State Space Systems With Time-Delays Analysis, Identification, and Applications

By Ya Gu and Chuanjiang Li

State Space Systems with Time-Delays Analysis, Identification and Applications covers the modeling, identification and control of industrial applications, including system identification, parameter estimation, dynamic simulation, nonlinear control, and other emerging techniques. The book introduces basic time-delay systems, architectures and control methods. Emphasis is placed on the mathematical analysis of these systems, identifying them, and applying them to practical engineering problems such as three independent water tank systems and distillation systems. This book contains a wide range of time-delay system identification methods that can help readers master the system controllers’ design methods.

• Presents the basic concepts of time delay systems stability analysis and classical time delay system identification methods
• Discusses the stability analysis of complex time delay systems
• Presents the identification of uncertain and unknown time delay systems
• Provides examples of industrial application

• Language: English
• Publisher: Academic Press
• Release date: Mar 25, 2023
• ISBN: 9780323972123

Ya Gu

She obtained her PhD from Jiangnan University, China in 2015. She was a visiting student of the University of Alberta in 2014 and a post doctor of the University of the West of England in 2019. Her main research interests are in the areas of system modelling, system identification, time delay system and control applications.

Book preview

Front Cover for State Space Systems With Time-Delays Analysis, Identification, and Applications - 1st edition - by Ya Gu, Chuanjiang Li

State Space Systems With Time-Delays Analysis, Identification, and Applications

Ya Gu

College of Information, Mechanical and Electrical Engineering, Shanghai Normal University, Shanghai, China

Chuanjiang Li

Shanghai Normal University, Shanghai, China

Table of Contents

Cover image

Title page

Copyright

About the authors

Acknowledgment

Introduction

1 An overview of each chapter

Chapter 1. One-unit state-delay identification

Abstract

1.1 Auxiliary model identification method for a unit time-delay system

1.2 Parameter and state estimation algorithm for one-unit state-delay system

1.3 Conclusions

References

Chapter 2. D-step state-delay identification

Abstract

2.1 State filtering and parameter estimation for d-step state delay

2.2 Identification and U-control of dual-rate state-space models with d-step state delay

2.3 Parameter estimation algorithm for d-step time-delay systems

2.4 Communicative state and parameters estimation for dual-rate state-space systems with time delays

2.5 Conclusion

References

Chapter 3. Multiple state-delay identification

Abstract

3.1 Parameter estimation and convergence for state-space model with time delay

3.2 Iterative parameter estimation for state-space model with multistate delays based on decomposition

3.3 Least squares-based iterative parameter estimation algorithm for multiple time delay

3.4 Two-stage least squares-based iterative parameter identification algorithm for time-delay systems

3.5 Conclusions

References

Chapter 4. Multivariable time-delay system identification

Abstract

4.1 Parameter estimation for a multivariable state space system with d-step delay

4.2 State filtering and parameter estimation for multivariable system

References

Chapter 5. Nonlinear time-delay system identification

Abstract

5.1 Parameter estimation for a Hammerstein state-space system with time delay

5.2 The bias compensation-based parameter and state estimation for a nonlinear system

5.3 Conclusion

Appendix

References

Chapter 6. Uncertain state delay systems identification

Abstract

6.1 State space model identification of multirate processes with uncertain time delay

6.2 Moving horizon estimation for multirate system with time-varying time delays

References

Index

Copyright

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About the authors

Ya Gu is an associate professor at the College of Information, Mechanical and Electrical Engineering, Shanghai Normal University, Shanghai, China. She received her PhD degree in automatic control from Jiangnan University, Wuxi, China, in 2015. She was a visiting PhD student from 2014 to 2015 at the University of Alberta, Edmonton, AB, Canada. She is the reviewer of Signal Processing, IET Control Theory & Applications, and Journal of the Franklin Institute. Her current research interests include model identification, parameter estimation, and adaptive control. E-mail: [e-mail].

Chuanjiang Li is a professor at the College of Information, Mechanical and Electrical Engineering, Shanghai Normal University, Shanghai, China. He obtained his BS degree from Henan University of Science and Technology, China, in 2000, MS degree from Henan University of Science and Technology, China, in 2003, and PhD degree from Shanghai University, China, in 2014. His current research interests are robot control, intelligent control theory, and application. E-mail: [e-mail].

Acknowledgment

This work was supported by the National Natural Science Foundation of China (No. 61903050), the Natural Science Foundation of Shanghai (No. 22ZR1445300).

I wish to express my sincere appreciation to the following professors who have encouraged and guided me throughout this book: Prof. Feng Ding, Jiangnan University, and Prof. Quanmin Zhu, University of the West of England.

Introduction

1 An overview of each chapter

System identification and parameter estimation are mostly applied to the mathematical model of the system with input and output variables. However, in the industrial system, many controlled objects need to be abstracted into state space models of the system, and in general, a physical phenomenon with time delay accompanies the controlled object, time delay is considered to be the most difficult to control, and the identification of delay system is also very difficult. In recent years, delay system has been an active research area, the application is very extensive especially in the industrial process, and it is still a research focus in the field of international process control. Therefore delay system is also the research hotspot in system identification. It is a subject worthy of further study to use system identification methods consciously and reasonably to identify time delay. This book focuses on the identification of state space system with time delay, and the proposed algorithms have theoretical value and application prospects. This book gives the detailed discussion and research, and the following results are obtained. A brief introduction of the major research contents and achievements is outlined in the following.

In Chapter 1, for the state space model with a unit time delay, by extending the state equation, using the properties of the shift operators, and eliminating some state variables, the state space system with time delay is converted into the form of the input–output expression, which is the identification model of the system. When the states are unmeasurable, they are replaced with their estimates, by using the parameter estimation and known input–output data to estimate states. This chapter proposes the auxiliary model–based least squares identification method, the stochastic gradient identification algorithm, and the least squares–based iterative parameter estimation algorithm for the state space model with a unit time delay. The simulation examples verify the effectiveness of the proposed algorithms.

In Chapter 2, for the state space model with d-step time delay, by using the least squares identification algorithm to estimate the parameters and states, first, the identification model is derived, the unknown noise and state variables in the information vectors are replaced with their estimates, and the parameters are estimated. Then, the parameter estimates are used to compute the states. For the multivariable state space model with d-step time delay, the number of variables is large, and the structure of the model is complex; while some single variable system identification methods can be applied to some of the multivariable systems, these identification algorithms for multivariable systems are not enough. To guarantee the identification accuracy of the algorithm, we also need to do some improvements to the original algorithm. This chapter mainly uses hierarchical identification theory to improve the computational efficiency of identification algorithm. Finally, simulation examples show the effectiveness of the proposed algorithms.

In Chapter 3, for the state space model with multistate delays, the number of delays is large, and the theoretical derivation of multistate delays is more complex than that of a unit time delay. This chapter uses hierarchical gradient–based iterative identification algorithm and hierarchical least squares–based iterative identification algorithm to estimate the parameters of the system. Decomposing the system into two subsystems reduces the calculation. For the dual-rate state space model with multistate delays, we use the data filtering technique; compared with the auxiliary model-based least squares algorithm, the filtering-based least squares algorithm can produce more accurate parameter estimates.

Chapter 4 considers the identification problem of the state space model with d-step delay for multivariable systems and presents a state estimation-based recursive least squares parameter identification algorithm by using the hierarchical identification principle. Combining the linear transformation and the property of the shift operator, a state space system is transformed into an equivalent canonical state space model, and its identification model is derived. Finally, an example is provided to validate the proposed theorems.

Chapter 5 researches parameter estimation problems for a Hammerstein input nonlinear system with state time delay. Combining the linear transformation and the property of the shift operator, the Hammerstein state space system is equivalent to a bilinear parameter identification model. The gradient-based and least squares–based iterative parameter estimation algorithms are used for identifying the state time-delay system, and the proposed iterative algorithms make full use of all data at each iteration, which can produce highly accurate parameter estimation. Finally, the example is provided to validate the proposed algorithms.

In Chapter 6, for the uncertain state space model with time delay, we mainly adopt two kinds of methods, one is the moving horizon estimation algorithm, and the basic idea is to derive the cost function and optimize the objective function; compared with the general Kalman filter algorithm, the proposed algorithm can simultaneously estimate the continuous state and discrete time delay. Another method is the expectation maximization algorithm, and it has E step and M step. Step E is used to calculate the expectation of the complete data which is often referred to as Q function, and step M is used to maximize the Q function. These two steps iterate until it converges. Finally, simulation examples are given.

Chapter 1

One-unit state-delay identification

Abstract

This chapter first considers parameter estimation problems for state-space systems with time delay. By means of the property of the shift operator, the state-space systems are transformed into input–output representations, and an auxiliary model identification method is presented to estimate the system parameters. Then, it derives a state estimation-based parameter identification algorithm for state-space systems with one-unit state delay. We derive the identification model of an observability canonical state-space system with one-unit state delay. The key is to replace the unknown states in the parameter estimation algorithm with their state estimates and to identify the parameters of state-space models.

Keywords

Parameter estimation; state-space models; time delay; auxiliary model identification; least squares; state estimation

1.1 Auxiliary model identification method for a unit time-delay system

State-space models have wide applications in many areas, for example, system modeling, system identification, signal processing, adaptive filtering, and control (Devakar and Lyengar, 2011). There exist many estimation methods for state-space models, such as the least squares methods, the auxiliary model identification methods, and the stochastic gradient methods (Ding et al., 2012).

It considers identification problems of time-delay control systems based on the auxiliary model identification idea. The auxiliary model method is a new-type parameter estimation one and can deal with identification problems with the information vector including unknown internal variables. It presents an auxiliary model-based identification algorithm of the input–output representations corresponding to state-space systems with time delays. The basic idea is, by means of the property of the shift operator, to transform the state-space model with a time delay into an input–output representation and then to identify the parameters of the input–output representation based on the auxiliary model identification technique. The proposed method has the advantage of handling the unmeasured variables in the information vector.

1.1.1 The input–output representation

Let us define some notations. represents the estimate of θ at time t; A ent X or X ≔ A stands for A is defined as X; the symbol I(In) stands