CLARKSON UNIVERSITY
IDENTIFICATION AND CONTROL FOR VIBRATION
SUPPRESSION OF A NONLINEAR AND TIME
VARYING SMART STRUCTURE
A DISSERTATION
BY
CHENGLI HE
DEPARTMENT OF MECHANICAL AND AERONAUTICAL ENGINEERING
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
(MECHANICAL ENGINEERING)
January 2004
Accepted by the Graduate School
__________________ __________________ Date Dean
2
The undersigned have examined the dissertation entitled
Identification and Control for Vibration Suppression of a Nonlinear and Time-
Varying Smart Structure
presented by Chengli He , a candidate for the degree of Doctor of Philosophy,
and hereby certify that it is worthy of acceptance.
Date ADVISOR Dr. Ratneshwar Jha EXAMINING COMMITTEE Dr. Goodarz Ahmadi Dr. James Carroll Dr. Sung P. Lin Dr. Alireza K. Ziarani
3
Identification and Control for Vibration Suppression of a
Nonlinear and Time Varying Smart Structure
(ABSTRACT)
Smart structure technology has found more and more applications in vibration control,
noise reduction, health monitoring, aerodynamic flow control, etc. Most smart structures,
due to their considerable flexibility, distributed sensors and actuators, require a relatively
high order model. The control system must also be capable of handling complexity,
uncertainty, nonlinearity, and variations with time. These demand the development of
suitable identification and control techniques for the application of smart structure.
Several identification and control techniques for active vibration control of nonlinear
and time-varying smart structures are developed and validated experimentally for active
vibration control of nonlinear and time-varying smart structures.
Three identification and modeling techniques, finite element/state space model,
controlled autoregressive integrated moving average model with augmented upper diagonal
identification for the adaptive parameter identification and neural network autoregressive
external input model with recursive Levenberg-Marquardt optimization method for neural
network online learning, are investigated. A simple effective controller, direct adaptive
neural network controller is developed and implemented experimentally for the active
vibration suppression. Two model based control systems, adaptive generalized predictive
control system based on controlled autoregressive integrated moving average model and
neural adaptive predictive control system based on neural network autoregressive external
input model, are studied. Experimental performances of each model-based controller are also
4
investigated and the comparison is made between the two adaptive generalized predictive
control systems. Linear quadratic regulator based on finite element/state space model is also
included to have a baseline for comparison.
Finite element/state space modeling approach is a cost-effective method for the
application of smart structures. There is no need to construct expensive experimental setup
before the finalization of the product. Direct adaptive neural network control is simple in
concept and implementation. With online adaptation, it can deal with the uncertainty and
time variation of smart structure. Without considering the control effort, the direct adaptive
neural network control is not an optimal controller. Adaptive generalized predictive control
and neural adaptive predictive control are optimal controllers, which take both the control
result and control effort into consideration. Experimental results show that, with a nonlinear
model representation of the smart structure, neural adaptive predictive control is more
effective than adaptive generalized predictive control, which is based on a linear model
(controlled autoregressive integrated moving average). However, with nonlinear
optimization involved, neural adaptive predictive control is much more computationally
expensive than adaptive generalized predictive control.
5
Contributions
The major contributions of this dissertation are the development and experimental
validation of several identification and control techniques for vibration suppression of
nonlinear and time-varying smart structures. A summary of the contributions is listed below:
1. Development of a direct adaptive neural network controller and experimental
implementation for application to nonlinear and time varying smart structures (New).
2. Modeling of smart structures based on finite element/state space technique
(Application).
3. Experimental implementation of adaptive generalized predictive control based on
controlled autoregressive integrated moving average model with augmented UD
identification for the vibration suppression of nonlinear and time-varying smart
structures (Application).
4. Development of neural adaptive predictive control system with recursive Levenberg-
Marquardt optimization for neural network online learning and experimental
implementation for the active vibration control of nonlinear time-varying smart
structures (New).
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Acknowledgements
Special thanks are due to my advisor, Dr. Ratneshwar Jha, for his guidance
throughout my work. I have sincerely appreciated his suggestions and knowledge, and his
support and understanding of my goals has been essential to this work.
I would like to thank my committee members, Dr. Goodarz Ahmadi, Dr. James
Carroll, Dr. Sung P. Lin, and Dr. Alireza K. Ziarani, for their invaluable suggestions and
knowledgeable insights. Thank you for all of your support, patience and encouragement.
I would also like to extend my thanks to my friends at Clarkson and my colleagues at
New York Power Authority, for their support.
Thanks to the Department of Mechanical and Aeronautical Engineering of Clarkson
University, the work contained herein would be impossible to accomplish without its
financial support.
Most of all, I am indebted to my parents and loving wife, Ying Liu. Their
understanding and compassion have truly enabled me to complete this dissertation.
Chengli He
New Rochelle, NY
Dec. 25, 2003
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Table of Contents
Abstract iii
Contributions v
Acknowledgments vi
Table of Contents vii
List of Figures xi
Nomenclature xvii
Abbreviations xxv
Chapter 1 Introduction 1
1.1 Background ………………….……………………………..……………………….1
1.2 Objective ………………………………………………..…………..........................3
1.3 Literature Review ………………………………………..………………………….4
1.4 Thesis Organization ……………………………………………….…….…………9
8
Chapter 2 Experimental Setup 11
2.1 Schematic Diagram of the Experimental Setup …………..………………………11
2.2 Experimental Hardware ……………………………………..…………………….12
2.3 Experimental Software …………………………………..…….…………………14
2.4 Piezoelectric Effect ………………………………………..………………………15
2.5 Structure Nonlinearity Test ………………………………..………………………16
2.6 Fourier amplitude of the Structure ……………...............………………………17
2.7 Test Signal Properties ………………………..…………….…………………….18
Chapter 3 Identification and Modeling of a Smart Structure 21
3.1 Finite Element/State Space Model …………………..……..……………………21
3.1.1 Modeling of Piezoelectric Actuator ……………..………..………………22
3.1.2 Structural Modal Analysis ………………………………..……………….25
3.1.3 Ranking of Vibration Modes ……………………………..………………28
3.1.4 State Space Model Formulation …………………………..……………….36
3.1.5 Discrete Time State Space Model …………….…………..………………39
3.2 Controlled AutoRegressive Integrated Moving Average Model ...……..…………40
3.2.1 CARIMA model ………………………………………..…………………40
3.2.2 Conventional Recursive Least Squares ……………..……………………..45
3.2.3 Augmented UD Identification …………..………..………………………46
3.3 Neural Network Based Model …………..………………….….………………….49
3.3.1 Artificial Neural Networks ………………..……………………………….50
9
3.3.2 Neural Network AutoRegressive eXternal Input Model …………..………51
3.3.3 BackPropagation Learning Rule for MLP ……………………..………….53
3.3.4 Online learning method …………..………………………………………..54
3.3.5 Recursive Levenberg-Marquardt Optimization Algorithm ………..………55
3.3.6 Matrix Inverse Calculation …………………………..…………………….56
Chapter 4 Direct Adaptive Neural Network Control 59
4.1 Introduction …………………………………………..………..………………….59
4.2 Direct Adaptive Neural Network Control Architecture …………..……………….60
4.3 DANNC Online Learning Algorithm …………...……………………………….61
4.4 Real Time Implementation of DANNC …………………………..……………….62
4.5 Experimental Results and Discussions ……………….……………………………64
4.6 Direct Inverse Neural Network Control …………………………………………67
4.7 Experimental Performance Comparison of DANNC and DINNC …..……………68
Chapter 5 Model Based Predictive Control 70
5.1 Introduction ………………………………………………………..……………….70
5.2 LQR Control System Design ………………..……………………………………..72
5.2.1 Discrete Linear-Quadratic State Feedback Regulator Design ………….…72
5.2.2 Prediction Estimator ……………………………..………..………………73
5.2.3 LQR Control System Architecture …………….….……………………….74
5.2.4 Experimental Results and Discussions …………………………………….75
5.3 Generalized Predictive Control Techniques ……….………………………………80
10
5.3.1 Cost Function …………..…………………………………………………..80
5.3.2 Selection of Horizons for the Performance Index ………………………….81
5.4 Adaptive Generalized Predictive Control …………………………………………82
5.5 Adaptive Generalized Predictive Control based on Augment UD Identification …84
5.5.1 Derivation of Control Law ………………………………….…………….84
5.5.2 Real Time Implementation of AGPC and Experimental Results ..……...…86
5.6 Neural Adaptive Predictive Control (NAPC) ………………...……..…………….92
5.6.1 Neural Adaptive Predictive Control Architecture ……………..…………..93
5.6.2 NNARX Representation of the Smart Structure Model ………..………….94
5.6.3 Derivation of Control Law …………………………..…………………….96
5.6.4 Real Time Implementation of NAPC and Experimental Results …….……97
5.7 Experimental Comparison of Adaptive Predictive Controllers ………..…………103
Chapter 6 Conclusions and Recommendations 111
6.1 Conclusions ………………………………………………….……………………111
6.2 Recommendations ……………………………………………..…………………116
Bibliography 118
Appendix A 128
Input/Output Formulation of the Equation of Motion………..…………………………128
Appendix B 130
List of Publications…………………………………………………...…………………130
11
List of Figures
Figure 2.1 Schematic diagram of experimental setup ….…………………………………….12
Figure 2.2 Experimental setup (a) original structure (b) plate added (c) tip mass added ...13
Figure 2.3 Dimensions of modification plate ………………………………………………..14
Figure 2.4 Piezoelectric actuator ……………………………………………………………..16
Figure 2.5 Natural frequencies of the structure ……………………………………………..18
Figure 2.6 Test signal properties (combined sine wave, 1st and 2nd mode) …….……………19
Figure 2.7 Test signal properties (0-50Hz white noise) ……………….…………………….20
Figure 3.1 Piezoelectric actuation ….……………………………….………………………..24
Figure 3.2 Mode contribution versus mode number for shaker input ………….…………...30
Figure 3.3 Mode contribution versus mode number for piezo input …….………………….31
Figure 3.4 Comparison of Bode plot of Reduced model by Peak Gain with Full model
(piezoelectric input)……………………..……………………………………………………31
Figure 3.5 Comparison of Bode plot of Reducted model by Peak Gain with Full model
(shaker input) ……………………….………………………………………………………..32
Figure 3.6 Diagonal of Balanced Gramian versus number of states ….…….………………..34
12
Figure 3.7 Comparison of Bode plot of Reduced Model by Balanced Reduction with Full
model (piezoelectric input) ………………………….……………………………………….34
Figure 3.8 Comparison of Bode plot of Reduced Model by Balanced Reduction with Full
model (shaker input) …………………………………………………………………………35
Figure 3.9 Neural Network AutoRegressive eXternal input model structure ………………..53
Figure 4.1 System diagram of direct adaptive Neural Network control system ……..………60
Figure 4.2 Main steps of DANNC learning ………………….………….…………………61
Figure 4.3 Schematic diagram of direct adaptive neural network controller …………..……61
Figure 4.4 Direct adaptive NN controller real time implementation block diagram …..……63
Figure 4.5 Controlled & uncontrolled response for the 1st mode sine disturbance input
(original structure) ………………………..……………………………………………….63
Figure 4.6 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(original structure) ……………….………..………………………..……………………….64
Figure 4.7 Controlled & uncontrolled response for the white noise disturbance input (original
structure) ………………………..………………………….……..…………………………64
Figure 4.8 Controlled & uncontrolled response for the sine wave disturbance change from 1st
to 2nd (original structure) …………………………………………..…..…………………….65
Figure 4.9 Controlled & uncontrolled response for the sine wave disturbance change from 2nd
to 1st (original structure) ………….………………………….………………………………65
Figure 4.10 Controlled & uncontrolled response for the 1st mode sine disturbance input (tip
mass added structure) …………………………………………..……………………………66
Figure 4.11 Controlled & uncontrolled response for the 2nd mode sine disturbance input (tip
mass added structure) ……………………….…………….…………………………………66
13
Figure 4.12 Controlled & uncontrolled response for the 1st mode sine disturbance input (plate
added structure) ……………………………..……………………………………………….67
Figure 4.13 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(plate added structure) ………………………………………………………………………..67
Figure 4.14 Controlled and uncontrolled Fourier amplitude of the structure ……..….……69
Figure 5.1 System diagram of LQR control .……………………..…..…….………………..75
Figure 5.2 Controlled & uncontrolled response for the 1st mode sine disturbance input (original
structure) …………………………………………………..…………………………………76
Figure 5.3 Controlled & uncontrolled response for the 2nd mode sine disturbance input (original
structure) …………………………………………………..………………..……………….76
Figure 5.4 Controlled & uncontrolled response for the white noise disturbance input (original
structure) …………………………………………………..…………………………………76
Figure 5.5 Controlled & uncontrolled response for the sine wave disturbance change from 1st
to 2nd mode (original structure)……………………………………………………………….77
Figure 5.6 Controlled & uncontrolled response for the sine wave disturbance change from 2nd
to 1st mode (original structure)………………………………………………………….…….77
Figure 5.7 Fourier amplitude to combined sine wave (1st and 2nd modes) disturbance input ..78
Figure 5.8 Controlled & uncontrolled response for the 1st mode sine disturbance input (tip
mass added structure) ……………………………………………….………………………79
Figure 5.9 Controlled & uncontrolled response for the 2nd mode sine disturbance input (tip
mass added structure) ………………………………………………..………………………79
Figure 5.10 Controlled & uncontrolled response for the 1st mode sine disturbance input (plate
added structure) …………………………………..……………………..…………………..79
14
Figure 5.11 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(plate added structure) ….………………….…………..…………………..………………..80
Figure 5.12 Block diagram of Adaptive Generalized Predictive Control ……..…………….83
Figure 5.13 Controlled & uncontrolled response for the 1st mode sine disturbance input
(original structure) ………………………………………………………..…………………87
Figure 5.14 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(original structure) …………………………………………………..………………………87
Figure 5.15 Controlled & uncontrolled response for the white noise disturbance input
(original structure) ……………………………….…………………..………………………87
Figure 5.16 Fourier amplitude to combined .sine wave (1st and 2nd modes) disturbance input
(original structure) ………………………………….……………..………………….88
Figure 5.17 Controlled & uncontrolled response for the sine wave disturbance change from
1st to 2nd mode (original structure) ………………………………………..………………….89
Figure 5.18 Controlled & uncontrolled response for the sine wave disturbance change from
2nd to 1st mode (original structure) ……………………………………..…………………….90
Figure 5.19 Controlled & uncontrolled response for the 1st mode sine disturbance input (plate
added structure) ……………………………………………………………………………..90
Figure 5.20 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(plate added structure) ……………………………………….………..……………………..90
Figure 5.21 Controlled & uncontrolled response for the 1st mode sine disturbance input (tip
mass added structure) …………………………………………………..……………………91
Figure 5.22 Controlled & uncontrolled response for the 2nd mode sine disturbance input (tip
mass added structure) ………………………………………………………………..………91
15
Figure 5.23 Block diagram of Neural Adaptive Predictive control system ….…..………….93
Figure 5.24 NNARX representation of smart structure model ………………………………95
Figure 5.25 Controlled & uncontrolled response for the 1st mode sine disturbance input
(original structure) ……………………………………………………..…………………….98
Figure 5.26 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(original structure) ……………………………………………………..…………………….99
Figure 5.27 Controlled & uncontrolled response for the white noise disturbance input
(original structure)…………………………….……………..……………………………….99
Figure 5.28 Response to combined .sine wave (1st and 2nd modes) disturbance input (original
structure)……………………………………………………………….…………………….100
Figure 5.29 Controlled & uncontrolled response for the sine wave disturbance change from
1st to 2nd mode (original structure)……………………………………………….….……….100
Figure 5.30 Controlled & uncontrolled response for the sine wave disturbance change from
2nd to 1st mode (original structure)…………………………..……………………….………101
Figure 5.31 Controlled & uncontrolled response for the 1st mode sine disturbance input (plate
added structure) ……………………………………………………………………….…….101
Figure 5.32 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(plate added structure) ……………………………………………………………………..102
Figure 5.33 Controlled & uncontrolled response for the 1st mode sine disturbance input (tip
mass added structure) ……………………………………………………………………….102
Figure 5.34 Controlled & uncontrolled response for the 2nd mode sine disturbance input (tip
mass added structure) ………………………………………………….……………………102
Figure 5.35 Controlled and uncontrolled Fourier amplitude of structure ……..………….105
16
Figure 5.36 Performance comparison of AGPC and NAPC ….……………………………106
Figure 5.37 Vibration reduction for combined sine wave disturbance input (original
structure) ………………………………..………………………………………………….107
Figure 5.38 Responses to disturbance change from 1st to 2nd mode frequency (plate added
structure) …………………………………………..………………………………………..108
Figure 5.39 Responses to tip mass attachment during experiment …….…………………...109
Figure 5.40 Power consumption for active vibration control using AGPC and NAPC …....110
Figure 6.1 Performance comparison of different controllers (1st mode) ……………….….112
Figure 6.2 Performance comparison of different controllers (2nd mode)…………………..113
Figure 6.3 Power consumption comparison of different controllers ……………………….114
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Nomenclature
Finite Element/State Space Modeling (Chapter 3.1)
1a Proportional damping const1
2a Proportional damping const2
b Width of the actuator
1b Point force location
2b Point moment location
31d Piezoelectric constant
ir Spatial coordinate
at Thickness of the actuator
bt Thickness of the host structure
z Modal state vector
λ Eigenvalue vector
η Principal coordinate
ζ Modal damping
iζ Percentage of the critical damping for thi mode
18
iω thi Eigenvalue
( )if t Forces in principal coordinate
0 ( )m t Point moment acting at the location r= 2b
0 ( )p t Point force at the location r= 1b
( )u t Forcing function acting on the structure
( )u k Control signal input.
( )v k Disturbance signal input
( )x t Displacement of a point on the structure at time t
( )x t Acceleration of a point on the structure at time t
( )x k Modal state vector of the structure system
( )y t Modal output
( )tη Modal displacement
( )tη Modal velocity
A System matrix
B Input matrix
C Output matrix
D Direct transmission matrix
dC Damping matrix
E Electric field
aE Elastic Modulus of the actuator
bE Elastic Modulus of the structure
F Output matrix
19
G Direct transmission matrix
H Disturbance input matrix
K Stiffness matrix
M Mass matrix
uN Number of inputs
mN Number of modes
yN Number of outputs
Q Modal input matrix
Ω Natural frequencies
V Applied voltage
W Modal output matrix
Γ Input matrix
Φ System matrix
Ψ Mass normalized eigenfunction matrix
ijψ thj Output row of the thi mode
ikψ thk Applied force of the thi mode
i jψ Eigenfunction i at location j
Controlled AutoRegressive Integrated Moving Average Model (Chapter 3.2)
( )e k Residual sequence of the parameter estimates
( )p k Covariance matrix
( )u k Plant input at the time k
20
( )fu k Filtered input
( )v k Disturbance term
( )y k Plant output at time step k
( )y k j∧
+ j step ahead prediction
( )fy k Filtered output
( )z k Innovation sequence of the parameter estimates
( )kγ Forgetting factor
ˆ( )kθ Predicted plant model parameter vector
( )n kϕ Augmented data vector
ˆ ( )n kθ Augmented parameter vector
jE Polynomials uniquely defined by the Diophantine equation
jF Polynomials uniquely defined by the Diophantine equation
I Identity matrix
,0jf The constant coefficient of the polynomial 1( )jF q−
1( )A q− Polynomials in the backward shift operator 1q−
1( )B q− Polynomials in the backward shift operator 1q−
1( )C q− Polynomials in the backward shift operator 1q−
( )L k Kalman gain vector
( )kξ Uncorrelated random sequence
∆ Difference operator which equals to 11 q−−
21
Neural Network Based System Identification (Chapter 3.3)
ke Error of the neural network output at step k
g Nonlinear function used to predict the output
dn Time delay
an Orders of the dynamic system
bn Orders of the dynamic system
kx Weight vector at step k
( )r k Reference input of the neural network
( )y k Neural network output
ϕ Regression vector
θ Vector containing the weights (and biases)
L Lower triangular factor
U Upper triangular factor
X Consists of all weights and bias of the neural network
( )H x Hessian matrix
( )J x Jacobian matrix
Direct Adaptive Neural Network Control (Chapter 4)
1b Bias of the first layer
2b Bias of the second layer
22
ke Neural network error at time step k
kt Targeted neural network output at time step k
ku Calculated control signal output at time step k
ku DANNC calculated control signal input
( )u k Plant control signal input
( )u k∧
Predicted plant control signal input
( )y k Plant output
P Neural network input vector
1W First layer weight matrix
2W Second layer weight matrix
ϕ Activation function for the hidden neurons ( tanhϕ = )
LQR Control System Design (Chapter 5.2)
x Estimation error
x Predicted state vector
u(k) Input vector
x(k) State vector
F Output matrix
G Direct transmission matrix
K Optimal state feedback gain
pL Prediction estimator gain vector
23
Q Symmetric state weighting function
R Symmetric control weighting function
S Infinite horizon solution discrete-time Riccati equation
Φ System matrix
Γ Input matrix
J(u) Cost function
Adaptive Generalized Predictive Control Techniques (Chapter 5.3)
λ Control weighting
u Control increment vector
optimalu Optimal control increment vector
( )y k j∧
+ Predicted system output
( 1)u k j∆ + − Control increment
J Cost function
1N Minimum costing horizon
2N Maximum costing horizon
uN Control horizon
Neural Adaptive Predictive Control (Chapter 5.6)
0a Outputs of input layer neurons
1a Outputs of hidden layer neurons
2a Output of output layer neurons
24
( )nα Step size
( )nf Search direction
1f Activation functions of the hidden layer
2f Activation functions of the output layer
dn Time delay
an Orders of the dynamic system
bn Orders of the dynamic system
( 1)u n + Current iteration of the sequence of future control input
( )u n Previous iteration of the sequence of previous control input
( 1)py k + One step ahead predicted output
1W Weights of hidden layer
2W Weights of output layer
J Cost function
( )G n Jacobina matrix
( )H n Hessian matrix
25
Abbreviations
AGPC Adaptive Generalized Predictive Control
AUDI Augmented UD Identification
ARMAX AutoRegressive Moving Average eXternal input
CARIMA Controlled AutoRegressive Integrated Moving Average
DANNC Direct Adaptive Neural Network Control
DINNC Direct Inverse Neural Network Control
DSP Digital Signal Processor
FEM Finite Element Method
FA Fourier Amplitude
GPC Generalized Predictive Control
LM Levenberg Marquardt
LQR Linear Quadratic Control
MIMO Multiple Input Multiple Output
MPC Model Predictive Control
NAPC Neural Adaptive Predictive Control
NN Neural Networks
26
NNARX Neural Network AutoRegressive eXternal input
OKID Observer/Kalman Filter Identification
RLS Recursive Least Square
RMS Root Mean Square
SS State Space
RTW Real Time Workshop
RT Real Time
SISO Single Input Single Output
UD Upper Diagonal
27
Chapter 1
Introduction
In this chapter, the background of current research is reviewed first and followed by the
objective of this dissertation. Then the currently available modeling and control techniques
for the application of smart structure, including summaries of previous work and past
accomplishments, are discussed briefly. Specific research areas which are lacking in the
available literature are outlined. Organization of this thesis is described at the end of this
chapter.
1.1 Background
A smart structure involves distributed sensors, actuators, and one or more processors
that can analyze the responses from the sensors and some control theory to command the
actuators to apply localized strain to minimize the response of the structure [1]. The current
development of smart structure technology has the potential to bring about a paradigm shift
in structural design philosophy, and develop a new generation of products and systems. This
emerging technology is multidisciplinary in nature involving structural dynamics, smart
materials (sensors and actuators), control systems (classical, modern, or neural network
based), and integrated design, analysis and fabrication. The smart structures technology
28
promises significant impact on diverse industries such as aerospace [2], automotive [3] and
civil infrastructure [4]. The lighter and more flexible structures and mechanical systems are
prone to low frequency vibration, which brings with it new challenges for dynamic control.
The many applications of smart structures include active vibration suppression, noise
reduction, aerodynamic flow control, aeroelastic stability enhancement, structural damage
mitigation and structural health monitoring.
Applications of smart structure technology are increasing rapidly. The major issues
need to be addressed are: appropriate control algorithms, mathematical modeling techniques
of smart structures, actuator stroke and reliable database of smart material characteristics [1].
Complex structures with multiple subsystems and a large number of distributed sensors and
actuators more likely exhibit nonlinearity and variations with time. Lin, et al. [5] identified
the changes in the plant dynamics with time, and actuator saturation (which introduces
nonlinearity in the plant) as the major performance limitations for the conventional control
system used. Denoyer and Henderson [6] discussed the need for developing adaptive control
system for on-orbit spacecraft structural control. Most smart structures, due to their
considerable flexibility, distributed sensors and actuators, require a relatively high order
model. Also, the complexity of formulating their vibration due to having several degree of
bending and twisting, accurate modeling of these structures is rather complicated. To realize
the full potential of the smart structures technology, the control system must be capable of
handling complexity, uncertainty, nonlinearity, and variations with time (either expected or
due to failure). For robust control, the objective performance and closed-loop stability must
be satisfied for various uncertainties or unmodelled dynamics [7]. This demands the
29
development of suitable modeling and control techniques for the application of smart
structure.
1.2 Objective
This dissertation focuses on developing and experimentally implementing the modeling
and control techniques for the vibration suppression of a smart structure. The main
objectives are:
1. Finite element/state space modeling technique for the application of smart
structure.
2. Controlled autoregressive integrated moving average (CARIMA) model with
augmented Upper Diagonal (UD) identification for adaptive identification of a
smart structure.
3. Neural network autoregressive external input (NNARX) model with Levenberg-
Marquardt recursive online learning algorithm.
4. Direct adaptive neural network control system development and experimental
implementation for vibration suppression of a smart structure.
5. Experimental implementation of linear quadratic regulator based on finite
element/state space model.
6. Experimental implementation of adaptive generalized predictive control system
based on CARIMA model with augmented UD identification.
30
7. Development and experimental implementation of neural adaptive predictive
control system based on NNARX model with Levenberg-Marquardt online
learning algorithm.
8. Comparison of different modeling and control techniques for the application of
smart structure.
1.3 Literature Review
One of the most important and challenging components of structural control system
design is the development of an accurate mathematical model of the structural system.
Accurate modeling of the dynamics of flexible smart structural systems is critical for a
variety of applications, including active vibration control and structural design optimization.
Analytical models are only available for structures with simple geometry and boundary
conditions. For more complicated structures, finite element and experimental methods are
used extensively.
Finite element modeling, without construction of experimental setup, is a cost effective
method for complex geometrical structures. Yazdani and his coworkers examined the
effectiveness of the experimental, analytical and finite element methods in modeling smart
structures, and found FEM to be very attractive [8]. Numerous researchers have proposed
finite element analysis and modeling method for the smart structures. Bisegna and Caruso
developed a new finite element formulation for the analysis of a plate having thin
31
piezoelectric actuators bonded on its upper and /or lower surfaces [9]. Lam and his
coworkers presented a FE model based on the classical laminated plate theory for the active
vibration control of a composite plate containing distributed piezoelectric sensors and
actuators [10]. Narayanan and Balamurugan used a shear-flexible nine-node shell finite
element derived from the field consistency approach for the active vibration control [11].
Although those analytical techniques showed good correlation with experimental data, they
can be difficult to implement even for simple structures. Due to the increasing interest in the
design of complex structures with piezoelectric actuators, major commercial FEM software
(ANSYS, NASTRAN, etc.) have incorporated or provided tools to create piezoelectric
elements. Pantling and Shin studied the active vibration control method and verification for
space truss using ANSYS Parametric Design Language [12]. Freed developed one and two-
dimensional finite element which include piezoelectric coupling for integration with
NASTRAN [13]. The structural model obtained by Finite Element Analysis is usually rather
big for complex systems. A reduced structural model which can represent the system
accurately enough is required. For the real time control purpose, an integrated approach is
needed to achieve the best control performance.
Experimental modeling, also called System Identification, is a black box method. It
can be modeled in either time or frequency domain. The system model can be in the form of
state space, a finite difference or neural network, etc. Ljung presents the theoretical
development of various techniques for system identification as well as convergence analysis
with advantages and disadvantages of several model types [14]. Juang describes several
structural modeling techniques based on input and output data, such as System Realization
Theory, Obsever/Kalman Filter Identification (OKID), etc [15]. The parameters of the
32
system model can be identified online or offline. The simplest and most intuitive method to
identify the numerical values of the parameters is batch least squares method [14]. This
method uses blocks of input and output data to perform the identification. Manning, et al.
developed a smart structure vibration control scheme using an ARMAX model of the
structure, and system identification was carried out in three phases, data collection, model
characterization and parameter estimation [16].
To have an online identification, a recursive least square (RLS) method [14] can be
used. Kvaternik and Juang [17] performed an evaluation of modern adaptive multi-input
multi-output (MIMO) control techniques for active stability augmentation and vibration
control of tiltrotor aircraft and showed the generalized predictive control (GPC) based MIMO
active control to be highly effective. However, the conventional RLS algorithm has a
number of shortcomings, such as poor numerical performance especially when implemented
on computers with finite precision [18]. Also, RLS algorithm is known to have optimal
properties when the parameters are time invariant, but it is unsuitable for tracking time-
varying parameters [19]. In order to improve estimation method, Bierman [20] proposed UD
factorization algorithm, which has a much better numerical performance than RLS.
However, the UD factorization algorithm has not been as widely used as RLS because it
appears to be more complicated to interpret and implement.
Niu et al. [18] proposed an augmented UD identification (AUDI) algorithm by
rearranging the data vectors and augmenting the covariance matrix of Bierman’s UD
factorization algorithm. The AUDI permits simultaneous and recursive identification of the
model parameters plus the loss function for all orders from 1 to n at each time step with
approximately the same calculation effort as nth order RLS and it has better numerical
33
properties. The augmented UD identification approach provides many features that are
particularly suitable for real time applications. It provides other information in addition to
the model parameters, such as model order and loss functions, parameter identifiability, noise
variance, and signal-to-noise ratio. However, no work has been reported using AUDI based
adaptive predictive control of smart structures. Unlike most of the process control
applications, smart structures have fast dynamics and, therefore, need efficient real time
application algorithms.
The neural networks have been applied for identification and control of dynamical
systems in many fields, including complex, practical systems such as robots, aircraft, arc
furnaces and steel rolling mills [21-22]. These examples show that neural networks are
capable of handling the difficulties of nonlinearity and uncertainty, which characterize
complex systems. Rivals and his coworkers pointed out that neural networks, especially for
the multi-layer perceptron, has been used for the block-box modeling of nonlinear dynamical
systems because of its universal approximation capability [23]. Nelles provides the
underlying principles of nonlinear system identification which includes nonlinear classical,
neural networks (NN), fuzzy models and optimization techniques [24]. Haykin describes the
fundamentals of neural networks [25]. Hagan and his coworkers present the computing
techniques for the application of neural networks [26]. Several researchers have studied NN
controllers for smart structures using numerical simulations [27-29] and experiments [30-37].
Adaptive control, which is simply a special type of nonlinear regulator, became popular
since 1970’s as the computing resource improved. Intuitively, an adaptive controller is thus a
controller that can modify its behavior in response to changes in the dynamics of the plant
and the character of the disturbances. In practice this implies that an adaptive controller is a
34
controller with adjustable parameters, which is tuned on-line according to some mechanism
in order to cope with time-variations in plant dynamics and changes in the environment.
Adaptive control can maintain consistent performance of a system in the presence of
uncertainty or unknown variation in plant parameters. Another advantage of adaptive control
is that it requires limited a priori knowledge of the plant to be controlled. A recent review of
the various adaptive control techniques is presented in [38]. Neural network based control
systems with on-line adaptation have the capability to cope with these challenges [21, 22].
With nonlinear optimization involved, adaptive neural network controller requires extensive
computation in real time. Efficient algorithms have to be used for neural network online
learning. Some of these works [35-37] include experimental validation of the on-line
adaptation capability. Spencer et al. [37] used a radial basis function NN for real-time,
closed-loop vibration control of piezo-actuated helicopter rotor blades. Adaptive neural
control for space structure vibration suppression was demonstrated by Davis et al. [35].
Youn et al. [36] used indirect model reference adaptive controller for vibration control of
composite beams subject to sudden delamination. Adaptive neural identification and control
capability is important since the environment, the structure or the system dynamics may
change with time. Few works have been done in this area, especial experimentally. The
rapid application of smart structure technology demands new methodology on the
development and real time implementation of neural network based adaptive control systems.
35
1.4 Thesis Organization
This dissertation consists of 6 chapters and 2 appendices. Chapter 1 is an introduction
of the research area addressed. A brief review of the application of smart structure and the
current research activities are presented thereafter. Currently available modeling and control
techniques for the vibration suppression of smart structure are also discussed in this chapter.
In Chapter 2, the experimental setup used in current research to validate the identification
and control techniques is described. Three identification techniques for the application of
smart structure, finite element/state space, controlled autoregressive integrated moving
average and neural network autoregressive external input model, are presented in Chapter 3.
Augmented UD identification for the adaptive parameter identification and recursive
Levenberg-Marquardt optimization method for the neural network online learning is also
discussed in this chapter. A direct adaptive neural network controller is developed in
Chapter 4, and implemented experimentally in the real time for the active vibration control of
smart structure. The experimental performance of direct adaptive neural network control is
compared with a direct inverse neural network controller, which does not have online
learning capability. In Chapter 5, two predictive control systems based on the identification
techniques discussed in Chapter 3, adaptive generalized predictive control system based on
augmented UD identification and neural adaptive predictive control system based on neural
network autoregressive external input model, are investigated. Experimental performances
of each model-based controller are also investigated and the comparison is made between the
two adaptive GPC systems. To have a baseline for comparison, linear quadratic control
(LQR) based on state space/finite element (SS/FE) model is also included in Chapter 5. In
36
Chapter 6, conclusions based on current research are made and the possible future research
directions are suggested.
Chapter 2
Experimental Setup
In this chapter, the experimental setup used to validate the modeling and control
techniques presented in this dissertation will be described.
2.1 Schematic Diagram of the Experimental Setup
Figure 2.1 shows the schematic diagram of the experimental setup. It comprises a thin
plate clamped rigidly at the base, which is free to move up and down on linear bearings. An
Electrodyne electromagnetic shaker (model AV-400) generates the excitation input for the
structure. The shaker is powered by Electrodyne model N-300 single channel amplifier with
a frequency range of 1.5 Hz to 22 kHz. Two ACX PZT actuators (QP10W) are bonded to the
37
surface of the plate at the root, which is considered the best location for controlling the
fundamental bending mode [39]. The actuator input is limited to ± 100 volts, which is well
within the range of the maximum permissible voltage. Two Kistler piezoceramic shear
accelerometers (model 8774A50) are connected to a Kistler signal conditioner that sends
vibration information through a low-pass filter to the PC through AD channel. The low
frequency cut-off for the accelerometers is 1 Hz. A 600 MHz PC is used for the data
acquisition, analysis and control. The signals are converted from Analog-to-Digital and
Digital-to-Analog using a Quanser 16-channel 12-bit AD/DA board. Only the tip
accelerometer is used in the current research and both actuators receive the same voltage.
Thus, we have a SISO (Single Input Single Output) control system.
Figure 2.1 Schematic diagram of experimental setup
38
2.2 Experimental Hardware
The structure to be controlled is a steel cantilever thin plate (dimensions, LxWxH are
30.16x8.573x0.0762cm), as shown in Fig. 2.2a.
(a)
(b) (c)
Figure 2.2 Experimental setup (a) original structure (b) tip mass added (c) plate added
To compare the experimental performance of each controller for time-varying systems,
modified structures are used. One modification is adding a plate to the original structure,
39
which basically increases the stiffness of the structure (Fig. 2.2b.). The other modification is
adding a mass near the tip (Fig. 2.2c), which decreases the natural frequencies. Figure 2.3
shows the dimensions of the modification plate.
Figure 2.3 Dimensions of modification plate
2.3 Experimental Software To implement the modeling and control techniques discussed in this dissertation,
commercially available software Matlab/Simulink/RTW developed by Mathworks is used.
RTW (Real Time Workshop) generates optimized, portable and customizable ANSI C code
from Simulink models. It can automatically build programs that execute in real time or non-
real time simulations. The generated code accelerates simulation and real time execution. S-
function is a C-MEX S-function that is treated identically by Simulink and the Real Time
Workshop. S-function uses a special calling syntax that enables the interaction with
Simulink equation solvers. S-functions allow customized blocks or algorithm to be added to
40
Simulink models. All the modeling and control techniques are implemented as an S-function
used in Simulink/RTW.
2.4 Piezoelectric Effect
The Piezoelectric effect was discovered by Pierre and Jacques Curie in 1880. It
remained a mere curiosity until the 1940s. Piezoelectric materials generate an electric
potential when stressed mechanically by a force. Conversely, under application of an electric
field across the thickness of the material, it elongates or shortens depending on the polarity.
A piezoelectric element is therefore capable of being used both as actuator and sensor. The
properties of certain crystals to exhibit electrical charges under mechanical loading was of no
practical use until very high input impedance amplifiers enabled engineers to amplify their
signals. In the 1950's, electrometer tubes of sufficient quality became available and the
piezoelectric effect was commercialized. Common types of piezoelectric actuators include
stack (axial displacement) and wafer (longitudinal displacement) shapes. In current research,
thin wafers made of lead zirconate titanate (PZT) are used as actuators. A basic schematic of
the piezoelectric actuator is shown in Figure 2.4. Two electrical leads are fixed to either side
of the actuator, forming the ends of a circuit. Applying a voltage across these leads (the 3-
direction) results in a corresponding change in length in the 1-direction. If attached to a host
structure, the piezoelectric actuators cause deformation of the structure when they receive
some voltage input because of the strain.
41
Figure2.4 Piezoelectric actuator
2.5 Structure Nonlinearity Test
The excitation voltage sent to the shaker amplifier produces large tip accelerations
showing a relatively large nonlinear response. To test the nonlinearity of the system, the
following steps were used [40].
1. Apply a zero input signal and wait for steady state to occur to investigate if there is a
DC-offset (D) (D=0 for current experimental system).
2. Apply two different input signal, 1( )u t and 2 ( )u t , where 1 2( ) ( )u t u tα= . Obtain the
corresponding steady state output 1( )y t and 2 ( )y t .
3. Calculate the ratio 2
1
( )( )( )
y t Dr ty t D
−=
−. For linear system, this ratio should equal to α all
the time. Use ( )maxi
r t ανα−
= as a “nonlinearity index” to measure the nonlinearity
of the system.
42
Two different input signals 1( ) sin( )u t tω= and 2 ( ) 2sin( )u t tω= were applied. The ratio
of the outputs for the two input signals is obtained as 2.38 and 3.05 for first and second
natural frequencies, respectively. This indicates nonlinearity of the system, especially for the
second mode.
2.6 Fourier Amplitude of the Structure Response
Because of the nonlinearity of the smart structure system, Fourier amplitude of the
structure response, instead of frequency response Function was used to find out the structure
properties in frequency domain.
The structure was excited by an impulse (generated through the shaker) and the tip
acceleration was measured to obtain the natural frequencies. Fig. 2.5 shows the natural
frequencies of the structure. The first two natural frequencies are 6.67 Hz and 40.4 Hz for
the original structure. Because of the plate added modification which basically increases the
stiffness of the structure, the natural frequencies change to 6.68 Hz and 42.6 Hz. For the tip
mass added case, the frequencies change to 5.33 Hz and 38.5 Hz because the tip mass
decreases the natural frequencies. These two modes are bending modes. The magnitude of
the tip acceleration is reduced due to the modifications, especially for the second mode with
plate added which stiffens the middle part of the structure as shown in Fig. 2.5.
43
Figure 2.5 Natural frequencies of the structure
2.7 Test Signal Properties
Various signals, namely 1st mode and 2nd mode sine wave disturbances, combined
sine wave and 0-50Hz white noise, are generated by the computer and sent to the shaker as
disturbance input for identification and control discussed in this thesis. Figure 2.6 shows an
example of the combined sine wave signal in time domain and frequency domain. Figure 2.7
shows an example of 0-50Hz white noise disturbance in time domain and frequency domain.
44
Figure 2.6 Test signal properties (combined sine wave, 1st and 2nd mode)
45
Figure 2.7 Test signal properties (0-50Hz white noise)
46
Chapter 3 Identification and Modeling of a Smart Structure
In this chapter, several identification and modeling techniques, including finite
element/state space method (FE/SS), controlled autoregressive integrated moving average
(CARIMA) model, and Neural Network based system identification, are presented.
Augmented UD identification (AUDI) method for the adaptive parameter identification and
recursive Levenberg-Marquardt optimization algorithm for the neural network online
learning, are also provided here. These identification and modeling techniques will be used
in Chapter 5 for active vibration control of smart structures.
3.1 Finite Element/State Space Method
Accurate modeling of the dynamics of the smart structure is crucial for active vibration
control. Analytical models are only available for structures with very simple geometry and
boundary conditions. For more complicated structures, which is most often the case in the
application of smart structures due to the large number of distributed sensors and actuators,
finite element analysis and experimental method are used extensively. The mass and
stiffness matrices resulting from finite element analysis are usually too big to be used directly
in real time control. It is useful to provide a model of the smart structure with as few state
variables as possible. In this section, a linear modeling approach based on finite
element/state space, will be introduced to model the smart structure using ANSYS, a
47
commercially available software. Four-node, elastic shell elements (ANSYS element
SHELL63) are used to model the structure. Shell elements typically have all nodes on a
single plane (or curvature), with a thickness extending out symmetrically from the central
plane. ANSYS provides two elements that have piezoelectric capabilities. But, the voltage is
defined as a degree of freedom, not a force input. It is inappropriate for this control oriented
application. In this effort, Euler-Bernoulli model discussed in the following section, instead
of finite element method, is used to model the piezoelectric force effect. To reduce the
model size constructed by finite element analysis, model reduction method is used to find out
the most significant modes. A small state space model is constructed based on the selected
eigenvalues and eigenvectors.
3.1.1 Modeling of Piezoelectric Actuator
Piezoelectric actuator is an important part of a smart structure system. There are a
number of models available to predict the interaction between induced strain actuators and
the substrate. For one dimensional structure, three modeling approaches are usually used
most often: block force model, uniform strain model and Euler-Bernoulli model [1]. Block
force model is the simplest one, the basic idea is to treat the actuator as if it is pinned with the
structure and produces a moment on the substrate when it expands or shrinks, thereby
bending it. This model can be inaccurate for low beam-actuator thickness ratio. Also, it does
not account for the bending stiffness of the actuator. The Euler-Bernoulli model is a
consistent strain model and does account for the bending stiffness of the actuator. This
model treats the actuator and the substrate as a composite structure and follows the
Bernoulli’s assumption: a plane section normal to the beam axis remains plane and normal to
48
the beam axis after bending. There is a linear distribution of strain in the actuator and the host
structure [1].
The moment equation for an actuator patch on one side of a structure is given by
Chaudhry and Rogers [41].
([( ) ( ) ]a bM EI EI κ= + (3.1)
where
3
( )12
aa a
btEI E= (3.2)
3
( )12
bb b
btEI E= (3.3)
2 2
6 ( 1)16 4 4b
T Tt T T T
κψ
ψ
+= ∆
+ + + + (3.4)
b
a
tTt
= (3.5)
b b
a a
t Et E
ψ = (3.6)
31 31a
Vd E dt
∆ = = (3.7)
at , bt shown in Fig. 3.1, are the thickness of the actuator and the host structure,
respectively,
aE and bE are the Elastic Modulus of the actuator and the structure, respectively,
b is the width of the actuator,
31d is the piezoelectric constant,
E is the electric field, and
49
V is the applied voltage.
thus, we have the following formula
31
2 2
6 ( 1)([( ) ( ) ] 16 4 4a b
b a
dT TM EI EI V Vt tT T T
αψ
ψ
+= + =
+ + + + (3.8)
where, α is a constant calculated from the above, which means the produced moment is
proportional to the applied voltage.
Figure 3.1 Piezoelectric actuation
3.1.2 Structural Modal Analysis
Modal analysis is a computationally elegant technique for modeling structural
dynamics. It is based on the eigenvalue and eigenvector information of a system. The
elegance and appeal of this technique is mainly due to its decoupling capability.
Consider a linear time-invariant flexible structure, which can be modeled as a second
order differential equation [42].
( ) ( ) ( ) ( )Mx t CDx t Kx t u t+ + = (3.9)
50
where,
M is the Mass matrix,
dC is the Damping matrix,
K is the Stiffness matrix, and
u(t), x(t) are the forcing function vector and displacement of the structure respectively.
To exactly predict the Damping matrix dC is impossible with the present state of art
[43]. But, for systems with the damping matrix dC associated with mass matrix M and
stiffness matrix K, such as proportional damping, these matrices can be diagonalized using
the mass normalized orthonormal eigenvectors as the columns of the transformation matrix.
First, for undamped system
( ) ( ) ( )Mx t Kx t u t+ = (3.10)
Let Ψ is the eigenvector matrix, which is mass normalized, λ is the eigenvalue vector
( ) 0K Mλ− Ψ = (3.11)
with the eigenfunction matrix Ψ , the spatial coordinate x can be changed into a new
coordinate, principal coordinate,η , using
x η= Ψ (3.12)
substitute into the above equation
( )M K u tη ηΨ + Ψ = (3.13)
premultiplying by 1−Ψ
51
1 1 1 ( )M K u tη η− − −Ψ Ψ +Ψ Ψ = Ψ (3.14)
using similarity transformation
1 T−Ψ = Ψ (3.15)
then, we have
( )T T TM K u tη ηΨ Ψ +Ψ Ψ = Ψ (3.16)
with T M IΨ Ψ = , 2( )TiK diag ωΨ Ψ =
2 ( )i i i if tη ω η+ = (3.17)
( )if t is the forces in principal coordinate.
with the assumption of proportional damping
1 2dC a M a K= + (3.18)
premultiply TΨ and postmultiplyΨ
1 22
1 2 i =a I+a diag( )
T T TdC a M a K
ω
Ψ Ψ = Ψ Ψ + Ψ Ψ (3.19)
thus
( ) ( ) ( ) ( )dMx t C x t Kx t u t+ + = (3.20)
becomes
52
2 21 2( ) ( )i i i i i ia a f tη ω η ω η+ + + = (3.21)
define
21 2( ) 2i i ia a ω ζ ω+ = (3.22)
we have
22 ( )i i i i i i if tη ζ ωη ω η+ + = (3.23)
where
iζ is the percentage of the critical damping for thi mode,
21 2
2i
ii
a a ωζω
+= (3.24)
For point moment and point force input (see Appendix B)
'2 0 1 0( ) ( ) ( ) ( ) ( )i if t b m t b p t= −Ψ +Ψ (3.25)
where
0 ( )p t is the point force at the location r= 1b ,
0 ( )m t is the point moment acting at the location r= 2b .
3.1.3 Ranking of vibration modes
The size of the structure model, obtained from finite element analysis, is usually very
large. For real time control purpose, a relatively small model is more desirable. The
53
dynamics of ANSYS model can be described well using a small percentage modes of
vibration, depending on what measure of ‘goodness’ is used.
For any mode, if the degree of freedom associated with the applied force has a zero
value, then the force applied at that degree of freedom cannot excite that mode, so the dc and
peak gain will also be zero. If the mode cannot be excited, then it has no effect on the
Fourier amplitude and can be eliminated. Similarly if the degree of freedom associated with
the output has a zero value, then no matter how much force is applied to that mode, there will
be no output. The dc and peak gains are zeros, and the mode can be eliminated because it
also has no effect on the Fourier amplitude [43].
A small state space model can be obtained in two steps, first by defining the
eigenvector elements for all modes for only the input and output degrees of freedom, and
second by analyzing the modal contributions to the overall response and sorting them to
decide which ones have the greatest contribution.
DC gain and peak gain ranking
The general transfer functions for undamped and damped systems are
i i2 2
1
mj ki
ik iF sψ ψψ
ω=
=+∑ (3.26)
i i2 2
1 2
mj ki
ik i i iF s sψ ψψζ ω ω=
=+ +∑ (3.27)
where
ijψ is the thj output row of the thi mode,
54
ikψ is the thk applied force of the thi mode,
iω is the thi eigenvalue.
which means, in general, every transfer function is made up of additive combinations of
single degree of freedom systems. To get the dc gain, substitute 0 0s j jω= = = in equation
3.28, we have
i i2
1
mj ki
ik iFψ ψψω=
=∑ (3.28)
which is same for undamped and damped systems.
To find peak gain, substitute is jω= into the damped system transfer function
i i i i2 2 2 2 2
1 1
i i2
1
2 2
= ( )2 2
m mj k j ki
i ik i i i i i i i
mj k
i i i i
F s s j
j dc gainj
ψ ψ ψ ψψζ ω ω ω ζ ω ω
ψ ψζ ω ζ
= =
=
= =+ + − + +
−=
∑ ∑
∑ (3.29)
If assume uniform damping, constantiζ ζ= = , there is no difference between dc gain and
peak gain rankings.
55
Figure 3.2 Mode contribution versus mode number for piezo input
Figure 3.2 shows the mode contribution versus mode number for piezo input. As we
can see from the figure, the first 6 highest peak gain contribution are mode 1,2,4,6,8,10.
Figure 3.3 shows the mode contribution versus mode number for shaker input. As we can see
from the graph, the first 6 highest peak gain contribution are also mode 1,2,4,6,8,10.
Figures3.4 and 3.5 show the comparison of Bode plot for the reduced model with full model
by peak gain reduction method. As it can be seen from these two figures, the reduced model
(obtained using the first 6 most significant modes) can represent full model (obtained using
all the vibration modes) very well.
56
Figure 3.3 Mode contribution versus mode number for shaker input
Figure 3.4 Comparison of Bode plot of Reduced model by Peak Gain
with Full model (piezo input)
57
Figure.3.5 Comparison of Bode plot of Reduced model by Peak Gain
with Full model (shaker input)
Balanced reduction
The controllability and observability can also be used for ranking the importance of
each mode, which involves calculating the controllability Gramian and observability
Gramian. In general, the controllability Gramian of a given mode has no relationship with
the observability Gramian. Balanced reduction simultaneously considers controllability and
observability in ranking and overcome the uncertainty of using controllability or
observability alone. If the system is normalized properly, the diagonal g of the joint
gramian can be used to reduce the model order. Because g reflects the combined
controllability and observability of individual states of the balanced model. Those states with
a small ( )g i can be deleted while retaining the most important input-output characteristics of
the original system. Consider the following system
58
x Ax Buy Cx Du= += +
(3.30)
with controllability Gramian cW , observability Gramian oW and state coordinate
transformation x Tx= , produced the following equivalent system
1
1
x TAT x TBuy CT x Du
−
−
= +
= + (3.31)
and transforms the Gramians to
1, T Tc c o oW TW T W T W T− −= = (3.32)
find a particular similarity transformation T such that
( )c oW W diag g= = (3.33)
where g is a vector containing the diagonal of the balanced gramian. See ref. [44-45] on
details on this algorithm.
We are now in a position to use the balanced system Gramian, either controllability or
observability, to decide which states are relatively important than others. Figure 3.6 shows
the diagonal of balanced Gramian verus number of states, which means two out of all the
states are very significant and contribute a lot to the overall response. The rest states are
relatively weak.
59
Figure 3.6 Diagonal of balanced Gramian versus number of states
Figure 3.7 Comparison of Bode plot of Reduced model by Balanced Reduction
with Full model (piezo input)
Figures3.7 and 3.8 show the comparison of bode plot of reduced model by balanced
reduction with full model, with piezo input and shaker input, respectively. Again, first 6
60
strongest states were used to have a fair comparison with dc gain and peak gain ranking. The
reduced model represents the system dynamics pretty well as it can be seen from the figures.
Figure 3.8 Comparison of Bode plot of Reduced model by Balanced Reduction
with Full model (shaker input)
One issue with the balanced reduction, unlike the dc gain or peak gain ranking, is that
it is difficult to identify the individual modes in the reduced system model. The system
matrix has to be looked at to identify which modes are included. For simple SISO model, it
can be ranked easily with dc gain or peak gain. But for MIMO system, it can be easily
handled with balanced reduction.
61
3.1.4 State Space Model Formulation
Having the differential equation in the principal coordinate, with the assumption of
point forces or moment as the input(s) and point displacement as the measured output(s), we
have the following state space model for the structural system [42]:
[ ]
2
0 I 0 -2
0
z z uQ
y W z Du
ζ⎡ ⎤ ⎡ ⎤
= +⎢ ⎥ ⎢ ⎥−Ω Ω ⎣ ⎦⎣ ⎦= +
(3.34)
where,
( )( )t
zt
ηη⎧ ⎫
= ⎨ ⎬⎩ ⎭
Modal state vector
1 2( ) ( ), ( ), , ( )uNu t u t u t u t= ⋅⋅⋅ Input
1 2( ) ( , ), ( , ), , ( , )
yN N Ny t x r t x r t x r t= ⋅⋅⋅ Modal output
1 2 , , , mNdiag ω ω ωΩ = ⋅⋅⋅ Natural frequencies
1,1 1,
,1 ,
u
m m u
N
N N N
Qψ ψ
ψ ψ
⋅⋅⋅⎡ ⎤= ⎢ ⎥
⋅⋅⋅⎢ ⎥⎣ ⎦ Modal input matrix
1,1 ,1
1, ,
m
y m y
N
N N N
Wψ ψ
ψ ψ
⋅⋅⋅⎡ ⎤= ⎢ ⎥
⋅⋅⋅⎢ ⎥⎣ ⎦ Modal output matrix
uN Number of inputs
mN Number of modes
yN Number of outputs
1 2( ) ( ), ( ), , ( )m
TNt t t tη η η η= ⋅⋅⋅ Modal displacement
1 2( ) ( ), ( ), , ( )m
TNt t t tη η η η= ⋅⋅⋅ Modal velocity
62
ir Spatial coordinate
1 2 , , , mNdiagζ ζ ζ ζ= ⋅⋅⋅ Modal damping
i jψ Eigenfunction i at location j
The output of the above state space formulation is the displacement. But, in many
practical applications, accurate measurement of the displacements (or velocities) is difficult
to achieve directly, especially for the base acceleration problem, since the foundation of the
structure is moving with the base. While, the accelerometers are readily available and
provide reliable measurements, so it is important to model the acceleration outputs instead of
displacement outputs. For most of the cases, the direct transmission matrix
D= [0]
thus
[ ]0 Wy z= (3.35)
2
0 I 0 -2
z z uQζ
⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥−Ω Ω ⎣ ⎦⎣ ⎦
(3.36)
so
2
0 0 -2
y z WQuW Wζ
⎡ ⎤= +⎢ ⎥− Ω Ω⎣ ⎦
(3.37)
In all, we have
2
2
0 I 0 -2
0 0 -2
z z uQ
y z WQuW W
ζ
ζ
⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥−Ω Ω ⎣ ⎦⎣ ⎦⎡ ⎤
= +⎢ ⎥− Ω Ω⎣ ⎦
(3.38)
63
which results in the modal acceleration output, define
2
0 I -2
Aζ
⎡ ⎤= ⎢ ⎥−Ω Ω⎣ ⎦
(3.39)
0
BQ⎡ ⎤
= ⎢ ⎥⎣ ⎦
(3.40)
2
0 0 -2
CW Wζ
⎡ ⎤= ⎢ ⎥− Ω Ω⎣ ⎦
(3.41)
D WQ= (3.42)
Obviously, the A, B, C, D matrices describe the flexible structures state space model,
which are the functions of the system parameters (natural frequencies, damping ratios, and
mode shapes). Finally, the state space model of flexible structure with acceleration output is
described as
( ) ( )( ) ( )
z A z B uy C z D u
θ θθ θ
= += +
(3.43)
where [ , , ]i i iθ ω ζ ψ=
64
3.1.5 Discrete Time State Space Model
Since most of the system identification routines use discrete time input/output and the
control algorithms are implemented on digital computer, it is desirable to transform the
continuous state space model into discrete state space model. This entails the evaluation of
( )
0( )
A T
T F
e
e B d
θ
γ θ γ
Φ =
Γ = ∫ (3.44)
Thus discrete state space model with acceleration as output is given as follows
( 1) ( ) ( )( ) ( ) ( )
x k x k u ky k Fx k Gu k
+ = Φ +Γ= +
(3.45)
where
( )x k is the modal state vector of the structure system,
( )u k is control signal input,
Φ ,Γ , F ,G are the system matrix, input matrix, output matrix and direct transmission
matrix, respectively.
If define ( )v k is the disturbance signal input and H is the disturbance input matrix, we have
the following general discrete state space model.
( 1) ( ) ( ) ( )( ) ( ) ( )
x k x k u k Hv ky k Fx k Gu k
+ = Φ +Γ += +
(3.46)
65
3.2 Controlled AutoRegressive Integrated Moving Average (CARIMA) Model A model of a system is description of its properties, suitable for a certain purpose. For
model based control system design purpose, the model needs not be a true and accurate
description of the system. It can just be an input and output mapping, which can predict the
output accurate enough for a given input. With a linear model representation of the system, it
is possible to find the analytical solution for the control signal input. A linear black box
model is provided here and Augmented UD Identification used for the adaptive parameter
identification will also be included.
3.2.1 CARIMA model
When considering regulation about a particular operating point, even a nonlinear plant
generally admits a locally linearized model [44]:
1 1( ) ( ) ( ) ( 1) ( )A q y k B q u k v k− −= − + (3.47)
where
( )y k is the plant output at time step k,
( )u k is the plant input at the time k,
( )v k is a disturbance term,
1( )A q− , 1( )B q− are the polynomials in the backward shift operator 1q− , and
1 11( ) 1 ... na
naA q a q a q− − −= + + + (3.48)
1 10 1( ) ... nb
nbB q b b q b q− − −= + + + (3.49)
( )v k is usually considered to be taking the Moving Average form.
1( ) ( ) ( )v k C q kξ−= (3.50)
66
where
1 11( ) 1 ... nc
ncC q c q c q− − −= + + +
( )kξ is an uncorrelated random sequence.
Thus, Controlled Auto-Regressive Moving Average (CARIMA) model is obtained:
1 1 1( ) ( ) ( ) ( 1) ( ) ( )A q y k B q u k C q kξ− − −= − + (3.51)
To model the non-stationary disturbance, such as random steps occurring at random
times and Brownian motion, the following model was used for the disturbance.
1( )( ) ( )C qv k kξ−
=∆
(3.52)
where
∆ is the difference operator which equals to 11 q−− .
Thus, CARIMA model was obtained as follows
1
1 1 ( )( ) ( ) ( ) ( 1) ( )C qA q y k B q u k kξ−
− −= − +∆
(3.53)
or
1 11 1( ) ( ) ( ) ( 1) ( )
( ) ( )A q y k B q u k k
C q C qξ− −
− −
∆ ∆= − + (3.54)
The term ∆ eliminates prediction errors caused by an inaccurate d.c. gain in the model and
removes dc. By ensuring that the degree of 1( )C q− is big enough, the roll –off of the filter
67
also reduces the component of prediction error caused by model mismatch which is often
large at high frequencies.
For simplicity, 1( )C q− is chosen to be 1, or 1 1( )C q− − is truncated and absorbed
into 1( )A q− , 1( )B q− polynomials. Thus,
1 1( ) ( ) ( ) ( ) ( )A q y k B q u k kξ− −∆ = ∆ + (3.55)
If using filtered signals from the plant I/O data
( ) ( )fy k y k= ∆ (3.56)
( ) ( )fu k u k= ∆ (3.57)
then the resulting overall plant model becomes
1 1( ) ( ) ( ) ( 1) ( )f fA q y k B q u k kξ− −= − + (3.58)
To find the j-step-ahead prediction ( )y k j∧
+ , the following identity was considered
1 1( ) ( )jj jI E q A q F q− − −= ∆ + (3.59)
where,
jE , jF are polynomials uniquely defined by the Diophantine equation given 1( )A q− and
the prediction interval j.
Recursion of the Diophantine equation to compute 1( )jE q− , 1( )jF q− was given by Clarke
[46-48] described as follows:
68
1 11 ,0( ) ( ) j
j j jE q E q f q− − −+ = + (3.60)
1 1 11 ,0( ) [ ( ) ( )]j j jF q q F q f A q− − −+ = − (3.61)
with the following initialization
1 1( ) ( )A q A q− −= ∆ (3.62)
11( ) 1E q− = (3.63)
1 11( ) [1 ( )]F q q A q− −= − (3.64)
where ,0jf is the constant coefficient of the polynomial 1( )jF q− .
Using the above identity, and defining 1 1 1( ) ( ) ( )jG q E q B q− − −= , we have
ˆ( | ) ( 1) ( )j jy k j k G u k j F y t+ = ∆ + − + (3.65)
The optimal predictor given measured output data up to time k and any give u(k+j) for
j>1, and ignoring the future noise sequence ( )k jξ + , is give as described in [46] by
1
ˆ( ) ( )j
i ji
y k j g u k j i p=
+ = ∆ + − +∑ , 21,....j N= (3.66)
where
1 2 ( 1) 10 1 2 ( 1)( ... ) ( ) ( ) ( )j
j j j j j j j jp G g g q g q g q u k F q y k− − − − −−= − − − − ∆ + (3.67)
which can be rewritten as
69
y Gu p= + (3.68)
where
[ ( ), ( 1),..., ( 1)]Tuu u k u k u k N= ∆ ∆ + ∆ + −
2ˆ ˆ ˆ ˆ[ ( 1), ( 2),..., ( )]Ty y k y k y k N= + + +
2[ ( 1), ( 2),..., ( )]Tp p k p k p k N= + + +
The matrix G is of dimension 2 2N N×
2 2
1
2 1
1 1
0 0 0
N N
gg g
G
g g g−
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
(3.69)
3.2.2 Conventional Recursive Least Squares
Conventional Recursive Least Squares method is used in nearly all practical adaptive
control, filtering, signal processing and prediction.
Assume the input and output data set from a system are available up to time step k as
(1), (1), (2), (2), ( ), ( )kZ u y u y u k y k= (3.70)
70
and the plant model described by
1 1 2( ) ( 1) ( ) ( 1) ( 2) ( ) ( )n ny k a y k a y k n b u k b u k b u k n kξ+ − + + − = − + − + + − + (3.71)
define
1 2 1 2[ , , , , , , , ]Tn na a a b b bθ =
( ) [ ( 1), ( 2), , ( ), ( 1), ( 2), , ( )]Th k y k y k y k n u k u k u k n= − − − − − − − − −
such that the model can be rewritten as
( ) ( ) ( )Ty k h k kθ ξ= + (3.72)
Given the above plant model and the data vector, the recursive least-squares estimate of the
process parameter vector θ is give by [14]
ˆ ˆ ˆ( ) ( 1) ( )[ ( ) ( ) ( 1)]Tk k L k y k h k kθ θ θ= − + − − (3.73)
where
( 1) ( )( )( ) ( ) ( 1) ( )T
P k h kL kk h k P k h kγ
−=
+ − (3.74)
1 ( 1) ( ) ( ) ( 1)( ) ( 1)( ) ( ) ( ) ( 1) ( )
T
T
P k h k h k P kP k P kk k h k P k h kγ γ
⎡ ⎤− −= − −⎢ ⎥+ −⎣ ⎦
(3.75)
where ( )kγ is a forgetting factor, ( )L k is Kalman gain vector and ( )p k is the covariance
matrix.
The corresponding loss function is given by
71
2
1
( ) ( ) ( 1) ( ) ( )k
k j
j
J k e j J k e k z kγ γ−
=
= = − +∑ (3.76)
where,
( )e k , ( )z k are the residual and innovation sequence of the parameter estimates respectively,
and are defined as
ˆ( ) ( ) ( ) ( )Te k z k h k kθ= − (3.77)
ˆ( ) ( ) ( ) ( 1)Tz k z k h k kθ= − − (3.78)
3.2.3 Augmented UD Identification
In the conventional recursive least squares algorithm, the covariance matrix ( )p k is
updated from ( 1)p k − . One major problem in applications is that this matrix may be ill-
conditioned, and it often leads to negative-definite P and to inaccurate result. In order to
achieve better numerical performance, Bierman proposed UD factorization algorithm [49],
which has a much better numerical performance than RLS. However, the UD factorization
algorithm has not been widely used as RLS because it appears to be more complicated to
interpret and to implement [50]. Augmented UD identification algorithm was initially
proposed by Niu and Fisher [18]. This algorithm is developed by rearranging the data
vectors and augmenting the covariance matrix of Bierman’s UD factorization algorithm.
AUDI permits simultaneous and recursive identification of the model parameters plus the
loss function for all orders from 1 to n at each time step with approximately the same
72
calculation effort as nth order RLS, with better numerical properties [51]. It is described as
follows,
An augmented data vector is defined as follows
TT
n
Tn
kykh
kynkunkykukykukyk
)]( )([
)](),(),(,),2(),2(),1(),1([)(
−=
−−−−−−−−−−−−−=ϕ(3.79)
The parameter vector is also rearranged in an analogous manner
2 2 1 1ˆ ( ) [ , , , , , , ]Tn n nk a b a b a bθ = (3.80)
subscripts ' 'n denotes the assumed maximum order of the model.
Thus, a new covariance matrix, Augmented Information Matrix (AIM), is defined as
follows
1
1
( ) ( ) ( )k
k j Tn n n
j
AIM k j jγ ϕ ϕ−
−
=
⎡ ⎤= ⎢ ⎥⎣ ⎦∑ (3.81)
decomposing ( )nAIM k into TUDU form, results
( ) ( ) ( ) ( )Tn n n nAIM k U k D k U k= (3.82)
where
73
0
1 1
n-1 1 n
ˆ1 (k-n) ˆ ˆ 1 ( 1) (k-n+1)
1 ˆ ˆˆ 1 ( 1) (k-1) ( )( )
n
n
k n
k kU k
α
θ α
θ α θ−
− +
−=
1
(2 1)
1 1 n+ ×
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ (2 1)n+
and
10 0 1 1( ) [ ( ), ( ), , ( ), ( ), ( )]n n n nD k diag J k n L k n J k n L k n J k−
− −= − − − − (3.83)
The symbols 0ˆ ˆˆ ˆ( ), , ( ), ( ), 1, , i i nk n k n i k n i i nα θ α θ− − + − + = in the ( )nU k matrix
represent column vectors, with dimensions from 1 to 2n respectively.
1
1 1
ˆ ( ) ( ) ( ) ( ) ( )k i k
k i j T k i jn i n i n i n i
i j
k i h j h j h j y jθ γ γ−−
− − − −− − − −
= =
⎡ ⎤− = ⎢ ⎥
⎣ ⎦∑ ∑ (3.84)
is the parameters estimates for the ( )n i th− order model ( 0,1, , 1i n= − ) and
1
1 1
ˆ ( ) ( ) ( ) ( ) ( )k i k
k i j T k i jn i n i n i n i
i j
k i j j j u jα γ ϕ ϕ γ ϕ−−
− − − −− − − −
= =
⎡ ⎤− = ⎢ ⎥
⎣ ⎦∑ ∑ (3.85)
The elements 0 0 1 1( ), ( ), , ( ), ( ), ( )n n nJ k n L k n J k n L k n J k− −− − − − of ( )nD k are all scalars,
with
74
2 2
1 1 1
ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ) ( )k k k
T Tn n n n n
j j jJ k y j k h j h j k y jθ θ
= = =
⎛ ⎞= − =⎜ ⎟
⎝ ⎠∑ ∑ ∑ (3.86)
and
2
1( ) ( )
k i
ij i
J k i y j−
= −
− = ∑ (3.87)
ˆ ( )n i k iα − − in ( )nU k and ( )n iL n i− − in ( )nD k are intermediate variables.
3.3 Neural Network Based System Identification
Many papers in literature currently available on system identification are focused on
dealing with models described by linear differential or difference equations. However,
motivated by the fact that the system may exhibit some kind of nonlinear behavior, there has
been a lot of research work on nonlinear system identification. One of the key techniques in
this effort is artificial neural networks. In this section, neural network based system
identification based on an efficient online learning algorithm, recursive Levenberg-
Marquardt optimization, is presented.
3.3.1 Artificial Neural Networks
Artificial neural networks (ANNs) are computational paradigms which implement
simplified models of their biological counterparts, that is, biological neural networks.
Although the initial intent of ANNs was to explore and reproduce human information
processing tasks such as speech, vision, and knowledge processing, ANNs also demonstrated
their superior capability for classification and function approximation problems. This has
great potential for solving complex problems such as systems control, data compression,
75
optimization problems, pattern recognition, and system identification. A neural network is a
powerful data modeling tool that is able to capture and represent complex input/output
relationships. The motivation for the development of neural network technology stemmed
from the desire to develop an artificial system that could perform "intelligent" tasks similar to
those performed by the human brain. Neural networks resemble the human brain in the
following two ways:
• A neural network acquires knowledge through learning.
• A neural network's knowledge is stored within inter-neuron connection strengths
known as synaptic weights.
The true power and advantage of neural networks lies in their ability to represent both
linear and non-linear relationships and in their ability to learn these relationships directly
from the data presented to them. Traditional linear models are simply inadequate when it
comes to modeling systems that contain non-linear characteristics. Properly formulated and
trained NN’s are capable of approximating any linear or nonlinear function to the desired
degree of accuracy [51].
A neural network can have several layers. Each layer has a weight matrix, a bias vector,
and an output vector. It is common for different layers to have different numbers of neurons.
A constant input 1 is fed to the biases for each neuron.
The most common neural network model is the MultiLayer Perceptron (MLP). This
type of neural network is known as a supervised network because it requires a desired output
in order to learn. The goal of this type of network is to create a model that correctly maps the
input to the output using historical data so that the model can then be used to produce the
output when the desired output is unknown. These networks are of feedforward type, wherein
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the effects of the input signals are propagated through the networks layer by layer.
Differences between the desired outputs (targets) and the network outputs give the errors.
The connection strengths (“weights”) and “biases” are updated during training (or, learning)
such that the network produces the desired output for the given input. The multilayer
perceptron type NN’s trained with back-propagation are compact and provide excellent
generalization (i.e., accurate outputs for inputs not encountered during training or learning).
3.3.2 Neural Network AutoRegressive eXternal Input Model
When using black-box method to identify the nonlinear dynamical systems, the
problem of selecting a suitable nonlinear model becoming difficult. MultiLayer Perceptron
(MLP) network discussed above is finding more and more application in those area because
of its universal mapping capability. To choose the model structure for MLP based black-box
model, two issues need to be addressed: first, the inputs to the network, and second, the
internal network architecture. A popular method is to reuse the input structure from the
linear models while letting the internal architecture be feedforward MLP network. This
approach is a natural extension of the well known linear model structure, and suitable for
control system design [39].
AutoRegressive eXternal (ARX) input model structure uses the previous inputs and
outputs as the basis to predict the future output. For Nonlinear AutoRegressive eXternal
(NARX) input model, the one-step-ahead predictor can be described as
( 1| ) ( ( 1), )py k g kθ ϕ θ+ = + (3.89)
77
where
θ is a vector containing the weights (and biases),
g is a nonlinear function used to predict the output,
ϕ is the regression vector.
The general form of regression vector for NARX model is
( 1) [ ( ), , ( 1- ),
( 1- ), , ( 2 - - )]a
Td d b
k y k y k n
u k n u k n n
ϕ + = ⋅ ⋅ ⋅ +
+ ⋅ ⋅ ⋅ + (3.90)
where dn is the time delay, and an , bn are orders of the dynamic system.
The function ( ( 1), )g kϕ θ+ can be any nonlinear function. If using neural network for the
nonlinear function, we will have a Neural Network AutoRegressive external (NNARX) input
model. Figure 3.10 shows the structure of NNARX model.
78
Figure 3.9 Neural Network AutoRegressive eXternal Input model structure
3.3.3 BackPropagation Learning Rule for MLP
The backpropagation algorithm is an extension of the LMS algorithm that can be used
to train multilayer neural networks. Both LMS and backpropagation are approximate
steepest descent algorithms that minimize squared error. The reason it is called
backpropagation is that the derivatives are computed first at the last layer of the network, and
then propagated backward through the network, using chain rule, to compute the derivatives
of the hidden layers [26]. For a given set of inputs to the network, outputs are computed for
each neuron in the first layer, and forwarded to the next layer. The signals propagate on a
layer-by-layer basis till the output layer is reached. The weights and biases remain
unchanged during the ‘forward pass’. The output of the network is compared with the
desired value, and the difference gives the error. The error represents the cost function, and
79
the weights and biases are updated to minimize it. The weights and biases are updated
during the ‘backward pass’ starting from the output layer, and recursively computing the
local gradient for each neuron. The training of the neural network is complete when the error
(or, change in the error) reduces to a predetermined small value.
3.3.4 Online learning method
In the online learning algorithm, only one example [u(t), y(t)], is given at a time and
then discarded after learning. So, it is less memory consuming and at the same time, it fits
well into more natural learning, where the learner receives new information at every moment
and should adapt to it, without having a large memory for storing old data. Apart from easier
feasibility and data handling, the most important advantage of on-line learning is its ability to
adapt to changing environments.
The on-line training algorithm was derived to minimize the criterion.
2 2( ) [ ( ) ( )]kt
F x e r k y k= = −∑ (3.91)
where
1 2[ , , , ]NX w w w= ⋅⋅⋅ , which consists of all weights and bias of the neural network,
ke is the error of the neural network output,
( )r k is the reference input of the neural network,
( )y k is the neural network output.
80
The computational performance of a neural network learning is largely based on the
minimization algorithm. There are many algorithms used for neural network training. The
selection of a minimization method can be based on several criteria such as: number of
iterations to a solution, computational costs and accuracy of the solution. In general, these
approaches are iteration intensive thus making real-time control difficult. Very few papers
address real-time implementation or the papers use plants that have a large time constant
[55]. It is shown that the Levernberg-Marquardt BackPropagation (LMBP) algorithm is the
fastest one for training multilayer networks of moderate size, even though it requires a matrix
inversion at each iteration [26]. It is very suitable for neural network training.
3.3.5 Recursive Levenberg-Marquardt Optimization Algorithm
The recursive Levenberg-Marquardt optimization algorithm is a variation of Newton’s
method that was designed for minimizing functions that are sums of squares of other
nonlinear functions, which significantly outperforms gradient descent and conjugate gradient
methods for medium sized problems. This algorithm is very well suited to neural network
training where the performance index is the mean squared error. Thus the weights and bias
of the NN are at time t adjusted according to
11 [ ( ) ( ) ] ( )T T
k k k k k k kx x J x J x I J x eλ −+ = − + (3.92)
In the above equation, Hessian matrix is approximated as
2( ) ( ) 2 ( ) ( )TH x F x J x J x= ∇ ≅ (3.93)
To guarantee the matrix H invertible, the following modification is used
G H Iλ= + (3.94)
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This algorithm has a very useful feature that as kλ is increased, it approaches the steepest
descent algorithm with small learning rate:
11 1( ) ( )
2T
k k k k kk k
x x J x e x F xλ λ+ ≅ − = − ∇ (3.95)
While as kλ is decreased to zero, the algorithm approaches Gauss-Newton. This algorithm
provides a nice compromise between the speed of Newton’s method and the guaranteed
convergence of steepest descent [26].
The Jacobian matrix for the NN training can be written as
1 2
( ) [ , ]N
F e e eJ XW w w w∂ ∂ ∂ ∂
= = ⋅⋅⋅∂ ∂ ∂ ∂
(3.96)
The LM algorithm requires computation of the Jacobian J matrix at each iteration step and
the inversion of TJ J square matrix.
3.3.6 Matrix Inverse Calculation
To find the search direction of Levenberg-Marquardt online learning algorithm, a linear
system of equations needs to be solved or a matrix inverse to be calculated at each iteration.
Efficient algorithms are required for fast learning of the neural network.
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Matrix Inverse Lemma
Matrix Inverse Lemma is an efficient algorithm to calculate the inverse of a matrix,
which stated that if a matrix A satisfies
1 1 TA B CD C− −= + (3.97)
then
1 1( )T TA B BC D C BC C B− −= − + (3.98)
let
Tk k kA J J Iλ= + (3.99)
1
k
B Iλ
= (3.100)
TkC J= (3.101)
D I= (3.102)
substituting A, B, C, D into the above equation, and for single output networks, we can
finally obtain
11 [ ]
TTk k
k k k kTk k k k
J Jx x I J eJ Jλ λ+ = − −
+ (3.103)
With this lemma, a lot of time is saved in computing the weights and bias change at each
iteration. [54].
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Cholesky Decomposition
Another efficient algorithm to calculate the inverse of a matrix is Cholesky
Decomposition. If a square matrix A happens to be symmetric and positive definite, then it
has a special, efficient, triangular decomposition. Cholesky Decomposition is about a factor
of two faster than alternative method for solving linear equations [55]. Instead of seeking
arbitrary lower and upper triangular factors L andU , Cholesky Decomposition constructs a
lower triangular matrix L whose transpose TL can itself serve as the upper triangular part,
which means
TLL A= (3.104)
, ,Ti j j iL L= (3.105)
and
1
2 1/ 2, , ,
1( )
i
i i i i i kk
L a L−
=
= −∑ (3.106)
1
, , , ,1,
1 ( )i
j i i j i k j kki i
L a L LL
−
=
= −∑ (3.107)
where 1, 2,...,j i i N= + +
After computation of the Cholesky factor, the search direction can be determined in a
two stage procedure employing simple forward and back substitutions.
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Chapter 4
Direct Adaptive Neural Network Control Introduction
The neural networks based control systems can be divided into two fundamentally
different categories:
• Direct Control, where the controller itself is a neural network.
• Indirect Control or Model based Control, the controller is not a neural network, but it
is based on a neural network model of the system.
In this chapter, direct neural network control is introduced and implemented in real
time for vibration control of smart structures. The direct inverse control was first developed
by Widrow [56], and is gaining popularity with the power of neural networks. More details
can be found in ref [57]. The direct inverse control seeks to model the plant inverse. The
controller appears in series with the plant. If the desired output is fed into the model of the
plant inverse, the output of the plant inverse model is the input to the plant, which has the
desired output. The neural networks used in the current research are known as MultiLayer
Perceptrons (MLP) as discussed in Chapter 3.
The design of the neural network controller follows the direct adaptive approach,
wherein the parameters of the controller (weights and biases) are directly adjusted to
minimize the output error. The architecture and training algorithm for the direct adaptive
neural network controller are presented next.
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Direct Adaptive Neural Network Control Architecture Direct Adaptive Neural Network Control (DANNC) uses the inverse of the process as
the controller, which is a popular method for neural network based control, especially useful
for plants having fast dynamics. Figure 4.1 shows the system diagram of direct adaptive
neural network control system.
Figure 4.1 System diagram of direct adaptive Neural Network control system
If a process can be described by
( 1) ( ( ),..., ( 1), ( ),..., ( ))y k g y k y k n u k u k m+ = − + − (4.1)
Then a neural network can be trained to simulate the inverse of the process as
1( ) ( ( 1), ( ),..., ( 1), ( 1),..., ( ))u k g y k y k y k n u k u k m−= + − + − − (4.2)
In direct adaptive control, the cost function is defined with respect to the real plant,
instead of on the basis of a model. The controller adapts in real time, also called online
learning.
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Figure 4.2 Main steps of DANNC learning
Figure 4.3 Schematic diagram of direct adaptive neural network controller
DANNC Online Learning Algorithm The direct controller itself is a neural network, which was trained through recursive
Levenberg-Marquardt online learning algorithm in real time starting with random weights
87
and biases. The following flow diagram describes the main steps of DANNC learning. The
computation is reduced by using a matrix inversion lemma discussed in chapter 3.
Real Time Implementation of DANNC
The direct adaptive neural network controller (Fig. 4.3) has five hidden neurons and one
output neuron, which gives the controller voltages. The inputs consist of time- delayed
values of control signal and target values. Since the external excitation is not accessible in
many practical situations, no disturbance signals are used for the controller. The tapped
delay operator, z-1, yields one time-step delayed version of the input signal, and thereby
builds short-term memory into the system. This feature transforms a static network into a
dynamic network whose output is a function of time [25]. The hidden layer uses the
hyperbolic tangent activation function, which limits the output to ± 1 for large values of the
activation potential. This ensures that the neural network controller signals remain bounded.
For the output neuron, the activation function is purely linear. Mathematically, the equation
for this network architecture is as follows.
( )2 1 1 2a W W P b bϕ= + + (4.3)
In Equation 4.3, a is the output of neural network, namely the control signal ku at time k, 1W
is the first layer weight matrix, 2W is the second layer weight matrix, P is the input, 1b is the
bias of the first layer, 2b is the bias of the second layer, and ϕ is the activation function for
the hidden neurons ( tanhϕ = ). The bias applies an affine transformation to the linear
combination of inputs and weights. The input vector is defined by the following equation.
1 2 1 2[ ; ; ; ; ]k k k k kP u u t t t− − − −= (4.4)
The target t for training the controller is obtained by summing the control signal u and the
plant output y. The error e for the controller is, therefore, equal to the plant output, which is
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minimized during the training. Since the plant output is the tip acceleration, active vibration
suppression is achieved.
k k k
k k k k
t y ue t u y= += − =
(4.5)
The DANNC Simulink/RTW real time implementation block diagram is shown as Fig.
4.4.
Figure 4.4 Direct adaptive NN controller real time implementation block diagram
Figure 4.5 Controlled & uncontrolled response for the 1st mode sine disturbance input
(Original Structure)
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Figure 4.6 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(Original structure)
Figure 4.7 Controlled & uncontrolled response for the white noise disturbance input
(Original structure)
Experimental Results and Discussions
The controller was trained online in real time at the sampling rate of 1000 Hz starting
with random weights and biases in the range of ±1. The neurocontroller weights and biases
were adjusted at every time step, which resulted in updated control signals.
Figures 4.5 to 4.7 show the uncontrolled and controlled responses of the plant for
various excitations generated by the shaker. These excitations (disturbances) included sine
wave at the first and second natural frequencies, and band-limited white noise (0-50 Hz).
The neurocontroller decreases the settling time for the impulse excitation to a very small
value, resulting in over 80 percent reduction (Fig. 4.5 to 4.6). For the 1st Mode sine wave
disturbances, the controller learning is completed within four cycles of oscillations (about 0.5
sec) and the root mean square (RMS) vibrations are reduced by 89 percent (Fig. 4.5). The
controller reduced vibration by 94 percent for the second mode. The controller was also
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tested with a number of white noise disturbances (0-50 Hz) and achieved RMS reductions of
about 60 percent (Fig. 4.7), which is considered satisfactory. The RMS computations also
consider the initial closed-loop response, which is somewhat larger than the initial open-loop
response (but much smaller than the steady state open-loop response), while the
neurocontroller is learning. The controller learning was performed starting from random
weights and biases to demonstrate its learning capability. In practice, the initial response can
be improved by training the controller using theoretical model or the actual system. The
RMS reductions are computed over a nine second time period, although the figures show the
responses for a few seconds only for the clarity of presentation. The reductions in the RMS
vibrations have a very significant effect on the fatigue life of a structure. In general, reducing
the vibrations by just ten percent doubles the fatigue life [32].
Figure 4.8 Controlled & uncontrolled response for the sine wave disturbance changing
from 1st to 2nd mode (Original structure)
Figure 4.9 Controlled & uncontrolled response for the sine wave disturbance changing
from 2nd to 1st mode (Original structure)
In many practical situations, the system dynamics or the external excitation changes
with time. A robust controller is therefore desired to maintain satisfactory performance with
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perturbations in the system. To clearly show the robustness of the neurocontroller, both
external disturbance and plant dynamics were varied during tests. The excitation frequency
was changed from first mode to second mode and vice versa after about four second. Figure
4.8 and 4.9 show that the controller adjusts its parameters quickly and continues to perform
very well even after such large changes in the excitation frequency. The controller responses
again show large RMS reductions.
Figure 4.10 Controlled & uncontrolled response for the 1st mode sine disturbance input
(Tip mass added structure)
Figure 4.11 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(Tip Mass added structure)
Modified Structure with plate added and tip mass added were used to test the
robustness of the neural controller with changes in the plant dynamics. The controller
performance is again excellent for the modified structure with RMS reductions of 85 percent
and 93 percent for excitations at the first and second natural frequency, respectively for the
tip mass added structure (Figs. 4.10 and 4.11), and 87% and 90% reduction respectively at
the first and second natural frequency for the plate added structure (Figs. 4.12 and 4.13).
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Figure 4.12 Controlled & uncontrolled response for the 1st mode sine disturbance input
(Plate added structure)
Figure 4.13 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(Plate added structure)
Direct Inverse Neural Network Control Adaptive neural network controller has an excellent experimental performance as
discussed above. To clearly bring out the advantages of adaptiveness, a Direct Inverse
Neural Network Controller (DINNC) was obtained by first training the controller as above
till the error was reduced to a satisfactory level and then freezing the weights so that the
controller did not change (adapt) further. With the fixed weights and biases, the DINNC is
much more computationally economical than DANNC.
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Experimental Performance Comparison of DANNC and DINNC Combined 1st and 2nd mode sine wave disturbance was used to compare the
performance of two direct neural controllers. Figure 4.14 (a-f) shows the controlled and
uncontrolled Fourier amplitude of the structure for DANNC and DINNC (1st and 2nd modes
are shown separately for clarity). The experiment was repeated five times for both
uncontrolled and controlled cases to obtain the average values and the uncertainty in the
results. For the original structure, the response at first natural frequency shows vibration
reductions of 18 dB, and 14 dB for DANNC and DINNC, respectively. At the second natural
frequency, vibration reductions are about 27dB and 24 dB respectively for DANNC and
DINNC. The maximum uncertainty of ± 0.7 dB was observed in these measurements. When
the structure is modified by adding a plate, DANNC shows 14 and 16 dB reductions for the
first and second mode, while DINNC, shows 11 dB to 16dB reductions. For the tip mass
added case, the performance trends are similar. Overall, the direct adaptive neural network
controller (DANNC) performs better for all cases. However, with online learning (nonlinear
optimization), the DANNC is much more computationally expensive than DINNC.
(a) (b)
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(c) (d)
(e) (f)
Figure 4.14 Controlled and uncontrolled Fourier amplitude of the structure.
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Chapter 5
Model Based Predictive Control
5.1 Introduction
The direct neural network controllers introduced in Chapter 4 is simple in concept
and effective, but it is not an optimal controller. It does not take the control effort into
consideration, and it cannot control inverse unstable systems. Model Based Predictive
Control (MBPC) has been used widely in many applications, especially in the process
control. They usually outperform PID or other direct controllers and are able to control non-
minimum phase, open-loop unstable, time delayed and Multi Input and Multi Output
(MIMO) system. The MBPC includes a variety of control methods that have the following
points in common.
1. A plant model is used to predict the plant output for a certain steps in future. Linear
models are most often used, this is because the possibility of analytic solution for the
future control trajectory for the unconstrained case.
2. A known future reference trajectory is used.
3. A future control sequence is calculated by minimizing a user defined cost function
which includes the predicted future outputs errors and the control effort increments.
4. At each instance, only the first control signal of the calculated control sequence is
applied and the rest are discarded (receding strategy).
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The generalized predictive control (GPC) system is known to control non-minimum
phase plants, open loop unstable plants and plants with variable or unknown dead time. It is
also robust with respect to modeling errors, over and under parameterization, and sensor
noise. Recently, an evaluation of modern adaptive multi-input multi-output (MIMO) control
techniques for active stability augmentation and vibration control of tiltrotor aircraft showed
GPC based MIMO active control to be highly effective [17]. In this chapter, two model-
based predictive control systems, adaptive generalized predictive control based on controlled
autoregressive integrated moving average with augmented UD identification, and neural
adaptive predictive control with recursive Levenberg-Marquardt online learning, will be
investigated and implemented in real time for the vibration control of smart structure.
Comparison will also be made to bring out the advantages and disadvantages of these model-
based controllers for the application of smart structure. Linear quadratic regulator based on
finite element/state space model is also included in this chapter to provide a baseline for
performance comparison.
5.2 LQR Control System Design
There have been major developments in the mathematical theory of multivariable
feedback systems which include the state space concept for system description and the notion
of mathematical optimization for the controller synthesis. Linear quadratic regulator (LQR)
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is one of those controllers which have been studied by researchers. In this section, a LQR
controller based on the integrated Finite Element/ State Space model discussed in Chapter 3
will be presented here and implemented experimentally for the vibration suppression of smart
structure.
5.2.1 Discrete Linear-Quadratic State Feedback Regulator Design The main idea of LQR optimal control is to find the optimal gain matrix K such that
the following state feedback law can be implemented [58].
u(k) Kx(k)= − (5.1)
where
u(k) is the optimal control,
x(k) is the system state vector.
The quadratic cost function
T T
n 1J(u) [x(k) Qx(k) u(k) Ru(k)]
∞
=
= +∑ (5.2)
is minimized to obtain the optimal control gain
T 1 TK(k) ( S(k 1) R) ( S(k 1) )−= Γ + Γ + Γ + Ω (5.3)
where
S is the infinite horizon solution of the associated discrete-time Riccati equation which
is determined by the following equation
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1T TS(k) S(k 1) S(k 1) S(k 1) R S(k 1) Q−
⎡ ⎤= Ω + − + Γ Γ + Γ + Γ + Ω+⎣ ⎦ (5.4)
and
S(k n) S(n)= = (5.5)
Ω is the system matrix,
Γ is the input matrix,
Q is symmetric state weighting function,
R is symmetric control weighting function.
5.2.2 Prediction Estimator LQR optimal controller is based on full state feedback, which is not possible for many
applications. A prediction estimator is needed to estimate the other states based on the
measured states.
Construct a model of the plant using the predicted state vector x(k)
x(k 1) x(k) u(k)+ = Φ +Γ (5.6)
If we define the error in the estimate as
x x x= − (5.7)
The dynamics of the estimator-error described by
x(k 1) x(k)+ = Φ (5.8)
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Feeding back the difference between the measured output and the estimated output and
constantly correcting the model with this error signal, the divergence should be minimized.
px(k 1) x(k) u(k) L (y(k) (Fx(k) Gu(k))+ = Φ +Γ + − + (5.9)
where
Φ is the system matrix,
Γ is the input matrix,
F is the output matrix,
G is the direct transmission matrix,
pL is the prediction estimator gain vector and can be obtained by the pole placement
method.
5.2.3 LQR Control System Architecture
Using the separation principle, the whole problem can be solved in two steps. First,
find the optimal control gain K assuming the full state feedback, and then construct the full
states using estimator. The system diagram is given as follows [58].
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Figure 5.1 System diagram of LQR control
5.2.4 Experimental Results and Discussions The finite element/state space based LQR control system was implemented in the real
time for the application of smart structure using the experimental setup discussed in chapter
2. These excitations (disturbances) include sine wave at the first and second natural
frequencies, and band-limited white noise (0-50 Hz). Figures 5.2 to 5.4 show the controlled
and uncontrolled response of the structure under different excitations, 1st mode, 2nd mode and
white noise. There is an 88% RMS vibration reduction for the 1st mode excitation and 92%
reduction for the second mode. For the white noise excitation, 54% RMS reduction is
achieved.
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Figure 5.2 Controlled & uncontrolled response for the 1st mode sine disturbance input
(Original structure)
Figure 5.3 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(Original structure)
Figure 5.4 Controlled & uncontrolled response for the white noise disturbance input
(Original structure)
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To test the robustness of the FE/SS based LQR controller, excitations changing from 1st
to 2nd mode and from 2nd mode to 1st were used. Figures5.5 and 5.6 show the controlled and
uncontrolled response for these excitations. Figure5.7 shows the controlled and uncontrolled
response of the original structure for the combined sine wave disturbance input in frequency
domain. The figure shows the average of 5 runs for both controlled and uncontrolled case,
and the uncertainty is about±0.7dB. As we can see from the figure, about 18dB and 24dB
reductions in the magnitude of the tip acceleration were achieved for the 1st mode and 2nd
mode respectively.
Figure 5.5 Controlled & uncontrolled response for the sine wave disturbance changing
from 1st to 2nd mode (Original structure)
Figure 5.6 Controlled & uncontrolled response for the sine wave disturbance changing
from 2nd to 1st mode (Original structure)
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Figure 5.7 Fourier amplitude to combined sine wave (1st and 2nd modes)
disturbance input
Modified structure, plate added and tip mass added, were also used as an indication of
changing plant dynamics to test the robustness of the controller. As shown in Figs. 5.8 and
5.9, the controller performance is excellent for the tip mass added structure with RMS
reductions of 82 percent and 92 percent for excitations at the first and second natural
frequency, respectively. For the plate added structure, 88% reduction was achieved at the
first natural frequency excitation (Fig. 5.10), and a relatively lower RMS reduction, which is
79%, for the second mode excitation because of the larger changing of structural dynamics
(Fig. 5.11).
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Figure 5.8 Controlled & uncontrolled response for the 1st mode sine disturbance input
(Tip mass added structure)
Figure 5.9 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(Tip mass added structure)
Figure 5.10 Controlled & uncontrolled response for the 1st mode sine disturbance input (Plate
added structure)
105
Figure 5.11 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(Plate added structure)
5.3 Generalized Predictive Control Techniques
Generalized Predictive Control (GPC) was first presented by Clarke in his classical
papers [46-48]. In Ref [46], he described the basic theory, algorithm and interpretations of
GPC. The expanded interpretations and additional filters are introduced into the GPC
algorithm in Ref [47] and the guaranteed theoretical stability was proved in Ref [48]. The
GPC algorithm uses a receding-horizon strategy to predict plant output over several steps
based on assumed future control inputs. It is known to control non-minimum phase plants,
open loop unstable plants and plants with variable or unknown dead time. It is also robust
with respect to modeling errors, over and under parameterization, and sensor noise. It has
been proved to be efficient, flexible, and successful in many applications.
5.3.1 Cost Function The GPC methodology minimizes a weighted sum (a quadratic function), which
includes the predicted future errors and the control signal increments.
2
1
2 21 2
1
( , , , ) ( ( ) ( )) ( 1)uNN
uj N j
N N N r k j y k j u k jλ λ∧
= =
= + − + + ∆ + −∑ ∑J (5.10)
106
where
( )r k j+ is the desired system output,
( )y k j∧
+ is the predicted system output,
( 1)u k j∆ + − is the control increment,
1N is the minimum costing horizon,
2N is the maximum costing horizon,
uN is the control horizon,
λ is a control-weighting .
The minimization is performed subject to the constraint that control increments ( )u k j∆ + is
zero, for uj N> .
5.3.2 Selection of Horizons for the Performance Index The choice of the parameters in the performance index has a large impact on the
performance of the control system. The term 1N is set to its usual value of 1 (with no loss of
stability if the dead-time of the plant is not exactly known). The maximum costing horizon
2N is also selected to be 1 since the plant model provides one-step-ahead prediction at each
sample time. Multiple-step-ahead prediction would require much larger computational
expense for the plant with fast dynamics. The control horizon is an important design
parameter since control increments are assumed to be zero after an interval uN , that is,
( 1) 0u k j∆ + − = for uj N> . The value of 1uN = is selected, which gives generally
107
acceptable control for open-loop stable plants. With the selection of the horizons, the
performance index is simplified as follows ( ( 1) 0ry k + = for vibration suppression).
[ ] [ ]2 2ˆ( 1) ( 1) ( ) ( 1)ry k y k u k u kλ= + − + + − −J (5.11)
5.4 Adaptive Generalized Predictive Control
In control theory, adaptive control is a scheme that changes its structure and /or
parameter to maintain a consistent performance despite environmental change. Adaptive
control is simply a special type of nonlinear regulator. Adaptive control become popular
since 1970’s as the computing resource improved. Adaptive control can maintain consistent
performance of a system in the presence of uncertainty or unknown variation in plant
parameters. Another advantage of adaptive control is that it requires limited a priori
knowledge of the plant to be controlled [59].
Adaptive generalized predictive control (AGPC) technique, which combines the
advantages of GPC and the adaptive plant model identification, has attracted lots of
attentions recently. The corner stone of this algorithm is the predictive plant model whose
parameters are estimated from online measurements. To realize real time adaptive predictive
control, especially for the application of smart structure because of the fast dynamics, the
adaptive system identification algorithm must be efficient (refer to Chapter 3 for detailed
adaptive modeling techniques). Figure 5.12 is the block diagram for the adaptive generalized
predictive control system. Just as the other general model based control systems, the AGPC
mainly consists of 4 parts, the actual plants, the adaptive plant model, the performance index
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optimization or cost function and the reference trajectory. If the system identification is done
online, the disturbance model will be reflected in the identified plant system model.
Therefore, there is no need to have a disturbance model separately [60].
Figure 5.12 Block diagram of Adaptive Generalized Predictive Control
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5.5 Adaptive Generalized Predictive Control Based on Augmented UD
Identification
Clarke [61] developed an adaptive predictive control algorithm using AUDI
identification method (instead of RLS) and showed its effectiveness through simulations.
Maniar et al. [62] evaluated the performance of the MIMO adaptive generalized predictive
control algorithm based on AUDI (with and without constraints) by experimental application
on a computer-interfaced, pilot-scale process. However, no work has been reported using
AUDI based adaptive predictive control of smart structures. Unlike most of the process
control applications, smart structures have fast dynamics and, therefore, need efficient real
time application algorithms. In this section, the effort to implement this algorithm in real
time and investigate the experimental performance of AUDI based adaptive generalized
predictive control in the vibration suppression of smart structures is discussed.
5.5.1 Derivation of Control Law The Augmented UD Identification has been discussed in Chapter 3. The GPC based
control law can be derived with the identified parameters for the CARIMA plant system
model.
The GPC approach uses a receding horizon strategy, where at each step k, calculates
the vector u comprising ( ), ( 1),..., ( 1)u k u k u k Nu∆ ∆ + ∆ + − by minimizing the cost function
J for the given 1 2 , , , uN N N λ , and the first element of vector u∆ is used and
( ) ( 1)u k u k u= − + ∆ is sent as the control signal to the plant.
Thus, the prediction equation can be refined as
110
y Gu p= + (5.12)
where
[ ( ),..., ( 1)]uu u k u k N= ∆ ∆ + − ,
1 1 2[ ( ), ( 1),..., ( )]Tp p k N p k N p k N= + + + + , and ( )p k j+ is simply the response of the
plant assuming that future controls equal to the previous control ( 1)u k − .
G is a 2 1( 1) uN N N− + × matrix with zero entries ijg for 1j i N− > .
so, the cost function can be rewritten
2
1
2 21 2
1
( , , , ) ( ( ) ) ( 1)uNN
uj N j
N N N r k j Gu p u k jλ λ= =
= + − − + ∆ + −∑ ∑J (5.13)
or
( ) ( )T Tr Gu p r Gu p u uλ= − − − − +J (5.14)
Its minimization implies the optimal control
1( ) ( )T Toptimalu G G I G r pλ −= + − (5.15)
1 1
1 1 1
2 2 2
1
1 1
1 1
0
0
u
N N
N N N
N N N N
g g
g g gG
g g g
−
+ −
− − +
⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
111
so that, the current control ( )u k is give by
( ) ( 1) ( )Toptimalu k u k g r p= − + − (5.16)
where the Tg is the first row of 1( )T TG G I Gλ −+
5.5.2 Real Time Implementation of AGPC and Experimental Results
The augmented UD identification based adaptive generalized predictive control system
is applied to the vibration suppression of a smart structure as described in the chapter 2. To
assess the performance of the control system, excitations at the first two natural frequencies
and band-limited white noise (covering the first two modes) were subsequently used. The
parameter vector of the CARIMA plant model was obtained in real time at the sampling
frequency of 1000 Hz starting from random small initial values, which results in the adaptive
prediction model. Since CARIMA model integrates the plant and the disturbance models
together, a relatively larger plant order (n = 9, after numerical experimentation) is used here
to capture plant dynamics adequately. The predicted (tip) acceleration was used to calculate
the performance index and to determine the best control signal that minimizes the
performance index.
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Figure 5.13 Controlled & uncontrolled response for the 1st mode sine disturbance input
(Original structure)
Figure 5.14 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(Original structure)
Figure 5.15 Controlled & uncontrolled response for the white noise disturbance input
(Original structure)
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Figures 5.13 to 5.15 show the uncontrolled and controlled responses of the plant for
several excitations generated by the shaker. The excitation voltage sent to the shaker
amplifier and the resulting tip accelerations are presented. The RMS reductions were
computed for a ten second time period. The figures show responses for smaller durations for
the clarity of presentation. For 1st and 2nd mode sine wave excitations, RMS reductions of
73% and 87% respectively were achieved. Even for band-limited white noise (0-50 Hz)
disturbance (Fig. 5.15), a large RMS reduction of 61% was observed.
Figure 5.16 Fourier amplitude to combined .sine wave (1st and 2nd modes) disturbance
input (Original structure)
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To analyze the response in the frequency domain, a combination of first and second
mode frequencies was used. The experiment was repeated five times for both uncontrolled
and controlled cases to obtain the average values and the uncertainty in the results. The
fourier amplitude (Fig. 5.16) shows average vibration reductions of 11 dB and 16.1 dB at
first and second natural frequencies, respectively. The maximum uncertainty of ± 1.3 dB was
observed in these measurements.
In many practical situations, the system dynamics or external excitations may change
with time for various reasons. A robust controller is therefore desired to maintain
satisfactory performance with perturbations in the system. To clearly show the robustness of
the adaptive generalized predictive control, the excitation frequency was changed from first
mode to second mode after 7 second (Fig. 5.17) and from second mode to first mode at about
3 second (Fig. 5.18). Figures show that the controller adjusts its parameters quickly and
continues to perform very well even after such large changes in the excitation frequency.
Figure 5.17 Controlled & uncontrolled response for the sine wave disturbance changing
from 1st to 2nd mode (Original structure)
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Figure 5.18 Controlled & uncontrolled response for the sine wave disturbance changing
from 2nd to 1st mode (Original structure)
Figure 5.19 Controlled & uncontrolled response for the 1st mode sine disturbance input
(Plate added structure)
Figure 5.20 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(Plate added structure)
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Figure 5.21 Controlled & uncontrolled response for the 1st mode sine disturbance input
(Tip mass added structure)
Figure 5.22 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(Tip mass added structure)
To test the experimental performance of adaptivity, modifications to the original
structure (Fig. 2.2a) are used, plated added modification (Fig. 2.2b) and tip mass added
modification (Fig. 2.2c). The controller was tested for the modified structures using sine
wave disturbances at the first two natural frequencies (Figs. 5.19 to 5.22). For plate added
case, first and second mode RMS reductions of 72% and 79% respectively were achieved.
The RMS vibrations amplitude at first and second natural frequencies decrease by 68% and
80% respectively for the tip mass added case. These vibration reductions are similar to those
for the original structure indicating that the developed controller is adaptive.
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5.6 Neural Adaptive Predictive Control
GPC was developed originally with linear plant prediction model which results in a
formulation that can be solved analytically [53]. Generalized predictive control for actively
controlling the swash plate of tiltrotor aircraft to enhance aeroelastic stability in the airplane
mode of flight is presented in [63], which uses a linear ARX model.
For nonlinear plants, the ability of the GPC to make accurate predictions can be
enhanced if a neural network is used to learn the dynamics of the plant instead of standard
nonlinear modeling techniques. The use of a neural network as the model affords embedding
plant nonlinearity and allows on-line adaptation. Soloway and Haley [53] developed an
efficient neural generalized predictive control (NGPC) algorithm using a Newton-Raphson
minimization algorithm. Haley [64] demonstrated the feasibility of controlling nonlinear
open loop unstable plant (magnetic levitation) using NGPC.
Pado and Damle [65] demonstrated random vibration suppression of a cantilevered
beam using NGPC. An NGPC based control system was used to reduce the tail buffeting of
the YF-17 aircraft model with PZT patches [66]. However, little work has been done with
on-line plant identification in the application of NGPC to smart structures. Adaptive
identification capability is important since the environment, the structure or the system
dynamics may change with time which is especially true for smart structures. In this section,
the neural adaptive predictive control (NAPC) system is presented. The NAPC system is
based on GPC framework and uses the neural network autoregressive external input model
with recursive Levenberg-Marquardt online learning algorithm discussed in chapter 3.
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5.6.1 Neural Adaptive Predictive Control (NAPC) Architecture The architecture of the NAPC system is shown in Fig. 5.23. It comprises the plant to
be controlled, the performance index optimization algorithm, and the adaptive NNARX plant
model, which is used to predict the output of the plant. The NAPC algorithm operates in two
modes, namely adaptive prediction and control. The adaptive prediction occurs between
samples when the performance index is minimized to calculate the next control input.
Figure 5.23 Block diagram of neural adaptive predictive control system
The algorithm is briefly outlined as follows.
1. Adaptively predict the output using the NNARX model, starting with the previous
calculated control input vector
2. Calculate a new control input vector, which minimizes the performance index.
3. Repeat steps 1 and 2 until the desired minimization is achieved or maximum iteration
is exceeded.
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4. Send the first value of the “best” control input vector to the plant as a new control
signal.
5. Repeat steps 1 and 2 until the desired minimization is achieved or maximum iteration
is exceeded.
5.6.2 NNARX Representation of the Smart Structure Model For any given nonlinear dynamical system, currently there is no systematic way to
determine the orders and the delay of the dynamic system. For current experimental setup
(see Fig. 2.2), , ,a b dn n n were chosen to be 3, 3, 1 respectively (after numerical
experimentation). The function ( (k 1), )g ϕ θ+ in equation 3.89 was chosen to be a feedforward
2-layer neural network with tapped time delays. Then the plant is represented by the
NNARX model shown in Fig. 5.24.
Since the identification is performed on-line in the presence of any disturbances acting
on the plant, hence no separate disturbance model is required [60]. The NNARX
representation of smart structure plant can be expressed by
2 2 2 2 1 2 2 1 1 0( 1) ( ) ( ) ( ( ))py k f a f W a f W W af+ = = = (5.17)
where 1f and 2f are activation functions of the hidden layer and the output layer,
respectively, as follows.
1( ) tanh( )f x x= , 2 ( )f x x= (5.18)
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The quantities 0 1 2, ,a a a are the outputs of input, hidden and output neurons, respectively.
The weights of hidden and output layers are given by 1W and 2W , respectively.
Figure 5.24 NNARX representation of smart structure plant
5.6.3 Derivation of control law It is necessary to apply an iterative search method to minimize the performance index
similar to the strategies used for neural network plant system identification.
( ) ( )( 1) ( ) n nu n u n fα+ = + (5.19)
where,
121
( 1)u n + and ( )u n specify the current and previous iteration of the sequence of future
control input respectively,
( )nα denotes the step size,
( )nf represents the search direction.
With the simplified performance index, Newton–Raphson optimization method [53] is
chosen for the minimization of the cost function. The Newton-Raphson update rule for
( 1)u n + is
12
( 1) ( ) ( ) ( )2J J
u n u n n nuu
−∂ ∂
+ = −∂∂
⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠
(5.20)
where the Jacobian is denoted as
( 1)
( ) ( ) 2 ( 1) 2 [ ( ) ( 1)]( )
y kJ pG n n y k u k u kpu u kρ
∂ +∂= = + + − −∂ ∂
(5.21)
and the Hessian denoted as
2( ) ( ) ( )2
2 2( 1) ( 1)2 2 ( 1) 22( ) ( )
J GH n n n
uu
y k y kp py kpu k u kρ
∂ ∂= =
∂∂
∂ + ∂ += + + +
∂ ∂
⎡ ⎤⎢ ⎥⎣ ⎦
(5.22)
122
The quantities( 1)
( )
y kpu k
∂ +
∂and
2 ( 1)2( )
y kp
u k
∂ +
∂ can be calculated from the NNARX plant system
model, which described by
( 1) 2 1 1 0 1[ ( ) 1]( )
y kp fu k
∂ +=
∂
i
W W a W (5.23)
2 ( 1) 2 1 1 0 1 2[ ( )( 1) ]2( )
y kp fu k
∂ +=
∂
iiW W a W (5.24)
where
1 1 2 1 1 1 21 ( ) 2 [1 ( ) ]f f f f f= − = − − , (5.25)
0 [ ( ) ( 1) ( 2) ( ) ( 1) ( 2)]Ta u k u k u k y k y k y k= − − − − , (5.26)
and 11W is the weight of hidden layer associated with first input ( )u k .
5.6.4 Real Time Implementation of NAPC and Experimental Results To assess the experimental performance of the NAPC system, excitations at the first
two natural frequencies and band-limited white noise (covering the first two modes) were
used. Due to the relatively large magnitude of the excitations used, the plant response (tip
acceleration) shows significant nonlinearity. The nonlinear NNARX plant model was trained
in real time at the sampling frequency of 500 Hz starting from random weights in the range
of 1± . The weights of the NNARX model were updated at each sampling time, which
resulted in the adaptive prediction model. The predicted (tip) acceleration was used to
calculate the performance index and to determine the best control signal which minimizes the
performance index.
123
Figures 5.25 to 5.27 show the uncontrolled and controlled responses of the plant for
several excitations generated by the shaker. The controller learns quickly within a few cycles
of oscillations to start reducing the vibrations. Since the reduction of vibrations in the RMS
sense has a very significant effect on the fatigue life of a structure, the RMS reductions were
computed for a ten second time period. For sine waves at the first and second natural
frequencies and band-limited white noise (0-50 Hz) excitations, RMS reductions of 94%,
94% and 70%, respectively, were achieved. The figures show the response for a few seconds
only for the clarity of presentation.
Figure 5.25 Controlled & uncontrolled response for the 1st mode sine disturbance input
(Original structure)
124
Figure 5.26 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(Original structure)
To analyze the response in the frequency domain, a combination of first and second
frequencies was used. The experiment was repeated seven times for both uncontrolled and
controlled cases to obtain the average values and the uncertainty in the results. The fourier
amplitude (Fig. 5.28) shows average vibration reductions of 22.6 dB and 29 dB at first and
second natural frequencies, respectively. The maximum uncertainty of ± 0.7 dB was
observed in these measurements.
Figure 5.27 Controlled & uncontrolled response for the white noise disturbance input
(Original structure)
125
Figure 5.28 Fourier amplitude to combined .sine wave (1st and 2nd modes) disturbance
input (Original structure)
Figure 5.29 Controlled & uncontrolled response for the sine wave disturbance changing from
1st to 2nd mode (Original structure)
126
Figure 5.30 Controlled & uncontrolled response for the sine wave disturbance changing
from 2nd to 1st mode (Original structure)
In many practical situations, the system dynamics or external excitations may change
with time for various reasons. A robust controller is therefore desired to maintain
satisfactory performance with perturbations in the system. To clearly show the robustness of
the NAPC, the excitation frequency was changed from first mode to second mode and vice
versa after about four second. Figures5.29 and 5.30 show that the controller adjusts its
parameters quickly and continues to perform very well even after such large changes in the
excitation frequency.
Figure 5.31 Controlled & uncontrolled response for the 1st mode sine disturbance input
(Plate added structure)
127
Figure 5.32 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(Plate added structure)
Figure 5.33 Controlled & uncontrolled response for the 1st mode sine disturbance input
(Tip mass added structure)
Figure 5.34 Controlled & uncontrolled response for the 2nd mode sine disturbance input
(Tip mass added structure)
128
To test the experimental performance of adaptivity, two modifications to the original
structure (Fig. 2.2a) are used, plated added modification (Fig. 2.2b) and tip mass added
modification (Fig. 2.2c). The controller was tested for the modified structures using sine
wave disturbances at the first two natural frequencies (Figs. 5.31 to 5.34). For plate added
case, first and second mode RMS reductions of 92% and 90% respectively were achieved.
The RMS vibration amplitudes at first and second natural frequencies decrease by 92% and
95% respectively for the tip mass added case. These vibration reductions are similar to those
for the original structure indicating that the developed controller is highly adaptive.
5.7 Experimental Comparison of Adaptive Predictive Controllers
Two adaptive generalized predictive control systems, one based on linear CARIMA
mode and the other based on nonlinear NNARX model, are applied to the vibration
suppression of a smart structure as described in the previous section. To bring out the
advantages and disadvantages of the two control systems, a comparison is made in this
section.
The excitation voltage sent to the shaker amplifier produces large tip accelerations
resulting in a nonlinear response. Combined sine wave (first two modes) disturbance is used
to evaluate the performance of the controllers. Both NNARX and CARIMA models
integrate the plant and disturbance together. To capture the nonlinearity and complexity
adequately, a relatively larger plant order (n = 9) is required for CARIMA model. With the
nonlinearity of the NNARX model, a small plant order is required (n =3). For neural
adaptive predictive control, the nonlinear plant model (NNARX) is trained in real time at the
129
sampling frequency of 500 Hz starting from random weights and biases in the range of ±1.
The weights and biases of the plant model are updated at each sampling time for adaptive
prediction. The parameter vector of the CARIMA plant model was obtained in real time at
the sampling frequency of 1000 Hz. The parameters of CARIMA model are identified by
augmented UD identification method, which results in the adaptive prediction model. The
predicted (tip) acceleration is used to calculate the performance index and to determine the
best control signal.
(a) (b)
130
(b) (d)
(e) (f)
Figure 5.35 Controlled and uncontrolled fourier amplitude of structure
Figure 5.35 (a-f) shows the controlled and uncontrolled fourier amplitude of the
structure for the two adaptive predictive controllers, wherein first and second modes are
131
shown separately for clarity. The experiment was repeated five times for both uncontrolled
and controlled cases to obtain the average values (shown in the figures) and the uncertainty
in the results. It is observed that for all three different cases (original structure, plate added
structure, and tip mass added structure), NAPC produces better results than AGPC.
(a) (b)
Figure 5.36 Performance comparison between AGPC and NAPC
Figure 5.36 shows a summary of the controller performances. For 1st mode, 11dB,
9.7dB, and 7.3dB vibration reductions were achieved for original, plated added and tip mass
added structure, respectively, with AGPC controller. The NAPC shows 22.6dB, 18dB and
13.7dB reductions for the three cases. For 2nd mode, 16.1dB, 11.4dB, and 13.1dB reductions
were achieved for original, plated added and tip mass added structure, respectively, with
AGPC controller. The NAPC produces 29dB, 18.7dB and 19.4dB reductions for the three
cases. The effect of controller adaptivity is clearly seen with the variations of the structure.
The maximum uncertainty of ± 0.7 dB for NAPC and ± 1.3dB for AGPC was observed in
these measurements.
132
Figure 5.37 Vibration reduction for combined sine wave disturbance input
(Original structure)
Figure 5.37 shows the controlled and uncontrolled response of the original structure to
the combined sine wave disturbance input in the time domain. The RMS vibration
reductions, computed for a ten second time period here, have a very significant effect on the
fatigue life of a structure. The figures show the response for one second only for the clarity
of presentation. The RMS reductions are 86% and 94% with AGPC and NAPC,
respectively.
To test for controller performance with time variation of excitations, the disturbance
frequency was changed from first mode to second mode (and vice versa) after several
seconds (Figure 5.38). The controller performances are similar to those discussed earlier,
133
wherein disturbance frequency remains constant. The RMS vibration reductions of 86% and
73% are obtained for AGPC and NAPC, respectively.
Figure 5.38 Responses to disturbance change from 1st to 2nd mode frequency.
(Plate added structure)
In many practical situations, the structure may change in real time. To test the
effectiveness of the two controllers in such situations, a tip mass was attached to the structure
(using Velcro) during experiment. A sine wave at the first natural frequency of the structure
with tip mass was used for excitation. The response is very small initially and grows in
magnitude due to resonance condition after attaching the tip mass (Fig. 5.39). Both
controllers damp the resonant vibrations significantly. For comparisons, RMS values
134
calculated during 13s to 14s, when the responses reach steady state, are used. The AGPC and
NAPC controllers show 72% and 83% RMS reductions, respectively.
(a)
(b)
(c)
Figure 5.39 Responses to tip mass attachment during experiment
135
The power consumption of the actuator is an important issue in the application of smart
structures. The average electric power consumption is computed by integrating the absolute
value of the instantaneous power over time as follows [38].
max
max
max
'
0
0max
0
t
t
t
p dt C dVp V dtt dtdt
= =∫∫
∫ (5.27)
The actuator capacitance is denoted C and V is the voltage supplied to the actuators.
Similar power consumption was observed for both controllers (Fig. 5.40). With combined
sine wave disturbance input, the power consumption of AGPC is 0.61mW, 1mW, 0.68mW
for original, plate added and tip mass added cases, respectively. The NAPC shows power
consumption of and 0.46mW, 1.2mW, and 0.53mW for the three cases, respectively.
Figure5.40 Power consumption for active vibration control using AGPC and NAPC
136
Chapter 6
Conclusions and Recommendations
6.1 Conclusions
In this dissertation, several modeling and control techniques, both conventional and
neural network based, were developed and implemented in real time for the active vibration
control of smart structures. Comparisons were made between direct adaptive neural network
control and direct inverse neural network control, adaptive generalized predictive control and
neural adaptive predictive control in Chapter 4.7 and 5.7, respectively. In this section, all the
controllers discussed in this dissertation, DANNC, DINNC, AGPC, NAPC, LQR, were
compared in terms of vibration reduction, power consumption and relative computation
effort. To make a fair comparison, all the controllers are tuned to have the best performance.
Figures 6.1 and 6.2 show the controlled and uncontrolled Fourier amplitude of the
structure for different controllers (1st and 2nd modes are shown separately for clarity). The
experiment was repeated five times for both uncontrolled and controlled cases to obtain the
average values and the uncertainty in the results. Overall, neural adaptive predictive
137
controller (NAPC) shows the best performance for all cases and the direct adaptive controller
(DANNC) performs second best. The direct inverse controller (DINNC) and LQR, which do
not have adaptive capability, perform at a substantially lower level. Adaptive generalized
predictive control (AGPC) has a relatively consistent performance because of adaptivity.
Tables 6.1 and 6.2 show the numerical values for performances of these controllers. In
performance index column, u represents plant control signal input, u∆ represents control
signal increment, y stands for tip acceleration and x stands for modal state vector.
Figure6.1 Performance comparison of different controllers (1st mode)
138
Figure 6.2 Performance comparison of different controllers (2nd mode)
Table 6.1 1st Mode FFT Amplitude (dB) Comparison (average of 5 runs)
Table 6.2 2nd Mode FFT Amplitude (dB) Comparison (average of 5 runs)
139
The average powers consumed by all controllers are also compared here. The
maximum power of 10 mW is consumed by DANNC and DINNC comes next which are not
optimal controllers and do not take power consumption into the minimization process. The
power requirements of other controllers, LQR, NAPC and AGPC are all less than 1 mW.
Figure 6.3 Power consumption comparison of different controllers
Based on the implementation effort, computation task, and experimental performance
(Appendix A), the following major conclusions are made:
• Finite element/state space (FE/SS) based linear quadratic regulator (LQR) is a cost
effective method for the application of smart structure. There is no need to build the
expensive actual structural system before the design is complete. Without online adaptation,
the FE/SS based LQR has some kind of derating for large change of structural dynamics in
spite of the robustness. For 2nd mode sine wave excitation, the percentage of vibration
reduction decreases from 92 to 79 with a plate modification. For combined (1st and 2nd
140
mode) sine wave disturbance excitation, the 2nd mode vibration reduction decreases from
24dB to 6dB.
• Direct adaptive neural network control is simple in concept and implementation. With
online adaptation, it can deal with the uncertainty and time variation of smart structure.
Without considering of the control effort, the direct adaptive neural network control is not an
optimal controller. The power consumed by DANNC is almost ten times higher than the
other controllers’ power consumption.
• Adaptive generalized predictive control, which is based on the GPC frame work, takes
both the vibration suppression and control effort into consideration. It can reduce the
vibration at low power consumption. With augmented UD identification, instead of
conventional recursive least squares method, AGPC approach provides many features that
are particularly suitable for real time applications. While, it seems difficult from experiment
for AGPC system to identify and control highly nonlinear system without derating because of
the linear model representation of the smart structural system. It has a relatively consistent
performance in spite of plant or disturbance change because of adaptivity.
• Neural adaptive predictive control is also an optimal controller based on GPC
framework. Experimental results show that, with a nonlinear NN model representation of the
smart structure, neural adaptive predictive control is more effective than adaptive generalized
predictive control which is based on a linear model. With nonlinear optimization involved,
neural adaptive predictive control is much more computationally expensive than adaptive
generalized predictive control. With current experimental setup, the data acquisition
frequency has to reduce from 2kHz to 1kHz to satisfy the computation time requirement.
141
6.2 Recommendations
It is hoped that this dissertation provides valuable information about the identification
and control techniques for the vibration suppression of smart structure. To extend the reach
of this work, a few recommendations on possible future work are suggested:
• Augmented UD Identification provides other information in addition to the model
parameters, such as model order and loss functions, parameter identifiability, noise variance,
and signal-to-noise ratio. More efficient algorithm and better performance may be achieved
with the utilization of this additional information.
• There are also several drawbacks for the neural network based system identification.
There is no systematic way with current state of art to determine the required structure to
model a given system and the difficulty of proving the convergence and stability, also the
properties of the model cannot be analyzed because the network representation is a black-box
model. Future research is required in this direction.
• With nonlinear optimization involved in neural network based adaptive system
identification, the computation work is extensive and difficult for the real time
implementation with current computation technology. A reduced computation task may be
achieved by using multirate for the data acquisition and online learning. The neural network
does not have to learning all the time after the initial learning is finished. The learning
process is triggered only when the system dynamics is changed.
• Current experiments are based on Windows Operation System, which is not a Real
Time Operation System (RTOS). A RTOS, Opal-RT or DSpace, for instance, is needed for
142
time critical application and better performance. With Dedicated RTOS, it is also possible to
find the exact computation time of each modeling and control technique, which will provide
very valuable information for the comparison of controlling and modeling techniques.
• Experimental Evaluation of Multi Input Multi Output (MIMO) systems can be
performed based on the modeling and control techniques discussed in this dissertation. A
much more efficient implementation method is needed for the MIMO system. Using DSP
platform or RTOS maybe the way to increase the implementation efficiency.
143
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153
Appendix A
Input/Output Formulation of the Equation of Motion
For one dimensional structure, when the beam is excited by a point force 0 ( )p t and a
point moment 0 ( )m t , the modal equation of motion is
20 0( ) 2 ( ) ( ) ( ) ( ) ( ) ( )m p
i i i i i i it t t Q r m t Q r p tη ζ ωη ω η+ + = + (C.1)
Where ( )miQ r , ( )p
iQ r are the generalized modal forcing functions due to the point moment
and point force inputs, respectively. The general format for the forcing function is
0
( ) ( ( ), ( )) ( ) ( )L
i i iQ r r F r r F r drψ ψ= = ∫ (C.2)
Where,
Ψ is the eigenvector matrix, which is mass normalized
For point force input 0 ( )p t acting at the location r= 1b , the forcing function can be described
as
0 1( , ) ( ) ( )f r t p t r bδ= − (C.3)
For point moment input 0 ( )m t acting at the location r=b, the forcing function is
154
'0 2( , ) ( ) ( )f r t m t r bδ= − (C.4)
Thus the generalized modal force function,
10( ) ( ) ( ) ( )
Lpi i iQ r r r a dr bψ δ ψ= − =∫ (C.5)
' '20
( ) ( ) ( ) ( )Lm
i i iQ r r r b dr bψ δ ψ= − − =∫ (C.6)
Finally
2 '2 0 1 0( ) 2 ( ) ( ) ( ) ( ) ( ) ( )i i i i i i it t t b m t b p tη ζ ωη ω η ψ ψ+ + = − + (C.7)
155
Appendix B
List of Publications
[1] Jha, R., and He, C., “Design and Experimental Validation of an Adaptive Neurocontroller
for Vibration Suppression,” Journal of Intelligent Material Systems and Structures, Vol.
14, No. 8, August 2003, pp. 497-506.
[2] Jha, R., and He, C., “Neural-Network-Based Adaptive Predictive Control for Vibration
Suppression of Smart Structures,” Smart Materials and Structures, Vol. 11, No. 6,
December 2002, pp. 909-916.
[3] He, C., and Jha, R., “Experimental Evaluation of Augmented UD Identification Based
Vibration Control of Smart Structures,” Journal of Sound and Vibration (In press).
[4] Jha, R., and He, C., “Adaptive Neurocontrollers for Vibration Suppression of Nonlinear
and Time Varying Structures,” Journal of Intelligent Material Systems and Structures (In
press).
156
[5] Jha, R., and He, C., “A Comparative Study of Neural and Conventional Adaptive
Predictive Controllers for Vibration Suppression,” Smart Materials and Structures (under
review).
[6] Jha, R., and He, C., “Design and Experimental Validation of Adaptive Neurocontroller
for Beam Vibration Suppression Using Piezoelectric Actuators,” IMECE2001/AD-23731,
2001 ASME International Mechanical Engineering Congress and Exposition, November
11-16, 2001, New York, NY.
[7] Jha, R., and He, C., “Neural Network Based Adaptive Predictive Control for Vibration
Suppression,” AIAA 2002-1540, 43rd AIAA/ASME/ASCE/AHS/ASC Structures,
Structural Dynamics and Materials Conference, April 22-25, 2002, Denver, CO.
[8] Jha, R., and He, C., “Adaptive Neurocontrollers for Vibration Suppression of Nonlinear
and Time Varying Structures,” 13th International Conference on Adaptive Structures and
Technologies (ICAST '02), October 7-9, 2002, Potsdam/Berlin, Germany.
[9] Jha, R., and He, C., “Neural and Conventional Adaptive Predictive Controllers for Smart
Structures,” AIAA-2003-1808, 44th AIAA/ASME/ASCE/AHS/ASC Structures,
Structural Dynamics and Materials Conference, 7-10 April 2003, Norfolk, VA.
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