTT3mnz

z

T4(z) = colkzTmatmn

zTT4kmnz

z (4.26)

As mentioned previously, we let x(t) represent the unbalanced state vector and z(t) the balanced one.For the purposes of our controller state balancing, we are interested in obtaining a transformation

such thatV+(z) = V+

(T(z)

)= 1

2zT z. (4.27)

However, note that our Taylor series approach calculates the gradient of the HJE solution and not thesolution itself. In general, we can write the HJE solution in terms of its gradient as follows:

V+(x) =∞∑k=2

1k ∇V+k(x) x(t). (4.28)

Thus, for the particular HJE solution approximation of Eq. (3.13), we have

V+(x) = V+2(x) + V+3(x) + V+4(x) + V+5(x)= 1

2∇V+2(x)x(t) + 13∇V+3(x)x(t) + 1

4∇V+4(x)x(t) + 15∇V+5(x)x(t)

(4.29)

Substituting the transformation of Eq. (4.25) into the storage function in Eq. (4.29) and grouping termsof the same order in z(t) yields a set of expressions for the balanced HJE solution. The notation wewill use here is meant to emphasize that the expressions for the HJE solution gradient are polynomialfunctions of x(t). Hence, the Taylor series approximation of the transformation in Eq. (4.25) must beinserted for each appearance of x(t) in the HJE solution. These expressions are given as follows.

The second-order terms in z(t) are

V+2(z) = 12∇V+2

(T1(z)

)T1(z). (4.30)

The third-order terms in z(t) are

V+3(z) = 12∇V+2

(T1(z)

)T2(z) + 1

2∇V+2

(T2(z)

)T1(z)

+ 13∇V+3

(T1(z),T1(z)

)T1(z).

(4.31)


The fourth-order terms in z(t) are

V+4(z) = 12∇V+2

(T1(z)

)T3(z) + 1

2∇V+2

(T2(z)

)T2(z) + 1

2∇V+2

(T3(z)

)T1(z)

+ 13∇V+3

(T1(z),T1(z)

)T2(z) + 1

3∇V+3

(T1(z),T2(z)

)T1(z)

+ 13∇V+3

(T2(z),T1(z)

)T1(z) + 1

4∇V+4

(T1(z),T1(z),T1(z)

)T1(z).

(4.32)

And finally, the fifth-order terms in z(t) are

V+5(z) = 12∇V+2

(T1(z)

)T4(z) + 1

2∇V+2

(T2(z)

)T3(z)

+ 12∇V+2

(T3(z)

)T2(z) + 1

2∇V+2

(T4(z)

)T1(z)

+ 13∇V+3

(T1(z),T1(z)

)T3(z) + 1

3∇V+3

(T1(z),T2(z)

)T2(z)

+ 13∇V+3

(T2(z),T1(z)

)T2(z) + 1

3∇V+3

(T2(z),T2(z)

)T1(z)

+ 13∇V+3

(T1(z),T3(z)

)T1(z) + 1

3∇V+3

(T3(z),T1(z)

)T1(z)

+ 14∇V+4

(T1(z),T1(z),T1(z)

)T2(z) + 1

4∇V+4

(T1(z),T1(z),T2(z)

)T1(z)

+ 14∇V+4

(T1(z),T2(z),T1(z)

)T1(z) + 1

4∇V+4

(T2(z),T1(z),T1(z)

)T1(z)

+ 15∇V+5

(T1(z),T1(z),T1(z),T1(z)

)T1(z).

(4.33)

In the following subsections, we will show how to obtain the state balancing transformation from theseexpressions.

4.3.1 First-Order Balancing Transformation

Based on the linear methods of state balancing, the first-order balancing transformation matrix T1 isdefined such that

TT1 P1T1 = 1

T−11 Q−1

1 T−T1 = −Σ2(4.34)

where P1 and Q1 are the stabilizing and antistabilizing solutions to the ARE of Eq. (3.22), respectively.Additionally, Σ = diagiσi, where σ2

i = λi(−P1Q−11 ) > 0 and σ1 ≥ . . . ≥ σnp

, with σi denoting theLQG-characteristic values of the (closed-loop) system [34]. The LQG-characteristic value σi provides ameasure of the relative importance of the balanced state zi in the closed-loop input-output map. Thus,balanced states with the smallest LQG-characteristic values have the least effect on the IO mapping,which will be useful later in this chapter when we discuss reducing the number of controller statevariables.

The transformation T1 in Eq. (4.34) is calculated using the following algorithm, which is based onLaub et al. [39]. First, compute a factorization of the two ARE solutions such that LPLTP = P1 andLQLTQ = −Q−1

1 . Second, compute the singular value decomposition (SVD) of the product LTPLQ =UΛVT . Then, the balancing transformation and its inverse are given by

T1 = LQVΛ−1

T−11 = UTLTP

(4.35)

The analytical nature of these first-order terms allows us to develop a recursive formulation of thehigher-order balancing expressions, as we shall see through the course of the next subsections.


4.3.2 Second-Order Balancing Transformation

The second-order state transformation term is obtained by equating Eq. (4.31) to zero. The derivationproceeds as follows:

V+3 = 12zTTT

1 P1colkzTT2kz

+ 1

2 rowkzTT2kz

P1T1z

+ 13 rowk

zTTT

1 P2kT1z

T1z

= 0.

(4.36)


zTTT1 P1colk

∑i

∑j

ziT(i,j)2k zj

+ 13zTTT

1 colk

∑i

∑j

zi(TT1 P2kT1)(i,j)zj

= 0. (4.37)

Canceling all the z’s and z’s and rearranging, the second-order state transformation term is given by

colkT

(i,j)2k

= − 1

3P−11 colk

(TT

1 P2kT1)(i,j). (4.38)

4.3.3 Third-Order Balancing Transformation

The third-order state transformation term is obtained by equating Eq. (4.32) to zero. The derivationproceeds as follows:

V+4 = 12zTTT

1 P1matmnzTT3mnz

z + 1

2zTmatnmzTT3mnz

P1T1z

+ 12 rows

zTT2sz

P1cols

zTT2sz

+ 1

4zTTT1 matst

zT (TT

1 P3stT1)z

T1z

+ 13 rows

zT (TT

1 P2sT1)z

colszTT2sz

+ 1

3 rowszT (TT

1 P2s)coltzTT2tz

T1z

+ 13 rows

rowt

zTT2tz

(P2sT1)z

T1z

= 0.

(4.39)


zT (TT1 P1)matmn

∑i

∑j ziT

(i,j)3mnzj

z

+ 12 rows

∑m

∑i zmT

(m,i)2s zi

P1colt

∑j

∑n zjT

(j,n)2t zn

+ 1

4zTTT1 matst

∑i

∑j zi(T

T1 P3stT1)(i,j)zj

T1z

+ 13 rows

∑m

∑i zm(TT

1 P2sT1)(m,i)zi

cols∑

j

∑n zjT

(j,n)2s zn

+ 1

3 rows∑

m

∑t zm(TT

1 P2s)(m,t)(∑

i

∑j ziT

(i,j)2t zj

)T1z

+ 13 rows

∑t

∑j

(∑m

∑i zmT

(m,i)2t zi

)(P2sT1)(t,j)zj

T1z = 0.

(4.40)


Rewriting columns and rows as matrices and rearranging, we have

zT (TT1 P1)matmn

∑i

∑j ziT

(i,j)3mnzj

z =

− 12zTmatms

∑i T

(m,i)2s zi

P1mattn

∑j zjT

(j,n)2t

z

− 14zTTT

1 matst∑

i

∑j zi(T

T1 P3stT1)(i,j)zj

T1z

− 13zTmatms

∑i(T

T1 P2sT1)(m,i)zi

matsn

∑j zjT

(j,n)2s

z

− 13zTmatms

∑t(T

T1 P2s)(m,t)

(∑i

∑j ziT

(i,j)2t zj

)T1z

− 13zTmatms

∑t

∑j

(∑i T

(m,i)2t zi

)(P2sT1)(t,j)zj

T1z.

(4.41)

Canceling all the z’s and z’s, this last expression becomes

(TT1 P1)matmn

T

(i,j)3mn

=

− 12matms

T

(m,i)2s

matst

P

(s,t)1

mattn

T

(j,n)2t

− 1

4matmsT

(s,m)1

matst

(TT

1 P3stT1)(i,j)

mattnT

(t,n)1

− 1

3matms

(TT1 P2sT1)(m,i)

matsn

T

(j,n)2s

− 1

3matms∑

t(TT1 P2s)(m,t)T

(i,j)2t

matsn

T

(s,n)1

− 1

3matms∑

t T(m,i)2t (P2sT1)(t,j)

matsn

T

(s,n)1

.

(4.42)

Therefore, the third-order state transformation term is given by

matmnT

(i,j)3mn

= −P−1

1 T−T1 matmn

12

∑s

∑t T

(m,i)2s P

(s,t)1 T

(j,n)2t

+ 14

∑s

∑t T

(s,m)1 (TT

1 P3stT1)(i,j)T (t,n)1

+ 13

∑s(T

T1 P2sT1)(m,i)T (j,n)

2s

+ 13

∑s

∑t(T

T1 P2s)(m,t)T

(i,j)2t T

(s,n)1

+ 13

∑s

∑t T

(m,i)2t (P2sT1)(t,j)T (s,n)

1

.

(4.43)

4.3.4 Fourth-Order Balancing Transformation

The fourth-order state transformation term is obtained by equating Eq. (4.33) to zero. The derivationproceeds as follows:

V+5 = 12zT TT

1 P1colk˘zT matmn

˘zT T4kmnz

¯z¯

+ 12rowp

˘zT T2pz

¯P1matqn

˘zT T3qnz

¯z

+ 12zT matkp

˘zT T3pkz

¯P1colq

˘zT T2qz

¯+ 1

2rowk

˘zT matmn

˘zT T4kmnz

¯z¯

P1T1z

+ 13rowp

˘zT TT

1 P2pT1z¯

matpn

˘zT T3pnz

¯z + 1

3rowp

˘zT TT

1 P2pcolq˘zT T2qz

¯¯colp

˘zT T2pz

¯+ 1

3rowp

˘rowq

˘zT T2qz

¯P2pT1z

¯colp

˘zT T2pz

¯+ 1

3rowp

˘rowq

˘zT T2qz

¯P2pcolr

˘zT T2rz

¯¯T1z

+ 13rowp

˘zT TT

1 P2pmatqj

˘zT T3qjz

¯z¯

T1z + 13rowp

˘zT matkq

˘zT T3qkz

¯P2pT1z

¯T1z

+ 14zT TT

1 matpq

˘zT TT

1 P3pqT1z¯

colq˘zT T2qz

¯+ 1

4zT TT

1 matpq

˘zT TT

1 P3pqcolr˘zT T2rz

¯¯T1z

+ 14zT TT

1 matpq

˘rowr

˘zT T2rz

¯P3pqT1z

¯T1z + 1

4rowp

˘zT T2pz

¯matpq

˘zT TT

1 P3pqT1z¯

T1z

+ 15rowp

˘zT TT

1 matqr

˘zT TT

1 P4pqrT1z¯

T1z¯

T1z

= 0.

(4.44)



zT (TT1 P1)colk

nPm

Pn zm

“Pi

Pj ziT

(i,j)4kmnzj

”zn

o+ 1

2rowp

nPk

Pm zkT

(k,m)2p zm

omatpq

nP

(p,q)1

omatqn

nPi

Pj ziT

(i,j)3qn zj

ocoln zn

+ 12zT matkp

nPm

Pi zmT

(m,i)3pk zi

omatpq

nP

(p,q)1

ocolq

nPj

Pn zjT

(j,n)2q zn

o+ 1

3rowp

˘Pk

Pm zk(TT

1 P2pT1)(k,m)zm¯

matpn

nPi

Pj ziT

(i,j)3pn zj

ocoln zn

+ 13rowp

nPq

Pk zk(TT

1 P2p)(k,q)“P

m

Pi zmT

(m,i)2q zi

”ocolp

nPj

Pn zjT

(j,n)2p zn

o+ 1

3rowp

nPq

Pi

“Pk

Pm zkT

(k,m)2q zm

”(P2pT1)(q,i) zi

ocolp

nPj

Pn zjT

(j,n)2p zn

o+ 1

3rowp

nPq

Pr

“Pk

Pm zkT

(k,m)2q zm

”P

(q,r)2p

“Pi

Pj ziT

(i,j)2r zj

”omatpn

nT

(p,n)1

ocoln zn

+ 13rowp

nPq

Pj

Pk zk(TT

1 P2p)(k,q)“P

m

Pi zmT

(m,i)3qj zi

”zj

omatpn

nT

(p,n)1

ocoln zn

+ 13rowp

nPk

Pq

Pj zk

“Pm

Pi zmT

(m,i)3qk zi

”(P2pT1)(q,j)zj

omatpn

nT

(p,n)1

ocoln zn

+ 14zT matkp

nT

(p,k)1

omatpq

˘Pm

Pi zm(TT

1 P3pqT1)(m,i)zi

¯colq

nPj

Pn zjT

(j,n)2q zn

o+ 1

4zT matkp

nT

(p,k)1

omatpq

nPm

Pr zm(TT

1 P3pq)(m,r)“P

i

Pj ziT

(i,j)2r zj

”omatqn

nT

(q,n)1

ocoln zn

+ 14zT matkp

nT

(p,k)1

omatpq

nPr

Pj

“Pm

Pi zmT

(m,i)2r zi

”(P3pqT1)(r,j)zj

omatqn

nT

(q,n)1

ocoln zn

+ 14rowp

nPk

Pm zkT

(k,m)2p zm

omatpq

nPi

Pj zi(T

T1 P3pqT1)(i,j)zj

omatqn

nT

(q,n)1

ocoln zn

+ 15rowp

nPk

Pq

Pr

Pj zkT

(q,k)1

`Pm

Pi zm(TT

1 P4pqrT1)(m,i)zi

´T

(r,j)1 zj

omatpn

nT

(p,n)1

ocoln zn = 0.

(4.45)

Rewriting rows as matrices and rearranging, we have

zT (TT1 P1)colk

nPm

Pn zm

“Pi

Pj ziT

(i,j)4kmnzj

”zn

o=

− 12zT matkp

nPm T

(k,m)2p zm

omatpq

nP

(p,q)1

omatqn

nPi

Pj ziT

(i,j)3qn zj

ocoln zn

− 12zT matkp

nPm

Pi zmT

(m,i)3pk zi

omatpq

nP

(p,q)1

ocolq

nPj

Pn zjT

(j,n)2q zn

o− 1

3zT matkp

˘Pm(TT

1 P2pT1)(k,m)zm¯

matpn

nPi

Pj ziT

(i,j)3pn zj

ocoln zn

− 13zT matkp

nPq(TT

1 P2p)(k,q)“P

m

Pi zmT

(m,i)2q zi

”ocolp

nPj

Pn zjT

(j,n)2p zn

o− 1

3zT matkp

nPq

Pi

“Pm T

(k,m)2q zm

”(P2pT1)(q,i) zi

ocolp

nPj

Pn zjT

(j,n)2p zn

o− 1

3zT matkp

nPq

Pr

“Pm T

(k,m)2q zm

”P

(q,r)2p

“Pi

Pj ziT

(i,j)2r zj

”omatpn

nT

(p,n)1

ocoln zn

− 13zT matkp

nPq

Pj(T

T1 P2p)(k,q)

“Pm

Pi zmT

(m,i)3qj zi

”zj

omatpn

nT

(p,n)1

ocoln zn

− 13zT matkp

nPq

Pj

“Pm

Pi zmT

(m,i)3qk zi

”(P2pT1)(q,j)zj

omatpn

nT

(p,n)1

ocoln zn

− 14zT matkp

nT

(p,k)1

omatpq

˘Pm

Pi zm(TT

1 P3pqT1)(m,i)zi

¯colq

nPj

Pn zjT

(j,n)2q zn

o− 1

4zT matkp

nT

(p,k)1

omatpq

nPm

Pr zm(TT

1 P3pq)(m,r)“P

i

Pj ziT

(i,j)2r zj

”omatqn

nT

(q,n)1

ocoln zn

− 14zT matkp

nT

(p,k)1

omatpq

nPr

Pj

“Pm

Pi zmT

(m,i)2r zi

”(P3pqT1)(r,j)zj

omatqn

nT

(q,n)1

ocoln zn

− 14zT matkp

nPm T

(k,m)2p zm

omatpq

nPi

Pj zi(T

T1 P3pqT1)(i,j)zj

omatqn

nT

(q,n)1

ocoln zn

− 15zT matkp

nPk

Pq

Pr

Pj T

(q,k)1

`Pm

Pi zm(TT

1 P4pqrT1)(m,i)zi

´T

(r,j)1 zj

omatpn

nT

(p,n)1

ocoln zn .

(4.46)


Summing across all indices yields

zT (TT1 P1)colk

nPm

Pn zm

“Pi

Pj ziT

(i,j)4kmnzj

”zn

o=

− 12zT colk

nPp

Pq

Pn

“Pm T

(k,m)2p zm

”P

(p,q)1

“Pi

Pj ziT

(i,j)3qn zj

”zn

o− 1

2zT colk

nPp

Pq

“Pm

Pi zmT

(m,i)3pk zi

”P

(p,q)1

“Pj

Pn zjT

(j,n)2q zn

”o− 1

3zT colk

nPp

Pn

“Pm(TT

1 P2pT1)(k,m)zm

”“Pi

Pj ziT

(i,j)3pn zj

”zn

o− 1

3zT colk

nPp

Pq(T

T1 P2p)(k,q)

“Pm

Pi zmT

(m,i)2q zi

”“Pj

Pn zjT

(j,n)2p zn

”o− 1

3zT colk

nPp

Pq

Pi

“Pm T

(k,m)2q zm

”(P2pT1)

(q,i) zi

“Pj

Pn zjT

(j,n)2p zn

”o− 1

3zT colk

nPp

Pn

“Pq

Pr

“Pm T

(k,m)2q zm

”P

(q,r)2p

“Pi

Pj ziT

(i,j)2r zj

””T

(p,n)1 zn

o− 1

3zT colk

nPp

Pn

“Pq

Pj(T

T1 P2p)(k,q)

“Pm

Pi zmT

(m,i)3qj zi

”zj

”T

(p,n)1 zn

o− 1

3zT colk

nPp

Pn

“Pq

Pj

“Pm

Pi zmT

(m,i)3qk zi

”(P2pT1)

(q,j)zj

”T

(p,n)1 zn

o− 1

4zT colk

nPp

Pq T

(p,k)1

“Pm

Pi zm(TT

1 P3pqT1)(m,i)zi

”“Pj

Pn zjT

(j,n)2q zn

”o− 1

4zT colk

nPp

Pq

Pn T

(p,k)1

“Pm

Pr zm(TT

1 P3pq)(m,r)

“Pi

Pj ziT

(i,j)2r zj

””T

(q,n)1 zn

o− 1

4zT colk

nPp

Pq

Pn T

(p,k)1

“Pr

Pj

“Pm

Pi zmT

(m,i)2r zi

”(P3pqT1)

(r,j)zj

”T

(q,n)1 zn

o− 1

4zT colk

nPp

Pq

Pn

“Pm T

(k,m)2p zm

”“Pi

Pj zi(T

T1 P3pqT1)

(i,j)zj

”T

(q,n)1 zn

o− 1

5zT matkp

nPk

Pq

Pr

Pj T

(q,k)1

“Pm

Pi zm(TT

1 P4pqrT1)(m,i)zi

”T

(r,j)1 zj

o∗

matpn

nT

(p,n)1

ocoln zn .

(4.47)

Canceling all the z’s and z’s, this last expression becomes

(TT1 P1)colk

nT

(i,j)4kmn

o=

− 12colk

nPp

Pq T

(k,m)2p P

(p,q)1 T

(i,j)3qn

o− 1

2colk

nPp

Pq T

(m,i)3pk P

(p,q)1 T

(j,n)2q

o− 1

3colk

nPp(TT

1 P2pT1)(k,m)T

(i,j)3pn

o− 1

3colk

nPp

Pq(T

T1 P2p)(k,q)T

(m,i)2q T

(j,n)2p

o− 1

3colk

nPp

Pq T

(k,m)2q (P2pT1)

(q,i)T(j,n)2p

o− 1

3colk

nPp

Pq

Pr T

(k,m)2q P

(q,r)2p T

(i,j)2r T

(p,n)1

o− 1

3colk

nPp

Pq(T

T1 P2p)(k,q)T

(m,i)3qj T

(p,n)1

o− 1

3colk

nPp

Pq T

(m,i)3qk (P2pT1)

(q,j)T(p,n)1

o− 1

4colk

nPp

Pq T

(p,k)1 (TT

1 P3pqT1)(m,i)T

(j,n)2q

o− 1

4colk

nPp

Pq

Pr T

(p,k)1 (TT

1 P3pq)(m,r)T

(i,j)2r T

(q,n)1

o− 1

4colk

nPp

Pq

Pr T

(p,k)1 T

(m,i)2r (P3pqT1)

(r,j)T(q,n)1

o− 1

4colk

nPp

Pq T

(k,m)2p (TT

1 P3pqT1)(i,j)T

(q,n)1

o− 1

5colk

nPp

Pq

Pr T

(q,k)1 (TT

1 P4pqrT1)(m,i)T

(r,j)1 T

(p,n)1

o.

(4.48)


Therefore, the fourth-order state transformation term is given by

colkT

(i,j)4kmn

= −P−1

1 T−T1 colk

12

∑p

∑q T

(k,m)2p P

(p,q)1 T

(i,j)3qn

+ 12

∑p

∑q T

(m,i)3pk P

(p,q)1 T

(j,n)2q

+ 13

∑p(T

T1 P2pT1)(k,m)T

(i,j)3pn

+ 13

∑p

∑q(T

T1 P2p)(k,q)T

(m,i)2q T

(j,n)2p

+ 13

∑p

∑q T

(k,m)2q (P2pT1)(q,i)T (j,n)

2p

+ 13

∑p

∑q

∑r T

(k,m)2q P

(q,r)2p T

(i,j)2r T

(p,n)1

+ 13

∑p

∑q(T

T1 P2p)(k,q)T

(m,i)3qj T

(p,n)1

+ 13

∑p

∑q T

(m,i)3qk (P2pT1)(q,j)T (p,n)

1

+ 14

∑p

∑q T

(p,k)1 (TT

1 P3pqT1)(m,i)T (j,n)2q

+ 14

∑p

∑q

∑r T

(p,k)1 (TT

1 P3pq)(m,r)T(i,j)2r T

(q,n)1

+ 14

∑p

∑q

∑r T

(p,k)1 T

(m,i)2r (P3pqT1)(r,j)T (q,n)

1

+ 14

∑p

∑q T

(k,m)2p (TT

1 P3pqT1)(i,j)T (q,n)1

+ 15

∑p

∑q

∑r T

(q,k)1 (TT

1 P4pqrT1)(m,i)T (r,j)1 T

(p,n)1

(4.49)

Now that we have obtained expressions for the nonlinear state balancing transformation, we shall findexpressions for its gradient.

4.4 Nonlinear Balancing Transformation Gradient

In the last section we derived the nonlinear state balancing transformation. Our goal here is to derive anexpression for the gradient of the state balancing transformation in Eq. (4.25). The state transformationgradient will be given by

∇T(z) =∂T(z)∂z

= ∇T1(z) +∇T2(z) +∇T3(z) +∇T4(z), (4.50)

where we define∇T1(z) = M0

∇T2(z) = matst M1stz∇T3(z) = matpq

zTM2pqz

∇T4(z) = matkm

rown

zTM3kmnz

z (4.51)

4.4.1 First-Order Balancing Transformation Gradient

The gradient of the first-order transformation term is simply given by

∇T1(z) = ∂∂z (T1z)

= T1

= M0.

(4.52)


4.4.2 Second-Order Balancing Transformation Gradient

The gradient of the second-order transformation term is

∇T2(z) = ∂∂z

(cols

zTT2sz

)= cols

∂∂z

(zTT2sz

)= cols

zT(T2s + TT

2s

)= cols

rowt

∑i zi(T2s + TT

2s

)(i,t)= matst

∑i zi(T2s + TT

2s

)(i,t)= matst M1stz

(4.53)

where M1st = rowiM1st,i and M1st,i =(T2s + TT

2s

)(i,t).4.4.3 Third-Order Balancing Transformation Gradient

The gradient of the third-order transformation term is

∇T3(z) = ∂∂z

(matpj

zTT3pjz

z)

=(∂∂zmatpj

zTT3pjz

)z + matpq

zTT3pqz

∂∂z (z)

= matpj∂∂z

(zTT3pjz

)z + matpq

zTT3pqz

1

= colp∑

j∂∂z

(zTT3pjz

)zj

+ matpq

zTT3pqz

= colp

∑j zT (T3pj + TT

3pj)zj

+ matpqzTT3pqz

= colp

rowq

∑i

∑j zi(T3pj + TT

3pj)(i,q)zj

+ matpq

zTT3pqz

= matpq

∑i

∑j zi(T3pj + TT

3pj)(i,q)zj

+ matpq

∑i

∑j ziT

(i,j)3pq zj

= matpq

∑i

∑j zi

[(T3pj + TT

3pj)(i,q) + T

(i,j)3pq

]zj

= matpqzTM2pqz

(4.54)

where M2pg = matijM (i,j)2pq and M

(i,j)2pq = (T3pj + TT

3pj)(i,q) + T

(i,j)3pq .

4.4.4 Fourth-Order Balancing Transformation Gradient

The gradient of the fourth-order transformation term is

∇T4(z) = ∂∂z

`colk

˘zT matnm

˘zT T4knmz

¯z¯´

= colk˘zT`matnm

˘zT T4knmz

¯+ matnj

˘∂∂z

ˆzT T4knjz

˜¯z + matnm

˘zT T4kmnz

¯´¯= colk

˘zT`matnm

˘zT T4knmz

¯+ matnj

˘zTˆT4knj + TT

4knj

˜¯z

+matnm

˘zT T4kmnz

¯´¯= colk

nzT“matnm

nPi

Pj ziT

(i,j)4knmzj

o+ matnm

nPi

Pj zi

hT

(i,m)4knj + T

(m,i)4knj

izj

o+matnm

nPi

Pj ziT

(i,j)4kmnzj

o”o= colk

nzT matnm

nPi

Pj zi

“T

(i,j)4knm + T

(i,m)4knj + T

(m,i)4knj + T

(i,j)4kmn

”zj

oo= colk

nrowm

nPn zn

Pi

Pj zi

“T

(i,j)4knm + T

(i,m)4knj + T

(m,i)4knj + T

(i,j)4kmn

”zj

oo= matkm

nPn zn

Pi

Pj zi

“T

(i,j)4knm + T

(i,m)4knj + T

(m,i)4knj + T

(i,j)4kmn

”zj

o= matkm

nrown

nPi

Pj zi

“T

(i,j)4knm + T

(i,m)4knj + T

(m,i)4knj + T

(i,j)4kmn

”zj

ozo

= matkm

˘rown

˘zT M3kmnz

¯z¯

(4.55)


where M3kmn = matijM (i,j)3kmn and M

(i,j)3kmn = T

(i,j)4knm + T

(i,m)4knj + T

(m,i)4knj + T

(i,j)4kmn.

Next, we will develop an expression for the inverse of the transformation gradient.

4.5 Inverse of Nonlinear Balancing Transformation Gradient

Our goal here is to derive an expression for the inverse of the state balancing transformation gradient.Given the difficulty in calculating an exact analytical expression for the inverse of a polynomial matrix,we will derive analytical expressions for approximations to the inverse of the gradient.

We shall denote the inverse of the transformation gradient by

S(z) = ∇T−1(z) = S0(z) + S1(z) + S2(z) + S3(z), (4.56)

where we defineS0(z) = S0

S1(z) = matpqS1pqzS2(z) = matpqzTS2pqzS3(z) = matpqrowrzTS3pqrzz

(4.57)

The transformation gradient inverse should satisfy the following relation:

∇T(z) · ∇T−1(z) = 1, (4.58)

where 1 denotes the identity matrix. We shall make use of this relation throughout the next subsectionsas a means of defining the successive terms in the gradient inverse.

4.5.1 Zeroth-Order Balancing Transformation Gradient Inverse

The inverse of the zeroth-order transformation gradient is simply given by

S0(z) = ∇T−11 (z) = T−1

1 = S0. (4.59)

4.5.2 First-Order Balancing Transformation Gradient Inverse

The inverse of the first-order transformation gradient is obtained by first substituting the first- andsecond-order terms from Eq. (4.50) and the zeroth- and first-order terms from Eq. (4.56) into Eq.(4.58):

[∇T1(z) +∇T2(z)] [S0(z) + S1(z)] = 1. (4.60)

Expanding the above and neglecting second-order terms in z(t), we obtain

∇T1(z)S0(z) +∇T1(z)S1(z) +∇T2(z)S0(z) = 1. (4.61)

Using the identity ∇T1(z)S0(z) = 1, this last equation becomes

∇T1(z)S1(z) +∇T2(z)S0(z) = 0. (4.62)


Rearranging terms yieldsS1(z) = −S0∇T2(z)S0. (4.63)

Thus, the approximate inverse of the first-order transformation gradient is given by

S1(z) = −S0∇T2(z)S0

= −matpsS(p,s)0 matst

∑iM1st,izimattqS(t,q)

0 = matpq

∑i

(∑s

∑t−S(p,s)

0 M1st,iS(t,q)0

)zi

= matpq S1pqz

(4.64)

where S1pq = rowiS1pq,i and S1pq,i = −∑s

∑t S

(p,s)0 M1st,iS

(t,q)0 . Note that this approximate inverse

expression only depends on the first-order gradient term and the analytical inverse of the zeroth-ordergradient term.

4.5.3 Second-Order Balancing Transformation Gradient Inverse

The inverse of the second-order transformation gradient is obtained by first substituting the first-, second-and third-order terms from Eq. (4.50) and the zeroth-, first- and second-order terms from Eq. (4.56)into Eq. (4.58):

[∇T1(z) +∇T2(z) +∇T3(z)] [S0(z) + S1(z) + S2(z)] = 1. (4.65)

Expanding the above and neglecting third- and higher-order terms in z(t), we obtain

∇T1(z)S0(z) +∇T1(z)S1(z) +∇T1(z)S2(z)+∇T2(z)S0(z) +∇T2(z)S1(z) +∇T3(z)S0(z) = 1.

(4.66)

Using the expression for S1(z) in Eq. (4.63) and the identity ∇T1(z)S0(z) = 1, this last equationbecomes

∇T1(z)S2(z)−∇T2(z)S0(z)∇T2(z)S0(z) +∇T3(z)S0(z) = 0. (4.67)

Rearranging terms yields

S2(z) = S0∇T2(z)S0∇T2(z)S0 − S0∇T3(z)S0. (4.68)

Thus, the approximate inverse of the second-order transformation gradient is given by

S2(z) = S0∇T2(z)S0∇T2(z)S0 − S0∇T3(z)S0

= matpsS

(p,s)0

matst

∑iM1st,izimattr

S

(t,r)0

matrk

∑jM1rk,jzj

matkq

S

(k,q)0

−matps

S

(p,s)0

matst

∑i

∑j ziM

(i,j)2st zj

mattq

S

(t,q)0

= matpq

∑i

∑j zi

(∑s

∑t

∑r

∑k S

(p,s)0 M1st,iS

(t,r)0 M1rk,jS

(k,q)0

−∑s

∑t S

(p,s)0 M

(i,j)2st S

(t,q)0

)zj

= matpq

zTS2pqz

(4.69)


where S2pq = matijS(i,j)2pq and

S(i,j)2pq =

∑s

∑t

∑r

∑k

S(p,s)0 M1st,iS

(t,r)0 M1rk,jS

(k,q)0 −

∑s

∑t

S(p,s)0 M

(i,j)2st S

(t,q)0 .

Note that this approximate inverse expression only depends on the first- and second-order gradient termsand the analytical inverse of the zeroth-order gradient term.

4.5.4 Third-Order Balancing Transformation Gradient Inverse

The inverse of the third-order transformation gradient is obtained by first substituting the first-, second-,third- and fourth-order terms from Eq. (4.50) and the zeroth-, first-, second- and third-order terms fromEq. (4.56) into Eq. (4.58):

[∇T1(z) +∇T2(z) +∇T3(z) +∇T4(z)] ∗[S0(z) + S1(z) + S2(z) + S3(z)] = 1.

(4.70)

Expanding the above and neglecting fourth- and higher-order terms in z(t), we obtain

∇T1(z)S0(z) +∇T1(z)S1(z) +∇T1(z)S2(z) +∇T1(z)S3(z)+∇T2(z)S0(z) +∇T2(z)S1(z) +∇T2(z)S2(z)+∇T3(z)S0(z) +∇T3(z)S1(z) +∇T4(z)S0(z) = 1.

(4.71)

Using the expressions for S1(z) and S2(z) in Eqs. (4.63) and (4.68), respectively, along with the identity∇T0(z)S0(z) = 1, this last equation becomes

∇T1(z)S3(z)−∇T3(z)S0(z)∇T2(z)S0(z)−∇T2(z)S0(z)∇T3(z)S0(z)+∇T2(z)S0(z)∇T2(z)S0(z)∇T2(z)S0(z) +∇T4(z)S0(z) = 0.

(4.72)

Rearranging terms yields

S3(z) = S0(z)∇T3(z)S0∇T2(z)S0 + S0∇T2(z)S0∇T3(z)S0

−S0∇T2(z)S0∇T2(z)S0∇T2(z)S0 − S0∇T4(z)S0.(4.73)


Thus, the approximate inverse of the third-order transformation gradient is given by

S3(z) = S0∇T3(z)S0∇T2(z)S0 + S0∇T2(z)S0∇T3(z)S0

−S0∇T2(z)S0∇T2(z)S0∇T2(z)S0 − S0∇T4(z)S0

= matps

nS

(p,s)0

omatst

nPi

Pj ziM

(i,j)2st zj

omattu

nS

(t,u)0

o∗

matuv

˘Pk M1uv,kzk

¯matvq

nS

(v,q)0

o+matps

nS

(p,s)0

omatst

˘Pi M1st,izi

¯mattu

nS

(t,u)0

o∗

matuv

nPj

Pk zjM

(j,k)2uv zk

omatvq

nS

(v,q)0

o−matps

nS

(p,s)0

omatst

˘Pi M1st,izi

¯mattu

nS

(t,u)0

omatuv

nPj M1uv,jzj

o∗

matvw

nS

(v,w)0

omatwm

˘Pk M1wm,kzk

¯matmq

nS

(m,q)0

o−matps

nS

(p,s)0

omatst

nPi

Pj

Pk ziT

(i,j)3stk zjzk

omattq

nS

(t,q)0

o= matpq

nPi

Pj

Pk zi

“Ps

Pt

Pu

Pv S

(p,s)0 M

(i,j)2st S

(t,u)0 M1uv,kS

(v,q)0

+P

s

Pt

Pu

Pv S

(p,s)0 M1st,iS

(t,u)0 M

(j,k)2uv S

(v,q)0

−Ps

Pt

Pu

Pv

Pw

Pm S

(p,s)0 M1st,iS

(t,u)0 M1uv,jS

(v,w)0 M1wm,kS

(m,q)0

− Ps

Pt S

(p,s)0 T

(i,j)3stk S

(t,q)0

”zjzk

o= matpq

nPk

“Pi

Pj ziS

(i,j)3pqkzj

”zk

o= matpq

˘rowk

˘zT S3pqkz

¯z¯

(4.74)

where S3pqk = matijS(i,j)3pqk and

S(i,j)3pqk =

∑s

∑t

∑u

∑v S

(p,s)0 M

(i,j)2st S

(t,u)0 M1uv,kS

(v,q)0

+∑s

∑t

∑u

∑v S

(p,s)0 M1st,iS

(t,u)0 M

(j,k)2uv S

(v,q)0

−∑s

∑t

∑u

∑v

∑w

∑m S

(p,s)0 M1st,iS

(t,u)0 M1uv,jS

(v,w)0 M1wm,kS

(m,q)0

−∑s

∑t S

(p,s)0 T

(i,j)3stk S

(t,q)0

Note that this approximate inverse expression only depends on the first-, second-, and third-order gra-dient terms and the analytical inverse of the zeroth-order gradient term.

We will next present analytical expressions for balancing the system matrices a(x), b(x) and c(x),the HJE solution gradient ∇V+(x), and the output injection gain G2(x).

4.6 Balanced Nonlinear Dynamics Matrix

Here we construct analytical expressions for the balanced nonlinear dynamics matrix a(z). From Eq.(4.8), the balanced nonlinear dynamics matrix is given by

a(z) = ∇T−1(z) a(T(z)

). (4.75)

The balanced nonlinear dynamics matrix in Eq. (4.75) is approximated to fourth-order as

a(z) = a1(z) + a2(z) + a3(z) + a4(z), (4.76)


wherea1(z) = A1z

a2(z) = colk

zT A2kz

a3(z) = matmn

zT A3mnz

z

a4(z) = colk

zTmatmn

zT A4kmnz

z

Substituting the transformation of Eq. (4.25) and its gradient inverse from Eq. (4.56) into Eq. (4.75)and collecting terms of the same order in z(t) yields a set of expressions for the balanced nonlineardynamics matrices. These expressions are given as follows. The first-order terms in z(t) are

a1(z) = S0(z) a1

(T1(z)

). (4.77)


a2(z) = S0(z) a2

(T1(z),T1(z)

)+ S0(z) a1

(T2(z)

)+ S1(z) a1

(T1(z)

). (4.78)


a3(z) = S0(z) a3

(T1(z),T1(z),T1(z)

)+ S0(z) a2

(T1(z),T2(z)

)+S0(z) a2

(T2(z),T1(z)

)+ S1(z) a2

(T1(z),T1(z)

)+S0(z) a1

(T3(z)

)+ S1(z) a1

(T2(z)

)+ S2(z) a1

(T1(z)

).

(4.79)

And finally, the fourth-order terms in z(t) are

a4(z) = S0(z) a4

(T1(z),T1(z),T1(z),T1(z)

)+ S0(z) a3

(T1(z),T1(z),T2(z)

)+S0(z) a3

(T1(z),T2(z),T1(z)

)+ S0(z) a3

(T2(z),T1(z),T1(z)

)+S1(z) a3

(T1(z),T1(z),T1(z)

)+ S0(z) a2

(T1(z),T3(z)

)+S0(z) a2

(T2(z),T2(z)

)+ S0(z) a2

(T3(z),T1(z)

)+ S1(z) a2

(T1(z),T2(z)

)+S1(z) a2

(T2(z),T1(z)

)+ S2(z) a2

(T1(z),T1(z)

)+ S0(z) a1

(T4(z)

)+S1(z) a1

(T3(z)

)+ S2(z) a1

(T2(z)

)+ S3(z) a1

(T1(z)

).

(4.80)

In the following subsections, we will show how to obtain the balanced nonlinear dynamics matricesfrom these expressions.

4.6.1 First-Order Balanced Dynamics Matrix

The balanced first-order dynamics matrix is given by

a1(z) = S0A1T1z = A1z. (4.81)


4.6.2 Second-Order Balanced Dynamics Matrix

The balanced second-order dynamics matrix is given by

a2(z) = S0coll˘zT TT

1 A2lT1z¯

+ S0A1coll˘zT T2lz

¯+ matkl S1klzA1T1z

= matkl

nS

(k,l)0

ocoll

nPi

Pj zi(T

T1 A2lT1)

(i,j)zj

o+matkl

n(S0A1)

(k,l)o

collnP

i

Pj ziT

(i,j)2l zj

o+matkl

˘Pi S1kl,izi

¯matlj

n(A1T1)

(l,j)o

colj zj= colk

nPl S

(k,l)0

“Pi

Pj zi(T

T1 A2lT1)

(i,j)zj

”o+colk

nPl(S0A1)

(k,l)“P

i

Pj ziT

(i,j)2l zj

”o+colk

nPi

Pj zi

“Pl S1kl,i(A1T1)

(l,j)”

zj

o= colk

nPi

Pj zi

“Pl S

(k,l)0 (TT

1 A2lT1)(i,j) +

Pl(S0A1)

(k,l)T(i,j)2l

+P

l S1kl,i(A1T1)(l,j)”

zj

o= colk

nzT A2kz

o

(4.82)

where A2k = matijA(i,j)2k and

A(i,j)2k =

∑l

S(k,l)0 (TT

1 A2lT1)(i,j) +∑l

(S0A1)(k,l)T (i,j)2l +

∑l

S1kl,i(A1T1)(l,j).

4.6.3 Third-Order Balanced Dynamics Matrix

The balanced third-order dynamics matrix is given by

a3(z) = S0A1matsq

˘zT T3sqz

¯z + S0matst

˘zT TT

1 A3stT1z¯

T1z

+S0cols˘zT TT

1 A2scolt˘zT T2tz

¯¯+ S0cols

˘rowt

˘zT T2tz

¯A2sT1z

¯+matps S1pszA1colt

˘zT T2tz

¯+ matps S1psz cols

˘zT TT

1 A2sT1z¯

+matps

˘zT S2psz

¯A1T1z

= matps

n(S0A1)

(p,s)o

matsq

nPi

Pj ziT

(i,j)3sq zj

oz

+matps

nS

(p,s)0

omatst

nPi

Pj zi(T

T1 A3stT1)

(i,j)zj

omattq

nT

(t,q)1

oz

+matps

nS

(p,s)0

ocols

nPi

Pt zi(T

T1 A2s)

(i,t)“P

j

Pq zjT

(j,q)2t zq

”o+matps

nS

(p,s)0

ocols

nPt

Pq

“Pi

Pj ziT

(i,j)2t zj

”(A2sT1)

(t,q)zq

o+matps

˘Pi S1ps,izi

¯matst

nA

(s,t)1

ocolt

nPj

Pq zjT

(j,q)2t zq

o+matps

˘Pi S1ps,izi

¯cols

nPj

Pq zj(T

T1 A2sT1)

(j,q)zq

o+matps

nPi

Pj ziS

(i,j)2ps zj

omatsq

n(A1T1)

(s,q)o

z

= matpq

nPi

Pj zi

“Ps(S0A1)

(p,s)T(i,j)3sq +

Ps

Pt S

(p,s)0 (TT

1 A3stT1)(i,j)T

(t,q)1

+P

s

Pt S

(p,s)0 (TT

1 A2s)(i,t)T

(j,q)2t +

Ps

Pt S

(p,s)0 T

(i,j)2t (A2sT1)

(t,q)

+P

s

Pt S1ps,iA

(s,t)1 T

(j,q)2t +

Ps S1ps,i(T

T1 A2sT1)

(j,q)

+P

s S(i,j)2ps (A1T1)

(s,q)”

zj

oz

= matpq

nzT A3pqz

oz

(4.83)


where A3pq = matijA(i,j)3pq and

A(i,j)3pq =

∑s(S0A1)(p,s)T (i,j)

3sq +∑s

∑t S

(p,s)0 (TT

1 A3stT1)(i,j)T (t,q)1

+∑s

∑t S

(p,s)0 (TT

1 A2s)(i,t)T(j,q)2t +

∑s

∑t S

(p,s)0 T

(i,j)2t (A2sT1)(t,q)

+∑s

∑t S1ps,iA

(s,t)1 T

(j,q)2t +

∑s S1ps,i(TT

1 A2sT1)(j,q)

+∑s S

(i,j)2ps (A1T1)(s,q).

4.6.4 Fourth-Order Balanced Dynamics Matrix

The balanced fourth-order dynamics matrix is given by

a4(z) = S0colpzTTT

1 matqrzTTT

1 A4pqrT1z

T1z

+ S0matpqzTTT

1 A3pqT1z

colqzTT2qz

+S0matpq

zTTT

1 A3pqcoltzTT2tz

T1z + S0matpq

rows

zTT2sz

A3pqT1z

T1z

+S1(z)matpqzTTT

1 A3pqT1z

T1z + S0colpzTTT

1 A2pmatsnzTT3snz

z

+S0colp

rowqzTT2qz

A2pcols

zTT2sz

+ S0colp

zTmatmq

zTT3qmz

A2pT1z

+S1(z)colp

zTTT

1 A2pcolszTT2sz

+ S1(z)colp

rows

zTT2sz

A2pT1z

+S2(z)colp

zTTT

1 A2pT1z

+ S0A1colqzTmatmn

zTT4qmnz

z

+S1(z)A1mattnzTT3tnz

z + S2(z)A1colq

zTT2qz

+ S3(z)A1T1z.

(4.84)This expression can be expanded to yield

a4(z) = matkp

nS

(k,p)0

ocolp

nPm

Pq zmT

(q,m)1 matqr

nPi

Pj

Ps

Pt ziT

(s,i)1 A

(s,t)4pqrT

(t,j)1 zj

o∗

matrn

nT

(r,n)1

ocoln zn

o+matkp

nS

(k,p)0

omatpq

nPm

Pi

Ps

Pt zmT

(s,m)1 A

(s,t)3pq T

(t,i)1 zi

ocolq

nPj

Pn zjT

(j,n)2q zn

o+matkp

nS

(k,p)0

omatpq

nPm

Ps

Pt zmT

(s,m)1 A

(s,t)3pq

“Pi

Pj ziT

(i,j)2t zj

”omatqn

nT

(q,n)1

ocoln zn

+matkp

nS

(k,p)0

omatpq

nPs

Pt

Pj

“Pm

Pi zmT

(m,i)2s zi

”A

(s,t)3pq T

(t,j)1 zj

omatqn

nT

(q,n)1

ocoln zn

+matkp

˘Pm S1kp,mzm

¯matpq

nPi

Pj

Ps

Pt ziT

(s,i)1 A

(s,t)3pq T

(t,j)1 zj

omatqn

nT

(q,n)1

ocoln zn

+matkp

nS

(k,p)0

ocolp

nPq

Ps

“Pm

Pi zmT

(m,i)2q zi

”A

(q,s)2p

“Pj

Pn zjT

(j,n)2s zn

”o+matkp

nS

(k,p)0

ocolp

nPm

Pq

Ps

Pn zmT

(q,m)1 A

(q,s)2p

“Pi

Pj ziT

(i,j)3sn zj

”zn

o+matkp

nS

(k,p)0

ocolp

nPm

Pq

Ps

Pn zm

“Pi

Pj ziT

(i,j)3qm zj

”A

(q,s)2p T

(s,n)1 zn

o+matkp

˘Pm S1kp,mzm

¯colp

nPi

Pq

Ps ziT

(q,i)1 A

(q,s)2p

“Pj

Pn zjT

(j,n)2s zn

”o+matkp

˘Pm S1kp,mzm

¯colp

nPs

Pt

Pn

“Pi

Pj ziT

(i,j)2s zj

”A

(s,t)2p T

(t,n)1 zn

o+matkp

nPm

Pi zmS

(m,i)2kp zi

ocolp

nPj

Pq

Ps

Pn zjT

(q,j)1 A

(q,s)2p T

(s,n)1 zn

o+matkp

nS

(k,p)0

omatpq

nA

(p,q)1

ocolq

nPm

Pn zm

“Pi

Pj ziT

(i,j)4qmnzj

”zn

o+matkp

˘Pm S1kp,mzm

¯matpt

nA

(p,t)1

omattn

nPi

Pj ziT

(i,j)3tn zj

ocoln zn

+matkp

nPm

Pi zmS

(m,i)2kp zi

omatpq

nA

(p,q)1

ocolq

nPj

Pn zjT

(j,n)2q zn

o+matkp

nPj

“Pm

Pi zmS

(m,i)3kpj zi

”zj

omatpq

nA

(p,q)1

omatqt

nT

(q,n)1

ocoln zn

(4.85)

Therefore, summing across all indices, the balanced fourth-order dynamics matrix is given by

a4(z) = colk

zTmatmn

zT A4kmnz

z

(4.86)


where A4kmn = matijA(i,j)4kmn and

A(i,j)4kmn =

Pp

Pq S

(k,p)0 A

(p,q)1 T

(i,j)4qmn +

Pp

Pq

Ps S

(k,p)0 T

(m,i)2q A

(q,s)2p T

(j,n)2s

+P

p

Pq

Ps S

(k,p)0 T

(q,m)1 A

(q,s)2p T

(i,j)3sn +

Pp

Pq

Ps S

(k,p)0 T

(i,j)3qm A

(q,s)2p T

(s,n)1

+P

p

Pq

Ps

Pt S

(k,p)0 T

(m,i)2s A

(s,t)3pq T

(t,j)1 T

(q,n)1

+P

p

Pq

Ps

Pt S

(k,p)0 T

(s,m)1 A

(s,t)3pq T

(i,j)2t T

(q,n)1

+P

p

Pq

Ps

Pt S

(k,p)0 T

(s,m)1 A

(s,t)3pq T

(t,i)1 T

(j,n)2q

+P

p

Pq

Pr

Ps

Pt S

(k,p)0 T

(q,m)1 T

(s,i)1 A

(s,t)4pqrT

(t,j)1 T

(r,n)1

+P

p

Pq S1kp,mA

(p,t)1 T

(i,j)3qn +

Pp

Pq

Ps

Pt S1kp,mT

(s,i)1 A

(s,t)3pq T

(t,j)1 T

(q,n)1

+P

p

Pq

Ps S1kp,mT

(q,i)1 A

(q,s)2p T

(j,n)2s +

Pp

Pq

Ps S1kp,mT

(i,j)2s A

(s,q)2p T

(q,n)1

+P

p

Pq S

(m,i)2kp A

(p,q)1 T

(j,n)2q +

Pp

Pq

Ps S

(m,i)2kp T

(q,j)1 A

(q,s)2p T

(s,n)1

+P

p

Pq S

(m,i)3kpj A

(p,q)1 T

(q,n)1

(4.87)

Next, we construct analytical expressions for the balanced nonlinear input matrix.

4.7 Balanced Nonlinear Input Matrix

Here we construct analytical expressions for the balanced input matrix b(z). Although it has beenassumed thus far in this thesis that the input matrix is constant, the nonlinear balancing transformationwill introduce nonlinear terms. From Eq. (4.8), the balanced input matrix is obtained according to therelation:

b(z) = ∇T−1(z) b(T(z)

)= ∇T−1(z) B. (4.88)

The balanced nonlinear input matrix in Eq. (4.88) is approximated to third-order as

b(z) = b0(z) + b1(z) + b2(z) + b3(z), (4.89)

whereb0(z) = B0

b1(z) = matst B1stzb2(z) = matst

zTB2stz

b3(z) = matst

rowr

zTB3strz

z

with B1st = rowiB1st,i.Substituting the transformation gradient inverse from Eq. (4.56) into Eq. (4.88) and collecting terms

of the same order in z(t) yields a set of expressions for the balanced nonlinear input matrices. Theseexpressions are given as follows. The zeroth-order terms in z(t) are

b0(z) = S0(z)B. (4.90)

The first-order terms in z(t) areb1(z) = S1(z)B. (4.91)

The second-order terms in z(t) areb2(z) = S2(z)B. (4.92)


And finally, the third-order terms in z(t) are

b3(z) = S3(z)B. (4.93)

In the following subsections, we will show how to obtain the balanced nonlinear input matrices fromthese expressions.

4.7.1 Zeroth-Order Balanced Input Matrix

The balanced zeroth-order input matrix is

b0(z) = S0B = B0. (4.94)

4.7.2 First-Order Balanced Input Matrix

The balanced first-order input matrix is

b1(z) = matpt S1ptzB

= matpt ∑i S1pt,izimattq

B(t,q)

= matpq

∑i

(∑t S1pt,iB

(t,q))zi

= matpq

B1pqz (4.95)

where B1pq = rowiB1pq,i and B1pq,i =∑t S1pt,iB

(t,q).

4.7.3 Second-Order Balanced Input Matrix

The balanced second-order input matrix is

b2(z) = matptzTS2ptz

B

= matpt∑

i

∑j ziS

(i,j)2pt zj

mattq

B(t,q)

= matpq

∑i

∑j zi

(∑t S

(i,j)2pt B

(t,q))zj

= matpq

zT B2pqz

(4.96)

where B2pq = matijB(i,j)2pq and B

(i,j)2pq =

∑t S

(i,j)2pt B

(t,q).

4.7.4 Third-Order Balanced Input Matrix

The balanced third-order input matrix is

b3(z) = matsp

rowrzTS3sprz

z

B

= matsp∑

r

∑i

∑j ziS

(i,j)3sprzjzr

matpt

B(p,t)

= matst

∑r

∑i

∑j zi

(∑p S

(i,j)3sprB

(p,t))zjzr

= matst

rowr

zT B3strz

z (4.97)

where B3str = matijB(i,j)3str and B

(i,j)3str =

∑p S

(i,j)3sprB

(p,t).Next, we construct analytical expressions for the balanced nonlinear output matrix.


4.8 Balanced Nonlinear Output Matrix

Here we construct analytical expressions for the balanced output matrix c(z). Once again, although ithas been assumed thus far in this thesis that the output matrix is constant, the nonlinear balancingtransformation will introduce nonlinear terms. From Eq. (4.8), the balanced output matrix is obtainedaccording to the relation:

c(z) = c(T(z)

)= C T(z). (4.98)

The balanced nonlinear output matrix in Eq. (4.98) is approximated to fourth-order as

c(z) = c1(z) + c2(z) + c3(z) + c4(z), (4.99)

wherec1(z) = C1z

c2(z) = colk

zT C2kz

c3(z) = matmn

zT C3mnz

z

c4(z) = colk

zTmatmn

zT C4kmnz

z

Substituting the transformation of Eq. (4.25) into Eq. (4.98) and collecting terms of the same orderin z(t) yields a set of expressions for the balanced nonlinear output matrices. These expressions aregiven as follows. The first-order terms in z(t) are

c1(z) = CT1(z). (4.100)

The second-order terms in z(t) arec2(z) = CT2(z). (4.101)

The third-order terms in z(t) arec3(z) = CT3(z). (4.102)


c4(z) = CT4(z). (4.103)

In the following subsections, we will show how to obtain the balanced nonlinear output matrices fromthese expressions.

4.8.1 First-Order Balanced Output Matrix

The balanced first-order output matrix is

c1(z) = CT1z = C1z. (4.104)


4.8.2 Second-Order Balanced Output Matrix

The balanced second-order output matrix is

c2(z) = CcoltzTT2tz= matkt

C(k,t)

colt

∑i

∑j ziT

(i,j)2t zj

= colk

∑i

∑j zi

(∑t C

(k,t)T(i,j)2t

)zj

= colk

zT C2kz

(4.105)

where C2k = matijC(i,j)2k and C

(i,j)2k =

∑t C

(k,t)T(i,j)2t .

4.8.3 Third-Order Balanced Output Matrix

The balanced third-order output matrix is

c3(z) = CmattqzTT3tqz

z

= matptC(p,t)

mattq

∑i

∑j ziT

(i,j)3tq zj

z

= matpq∑

i

∑j zi

(∑t C

(p,t)T(i,j)3tq

)zj

z

= mattn

zT C3pqz

z

(4.106)

where C3pq = matijC(i,j)3pq and C

(i,j)3pq =

∑t C

(p,t)T(i,j)3tq .

4.8.4 Fourth-Order Balanced Output Matrix

The balanced fourth-order output matrix is

c4(z) = CcolszTmatmn

zTT4smnz

z

= matksC(k,s)

cols

∑m

∑n zm

(∑i

∑j ziT

(i,j)4smnzj

)zn

= colk

∑s C

(k,s)(∑

m

∑n

∑i

∑j zmziT

(i,j)4smnzjzn

)= colk

∑m

∑n

∑i

∑j zmzi

(∑s C

(k,s)T(i,j)4smn

)zjzn

= colk

zTmatmn

zT C4kmnz

z

(4.107)

where C4smn = matijC(i,j)4smn and C

(i,j)4kmn =

∑s C

(k,s)T(i,j)4smn.

Next, we construct analytical expressions for the balanced HJE solution gradient.

4.9 Balanced HJE Solution Gradient

Here we construct analytical expressions for the balanced nonlinear stabilizing HJE solution gradient∇V+(z). From Eq. (4.8), the balanced HJE solution gradient is given by

∇V+(z) = ∇V+

(T(z)

)∇T(z). (4.108)


The balanced nonlinear stabilizing HJE solution gradient in Eq. (4.108) is approximated to fourth-orderas

∇V+(z) = ∇V+2(z) +∇V+3(z) +∇V+4(z) +∇V+5(z), (4.109)

where∇V+2(x) = zT P1

∇V+3(x) = rowk

zT P2kz

∇V+4(x) = zTmatnm

zT P3nmz

∇V+5(x) = rowk

zTmatnm

zT P4knmz

z

Substituting the transformation of Eq. (4.25) and its gradient from Eq. (4.50) into Eq. (4.108) andcollecting terms of the same order in z(t) yields a set of expressions for the balanced nonlinear HJEsolution gradient. These expressions are given as follows. The first-order terms in z(t) are

∇V+2(z) = ∇V+2

(T1(z)

)∇T1(z). (4.110)


∇V+3(z) = ∇V+2

(T1(z)

)∇T2(z) +∇V+2

(T2(z)

)∇T1(z) +∇V+3

(T1(z),T1(z)

)∇T1(z). (4.111)


∇V+4(z) = ∇V+2

(T1(z)

)∇T3(z) +∇V+2

(T2(z)

)∇T2(z) +∇V+2

(T3(z)

)∇T1(z)+∇V+3

(T1(z),T1(z)

)∇T2(z) +∇V+3

(T1(z),T2(z)

)∇T1(z)+∇V+3

(T2(z),T1(z)

)∇T1(z) +∇V+4

(T1(z),T1(z),T1(z)

)∇T1(z).

(4.112)


∇V+5(z) = ∇V+2

(T1(z)

)∇T4(z) +∇V+2

(T2(z)

)∇T3(z) +∇V+2

(T3(z)

)∇T2(z)+∇V+2

(T4(z)

)∇T1(z) +∇V+3

(T1(z),T1(z)

)∇T3(z)+∇V+3

(T1(z),T2(z)

)∇T2(z) +∇V+3

(T2(z),T1(z)

)∇T2(z)+∇V+3

(T1(z),T3(z)

)∇T1(z) +∇V+3

(T2(z),T2(z)

)∇T1(z)+∇V+3

(T3(z),T1(z)

)∇T1(z) +∇V+4

(T1(z),T1(z),T1(z)

)∇T2(z)+∇V+4

(T1(z),T1(z),T2(z)

)∇T1(z) +∇V+4

(T1(z),T2(z),T1(z)

)∇T1(z)+∇V+4

(T2(z),T1(z),T1(z)

)∇T1(z) +∇V+5

(T1(z),T1(z),T1(z),T1(z)

)∇T1(z).(4.113)

In the following subsections, we will show how to obtain the balanced nonlinear HJE solution gradientfrom these expressions.

4.9.1 First-Order Balanced HJE Solution Gradient

The balanced first-order HJE solution gradient is

∇V+2(z) = zTTT1 P1M0 = zT P1. (4.114)


4.9.2 Second-Order Balanced HJE Solution Gradient

The balanced second-order HJE solution gradient is

∇V+3(z) = zT TT1 P1mattkM1tkz+ rowtzT T2tzP1M0 + rowtzT TT

1 P2tT1zM0

= rowizimatit

n(TT

1 P1)(i,t)o

mattk

nPj M1tk,jzj

o+rowt

nPi

Pj ziT

(i,j)2t zj

omattk

n(P1M0)

(t,k)o

+rowt

nPi

Pj zi(T

T1 P2tT1)

(i,j)zj

omattk

nM

(t,k)0

o= rowk

nPi

Pj zi

“Pt(T

T1 P1)

(i,t)M1tk,j +P

t T(i,j)2t (P1M0)

(t,k)

+P

t(TT1 P2tT1)

(i,j)M(t,k)0

”zj

o= rowk

nzT P2kz

o(4.115)

where P2k = matijP (i,j)2k and

P(i,j)2k =

∑t

(TT1 P2tT1)(i,j)M (t,k)

0 +∑t

T(i,j)2t (P1M0)(t,k) +

∑t

(TT1 P1)(i,t)M1tk,j .

4.9.3 Third-Order Balanced HJE Solution Gradient

The balanced third-order HJE solution gradient is

∇V+4(z) = zT TT1 matstzT TT

1 P3stT1zM0 + zT matpszT T3spzP1M0

+rowszT TT1 P2scoltzT T2tzM0 + rowsrowtzT T2tzP2sT1zM0

+rowszT TT1 P2sT1zmatsqM1sqz+ rowszT T2szP1mattqM1tqz

+zT TT1 P1matsqzT M2sqz

= zT matps

n(TT

1 )(p,s)o

matst

nPi

Pj zi(T

T1 P3stT1)

(i,j)zj

omattq

nM

(t,q)0

o+zT matps

nPi

Pj ziT

(i,j)3sp zj

omatsq

n(P1M0)

(s,q)o

+rows

nPp

Pt zp(TT

1 P2s)(p,t)

“Pi

Pj ziT

(i,j)2t zj

”omatsq

nM

(s,q)0

o+rows

nPt

Pj

“Pp

Pi zpT

(p,i)2t zi

”(P2sT1)

(t,j)zj

omatsq

nM

(s,q)0

o+rows

nPp

Pi zp(TT

1 P2sT1)(p,i)zi

omatsq

nPj M1sq,jzj

o+rows

nPp

Pi zpT

(p,i)2s zi

omatst

nP

(s,t)1

omattq

nPj M1tq,jzj

o+zT matps

n(TT

1 P1)(p,s)

omatsq

nPi

Pj ziM

(i,j)2sq zj

o= zT matpqzT P3pqz

(4.116)

where P3pq = matijP (i,j)3pq and

P(i,j)3pq =

∑s

∑t(T

T1 )(p,s)(TT

1 P3stT1)(i,j)M (t,q)0 +

∑s T

(i,j)3sp (P1M0)(s,q)

+∑s

∑t(T

T1 P2s)(p,t)T

(i,j)2t M

(s,q)0 +

∑s

∑t T

(p,i)2t (P2sT1)(t,j)M (s,q)

0

+∑s(T

T1 P2sT1)(p,i)M1sq,j +

∑s

∑t T

(p,i)2s P

(s,t)1 M1tq,j

+∑s(T

T1 P1)(p,s)M (i,j)

2sq .


4.9.4 Fourth-Order Balanced HJE Solution Gradient

The balanced fourth-order HJE solution gradient is

∇V+5(z) = rowkzTTT

1 matpqzTTT

1 P4kpqT1z

T1z

M0

+rowk

rowpzTT2pz

P2kcolq

zTT2qz

M0

+rowkzTTT

1 P2kmatqnzTT3qnz

z

M0

+rowkzTmatmp

zTT3pmz

P2kT1z

M0

+zTTT1 matpk

zTTT

1 P3pkcolszTT2sz

M0

+zTTT1 matpk

rowq

zTT2qz

P3pkT1z

M0

+rowpzTT2pz

matpk

zTTT

1 P3pkT1z

M0

+rowkzTmatmn

zTT4kmnz

z

P1M0

+zTTT1 matpk

zTTT

1 P3pkT1z

matkr M1krz+zTmatmp

zTT3pmz

P1matkr M1krz

+rowkzTTT

1 P2kcolqzTT2qz

matkr M1krz

+rowk

rowpzTT2pz

P2kT1z

matkr M1krz

+rowkzTTT

1 P2kT1z

matkrzTM2krz

+rowk

zTT2kz

P1matpr

zTM2prz

+zTTT

1 P1rowr

matqnzTM3rqnz

z.

(4.117)


∇V+5(z) = rowm zmmatmp

nT

(p,m)1

omatpr

nP

(p,r)1

orowr

nPn

“Pi

Pj ziM

(i,j)3rqnzj

”zn

o+rowk

nPm

Pi zmT

(m,i)2k zi

omatkp

nP

(k,p)1

omatpr

nPj

Pn zjM

(j,n)2pr zn

o+rowm zmmatmp

nPi

Pj ziT

(i,j)3pm zj

omatpk

nP

(p,k)1

omatkr

˘Pn M1kr,nzn

¯+rowk

nPm

Pn zm

“Pi

Pj ziT

(i,j)4kmnzj

”zn

omatkp

nP

(k,p)1

omatpr

nM

(p,r)0

o+rowk

nPp

Pq

“Pm

Pi zmT

(m,i)2p zi

”P

(p,q)2k

“Pj

Pn zjT

(j,n)2q zn

”omatkr

nM

(k,r)0

o+rowk

nPm

Pp

Pq

Pn zmT

(p,m)1 P

(p,q)2k

“Pi

Pj ziT

(i,j)3qn zj

”zn

omatkr

nM

(k,r)0

o+rowk

nPm

Pp

Pq

Pn zm

“Pi

Pj ziT

(i,j)3pm zj

”P

(p,q)2k T

(q,n)1 zn

omatkr

nM

(k,r)0

o+rowk

nPm

Pp

Pq zmT

(p,m)1 P

(p,q)2k

“Pi

Pj ziT

(i,j)2q zj

”omatkr

˘Pn M1kr,nzn

¯+rowk

nPp

Pq

Pj

“Pm

Pi zmT

(m,i)2p zi

”P

(p,q)2k T

(q,j)1 zj

omatkr

˘Pn M1kr,nzn

¯+rowk

nPm

Pp

Pq

Pi zmT

(p,m)1 P

(p,q)2k T

(q,i)1 zi

omatkr

nPj

Pn zjM

(j,n)2kr zn

o+rowm zmmatmp

nT

(p,m)1

omatpk

nPi

Pq

Ps

Pj ziT

(q,i)1 P

(q,s)3pk T

(s,j)1 zj

o∗

matkr

˘Pn M1kr,nzn

¯+rowm zmmatmp

nT

(p,m)1

omatpk

nPi

Pq

Ps ziT

(q,i)1 P

(q,s)3pk

“Pj

Pn zjT

(j,n)2s zn

”o∗

matkr

nM

(k,r)0

o+rowm zmmatmp

nT

(p,m)1

omatpk

nPq

Ps

Pn

“Pi

Pj ziT

(i,j)2q zj

”P

(q,s)3pk T

(s,n)1 zn

o∗

matkr

nM

(k,r)0

o+rowp

nPm

Pi zmT

(m,i)2p zi

omatpk

nPj

Pq

Ps

Pn zjT

(q,j)1 P

(q,s)3pk T

(s,n)1 zn

o∗

matkr

nM

(k,r)0

o+rowk

nPm

Pp

Pq

Pn zmT

(p,m)1

“Pi

Pj

Ps

Pt ziT

(s,i)1 P

(s,t)4kpqT

(t,j)1 zj

”T

(q,n)1 zn

o∗

matkr

nM

(k,r)0

o.

(4.118)


Therefore, summing across all indices, the balanced fourth-order HJE solution gradient is given by

∇V+5(z) = rowk

zTmatnm

zT P4knmz

z

(4.119)

where P4kmn = matijP (i,j)4kmn and

P(i,j)4kmn =

∑p T

(p,m)1 P

(p,r)1 M

(i,j)3rqn +

∑k

∑p T

(m,i)2k P

(k,p)1 M

(j,n)2pr +

∑k

∑p T

(i,j)3pmP

(p,k)1 M1kr,n

+∑k

∑p T

(i,j)4kmnP

(k,p)1 M

(p,r)0 +

∑k

∑p

∑q T

(p,m)1 P

(p,q)2k T

(q,i)1 M

(j,n)2kr

+∑k

∑p

∑q T

(p,m)1 P

(p,q)2k T

(i,j)2q M1kr,n +

∑k

∑p

∑q T

(m,i)2p P

(p,q)2k T

(q,j)1 M1kr,n

+∑k

∑p

∑q T

(p,m)1 P

(p,q)2k T

(i,j)3qn M

(k,r)0 +

∑p

∑q T

(m,i)2p P

(p,q)2k T

(j,n)2q M

(k,r)0

+∑k

∑p

∑q T

(i,j)3pmP

(p,q)2k T

(q,n)1 M

(k,r)0 +

∑k

∑p

∑q

∑s T

(p,m)1 T

(q,i)1 P

(q,s)3pk T

(s,j)1 M1kr,n

+∑k

∑p

∑q

∑s T

(p,m)1 T

(q,i)1 P

(q,s)3pk T

(j,n)2s M

(k,r)0

+∑k

∑p

∑q

∑s T

(p,m)1 T

(i,j)2q P

(q,s)3pk T

(s,n)1 M

(k,r)0

+∑k

∑p

∑q

∑s T

(m,i)2p T

(q,j)1 P

(q,s)3pk T

(s,n)1 M

(k,r)0

+∑k

∑p

∑q

∑s

∑t T

(p,m)1 T

(s,i)1 P

(s,t)4kpqT

(t,j)1 T

(q,n)1 M

(k,r)0 .

(4.120)Next, we construct analytical expressions for the balanced output injection gain.

4.10 Balanced Output Injection Gain

Here we construct analytical expressions for the balanced nonlinear output injection gain G2(x). Recallthat the nonlinear output injection gain is obtained by solving Eq. (3.73), repeated here for convenience:

∇r2e(x)G2(x) = γ2β−1cT (x), (4.121)

where r2e(x) = β−2V+(x) + γ2V−(x) is the combined HJE solution described in Sec. 3.2.From Eq. (4.8), the balanced output injection gain is obtained according to the relation

G2(z) = ∇T−1(z)G2

(T(z)

). (4.122)

We can therefore construct the balanced nonlinear output injection gain by substituting the transforma-tion of Eq. (4.25) and its gradient inverse from Eq. (4.56) into Eq. (4.122) and collecting terms of thesame order in z(t). Alternatively, we can obtain an expression identical to Eq. (4.121) in terms of thebalanced states z(t) and solve for G2(z) normally using the method of the previous chapter. We shalladopt this alternate approach for the sake of simplicity. It will be shown that an expression identicalto Eq. (4.121) can be obtained in terms of the balanced states z(t). Then, the approach used in theprevious chapter will be applied to obtain an expression for the balanced output injection gain G2(z).

Based on the definitions and derivations from Section 3.2, it is easy to see that the balanced combinedHJE solution gradient is given by

∇r2e(z) = ∇r2e(T(z)

)∇T(z)= β−2∇V+

(T(z)

)∇T(z) + γ2∇V−(T(z)

)∇T(z)= β−2∇V+(z) + γ2∇V−(z).

(4.123)


The balanced combined HJE solution gradient can be approximated as follows:

∇r2e(z) = ∇r2e,2(z) +∇r2e,3(z) +∇r2e,4(z) +∇r2e,5(z), (4.124)

where∇r2e,2(x) = zT R1

∇r2e,3(x) = rowk

zT R2kz

∇r2e,4(x) = zTmatnm

zT R3nmz

∇r2e,5(x) = rowk

zTmatnm

zT R4knmz

z

Using Eqs. (4.8) and (4.123), we can readily rewrite Eq. (4.121) as

[∇r2e(T(z))∇T(z)][∇T−1(z)G2(T(z))

]= γ2β−1cT (T(z)) (4.125)

or∇r2e(z)G2(z) = γ2β−1cT (z). (4.126)

It is now possible to apply the approach used in the previous chapter to solve Eq. (4.126) for the balancednonlinear output injection gain G2(z). There is one difference, however. In the previous chapter we onlyhad to extract the leading xT from a linear output matrix c(x). However, due to the nonlinear balancingtransformation, we must now handle a nonlinear output matrix c(z). As in the previous chapter, theapproach to solving for G2(z) involves two steps. First, we extract and cancel the leading zT from bothsides of Eq. (4.126). Then, we invert the resulting left-hand square matrix numerically.

Extracting the leading zT from the balanced combined HJE solution gradient of Eq. (4.124), weobtain

∇r2e(z) = zTΨ(z), (4.127)

where we can define the Taylor series approximation of the term Ψ(z) as follows:

Ψ(z) = Ψ0(z) + Ψ1(z) + Ψ2(z) + Ψ3(z), (4.128)

withΨ0(z) = Ψ0

Ψ1(z) = matstΨ1stzΨ2(z) = matstzTΨ2stzΨ3(z) = matstrowkzTΨ3stkzz

(4.129)

Here, Ψ0 is a square matrix with Ψ(i,j)0 = R

(i,j)1 , Ψ1st = rowiΨ1st,i is a row matrix with Ψ1st,i = R

(s,i)2t ,

Ψ2st is a square matrix with Ψ(i,j)2st = R

(i,j)3st , and Ψ3stk is a square matrix with Ψ(i,j)

3stk = R(i,j)4tsk. See

Chapter 3 for a more detailed derivation.The balanced nonlinear output matrix was given by Eq. (4.99). However, here we require the

transpose of this matrix. The transpose of the balanced nonlinear output matrix in Eq. (4.99) is givenby

cT (z) = cT1 (z) + cT2 (z) + cT3 (z) + cT4 (z), (4.130)


wherecT1 (z) = zT CT

1

cT2 (z) = rowt

zT C2tz

cT3 (z) = matst

zT C3tsz

z

cT4 (z) = rowt

zTmatsk

zT C4tskz

z (4.131)

Comparing Eqs. (4.124) and (4.130), we see that they are of the same form. It is thus straightforwardto apply the same method to extract the leading zT from cT (z).

Therefore, extracting the leading zT from the transpose of the balanced nonlinear output matrix ofEq. (4.130) leads to

cT (z) = zTΦ(z), (4.132)

where we can define the Taylor series approximation of the term Φ(z) as follows:

Φ(z) = Φ0(z) + Φ1(z) + Φ2(z) + Φ3(z), (4.133)

withΦ0(z) = Φ0

Φ1(z) = matstΦ1stzΦ2(z) = matstzTΦ2stzΦ3(z) = matstrowkzTΦ3stkzz

(4.134)

The first-order term in z(t) iscT1 (z) = zT CT

1 = zTΦ0 (4.135)

where Φ0 = matijΦ(i,j)0 is a rectangular matrix with Φ(i,j)

0 = C(j,i)1 . The second-order term in z(t) is

cT2 (z) = rowt

zT C2tz

= rowt∑

s

∑i zsC

(s,i)2t zi

= rowt

zT cols

∑i C

(s,i)2t zi

= zTmatst

∑i C

(s,i)2t zi

= zTmatst Φ1stz

(4.136)

where Φ1st = rowiΦ1st,i is a row matrix with Φ1st,i = C(s,i)2t . The third-order term in z(t) is

cT3 (z) = zTmatst

zT C3tsz

= zTmatstzTΦ2stz

(4.137)

where Φ2st = matijΦ(i,j)2st is a square matrix with Φ(i,j)

2st = C(i,j)3ts . And finally, the fourth-order term in


z(t) becomescT4 (z) = rowt

zTmatsk

zT C4tskz

z

= rowt∑

s

∑k zs

(∑i

∑j ziC

(i,j)4tskzj

)zk

= rowt

zT cols

∑k

(∑i

∑j ziC

(i,j)4tskzj

)zk

= zTmatst

∑k

(∑i

∑j ziC

(i,j)4tskzj

)zk

= zTmatst

rowk

∑i

∑j ziC

(i,j)4tskzj

z

= zTmatst

rowkzTΦ3stkz

z

(4.138)

where Φ3stk = matijΦ(i,j)3stk is a square matrix with Φ(i,j)

3stk = C(i,j)4tsk .

Therefore, canceling the leading zT from both sides of Eq. (4.126) yields

Ψ(z)G2(z) = γ2β−1Φ(z). (4.139)

Inverting the square matrix Ψ(z) numerically, the balanced nonlinear output injection gain is given by

G2(z) = γ2β−1 [Ψ(z)]−1 Φ(z). (4.140)

Now that all the main controller terms have been balanced, we examine how to construct the differentreduced-state-order controllers.

4.11 Reduced-State-Order Nonlinear Controllers

Having balanced all the matrices required to construct each of the balanced nonlinear controllers, we nowexamine how to reduce the number of controller states in each of these matrices. For simplicity, we shalldetermine which states to eliminate based solely on the linear LQG-characteristic values. It was seen inEq. (4.34) that Σ = diagiσi for i = 1, . . . , np, where σ2

i = λi(−P1Q−1

1

)> 0 and σ1 ≥ . . . ≥ σnp , with

σi denoting the LQG-characteristic values of the (closed-loop) system. The LQG-characteristic value σiprovides a measure of the relative importance of the balanced state zi in the closed-loop input-outputmap. Thus, balanced states with the smallest LQG-characteristic values have the least effect on themapping.

Therefore, pick some scalar 1 < τ < np such that στ > στ+1 and partition Σ as

Σ =

[Σ1 00 Σ2

], (4.141)

where Σ1 = diagσ1, . . . , στ and Σ2 = diagστ+1, . . . , σnp. The nonlinear controller state space

matrices will then be partitioned to conform with the partitioning of Σ. We define zcr(t) as the balancedcontroller state vector corresponding to Σ1 and use the subscript (·)r to denote reduced-state-orderterms. Additionally, note that although we will be reducing the number of controller states, the inputand output dimensions of the different controllers will remain unchanged.


4.11.1 Reduced-State-Order Nonlinear H∞ Controller

The reduced-state-order nonlinear dynamics matrix is given by

ar(zcr) = a1r(zcr) + a2r(zcr) + a3r(zcr) + a4r(zcr), (4.142)

wherea1r(zcr) = A1,11zcra2r(zcr) = colp

zTcrA2p,11zcr

a3r(zcr) = matpq

zTcrA3pq,11zcr

zcr

a4r(zcr) = colp

zTcrmatqs

zTcrA4pqs,11zcr

zcr (4.143)

with indices p, q, s ∈ 1, . . . , τ. The nonlinear dynamics matrices are thus partitioned as

A1 =

[A1,11 A1,12

A1,21 A1,22

]A2p =

[A2p,11 A2p,12

A2p,21 A2p,22

]

A3pq =

[A3pq,11 A3pq,12

A3pq,21 A3pq,22

]A4pqs =

[A4pqs,11 A4pqs,12

A3pqs,21 A4pqs,22

] (4.144)

The reduced-state-order nonlinear input matrix is given by

br(zcr) = b0r(zcr) + b1r(zcr) + b2r(zcr) + b3r(zcr), (4.145)

whereb0r(zcr) = B0,1

b1r(zcr) = matpq

B1pq,11zcr

b2r(zcr) = matpq

zTcrB2pq,11zcr

b3r(zcr) = matpq

rows

zTcrB3pqs,11zcr

zcr (4.146)

with indices p, s ∈ 1, . . . , τ and q ∈ 1, . . . ,mp. The nonlinear input matrices are thus partitioned as

B0 =

[B0,1

B0,2

]B1pq =

[B1pq,1 B1pq,2

]B2pq =

[B2pq,11 B2pq,12

B2pq,21 B2pq,22

]B3pqs =

[B3pqs,11 B3pqs,12

B3pqs,21 B3pqs,22

] (4.147)

The reduced-state-order nonlinear output matrix is given by

cr(zcr) = c1r(zcr) + c2r(zcr) + c3r(zcr) + c4r(zcr), (4.148)

wherec1r(zcr) = C1,1zcrc2r(zcr) = colp

zTcrC2p,11zcr

c3r(zcr) = matpq

zTcrC3pq,11zcr

zcr

c4r(zcr) = colp

zTcrmatqs

zTcrC4pqs,11zcr

zcr (4.149)


with indices p ∈ 1, . . . , pp and q, s ∈ 1, . . . , τ. The nonlinear output matrices are thus partitioned as

C1 =[C1,1 C1,2

]C2p =

[C2p,11 C2p,12

C2p,21 C2p,22

]

C3pq =

[C3pq,11 C3pq,12

C3pq,21 C3pq,22

]C4pqs =

[C4pqs,11 C4pqs,12

C4pqs,21 C4pqs,22

] (4.150)

The reduced-state-order nonlinear HJE solution gradient is given by

∇Vr(zcr) = ∇V1r(zcr) +∇V2r(zcr) +∇V3r(zcr) +∇V4r(zcr). (4.151)

where∇V1r(zcr) = zTcrP1,11

∇V2r(zcr) = rowp

zTcrP2p,11zcr

∇V3r(zcr) = zTcrmatpq

zTcrP3pq,11zcr

∇V4r(zcr) = rowp

zTcrmatqs

zTcrP4pqs,11zcr

zcr (4.152)

with indices p, q, s ∈ 1, . . . , τ. The nonlinear HJE solution gradient matrices are thus partitioned as

P1 =

[P1,11 P1,12

P1,21 P1,22

]P2p =

[P2p,11 P2p,12

P2p,21 P2p,22

]

P3pq =

[P3pq,11 P3pq,12

P3pq,21 P3pq,22

]P4pqs =

[P4pqs,11 P4pqs,12

P4pqs,21 P4pqs,22

] (4.153)

The reduced-state-order nonlinear terms in the output injection gain are given by

Ψr(zcr) = Ψ0r(zcr) + Ψ1r(zcr) + Ψ2r(zcr) + Ψ3r(zcr)Φr(zcr) = Φ0r(zcr) + Φ1r(zcr) + Φ2r(zcr) + Φ3r(zcr)

(4.154)

where

Ψ0r(zcr) = Ψ0,11 Φ0r(zcr) = Φ0,1

Ψ1r(zcr) = matpq Ψ1pq,1zcr Φ1r(zcr) = matps Φ1ps,1zcrΨ2r(zcr) = matpq

˘zT

crΨ2pq,11zcr

¯Φ2r(zcr) = matps

˘zT

crΦ2ps,11zcr

¯Ψ3r(zcr) = matpq

˘rowt

˘zT

crΨ3pqt,11zcr

¯zcr

¯Φ3r(zcr) = matps

˘rowt

˘zT

crΨ3pst,11zcr

¯zcr

¯ (4.155)

with indices p, q, t ∈ 1, . . . , τ and s ∈ 1, . . . , pp. The nonlinear terms in the output injection gain arethus partitioned as

Ψ0 =

[Ψ0,11 Ψ0,12

Ψ0,21 Ψ0,22

]Ψ1pq =

[Ψ1pq,1 Ψ1pq,2

]Ψ2pq =

[Ψ2pq,11 Ψ2pq,12

Ψ2pq,21 Ψ2pq,22

]Ψ3pqt =

[Ψ3pqt,11 Ψ3pqt,12

Ψ3pqt,21 Ψ3pqt,22

] (4.156)


and

Φ0 =

[Φ0,1

Φ0,2

]Φ1ps =

[Φ1ps,1 Φ1ps,2

]Φ2ps =

[Φ2ps,11 Φ2ps,12

Φ2ps,21 Φ2ps,22

]Φ3pst =

[Φ3pst,11 Φ3pst,12

Φ3pst,21 Φ3pst,22

] (4.157)

The reduced-state-order nonlinear output injection gain is therefore given by

G2r(zcr) = γ2β−1Ψ−1r (zcr)Φr(zcr). (4.158)

The resulting reduced-state-order nonlinear H∞ controller is

Kr :

zcr = ar(zcr)− br(zcr)bTr (zcr)∇V T+r(zcr) + G2r(zcr)

[β−1cr(zcr) + βy2

]u2(zcr) = β−2bTr (zcr)∇V T+r(zcr)

(4.159)

4.11.2 Reduced-State-Order Nonlinear H∞ Loop Shaping Controller

The process of reducing the number of controller states for the H∞ loop shaping controller is nearlyidentical to the derivation of the previous subsection. The primary difference is that the balancedcontroller state vector zc is now an estimate of the augmented vector x(t) in Eq. (3.85), which includesboth the plant and filter states. A result of this is that the original balanced controller state zc(t) nowhas dimensions of nc = np + nF , where nF is the state dimension of the loop shaping filter. However,after the controller state-order-reduction, the controller will have dimensions nc = τ . It should benoted here that the dimensions of the loop shaping filter F do not change. Therefore, the augmentedreduced-state-order loop shaping controller has dimensions τ + nF .

The resulting augmented reduced-state-order nonlinear H∞ loop shaping controller is

Kr :

zcr = ar(zcr)− br(zcr)b

T

r (zcr)∇V T+r(zcr) + G2r(zcr)[β−1cr(zcr) + βy2

]u2(zcr) = β−2b

T

r (zcr)∇V T+r(zcr)xF (zcr,xF ) = aF (xF ) + bF (xF )uF (zcr)u2(zcr,xF ) = cF (xF ) + dF (xF )uF (zcr)

(4.160)

4.11.3 Reduced-State-Order Nonlinear Weighted Mixed Sensitivity Controller

We now examine how to reduce the number of controller states for the nonlinear weighted mixed sensi-tivity controller. It was noted earlier in this chapter how the process of balancing the nonlinear terms aresimilar to the H∞ controller terms. The similarities with regards to state-order-reduction are the samehere. The primary difference is that the controller state vector zc(t) is now an estimate of the augmentedvector x(t) in Eq. (3.98), which includes the plant states and the states from both weighting functions.A result of this is that the controller state zc(t) now has dimensions of nc = np + nw1 + nw2, where nw1

is the state dimension of the tracking error weighting function and nw2 is the state dimension of thecontrol effort weighting function. However, after the controller state-order-reduction, the controller willhave dimensions nc = τ .

In particular, the method for reducing the dynamics matrix a(zc) is identical to the method for a(zc),except for the matrix dimensions. Similarly, reducing the input matrices b1(zc) and b2(zc) is identicalto the method for b(zc), and reducing the output matrix c2(zc) is identical to the method for c(zc).


Additionally, recall that the matrices d12(zc), d21(zc) and e(zc) are constant for our purposes. Thismeant that they do not require balancing or state-order-reduction.

The resulting reduced-state-order nonlinear weighted mixed sensitivity controller is

Kr :

zcr = ar(zcr) + b1r(zcr)y0(zcr) + b2r(zcr)u(zcr)

y0(zcr) = D−1

21

[y(t)− c2r(zcr)

]u(zcr) = −E

−1[DT

12c1r(zcr) + bT

2r(zcr)∇V T+r(zcr)] (4.161)

4.12 Chapter Summary

This chapter focused on developing practical nonlinear controller balancing and state-order-reductiontechniques. We began by stating a few important definitions and by examining how a nonlinear statetransformation can be applied to balance the three nonlinear controllers described in the previous chap-ter. Then, we proceeded to derive the nonlinear state balancing transformation, its gradient, and anapproximate inverse of its gradient. After the transformation was obtained, we showed how it can beapplied to the different terms that comprise the three nonlinear control systems. Finally, we discussedhow to explicitly construct the three reduced-state-order control systems.

Chapter 5

Simulation Results

In this chapter, we shall present numerical simulation results using the control methods developed inthis thesis. We will make use of three separate problems to demonstrate and emphasize different aspectsof the control synthesis and state-order-reduction methods discussed in Chapters 3 and 4. In particular,the three problems considered here are the spacecraft attitude control problem, the spacecraft formationflying problem, and the problem of controlling a nonlinear mass-spring chain. The attitude controlproblem will be used primarily to compare the H∞ control method with existing methods from theliterature. The formation flying problem will focus on comparing the H∞ and loop shaping controlmethods, as well as the effects of state-order-reduction. And finally, the nonlinear mass-spring problemwill study the effects of state-order-reduction further. We shall examine each of these problems in turn.

5.1 Attitude Control Problem

The attitude control problem is critical for most satellite applications and has thus attracted extensiveinterest. A prime example is the Hubble Space Telescope, which requires very precise pointing for theobservation of stellar phenomena. While many control methods have been developed to address thisproblem, most are concerned primarily with the optimality of attitude maneuvers [24, 58, 65, 66]. Incontrast, the nonlinear control methods developed in this thesis focus more on robustness. Therefore,in this section we will compare the robustness and performance properties of several existing methodsfrom the literature with two control methods discussed in Chapter 3 of this thesis.

First, we present the nonlinear attitude dynamics of a rigid spacecraft. Then, we provide an overviewof several attitude control methods from the existing literature. Then, we compare the existing and newcontrol methods through numerical simulations. This section concludes with a summary of the results.

5.1.1 Attitude Dynamics

The attitude dynamics of a rigid spacecraft are given by Euler’s equation:

ω = −I−1ω×Iω + I−1u, (5.1)

where ω = [ω1 ω2 ω3]T are the body angular velocities, I is the inertia tensor, and u1 = [u1 u2 u3]T

are the body torques. The notation ω× is the matrix representation of the cross product and is defined

89

Chapter 5. Simulation Results 90

as

ω× =

0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

. (5.2)

While many representations are possible to define the spacecraft attitude kinematics, the ModifiedRodriguez Parameters (MRPs) are chosen here for two reasons. First, they are polynomial in thestates, which fits nicely with the present controller synthesis approach, and second, they possess neithersingularities nor norm constraints when used in conjunction with the shadow parameters [57]. The MRPvector σ = [σ1 σ2 σ3]T can be defined in terms of the principal rotation axis e = [e1 e2 e3]T andprincipal rotation angle Φ of Euler’s theorem according to

σ = e tan (Φ/4). (5.3)

The attitude kinematics using MRPs are defined by

σ = 12

[12 (1− σTσ)1 + σ× + σσT

]ω. (5.4)

Upon closer inspection of Eq. (5.3), it is seen that the MRPs encounter a singularity for rotations ofΦ = ±2π rad. This corresponds to a complete rotation in either direction about the principal axis. Tocircumvent this, another set of MRPs, called the shadow parameters and denoted by σS , are used inconjunction with the regular MRPs. By switching from one set to the other at rotations of Φ = ±π rad,it is possible to avoid any singularities. The parameter switching occurs on the surface defined by

σTσ = σTSσS = 1. (5.5)

The kinematics are identical for both the regular and the shadow parameters. However, when theswitching surface is encountered, both the MRPs and their rates must be converted from one set to theother. This can be accomplished with the following relations:

σS = − σ

(σTσ)+

1 + (σTσ)2(σTσ)2

σσTω, σS = − σ

(σTσ). (5.6)

More details can be found in the text by Schaub and Junkins [57].Defining the state vector x = [ω1 ω2 ω3 σ1 σ2 σ3]T and grouping terms of the same order, the

attitude dynamics and kinematics can be expressed asx(t) = a1(x) + a2(x) + a3(x) + Bu1(t)y1(t) = Cx

(5.7)

where

a1(x) = A1x =

[0 0141 0

][ω

σ

], B =

[I−1

0

], C = 1, (5.8)

and the second- and third-order terms are given by

a2(x) =

[−I−1ω×Iω

12σ×ω

], a3(x) =

[0(

12σσ

T − 14σ

Tσ1)ω

], (5.9)


respectively. It should be noted that the attitude kinematics and dynamics given by Eqs. (5.7) – (5.9)are exact. Next, we will present several existing attitude control methods from the literature.

5.1.2 Existing Control Methods

We now provide a brief overview of the controller synthesis methods from the literature to be used forcomparisons. The methods to be used are the linear and nonlinear Proportional-Derivative (PD) lawsof Tsiotras [66], the open-loop (OL) optimal control method by Schaub, Junkins and Robinett [58], theclosed-loop (CL) optimal nonlinear method of Tewari [65], and the sum of squares (SOS) approach ofGollu and Rodrigues [24]. The interested reader is referred to the appropriate literature for a moredetailed exposition of these methods.

Proportional-Derivative Controllers

Tsiotras [66] developed two Proportional-Derivative (PD) control laws for the attitude control problem.The linear PD controller is given by

u2(t) = ω + 2σ, (5.10)

while the nonlinear version is given by

u2(t) = ω + 2(1 + σTσ)σ. (5.11)

Open-Loop Optimal Control

The optimal control method presented by Schaub, Junkins and Robinett [58] is designed to minimizethe cost function

J = 12K1g(σ(tf )) + 1

2ωT (tf )K2ω(tf )︸︷︷︸

φ(tf )

+∫ tf

t0

12

[K3g(σ) + ωTK4ω + uT2 Ru2

]︸︷︷︸p(x,u2,t)

dt, (5.12)

where K1 and K3 are scalar weights, K2, K4 and R are weighting matrices, and

g(σ) = 4σTσ

(1 + σTσ)2. (5.13)

The Hamiltonian relating to this optimal control problem is defined as

H = p(x,u, t) + ΛT [a(x)−Bu2(t)] , (5.14)

where we have used the plant model in Eq. (2.7) and the second relation of Eq. (2.6) with y1 = 0. Thecostates, denoted by Λ, have dynamics

Λ = −∂H∂x

, Λ(tf ) = 0. (5.15)

Note that the costates are specified at some final time tf , not the initial time. The optimal control lawfor this problem is determined from

∂H

∂u2= 0, (5.16)


which yieldsu2(t) = R−1BTΛ. (5.17)

The primary disadvantage of this method from a practical point of view is that it requires the solutionof a two-point boundary value problem (TPBVP) and results in an open-loop control strategy. For thesimulation results presented below, the following weighting parameters are used: K1 = 5.0, K2 = 5.0 1,K3 = 1.0, K4 = 1, and R = 1. The maneuver is optimized for a final time tf = 60 s.

Closed-Loop Optimal Controller

The optimal control method presented by Tewari [65] is based on obtaining an exact analytical solutionto the Hamilton-Jacobi equation. Consider the HJE (3.7) with parameters

A(x) = a(x)R(x) = BR−1BT

Q(x) =

[x

x[2]

]T [Q11 Q12

Q21 Q22

]︸︷︷︸

Q

[x

x[2]

](5.18)

where the inertia matrix is defined as I = diagiIi, R = diagiRi is symmetric and positive definite,Q is symmetric and positive semidefinite, and x[2] = [ω2

1 ω22 ω2

3 σ21 σ2

2 σ23 ]T . It is assumed that V (x)

has the same form as Q(x); thus,

V (x) =12

[x

x[2]

]T [P11 P12

P21 P22

]︸︷︷︸

P

[x

x[2]

], (5.19)

where P is symmetric and positive definite. The state feedback controller is given by

u2(t) = R−1BT∇V T (x) (5.20)

where∇V T (x) = P11x + 2diagixiP12x + P12x[2] + 2diagixiP22x[2]. (5.21)

The matrix P is obtained as follows. First, P11 is calculated from the algebraic Riccati equation(3.22) corresponding to the parameters in Eq. (5.18), that is, with R in Eq. (3.22) replaced by BR−1BT

and Q2 in Eq. (3.22) replaced by Q11. Then, Eqs. (3.20), with k ∈ 3, . . . , 6, are solved simultaneously


for the remaining unknowns. The resulting non-zero elements of the matrices P12 and P22 are as follows:

P 1112 = I2

1R1(2Q1112 + k2P

1211 + k3P

1311 )/(6P 11

11 )P 16

12 = I21R1Q

1612/(2P

1111 )

P 2212 = I2

2R2(Q1212 − 2k2P

2211 )/(3P 22

11 )P 25

12 = I22R2(2Q25

12 − 3P 2212P

2511 )/(2P 22

11 )P 33

12 = I23R3Q

3312/(3P

3311 )

P 1122 = [I2

1R1Q1122 − 9(P 11

12 )2]/(2P 1111 )

P 1222 = Q12

22/[P1111 /(I

21R1) + P 22

11 /(I22R2)]

P 1322 = Q13

22/[P1111 /(I

21R1) + P 33

11 /(I23R3)]

P 2222 = [I2

2R2Q2222 − 9(P 22

12 )2]/(2P 2211 )

P 2322 = Q23

22/[P3311 /(I

23R3) + P 22

11 /(I22R2)]

P 3322 = [I2

3R3Q3322 − 9(P 33

12 )2]/(2P 3311 )

(5.22)

where k1 = (I2 − I3)/I1, k2 = (I3 − I1)/I2 and k3 = (I1 − I2)/I3.For the simulation results presented below, the following weighting parameters are used [65]:

R = diagi = I−2i , Q11 = (0.01)1,

Q12 =

[(0.1)1 0

0 0

], Q22 =

P 11

11 P 1211 P 13

11

P 2111 P 22

11 P 2311

P 3111 P 32

11 P 3311

0

0 0

. (5.23)

We now make a few remarks regarding the characteristics of the synthesis method of Tewari [65]. Itshould be quite evident that the overall concept behind the method of Tewari is similar to that of thisthesis. Like the control methods presented in this thesis, Tewari’s closed-loop optimal method results ina polynomial feedback controller. However, while our method uses a recursive approach to obtaining theHJE solution gradient at each order, Tewari uses a fixed-order approximation and solves for all the termsin P12 and P22 simultaneously. Moreover, forcing the matrix P in Eq. (5.19) to be positive definiteis more restrictive than the conditions used in our method. Additionally, Tewari’s solution is derivedspecifically for the attitude control problem and furthermore is predicated on the assumption that theinertia matrix is diagonal. Although it is theoretically possible to find a coordinate transformation suchthat the inertia matrix is diagonal, in practice there are always uncertainties present and the choice ofbody frame is not always arbitrary.

In contrast, the method developed in the present thesis is applicable to a much wider class of systems,including the attitude problem with non-diagonal inertia matrices. Moreover, even for a diagonal inertiamatrix, we note that the stability of Tewari’s controller is dependent on the values of the inertia.For example, his controller is completely unstable for the initial conditions specified below when I =diag10, 20, 30 kg·m2. Due to this limitation, the inertia matrix used in the simulation results of thenext subsection is taken from Tewari such that a suitable comparison can be made. We note that themethods developed in this thesis do indeed work for the situation where Tewari’s controller does not. Inparticular, Figures 5.1 and 5.2 present results using the fourth-order H∞ controller from this thesis. TheMRP switching can clearly be seen in Figure 5.1, where we also note that it requires several rotationsfor the controller to sufficiently slow down the spacecraft. This is due to the small torques applied, as


seen in Figure 5.2.

0 50 100 150 200Time, t [s]

−1.0

−0.5

0.0

0.5

1.0M

odifi

edR

odrig

ues

Par

amet

ers,σ σ1

σ2σ3

Figure 5.1: MRP trajectories for I = diag10, 20, 30 kg·m2 with 4th-order H∞ control system

0 50 100 150 200Time, t [s]

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

Con

trolT

orqu

e,u

2[N·m

]

u1u2u3

Figure 5.2: Control torques for I = diag10, 20, 30 kg·m2 with 4th-order H∞ control system


Sum of Squares Controller

A multivariate polynomial p(x) is a sum of squares (SOS) if there exist some polynomials fi(x), i ∈1, . . . ,M, such that

p(x) =M∑i=1

f2i (x). (5.24)

The SOS controller synthesis approach relaxes the search for positive definite functions to a search forSOS functions. It should be noted, however, that the use of sums of squares is conservative, since p(x)being SOS implies that p(x) ≥ 0, while the converse is not true in general. Additionally, it can be shownthat, if vTP(x)v is SOS for v ∈ RN , then P(x) ≥ 0 for all x ∈ Rn [54].

In applying the SOS controller synthesis method to the attitude control problem, we first rewrite thestate space system in Eq. (2.7) as

x(t) = A(x)x−Bu2(t), (5.25)

where we have set u0(t) = 0 and used the fact that the matrix b(x) is constant for this problem. Notethat the matrix A(x) in Eq. (5.25) is not unique. The SOS state feedback controller for this problem isgiven by [24]

u2(t) = −K(x)P−1x, (5.26)

with Lyapunov function given byV (x) = xTP−1x. (5.27)

The time derivative of this Lyapunov function along the trajectories of the system in Eq. (5.25) withthe controller of Eq. (5.26) is

V (x) = [A(x)x + BK(x)P−1x]TP−1x + xTP−1[A(x)x + BK(x)P−1x]. (5.28)

Using the change of variables x = Pv, this last equation can be written as

V (x) = vT [PAT (x) + A(x)P + KT (x)BT + BK(x)]v. (5.29)

The conditions V (x) > 0 and V (x) < 0 can be replaced by the conditions P is positive definite and−V (x) is SOS. The second of these conditions will be strengthened to −[V (x) + ε(x)] being SOS, whereε(x) is some SOS function. This semidefinite programming (SDP) problem can then be written asfollows:

find P, K(x), ε(x)s.t. P = PT > 0

ε(x) is SOS−vT [PAT (x) + A(x)P + KT (x)BT + BK(x) + ε(x)1]v is SOS

(5.30)

This optimization problem can be solved using the SOSTOOLS software package [52].

5.1.3 Simulation Results and Comparisons

We now present simulation results comparing the existing methods from the literature discussed in theprevious subsection with the H∞ control method presented earlier in this thesis. The purpose of thesecomparisons is to examine the effects of different disturbances and uncertainties on the performance


and robustness of the various controllers. In particular, we include gravity-gradient and geomagnetictorques, as well as unmodeled first- and second-order actuator dynamics and actuation time-delays. Inaddition to these comparisons, we will make use of the gap metric to characterize the difference in theinput-output (IO) map of the system induced by the unmodeled actuator dynamics.

Simulation Parameters

We now define the various parameters pertinent to the numerical simulations and then proceed with thedifferent comparisons. The satellite is in a circular orbit with an inclination of i = 87 and a longitudeof the ascending node of Ω = 0. The initial value of the argument of latitude is zero. The altitude is700 km and we take Re = 6378.14 km for the Earth’s radius. These orbital parameters will be used todetermine the gravity-gradient and geomagnetic disturbance torques acting on the spacecraft. For thepurposes of the geomagnetic disturbance torque, the satellite is assumed to generate a magnetic dipole ofm = [0.1 0.1 0.1]T A·m2. The satellite inertia tensor is given by I = diag10.0, 6.3, 8.5 kg·m2. In allcomparisons, we consider the regulation problem only, hence y0 = 0. All simulations will be performedfor one orbit using a 4th-order Runge-Kutta numerical integration method with a step-size ∆t = 0.01 s.

In the case of the disturbance rejection comparisons, the simulations start from the desired atti-tude and we compare the ability of the different controllers to maintain that attitude. For all othercomparisons, the initial states are chosen from Schaub, Junkins and Robinett [58] to be ω(0) =[1.4 0.9 0.8]T rad/s and σ(0) = [0.87 0 0]T . These initial conditions are such that the satelliteis oriented almost π rad from the desired attitude with large angular velocities moving it towards thisupside-down attitude. The H∞ controllers are designed with γ = 4.0, which was chosen to satisfy thelinear version of the condition in Eq. (3.65).

In order to gauge the performance and properly compare the various control systems, we will makeuse of several different metrics. When performing comparisons of the disturbance rejection capabilitiesand the robustness to unmodeled actuator dynamics, we will make use of the RMS tracking error andcontrol effort. The RMS tracking error is given by

Erms =

[1To

∫ To

0

yT2 (t)y2(t) dt

](1/2)

(5.31)

and the RMS control effort is given by

Trms =

[1To

∫ To

0

uT2 (t)u2(t) dt

](1/2)

(5.32)

where y2 = y0 − y1 = −y1 and To is the orbital period.When performing comparisons of the robustness of the various control systems to actuation time-

delay, we will not use the above two metrics. Since our objective will be to simply determine themaximum allowable time-delay such that each control system remains stable over the course of oneorbit, the metric used for these comparisons will simply be the value of the time-delay itself.


Disturbance Rejection

We begin by comparing the controllers presented in this thesis with the methods from the literaturewith regards to disturbance rejection. The two disturbances considered here are the gravity-gradientand geomagnetic torques. For the purposes of this comparison, the simulations start from the desiredattitude (i.e., y2 = 0) and we compare the ability of the different controllers to maintain that attitudeover one complete orbit. In order to make these comparisons even more meaningful, we scale the inputto the plant such that the tracking error is the same across all control methods. We are then able tocompare the effort required by each control scheme. We use the linear H∞ controller as the referenceerror profile, with Erms = 1.081× 10−5. Tables 5.1 and 5.2 present values of the RMS control effort forthe different controllers.

Table 5.1: Disturbance rejection results for existing controllersPD PD Optimal SOS

(linear) (nonlinear) (closed-loop)Trms [N · m] (×10−6) 3.984 3.984 3.979 3.979

Table 5.2: Disturbance rejection results for H∞ controllersH∞ H∞ H∞ H∞

(1st-order) (2nd-order) (3rd-order) (4th-order)Trms [N · m] (×10−6) 3.980 3.980 3.980 3.980

The (open-loop) optimal control method of Schaub, Junkins and Robinett [58] is not included in thetable because it is completely unable to reject any disturbances, which is entirely due to its open-loopnature. There is no apparent difference in performance between the linear and nonlinear PD controllers,nor between any of the nonlinear H∞ controllers developed using the present method. Overall, the newH∞ controllers present no significant improvement in comparison to the other methods with regards todisturbance rejection. This is due to the very small magnitudes of these disturbance torques, which thelinear control system can very easily overcome.

Robustness to Unmodeled Actuator Dynamics

The robustness properties of the different controllers are now examined with regards to unmodeledactuator dynamics. For the purposes of this study, we make use of first- and second-order actuatormodels. In practice the actuator dynamics may be far more complex than the ones used here. However,we use these simple models here for the purposes of studying the capabilities of the different controlmethods. Evidently, as the actuator bandwidth decreases, it becomes harder for the controller to stabilizethe system. Thus, we are able to infer the relative robustness properties of the different controllers byexamining the figures below. In particular, the farther a line reaches towards the left-hand side of agraph, the more robust that controller is with regards to the unmodeled actuator dynamics. As we shallsee, the present H∞ controllers are always more robust than the other methods.

Unmodeled 1st-Order Actuator Dynamics

We begin by examining the robustness of the controllers with regards to unmodeled first-order actuatordynamics. This shall be accomplished by including the following first-order dynamics model in each


component (i ∈ 1, 2, 3) of the controller output:

x = −ωbx+ ωbu2,i

yi = x(5.33)

where ωb is the actuator bandwidth. Figures 5.3 and 5.4 show Erms and Trms, respectively, as a functionof the actuator bandwidth. It is noted from these figures that all four H∞ controllers provide nearlythe same tracking error and control effort. Moreover, while the closed-loop optimal control law providesbetter tracking error compared with the H∞ controllers, the trade-off is that it requires more controleffort. Similarly, the SOS controller yields very low tracking error at the expense of greater controleffort. The two PD laws, on the other hand, have a higher tracking error and control effort than theother methods. Note that the linear PD law performs better than its nonlinear counterpart; this wasobserved by Tsiotras in his original work [66]. The four H∞ controllers demonstrate better robustnessproperties compared to the other controllers with regards to the unmodeled actuator dynamics. However,it is not possible to discern any difference in robustness between the different order H∞ controllers fromthese figures.

10−1 100 101 102

Actuator bandwidth, ωb [rad/s]

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

RM

Stra

ckin

ger

ror,Erm

s

H∞ (1st-order)H∞ (2nd-order)H∞ (3rd-order)H∞ (4th-order)OptimalPD (linear)PD (nonlinear)SOS

Figure 5.3: RMS tracking error with respect to 1st-order actuator bandwidth

Unmodeled 2nd-Order Actuator Dynamics

The robustness properties of the different controllers are now examined with regards to unmodeledsecond-order actuator dynamics. This shall be accomplished by including the following second-order


10−1 100 101 102


0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

RM

Sco

ntro

leffo

rt,T

rms

[N·m

]H∞ (1st-order)H∞ (2nd-order)H∞ (3rd-order)H∞ (4th-order)OptimalPD (linear)PD (nonlinear)SOS

Figure 5.4: RMS control effort with respect to 1st-order actuator bandwidth

dynamics model in each component (i ∈ 1, 2, 3) of the controller output:[x1

x2

]=

[0 1−ω2

b −2ζωb

][x1

x2

]+

[0ω2b

]u2,i

yi = x1

(5.34)

where ζ is the actuator damping ratio and ωb is the actuator bandwidth. All simulations were performedwith a damping ratio ζ = 0.5. Figures 5.5 and 5.6 show Erms and Trms, respectively, as a function ofthe actuator bandwidth. It is seen from these figures that the various control methods follow the sametrends as in the case of the first-order actuator dynamics. In particular, the four H∞ controllers aremore robust than the other control methods. However, once again, it is not possible to discern anydifference in robustness between the different order H∞ controllers from these figures.

Robustness in the Gap Metric (revisited)

We now make use of the gap metric to characterize the difference in the input-output (IO) map ofthe system induced by the unmodeled actuator dynamics. However, since we cannot calculate the gapbetween two nonlinear systems, we calculate the gap metric for the linearized systems only. As theactuator bandwidth and damping ratio change, the value of the gap metric will also vary. Figure 5.7shows the value of the gap metric with respect to the first-order actuator bandwidth. Figure 5.8 showsthe value of the gap metric with respect to the second-order actuator bandwidth for the damping ratioζ = 0.5 used in the above numerical simulations. As the damping ratio varies, the curve of this figuremoves to the left or right slightly, although there does not appear to be any discernible trend. Moreover,it should be noted that as the actuator bandwidth approaches infinity, the effects of the actuator in thecontroller input-output map becomes negligible. This is to be expected and is seen in both Figures 5.7


10−1 100 101 102


0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

RM

Stra

ckin

ger

ror,Erm

sH∞ (1st-order)H∞ (2nd-order)H∞ (3rd-order)H∞ (4th-order)OptimalPD (linear)PD (nonlinear)SOS

Figure 5.5: RMS tracking error with respect to 2nd-order actuator bandwidth

10−1 100 101 102


0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

RM

Sco

ntro

leffo

rt,T

rms

[N·m

]

H∞ (1st-order)H∞ (2nd-order)H∞ (3rd-order)H∞ (4th-order)OptimalPD (linear)PD (nonlinear)SOS

Figure 5.6: RMS control effort with respect to 2nd-order actuator bandwidth

and 5.8, where the gap metric approaches zero with increasing actuator bandwidth.We now return for a moment to the question of controller robustness in the gap metric. From Eq.


10−4 10−3 10−2 10−1 100 101 102 103


0.0

0.2

0.4

0.6

0.8

1.0

Gap

met

ric,ρ

g

Figure 5.7: Gap metric with respect to 1st-order actuator bandwidth

10−4 10−3 10−2 10−1 100 101 102 103


0.0

0.2

0.4

0.6

0.8

1.0

Gap

met

ric,ρ

g

Figure 5.8: Gap metric with respect to 2nd-order actuator bandwidth (with damping ratio ζ = 0.5)

(2.49) we have the following small-gain-type criterion:

ρg(P ,P ′) ‖ΠN‖M‖∞︸︷︷︸≤ γ

< 1. (5.35)


In the numerical simulations performed with γ = 4, it was determined that the closed-loop system isstable in the presence of the unmodeled second-order actuator with bandwidth as low as ωb = 0.3 rad/s.With this actuator bandwidth, the calculation of the gap between the linear attitude dynamics withand without the actuator results in a value of ρg = 0.522, which does not satisfy the condition of Eq.(5.35). This could be explained by the conservativeness of the small-gain criterion. However, it can alsobe attributed to the nonlinear effects of the dynamics not taken into account in the gap calculation here.This emphasizes the need for a method to calculate the gap metric for nonlinear systems.

Robustness to Actuation Time-Delay

The robustness properties of the different controllers are now examined with regards to a time-delay inthe actuation. Such a delay could represent the finite time required by a satellite on-board computer totake the sensor measurements and calculate the required control signal. The time-delay is made equal toan integer multiple of the numerical integration step-size ∆t. Tables 5.3 and 5.4 indicate the maximumallowable time-delay, hmax, for each controller such that the desired attitude maneuver is achieved withinone orbit.

Table 5.3: Robustness to actuation time-delay for existing controllersPD PD Optimal Optimal SOS

(linear) (nonlinear) (OL) (CL)hmax [s] 1.69 1.09 0.00 1.41 0.81

Table 5.4: Robustness to actuation time-delay for H∞ controllersH∞ H∞ H∞ H∞

(1st-order) (2nd-order) (3rd-order) (4th-order)hmax [s] 3.93 3.94 3.97 3.98

As can be seen from these results, the four H∞ controllers are more robust with regards to thiseffect than the other control methods. In particular, the present fourth-order H∞ controller is the mostrobust. We also note that the open-loop method of Schaub, Junkins and Robinett [58] has no robustnessto this effect.

Summary of Results

The above numerical simulations highlight several important results. In particular, we saw that thenonlinear control systems developed using the methods in this thesis are much more robust to uncertaintythan the prevalent control methods from the literature for this type of problem. We also emphasizedthe fact that robustness in the gap metric is conservative and that there is a need to develop methodsfor calculating this property for nonlinear systems.

5.2 Formation Flying Problem

In recent years, the idea of replacing single-purpose, monolithic satellites with a large number of smaller,cheaper satellites has grown in popularity. Consequently, improved methods are needed to control theseformations of satellites with a certain degree of accuracy and robustness. Linear control systems are


usually developed based on the relative equations of motion obtained through linearization about thechief satellite’s orbit [12, 29]. Instead, we shall apply our robust nonlinear control methods to thisproblem using higher-order approximations of the true formation flying dynamics.

First, we present the nonlinear dynamics for the relative motion of two spacecraft. Then, we comparethe performance and robustness of two new control methods through numerical simulations. This sectionconcludes with a summary of the results.

5.2.1 Formation Flying Dynamics

Consider two satellites in close proximity orbiting the Earth at low altitudes. The reference satellitewill be designated the chief, while the other is the deputy. It is assumed that the chief satellite’s orbitis circular. Whereas the nonlinear relative equations of motion have typically been linearized aboutthe chief satellite’s orbit [12, 29], we shall approximate the relative motion equations up to fourth-order. Additionally, because we are interested in examining the robustness of our nonlinear control laws,the relative motion equations will be developed under the assumption that there are no perturbationaccelerations acting on the satellites. However, our exact model for simulation purposes will include theJ2 perturbation acceleration on both the chief and deputy, as well as continuous thrusting on the deputysatellite for control purposes.

The equation of motion for each satellite is given by

R−→ = − µ

R3R−→+ u−→, (5.36)

where R−→ is the geocentric inertial position vector of a generic satellite, R = ‖R−→‖ is the magnitude ofthis vector, and u−→ includes all exogenous inputs (e.g., orbital perturbations and control thrusts). Theinertial frame shall be denoted F−→i (the vectrix notation is explained in Hughes [31]). Thus, we canexpress the inertial position vector of the chief satellite by R−→c = F−→T

i Rc, that of the deputy satellite byR−→d = F−→T

i Rd, and the position vector of the deputy relative to the chief by δr−→ = R−→d − R−→c = F−→Ti δri.

These position vectors are all shown in Fig. 5.9.We now define a rotating local-vertical-local-horizontal (LVLH) frame, denoted by F−→c, which moves

with the chief satellite. This frame has its x-axis in the radial direction, its z-axis normal to the orbit, andits y-axis in the along-track direction, thus completing a right-handed coordinate system. The inertialposition vectors can be expressed in this rotating frame as R−→c = F−→T

c rc and R−→d = F−→Tc rd. Similarly, the

relative position vector can be written as δr−→ = F−→Tc δrc with δrc = CT

icδri, where the rotation matrixfrom F−→i to F−→c is defined by

Cic =[

Rc

‖Rc‖H×c Rc

‖H×c Rc‖Hc

‖Hc‖]

(5.37)

and Hc is the chief’s angular momentum (per unit mass). Expressing all vectors in the rotating chiefframe, we have

rc =

Ro00

, δrc =

xyz

, rd = rc + δrc =

Ro + x

y

z

, ω =

00ωo

, (5.38)

where rc = Ro is the chief satellite’s radial distance and ωo =√µ/R3

o is its mean rotation rate, whichare both constant due to the assumption of a circular orbit. The deputy satellite’s radial distance is


i−→2

i−→3

i−→1

R−→d

R−→c

δr−→

Deputysatellite

Chiefsatellite

c−→2

c−→1

c−→3Inertialframe

Chiefframe

Chieforbit

Figure 5.9: Formation flying dynamics

given by rd = [(Ro + x)2 + y2 + z2]1/2.Taking derivatives in the rotating frame F−→c, the two-body equation of motion (5.36) for the deputy

yieldsrd + 2ω×rd + ω×rd + ω×ω×rd = − µ

r3drd + u (5.39)

or, expanding terms,x− 2ωoy − ω2o(Ro + x)

y + 2ωox− ω2oy

z

=−µ

[(Ro + x)2 + y2 + z2]3/2

Ro + x

y

z

+

uxuyuz

. (5.40)

The nonlinear term r−3d = [(Ro + x)2 + y2 + z2]−3/2 shall be approximated by a fourth-order Taylor

series about the chief satellite’s nominal orbital radius Ro.Defining the state vector x = [x y z x y z]T and rearranging terms, the fourth-order approximation

of the deputy satellite’s motion relative to the chief, expressed in the chief’s LVLH frame, is given byx(t) = a1(x) + a2(x) + a3(x) + a4(x) + Bu1

y(t) = Cx(5.41)

where

a1(x) = A1x =

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

3ω2o 0 0 0 2ωo 0

0 0 0 −2ωo 0 00 0 −ω2

o 0 0 0

x

y

z

x

y

z

, B =

[0

1

], C =

[1], (5.42)


the second- and third-order terms are given by

a2(x) = − 3µR4o

000

x2 − 12y

2 − 12z

2

−xy−xz

, a3(x) = − 3µ

R5o

000

− 43x

3 + 2xy2 + 2xz2

2x2y − 12y

3 − 12yz

2

2x2z − 12y

2z − 12z

3

, (5.43)

respectively, and the fourth-order term is given by

a4(x) = − 3µR6o

000

53x

4 + 58y

4 + 58z

4 − 5x2y2 − 5x2z2 + 54y

2z2

− 103 x

3y + 52xy

3 + 52xyz

2

− 103 x

3z + 52xy

2z + 52xz

3

. (5.44)

Next, we present simulation results for this problem that emphasize different aspects of the nonlinearcontrol methods developed in previous chapters of this thesis.


We now present simulation results for the formation flying problem using the nonlinear H∞ and loopshaping controllers developed in this thesis. First, we examine the effects of a simple loop shaping filteron performance. And second, we study the variations in performance and robustness that result fromapplying state-order-reduction to the H∞ control system.


We now define the various parameters pertinent to the numerical simulations and then proceed with thedifferent comparisons. The chief satellite is in a circular orbit with an inclination of i = π/3 rad and alongitude of the ascending node of Ω = π/3 rad. The initial value of the argument of latitude is zero. Thealtitude is 700 km and we take Re = 6378.14 km for the Earth’s radius. These orbital parameters will beused to determine the gravitational perturbations acting on the two spacecraft. In particular, we includethe gravitational perturbation term J2 resulting from the Earth’s asphericity. The simulation model forthe motion of each satellite is the two-body problem under the action of the asphericity perturbations.Additionally, the deputy satellite will be acted upon by continuous thrusting in accordance with theoutput from the control system. Note that, although the present synthesis methods focus on continuous-time control signals, the thrusting could be implemented using Pulse-Width Modulation (PWM). Allsimulations will be performed for ten orbits using a 4th-order Runge-Kutta numerical integration methodwith a step-size ∆t = 0.1 s.

The desired trajectory for the relative motion corresponds to a projected circular orbit (PCO):


y0 = [xd yd zd]T withxd(t) = 1

2A sin(ωot)yd(t) = A cos(ωot)zd(t) = A sin(ωot)

(5.45)

It is well known that this is an exact solution of the linear relative motion model x = A1x, which isknown as the Hill-Clohessy-Wiltshire (HCW) equations [12, 29]. In addition to forming the referencesignal, the PCO solution will be used to obtain initial conditions Rd(0) and Rd(0) for the deputy satellite.Simulations are performed with a PCO reference radius of A = 1000 m.

The H∞ controllers are designed with γ = 4.0, which was chosen to satisfy the linear version of thecondition in Eq. (3.65). The loop shaping controllers were designed with γ = 15.0.

In order to gauge the performance and properly compare the different controllers, we make use oftwo metrics. The RMS tracking error is given by

Erms =

[1

10To

∫ 10To

0

yT2 (t)y2(t) dt

](1/2)

(5.46)


∆vrms =

[1

10To

∫ 10To

0

uT2 (t)u2(t) dt

](1/2)

(5.47)

where y2 = y0 − y1 and To is the orbital period.

Disturbance Rejection using Loop Shaping Techniques

Here we examine disturbance rejection using a simple H∞ loop shaping control system. The simplestpossible shaping function is a constant direct feedthrough: aF (xF ) = 0, bF (xF ) = 0, cF (xF ) = 0, anddF (xF ) = D1, where 1 is the identity matrix. It should be obvious that, when the scalar D is unity,the results of the loop shaping controller are identical to those of an H∞ controller with the same valueof γ.

The behaviour of the controller performance metrics as a function of the scalar D is shown in Figures5.10 and 5.11. It is interesting to note that the tracking error varies inversely with D whereas the controleffort exhibits relatively little variation with D. This can be attributed to the orbital perturbations,which are roughly balanced by the control acceleration independently of D. The corresponding graphswere generated for the second-, third- and fourth-order controllers and found to be identical to the first-order case. The controller metrics in these cases agreed with the first-order controller to all significantfigures. Although the higher-order terms significantly improve the accuracy of the (open-loop) relativemotion approximation, the present results indicate that the first-order control system is more thanadequate at compensating for the nonlinear motion effects and perturbations.

The tracking errors −y2 = y1−y0 = [∆x ∆y ∆z]T = [x− xd y− yd z− zd]T are shown in Figure5.12 for D = 10−4. The tracking errors exhibit the same shape when using different values of D, buthave different magnitudes.


10−4 10−3 10−2 10−1 100 101

D

10−6

10−5

10−4

10−3

10−2

10−1

100

Erm

s

Figure 5.10: RMS tracking error vs. loop shaping parameter

10−4 10−3 10−2 10−1 100 101

D

0.5

0.6

0.7

0.8

0.9

1.0

1.1

∆v rms

[m/s

]

×10−5

Figure 5.11: RMS control effort vs. loop shaping parameter

State-Order-Reduction Comparisons

Here we examine the effects of state-order-reduction on the performance and robustness of the H∞control method. Table 5.5 presents values of the tracking error and control effort for different numbers


0 1 2 3 4 5 6 7 8 9 10−0.10−0.05

0.000.050.100.150.200.25

∆x

[m]

0 1 2 3 4 5 6 7 8 9 10−0.15−0.10−0.05

0.000.050.100.15

∆y

[m]

0 1 2 3 4 5 6 7 8 9 10Number of orbits

−0.08−0.06−0.04−0.02

0.000.020.040.060.08

∆z

[m]

Figure 5.12: Tracking errors for D = 10−4

of controller states nc with the first-order H∞ controller. The performance metrics for the second-, third-and fourth-order nonlinear controllers agreed with the first-order case to all significant figures. Again,the present results indicate that the first-order control system is more than adequate at compensatingfor the nonlinear motion effects.

Table 5.5: State-order-reduction comparisonsnc Erms [×10−6] ∆vrms [×10−6] ρg(K,Kr)6 14.546911 6.087 0.05 9.587493 6.090 0.31024 7.044141 6.092 0.31023 6.259388 6.093 0.31022 N/A N/A 0.7112

It is interesting to note from these results that, while the control effort increases as fewer controllerstates are used, the tracking error decreases. This phenomenon has been observed in other work [5, 11,36, 37, 62, 63]. The controller is entirely unable to stabilize the system when nc = 2. The positiontracking errors −y2 = y1 − y0 = [∆x ∆y ∆z]T = [x − xd y − yd z − zd]T are shown in Figures5.13 and 5.14 for the first-order H∞ controller with nc = 6 and nc = 3, respectively. The correspondinggraphs were generated for the second-, third- and fourth-order controllers and found to be identical tothe first-order case, which agrees with above observations on the effectiveness of the first-order controlsystem for this problem.

The gap metric between the full- and reduced-state-order controllers is also shown in Table 5.5 forthe linear H∞ controller. It can be seen from these results that instability in the closed-loop system iscorrelated to a significant increase in the gap metric between the full- and reduced-state-order controllers


0 2 4 6 8 10−1.0−0.5

0.00.51.01.52.02.5

∆x

[m]

×10−5

0 2 4 6 8 10−1.5−1.0−0.5

0.00.51.01.5

∆y

[m]

×10−5

0 2 4 6 8 10Number of orbits

−1.0

−0.5

0.0

0.5

1.0

∆z

[m]

×10−5

Figure 5.13: Tracking errors for 1st-order H∞ controller with nc = 6

0 1 2 3 4 5 6 7 8 9 10−0.4−0.2

0.00.20.40.60.81.0

∆x

[m]

×10−5

0 1 2 3 4 5 6 7 8 9 10

−4

−20

2

4

6

∆y

[m]

×10−6

0 1 2 3 4 5 6 7 8 9 10Number of orbits

−3−2−1

01234

∆z

[m]

×10−6

Figure 5.14: Tracking errors for 1st-order H∞ controller with nc = 3

when nc = 2. Recall that simultaneous perturbations in the plant and controller were examined in Chap-ter 2. More specifically, it was shown that there is a trade-off between plant and controller uncertainty.


It was even possible to define an upper bound on the tolerable plant uncertainty for a given controlleruncertainty (refer to Eq. (2.48)). Thus, the loss in stability observed here when the number of controllerstates is reduced to nc = 2 can be explained by a significant reduction in the allowable plant uncertaintycaused by an increase in the controller uncertainty. In general, a reduction in the number of controllerstates will result in a decrease in the tolerable plant uncertainty, eventually leading to instability of theclosed-loop system.

Summary of Results

The above numerical simulations highlight several important results. We saw that even a simple loopshaping function can dramatically affect the performance metrics. Moreover, it appears that the linearcontrol system is more than adequate at compensating for the nonlinear motion effects for this specificproblem. Additionally, we observed that state-order-reduction also has an effect on the performancemetrics and robustness characteristics of the closed-loop systems.

5.3 Nonlinear Mass-Spring Chain Problem

The third and final problem we consider in this chapter is the control of a nonlinear mass-spring chain.Although this is not directly a space systems application, such a model could be considered an ap-proximation of a large flexible space structure. The objective for the mass-spring chain problem is tocontrol the position and velocity of the tip mass while simultaneously suppressing vibrations along theentire chain. The dynamics for this problem can be derived with an arbitrary number of masses andsprings. This can result in a very large number of plant and controller states. Hence, we will primarilybe interested in comparing results using the controller state-order-reduction techniques developed in theprevious chapter.

First, we present the equations of motion for a nonlinear mass-spring chain. Then, we compareH∞ control systems using the nonlinear state-order-reduction method developed in the previous chapterthrough numerical simulations. This section concludes with a summary of the results.

5.3.1 Nonlinear Mass-Spring Chain Dynamics

Consider the mass-spring chain shown in Figure 5.15. We denote the position of the ith mass with thegeneralized coordinate qi(t), where i ∈ 1, . . . , N and N denotes the total number of masses. Thecontrol input u(t) and the measured output y(t) shall be collocated at the tip mass mN . The springsin the chain will include two stiffness components. The first is a linear term of the form k`,i∆x, wherek`,i is the linear stiffness of the ith spring and ∆x is the change in elongation of the spring as measuredfrom its equilibrium position. The second component is a cubic term of the form kn`,i(∆x)3, where kn`,iis the nonlinear stiffness of the ith spring. We define q0 = 0 as the fixed reference point at the base ofthe chain.

We shall derive the equations of motion using a Lagrangian formulation [31]. Defining the Lagrangianas L = T − V , where T is the total kinetic energy of the system and V is the total potential energy,Lagrange’s equation is given by:

d

dt

(∂L

∂q

)− ∂L

∂q= Q (5.48)


m1

q1

m2

q2q0

mN

qN

k1 k2 k3 kN

u(t)

y(t)

Figure 5.15: Mass-spring chain

where Q is the vector of input forces. For the system shown in Figure 5.15, the total kinetic energy isgiven by

T =N∑i=1

12miq

2i (5.49)

and the total potential energy of the system is given by

V =N∑i=1

12k`,i (qi − qi−1)2 + 1

4kn`,i (qi − qi−1)4 . (5.50)

Note that the potential energy contains components resulting from each of the linear and cubic springstiffness terms.

The resulting general equation of motion for such a system is given by

Mq + K`q + Kn`(q) = eNu(t)y(t) = eTNq

(5.51)

where M = diagimi is the mass matrix, K` is a tridiagonal matrix containing only linear stiffnessterms, and Kn` is a column matrix containing all the nonlinear stiffness terms. The elementary vectorei ∈ RN×1 consists of all zero elements except for a single 1 in the ith entry. Note that the mass matrixsatisfies M = MT > 0 and the linear stiffness matrix satisfies K` = KT

` > 0.Rewriting this second-order differential equation in state space forms yields[

q

q

]︸︷︷︸

x

=

[0 1

−M−1K` 0

]︸︷︷︸

A1

[q

q

]︸︷︷︸

x

+

[0

−M−1Kn`(q)

]︸︷︷︸

a3(x)

+

[0

M−1eN

]︸︷︷︸

B

u(t)

y(t) =[eTN 0

]︸︷︷︸

C

[q

q

] (5.52)

Next, we will demonstrate the effectiveness of our nonlinear state-order-reduction technique throughnumerical simulations.


We now present simulation results using the new nonlinear H∞ control method described in this thesis.The simulations shall compare different polynomial-order H∞ controllers, as well as study the effects ofnonlinear controller state-order-reduction on two opposing performance metrics.



We now define the various parameters pertinent to the numerical simulations and then proceed withthe different comparisons. For the purposes of our simulations, we shall define the chain with N = 15masses and springs. The plant model will therefore have np = 30 states. In order to simplify matters,we shall assume that all the masses and springs are identical, hence mi = m, k`,i = k` and kn`,i = kn`

for all i ∈ 1, . . . , N. We use the values m = 2 kg, k` = 4 N/m and kn` = 1 N/m3. All simulations willbe performed for a duration of 300 s using a 4th-order Runge-Kutta numerical integration method witha step-size ∆t = 0.001 s.

The control objective for this problem is to regulate the system output (i.e., y0 = 0). The H∞controllers are designed with γ = 1.8, which was chosen to satisfy the linear version of the condition inEq. (3.65).

In order to gauge the performance and properly compare the different controllers, we make use oftwo metrics. The RMS tracking error is given by

Erms =

[1To

∫ To

0

yT2 (t)y2(t) dt

](1/2)

(5.53)


Frms =

[1To

∫ To

0

uT2 (t)u2(t) dt

](1/2)

(5.54)

where y2 = y0 − y1 = −y1 and To is the total simulation period.

State-Order-Reduction Comparisons

Here we examine the effects of state-order-reduction on the performance of the H∞ control method.Figures 5.16 and 5.17 show the variation of the RMS tracking error and control effort for different numbersof controller states nc. The performance metrics for the first- and second-polynomial-order controllersbehave identically with regards to state-order-reduction. The third-polynomial-order controller, on theother hand, behaves much differently.

The first main point to note from these figures is the trade-off between the two performance metricswith controller state-order-reduction. In particular, the controllers exhibit a significant increase in RMStracking error and a small decrease in RMS control effort as fewer controller states are used. Althoughthis is opposite to the trend observed in the formation flying results, it does highlight the trade-offbetween the two opposing performance metrics with changing controller state-order. These two sets ofresults also indicate that the trend in performance trade-off is problem-dependent.

The second main point to note from Figures 5.16 and 5.17 is that the performance metrics for thereduced-state-order controllers are lower for the third-polynomial-order controller than for the first- andsecond-polynomial-order controllers. In fact, there is a significant jump in the performance metricsonce the controller state order changes from the full-state-order case. This is likely because the third-polynomial-order controller captures the complete nonlinear dynamics of the system. It is interesting tonote, however, that the full-state-order controllers all perform identically. This last point was observedin the other problems examined in this chapter.


10 15 20 25 30Number of Controller States,nc

10−2

10−1

100

RM

STr

acki

ngE

rror

,Erm

sH∞ (1st-order)H∞ (2nd-order)H∞ (3rd-order)

Figure 5.16: RMS tracking error vs. number of controller states

10 15 20 25 30Number of Controller States,nc

10−2

10−1

RM

SC

ontro

lEffo

rt,F

rms

H∞ (1st-order)H∞ (2nd-order)H∞ (3rd-order)

Figure 5.17: RMS control effort vs. number of controller states

One characteristic of the reduced-state-order controllers that is not observed here is the trade-offbetween performance and robustness. As just discussed, the third-polynomial-order controllers perform


better than their first- and second-polynomial-order counterparts for all reduced-state-order cases. It isimpossible, however, to quantify the change in robustness in the gap metric for such a nonlinear systemat this point.

Summary of Results

The above numerical simulations highlight several important results. First, we saw that the performancemetrics vary with the number of controller states. In particular, there is a trade-off between the twoopposing performance metrics. And second, it was seen that the third-polynomial-order state-order-reduction technique effectively accounted for the nonlinear plant dynamics. This led to a significantimprovement in performance metrics when compared to the lower-polynomial-order reduced-state-ordercontrol systems.

5.4 Computational Costs

We now discuss briefly an estimate of the computational costs associated with the algorithms developedin this thesis. The estimates provided here are based on a naive computational approach (i.e., nooptimization at all) and are meant to represent a worst-case cost only. Note that the computation costassociated with evaluating a product is greater than that of an addition. Hence, we make use of thesymbols p and a to represent the costs of a product and an addition, respectively.

Table 5.6 shows the number of products and additions required to evaluate each term in the Taylorexpansion of the system dynamics a(x) as given by Eq. (3.11), where n represents the dimension ofthe (square) matrices and vectors. Note that each entry in the table is the cost of associated with thatindividual term, not the cumulative cost.

Table 5.6: Computational CostsTerm p aa1(x) n2 n2 − na2(x) n3 + n2 n3 − na3(x) n4 + n3 + n2 n4 − na4(x) n5 + n4 + n3 + n2 n5 − n

Thus, as the order of approximation increases by one order, so too does the computational costassociated with evaluating that additional term. Moreover, the computational cost is always one orderhigher than the order of the additional approximation term.

5.5 Chapter Summary

This chapter made use of numerical simulations to demonstrate the effectiveness of the controller synthe-sis methods developed in this thesis. The effectiveness of the state-order-reduction technique developedin this thesis was also demonstrated. We began with the spacecraft attitude control problem, where wecompared the performance and robustness of our methods with existing techniques from the literature.Whereas the existing methods focus on performance, it is shown that the present approach results inbetter robustness properties. Then, the satellite formation flying problem demonstrated how even asimple loop shaping function can affect performance and robustness. This problem was also used to


show that the inherent trade-off between performance and robustness changes when applying state-order-reduction. A nonlinear mass-spring chain was used to further demonstrate the effects of nonlinearstate-order-reduction. Finally, a brief discussion on the computational costs of the algorithms developedin this thesis was provided.

Chapter 6

Conclusions and Future Work

The primary focus of this thesis was the development of robust nonlinear feedback control methods.In particular, we developed explicit solutions to the nonlinear H∞, loop shaping, and weighted mixedsensitivity control problems. It was shown how each of these controllers can be constructed from thesolutions to Hamilton-Jacobi equations with a particular structure. However, one of the primary issuesoutstanding in the literature is an analytical method for obtaining exact nonlinear solutions to the HJE. Amethodology was therefore developed in this thesis for constructing analytical expressions to approximatesolutions of the HJE. The methods developed result in (sub-)optimal robust nonlinear controllers. Indeveloping these approximate HJE solutions, certain assumptions were be made regarding the class ofnonlinear systems used so as to simplify the ensuing derivations.

Additionally, it is often desirable or necessary to develop control systems with fewer state variablesthan the plant model. Thus, this thesis also considered the development of nonlinear balancing andstate-order-reduction techniques. In particular, we developed a methodology for constructing balancingstate transformations using the same new Taylor series approach. In fact, it was shown that this statetransformation can be developed using the nonlinear solution to the same HJE used for constructing thethree nonlinear control systems.

Due to the difficulties associated with proving the robustness and performance characteristics ofnonlinear control systems, the effectiveness of the methods developed in this thesis was demonstratedvia numerical simulations. In contrast with most of the methods used for comparisons, which weredeveloped to address specific problems, the methods developed in this thesis are more general and thuscan be applied to a broad class of systems.

6.1 Summary of Contributions

The following is a summary of the major contributions of this thesis:

1. Developed a framework for obtaining approximate solutions to a Hamilton-Jacobi equation (HJE):

(a) Approximated nonlinear system dynamics and HJE solution gradient via Taylor series expan-sions by using a new structure;

(b) Constructed a sequential set of equations for successively higher orders of approximation tothe HJE solution gradient;

116

Chapter 6. Conclusions and Future Work 117

(c) Developed analytical expressions for each order of approximation of the HJE solution gradient;

(d) Demonstrated this methodology for obtaining both the stabilizing and antistabilizing HJEsolution gradients.

2. Developed a synthesis method for nonlinear feedback control systems using analytical expressionsfor the HJE solution gradient:

(a) Designed explicit solutions to three different robust nonlinear control methods:

i. H∞ control problem

ii. H∞ loop shaping control problem

iii. weighted mixed sensitivity control problem

(b) All three control methods used a HJE with a particular structure;

(c) All three control methods yielded robust nonlinear dynamic output feedback control systems;

(d) All three control systems provided stability and disturbance attenuation over a neighbourhoodof the equilibrium that is larger than existing linear methods.

3. Developed a method for obtaining and applying a nonlinear controller state balancing transforma-tion:

(a) Developed an explicit solution technique for obtaining a nonlinear state balancing transfor-mation using a Taylor series expansion of the HJE solution;

(b) Devised the first technique in the literature for completely balancing a nonlinear controlsystem;

(c) Developed a single systematic methodology for balancing all matrices used to construct thethree nonlinear control systems;

(d) Developed a methodology for constructing reduced-state-order nonlinear control systems thatapplies to all three control methods;

(e) Demonstrated via numerical simulations that state-order-reduction induces a trade-off be-tween performance and robustness.

4. Demonstrated the novel methods developed in this thesis using space applications:

(a) Spacecraft attitude control;

(b) Spacecraft formation flying control;

(c) Nonlinear mass-spring chain.

6.2 Recommendations for Future Work

Despite the many contributions just mentioned, there is still much room for improvement. The followingtopics are proposed for future work:

1. Sign-definiteness of multivariate polynomials: In the present work, it was simply assumedthat the polynomial solutions obtained were sign-definite. This argument could perhaps be sup-ported by the fact that the higher-order solutions are built upon the sign-definite solutions to

Chapter 6. Conclusions and Future Work 118

the algebraic Riccati equation. This is not strictly true, however, and conditions for which sign-definiteness is guaranteed are still required.

2. Approximation effects on performance and robustness: The use of approximate HJE solu-tions in the synthesis of nonlinear controllers and state balancing has been shown here to have aneffect on the performance and robustness of a control system. Once methods for quantifying theseproperties exist for nonlinear systems, this problem should be analyzed further.

3. Minimum approximation order: The present work developed Taylor series approximationsup to fourth-order. However, for certain types of problems, a lower (or higher) polynomial ordermay be necessary to construct control systems that effectively capture the system nonlinearities.Again, addressing this problem requires methods for quantifying certain characteristics of nonlinearsystems.

4. Quantification of performance-robustness trade-off: The trade-off between performanceand robustness for both linear and nonlinear systems has long been an open problem in controltheory. With an accumulation of evidence suggesting that the balance of this trade-off shifts whenusing state balancing and reduction techniques, this problem is even more important.

5. Effects of nonlinearities on state-order-reduction: It is not clear from existing literatureor the present work how the trade-off between performance and robustness of reduced-state-ordercontrollers is affected by different types of nonlinearities in the system model. Hence, the interactionof different system nonlinearities on the state-order-reduction problem should be explored further.

6. Improve computational efficiency: The structures developed in the present work are effectivefor constructing nonlinear HJE solutions, but are not computationally efficient. The computationalcosts associated with the structures and algorithms developed was only discussed very briefly inthis thesis and warrants further examining. More efficient structures or algorithms are necessaryto make these approximation methods more practical and this should be explored in the future.

Although the present manuscript contributes much to our overall understanding of nonlinear controlsystems, the author fully acknowledges that there exist certain limitations with his methods and thatthere is much room for improvement. It is hoped that the present work may be used as a stepping stonefor future developments.

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