Identification of dynamic systems: An introduction with applications

Identification of Dynamic Systems

Rolf Isermann · Marco Munchhof ¨

Identification of Dynamic

Systems

An Introduction with Applications

123

Prof. Dr. -Ing. Dr. h.c. Rolf Isermann

Technische Universitat Darmstadt ¨

Institut fur Automatisierungstechnik ¨

Landgraf-Georg-Straße 4

64283 Darmstadt

Germany

Dr. -Ing. Marco Munchhof, M.S./SUNY ¨

Technische Universitat Darmstadt ¨

Institut fur Automatisierungstechnik ¨

Landgraf-Georg-Straße 4

64283 Darmstadt

Germany

[email protected]

c Springer-Verlag Berlin Heidelberg 2011

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Preface

For many problems of the design, implementation, and operation of automatic con￾trol systems, relatively precise mathematical models for the static and dynamic be￾havior of processes are required. This holds also generally in the areas of natural

sciences, especially physics, chemistry, and biology, and also in the areas of medical

engineering and economics. The basic static and dynamic behavior can be obtained

by theoretical or physical modeling, if the underlying physical laws (first principles)

are known in analytical form. If, however, these laws are not known or are only par￾tially known, or if significant parameters are not known precisely enough, one has

to perform an experimental modeling, which is called process or system identifica￾tion. Then, measured signals are used and process or system models are determined

within selected classes of mathematical models.

The scientific field of system identification was systematically developed since

about 1960 especially in the areas of control and communication engineering. It is

based on the methods of system theory, signal theory, control theory, and statistical

estimation theory and was influenced by modern measurement techniques, digital

computations and the need for precise signal processing, control, and automation

functions. The development of identification methods can be followed in wide spread

articles and books. However, a significant influence had the IFAC-symposia on sys￾tem identification, which were since 1967 organized every three years around the

world, in 2009 a 15th time in Saint-Malo.

The book is intended to give an introduction to system identification in an easy

to understand, transparent, and coherent way. Of special interest is an application￾oriented approach, which helps the user to solve experimental modeling problems. It

is based on earlier books in German, published in 1971, 1974, 1991 and 1992, and

on courses taught over many years. It includes own research results within the last

30 years and publications of many other research groups.

The book is divided into eight parts. After an introductory chapter and a chapter

on basic mathematical models of linear dynamic systems and stochastic signals, part

I treats identification methods with non-parametric models and continuous time sig￾nals. The classical methods of determining frequency responses with non-periodic

VI Preface

and periodic test signals serve to understand some basics of identification and lay

ground for other identifications methods.

Part II is devoted to the determination of impulse responses with auto- and cross￾correlation functions, both in continuous and discrete time. These correlation meth￾ods can also be seen as basic identification methods for measurements with stochas￾tic disturbances. They will later appear as elements of other estimation methods and

allow directly the design of binary test signals.

The identification of parametric models in discrete time like difference equations

in Part III is based mainly on least squares parameter estimation. These estimation

methods are first introduced for static processes, also known as regression analysis,

and then expanded to dynamic processes. Both, non-recursive and recursive param￾eter estimation methods are derived and various modifications are described, like

methods of extended least squares, total least squares, and instrumental variables.

The Bayes and maximum likelihood methods yield a deeper theoretical background,

also with regard to performance bounds. Special chapters treat the parameter estima￾tion of time-variant processes and under closed-loop conditions.

Part IV now looks at parameter estimation methods for continuous-time models.

First parameter estimation is extended to measured frequency responses. Then, the

parameter estimation for differential equations and subspace methods operating with

state variable filters are considered.

The identification of multi-variable systems (MIMO) is the focus of Part V. First

basic structures of linear transfer functions and state space models are considered.

This is followed by correlation and parameter estimation methods, including the

design of special uncorrelated test signals for the simultaneous excitation of sev￾eral inputs. However, sometimes it is easier to identify single-input multiple outputs

(SIMO) processes sequentially.

Of considerable importance for many complex processes is the identification of

non-linear systems, treated in Part VI. Special model structures, like Volterra series,

Hammerstein- and Wiener-models allow applying parameter estimation methods di￾rectly. Then, iterative optimization methods are treated, taking into account multi￾dimensional, non-linear problems. Powerful methods were developed based on non￾linear net models with parametric models like neural networks and their derivations

and look-up tables (maps) as non-parametric representations. Also, extended Kalman

filters can be used.

Some miscellaneous issues, which are common to several identification methods,

are summarized in Part VII, as e.g. numerical aspects, practical aspects of parameter

estimation and a comparison of different parameter estimation methods.

Part VIII then shows the application of several treated identification methods to

real processes like electrical and hydraulic actuators, machine tools and robots, heat

exchangers, internal combustion engines and the drive dynamic behavior of automo￾biles.

The Appendix as Part IX then presents some mathematical aspects and a de￾scription of the three mass oscillator process, which is used as a practical example

throughout the book. Measured data to be used for applications by the reader can be

downloaded from the Springer web page in the Internet.

Preface VII

The wide topics of dynamic system identification are based on the research per￾formed by many experts. Because some early contributions lay the ground for many

other developments we would just like to mention a few authors from early semi￾nal contributions. The determination of characteristic parameters of step responses

was published by V. Strejc (1959). First publications on frequency response measure￾ment with orthogonal correlation go back to Schaefer and Feissel (1955) and Balchen

(1962). The field of correlation methods and ways to design pseudo-random-binary

signals was essentially brought forward by e.g. Chow, Davies (1964), Schweitzer

(1966), Briggs (1967), Godfrey (1970) and Davies (1970). The theory and appli￾cation of parameter estimation for dynamic processes was around 1960 until about

1974 essentially promoted by works of J. Durbin, R.C.K. Lee, V. Strejc, P. Eykhoff,

K.J. Åström, V. Peterka, H. Akaike, P. Young, D.W. Clarke, R.K. Mehra, J.M.

Mendel, G. Goodwin,

This was followed by many other contributions to the field which are cited in the

respective chapters, see also Table 1.3 for an overview over the literature in the field

of identification.

The authors are also indebted to many contributions for developing and applying

identifications methods from researchers at our own group since 1973 until now, like

M. Ayoubi, W. Bamberger, U. Baur, P. Blessing, H. Hensel, R. Kofahl, H. Kurz,

K.H. Lachmann, O. Nelles, K.H. Peter, R. Schumann, S. Toepfer, M. Vogt, and R.

Zimmerschied. Many other developments with regard to special dynamic processes

are referenced in the chapters on applications.

The book is dedicated as an introduction to system identification for undergrad￾uate and graduate students of electrical and electronic engineering, mechanical and

chemical engineering and computer science. It is also oriented towards practicing

engineers in research and development, design and production. Preconditions are ba￾sic undergraduate courses of system theory, automatic control, mechanical and/or

electrical engineering. Problems at the end of each chapter allow to deepen the un￾derstanding of the presented contents.

Finally we would like to thank Springer-Verlag for the very good cooperation.

Darmstadt, Rolf Isermann

June 2010 Marco Münchhof

L. Ljung , and T. S derstr m. ö ö

Contents

1 Introduction ................................................... 1

1.1 Theoretical and Experimental Modeling . . ...................... 1

1.2 Tasks and Problems for the Identification of Dynamic Systems . . . . 7

1.3 Taxonomy of Identification Methods and Their Treatment in This

Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Overview of Identification Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4.1 Non-Parametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4.2 Parametric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.4.3 Signal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.5 Excitation Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.6 Special Application Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.6.1 Noise at the Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.6.2 Identification of Systems with Multiple Inputs or Outputs . . 23

1.7 Areas of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.7.1 Gain Increased Knowledge about the Process Behavior . . . . 24

1.7.2 Validation of Theoretical Models . . . . . . . . . . . . . . . . . . . . . . . 25

1.7.3 Tuning of Controller Parameters . . . . . . . . . . . . . . . . . . . . . . . . 25

1.7.4 Computer-Based Design of Digital Control Algorithms . . . . 25

1.7.5 Adaptive Control Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.7.6 Process Supervision and Fault Detection . . . . . . . . . . . . . . . . . 26

1.7.7 Signal Forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.7.8 On-Line Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.8 Bibliographical Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Mathematical Models of Linear Dynamic Systems and Stochastic

Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1 Mathematical Models of Dynamic Systems for Continuous Time

Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.1 Non-Parametric Models, Deterministic Signals . . . . . . . . . . . 34

X Contents

2.1.2 Parametric Models, Deterministic Signals . . . . . . . . . . . . . . . . 37

2.2 Mathematical Models of Dynamic Systems for Discrete Time Signals 39

2.2.1 Parametric Models, Deterministic Signals . . . . . . . . . . . . . . . . 39

2.3 Models for Continuous-Time Stochastic Signals . . . . . . . . . . . . . . . . . 45

2.3.1 Special Stochastic Signal Processes . . . . . . . . . . . . . . . . . . . . . 51

2.4 Models for Discrete Time Stochastic Signals . . . . . . . . . . . . . . . . . . . . 54

2.5 Characteristic Parameter Determination . . . . . . . . . . . . . . . . . . . . . . . . 58

2.5.1 Approximation by a First Order System . . . . . . . . . . . . . . . . . 59

2.5.2 Approximation by a Second Order System . . . . . . . . . . . . . . . 60

2.5.3 Approximation by nth Order Delay with Equal Time

Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

2.5.4 Approximation by First Order System with Dead Time . . . . . 68

2.6 Systems with Integral or Derivative Action . . . . . . . . . . . . . . . . . . . . . 69

2.6.1 Integral Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.6.2 Derivative Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Part I IDENTIFICATION OF NON-PARAMETRIC MODELS IN THE

FREQUENCY DOMAIN — CONTINUOUS TIME SIGNALS

3 Spectral Analysis Methods for Periodic and Non-Periodic Signals. . . . 77

3.1 Numerical Calculation of the Fourier Transform . . . . . . . . . . . . . . . . . 77

3.1.1 Fourier Series for Periodic Signals . . . . . . . . . . . . . . . . . . . . . . 77

3.1.2 Fourier Transform for Non-Periodic Signals . . . . . . . . . . . . . . 78

3.1.3 Numerical Calculation of the Fourier Transform . . . . . . . . . . 82

3.1.4 Windowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3.1.5 Short Time Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.2 Wavelet Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.3 Periodogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4 Frequency Response Measurement with Non-Periodic Signals . . . . . . . 99

4.1 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2 Fourier Transform of Non-Periodic Signals . . . . . . . . . . . . . . . . . . . . . 100

4.2.1 Simple Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.2.2 Double Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.2.3 Step and Ramp Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.3 Frequency Response Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.4 Influence of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Contents XI

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5 Frequency Response Measurement for Periodic Test Signals . . . . . . . . 121

5.1 Frequency Response Measurement with Sinusoidal Test Signals . . . 122

5.2 Frequency Response Measurement with Rectangular and

Trapezoidal Test Signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.3 Frequency Response Measurement with Multi-Frequency Test

Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.4 Frequency Response Measurement with Continuously Varying

Frequency Test Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.5 Frequency Response Measurement with Correlation Functions . . . . . 129

5.5.1 Measurement with Correlation Functions . . . . . . . . . . . . . . . . 129

5.5.2 Measurement with Orthogonal Correlation . . . . . . . . . . . . . . . 134

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Part II IDENTIFICATION OF NON-PARAMETRIC MODELS WITH

CORRELATION ANALYSIS — CONTINUOUS AND DISCRETE TIME

6 Correlation Analysis with Continuous Time Models . . . . . . . . . . . . . . . . 149

6.1 Estimation of Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.1.1 Cross-Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

6.1.2 Auto-Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

6.2 Correlation Analysis of Dynamic Processes with Stationary

Stochastic Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

6.2.1 Determination of Impulse Response by Deconvolution . . . . . 154

6.2.2 White Noise as Input Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

6.2.3 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6.2.4 Real Natural Noise as Input Signal . . . . . . . . . . . . . . . . . . . . . . 161

6.3 Correlation Analysis of Dynamic Processes with Binary Stochastic

Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.4 Correlation Analysis in Closed-Loop . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7 Correlation Analysis with Discrete Time Models . . . . . . . . . . . . . . . . . . . 179

7.1 Estimation of the Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . 179

7.1.1 Auto-Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

7.1.2 Cross-Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.1.3 Fast Calculation of the Correlation Functions . . . . . . . . . . . . . 184

7.1.4 Recursive Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

XII Contents

7.2 Correlation Analysis of Linear Dynamic Systems . . . . . . . . . . . . . . . . 190

7.2.1 Determination of Impulse Response by De-Convolution . . . . 190

7.2.2 Influence of Stochastic Disturbances . . . . . . . . . . . . . . . . . . . . 195

7.3 Binary Test Signals for Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . 197

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Part III IDENTIFICATION WITH PARAMETRIC MODELS — DISCRETE TIME

SIGNALS

8 Least Squares Parameter Estimation for Static Processes . . . . . . . . . . . 203

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

8.2 Linear Static Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

8.3 Non-Linear Static Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

8.4 Geometrical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

8.5 Maximum Likelihood and the Cramér-Rao Bound . . . . . . . . . . . . . . . 215

8.6 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

8.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

9 Least Squares Parameter Estimation for Dynamic Processes . . . . . . . . 223

9.1 Non-Recursive Method of Least Squares (LS). . . . . . . . . . . . . . . . . . . 223

9.1.1 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

9.1.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

9.1.3 Covariance of the Parameter Estimates and Model

Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

9.1.4 Parameter Identifiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

9.1.5 Unknown DC Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

9.2 Spectral Analysis with Periodic Parametric Signal Models . . . . . . . . 257

9.2.1 Parametric Signal Models in the Time Domain. . . . . . . . . . . . 257

9.2.2 Parametric Signal Models in the Frequency Domain . . . . . . . 258

9.2.3 Determination of the Coefficients . . . . . . . . . . . . . . . . . . . . . . . 259

9.2.4 Estimation of the Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 261

9.3 Parameter Estimation with Non-Parametric Intermediate Model . . . . 262

9.3.1 Response to Non-Periodic Excitation and Method of Least

Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

9.3.2 Correlation Analysis and Method of Least Squares

(COR-LS) 264

9.4 Recursive Methods of Least Squares (RLS) . . . . . . . . . . . . . . . . . . . . . 269

9.4.1 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

9.4.2 Recursive Parameter Estimation for Stochastic Signals . . . . . 276

9.4.3 Unknown DC values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

9.5 Method of weighted least squares (WLS) . . . . . . . . . . . . . . . . . . . . . . . 279

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Contents XIII

9.5.1 Markov Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

9.6 Recursive Parameter Estimation with Exponential Forgetting . . . . . . 281

9.6.1 Constraints and the Recursive Method of Least Squares . . . . 283

9.6.2 Tikhonov Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

9.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

10 Modifications of the Least Squares Parameter Estimation . . . . . . . . . . 291

10.1 Method of Generalized Least Squares (GLS) . . . . . . . . . . . . . . . . . . . . 291

10.1.1 Non-Recursive Method of Generalized Least Squares (GLS) 291

10.1.2 Recursive Method of Generalized Least Squares (RGLS) . . . 294

10.2 Method of Extended Least Squares (ELS) . . . . . . . . . . . . . . . . . . . . . . 295

10.3 Method of Bias Correction (CLS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

10.4 Method of Total Least Squares (TLS) . . . . . . . . . . . . . . . . . . . . . . . . . . 297

10.5 Instrumental Variables Method (IV) . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

10.5.1 Non-Recursive Method of Instrumental Variables (IV) . . . . . 302

10.5.2 Recursive Method of Instrumental Variables (RIV) . . . . . . . . 305

10.6 Method of Stochastic Approximation (STA) . . . . . . . . . . . . . . . . . . . . 306

10.6.1 Robbins-Monro Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

10.6.2 Kiefer-Wolfowitz Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

10.7 (Normalized) Least Mean Squares (NLMS) . . . . . . . . . . . . . . . . . . . . . 310

10.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

11 Bayes and Maximum Likelihood Methods . . . . . . . . . . . . . . . . . . . . . . . . 319

11.1 Bayes Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

11.2 Maximum Likelihood Method (ML) . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

11.2.1 Non-Recursive Maximum Likelihood Method . . . . . . . . . . . . 323

11.2.2 Recursive Maximum Likelihood Method (RML) . . . . . . . . . . 328

11.2.3 Cramér-Rao Bound and Maximum Precision . . . . . . . . . . . . . 330

11.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

12 Parameter Estimation for Time-Variant Processes . . . . . . . . . . . . . . . . . 335

12.1 Exponential Forgetting with Constant Forgetting Factor . . . . . . . . . . 335

12.2 Exponential Forgetting with Variable Forgetting Factor . . . . . . . . . . . 340

12.3 Manipulation of Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

12.4 Convergence of Recursive Parameter Estimation Methods . . . . . . . . . 343

12.4.1 Parameter Estimation in Observer Form . . . . . . . . . . . . . . . . . 345

12.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

XIV Contents

13 Parameter Estimation in Closed-Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

13.1 Process Identification Without Additional Test Signals . . . . . . . . . . . 354

13.1.1 Indirect Process Identification (Case a+c+e) . . . . . . . . . . . . . . 355

13.1.2 Direct Process Identification (Case b+d+e) . . . . . . . . . . . . . . . 359

13.2 Process Identification With Additional Test Signals . . . . . . . . . . . . . . 361

13.3 Methods for Identification in Closed Loop . . . . . . . . . . . . . . . . . . . . . . 363

13.3.1 Indirect Process Identification Without Additional Test

Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

13.3.2 Indirect Process Identification With Additional Test Signals . 364

13.3.3 Direct Process Identification Without Additional Test Signals 364

13.3.4 Direct Process Identification With Additional Test Signals . . 364

13.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

Part IV IDENTIFICATION WITH PARAMETRIC MODELS — CONTINUOUS

TIME SIGNALS

14 Parameter Estimation for Frequency Responses . . . . . . . . . . . . . . . . . . . 369

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

14.2 Method of Least Squares for Frequency Response Approximation

(FR-LS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370

14.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

15 Parameter Estimation for Differential Equations and Continuous

Time Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

15.1 Method of Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

15.1.1 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

15.1.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382

15.2 Determination of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

15.2.1 Numerical Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383

15.2.2 State Variable Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

15.2.3 FIR Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

15.3 Consistent Parameter Estimation Methods . . . . . . . . . . . . . . . . . . . . . . 393

15.3.1 Method of Instrumental Variables. . . . . . . . . . . . . . . . . . . . . . . 393

15.3.2 Extended Kalman Filter, Maximum Likelihood Method . . . . 395

15.3.3 Correlation and Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . 395

15.3.4 Conversion of Discrete-Time Models . . . . . . . . . . . . . . . . . . . . 398

15.4 Estimation of Physical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399

15.5 Parameter Estimation for Partially Known Parameters . . . . . . . . . . . . 404

15.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406