identification and control for vibration suppression …people.clarkson.edu/~rjha/research/phd...

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).

6

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

7

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


(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


mass added structure) ……………………………………………….………………………79


mass added structure) ………………………………………………..………………………79


added structure) …………………………………..……………………..…………………..79

14


(plate added structure) ….………………….…………..…………………..………………..80

Figure 5.12 Block diagram of Adaptive Generalized Predictive Control ……..…………….83


(original structure) ………………………………………………………..…………………87


(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


2nd to 1st mode (original structure) ……………………………………..…………………….90


added structure) ……………………………………………………………………………..90


(plate added structure) ……………………………………….………..……………………..90


mass added structure) …………………………………………………..……………………91


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


(original structure) ……………………………………………………..…………………….98


(original structure) ……………………………………………………..…………………….99


(original structure)…………………………….……………..……………………………….99

Figure 5.28 Response to combined .sine wave (1st and 2nd modes) disturbance input (original

structure)……………………………………………………………….…………………….100


1st to 2nd mode (original structure)……………………………………………….….……….100


2nd to 1st mode (original structure)…………………………..……………………….………101


added structure) ……………………………………………………………………….…….101


(plate added structure) ……………………………………………………………………..102


mass added structure) ……………………………………………………………………….102


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

76

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)

81

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.

82

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].

83

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.

84

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.

85

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.

86

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

88

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


(Original Structure)

89


(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

90

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)


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

91

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.


(Tip mass added structure)


(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).

92


(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.

93

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)

94

(c) (d)

(e) (f)

Figure 4.14 Controlled and uncontrolled Fourier amplitude of the structure.

95

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).

96

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)

97

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

98

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)

99

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].

100

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.

101







102

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.





103

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).

104





Figure 5.10 Controlled & uncontrolled response for the 1st mode sine disturbance input (Plate

added structure)

105



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

108

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

109

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.

112







113


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)

114

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.



115







116





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.

117

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.

118

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.

119

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)

120

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


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.



124



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.



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



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.



127







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


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.


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.

identification and control for vibration suppression …people.clarkson.edu/~rjha/research/phd...

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