This monograph is the first of its kind to present innovative research results on truncated predictor feedback (TPF) designs for general linear systems with input delay. Beginning with a brief review of time delay systems, the first half of the book focuses on TPF with a constant feedback parameter. Both state feedback and output feedback are considered. It is established that TPF achieves stabilization in the presence of an arbitrarily large bounded delay if the open loop system is not exponentially unstable. Examples are presented to illustrate that TPF may fail to stabilize an exponentially unstable system when the delay is sufficiently large. Bounds on the delay are then established under which stabilization can be achieved. The second half of the book explores variations of the TPF laws designed with a non-constant feedback parameter to accommodate unknown delays and improve closed-loop performance. The authors employ a step-by-step approach to presenting the ultimate result on a completely delay-independent feedback law.
Truncated Predictor Based Feedback Designs for Linear Systems with Input Delay will appeal to control engineers, control theorists, and graduate students studying control systems. This volume will also be a valuable resource for engineers and applied mathematicians interested in dynamic systems with time delays.
Author(s): Yusheng Wei, Zongli Lin
Series: Control Engineering
Publisher: Birkhäuser
Year: 2020
Language: English
Pages: 349
City: Cham
Preface
Contents
Notation
1 Introduction
1.1 Introduction to Time Delay Systems
1.1.1 Examples of Time Delay Systems
1.1.1.1 A Predator–Prey Model
1.1.1.2 The Distribution of Primes
1.1.1.3 A Traffic Flow Model
1.1.2 Delay Differential Equations
1.1.3 The Initial Condition, the Cauchy Problem, and the Step Method
1.2 Stability of Time Delay Systems
1.2.1 Stability Definitions
1.2.2 Lyapunov Stability Theorems
1.3 Control Systems with Time Delays
1.3.1 Input and State Delays
1.3.2 An Overview of Stabilization of Time Delay Systems
1.4 Predictor Feedback
1.4.1 Linear Systems with a Single Input Delay
1.4.2 Linear Systems with Multiple Input Delays
1.4.3 Linear Systems with Input and State Delays
1.5 Discrete-Time Systems with Delay
1.5.1 Delay Difference Equations
1.5.2 Stability of Delay Difference Equations
1.5.3 Predictor Feedback
1.6 Notes and References
2 Truncated Predictor Feedback for Continuous-TimeLinear Systems
2.1 Introduction
2.2 The Eigenstructure Assignment Based Design
2.2.1 Low Gain Feedback Design
2.2.2 Truncated Predictor State Feedback Design
2.2.3 Truncated Predictor Output Feedback Design
2.2.4 A Numerical Example
2.3 The Lyapunov Equation Based Design
2.3.1 Low Gain Feedback Design
2.3.2 Truncated Predictor State Feedback Design
2.3.3 Truncated Predictor Output Feedback Design
2.3.4 A Numerical Example
2.4 Conclusions
2.5 Notes and References
3 Truncated Predictor Feedback for Discrete-Time Linear Systems
3.1 Introduction
3.2 The Eigenstructure Assignment Based Design
3.2.1 Low Gain Feedback Design
3.2.2 Truncated Predictor State Feedback Design
3.2.3 Truncated Predictor Output Feedback Design
3.2.4 A Numerical Example
3.3 The Lyapunov Equation Based Design
3.3.1 Low Gain Feedback Design
3.3.2 Truncated Predictor State Feedback Design
3.3.3 Truncated Predictor Output Feedback Design
3.3.4 A Numerical Example
3.4 Conclusions
3.5 Notes and References
4 Truncated Predictor Feedback for General Linear Systems
4.1 Introduction
4.2 Continuous-Time Systems
4.2.1 Truncated Predictor State Feedback Design
4.2.2 Truncated Predictor Output Feedback Design
4.2.3 A Numerical Example
4.3 Discrete-Time Systems
4.3.1 Truncated Predictor State Feedback Design
4.3.2 Truncated Predictor Output Feedback Design
4.3.3 A Numerical Example
4.4 Conclusions
4.5 Notes and References
5 Delay Independent Truncated Predictor Feedback for Continuous-Time Linear Systems
5.1 Introduction
5.2 Delay Independent Truncated Predictor State Feedback Design
5.2.1 Preliminaries
5.2.2 Stability Analysis
5.2.3 Numerical Examples
5.3 Improvement on the Closed-Loop Performance
5.3.1 Time-Varying Low Gain Feedback Design
5.3.2 PDE-ODE Cascade Representation
5.3.3 Direct Stability Analysis
5.3.4 Convergence Rate Analysis
5.3.5 A Numerical Example
5.4 Delay Independent Truncated Predictor Output Feedback Design
5.4.1 Feedback Design
5.4.2 Stability Analysis
5.4.3 Numerical Examples
5.5 Conclusions
5.6 Notes and References
6 Delay Independent Truncated Predictor Feedback for Discrete-Time Linear Systems
6.1 Introduction
6.2 Delay Independent Truncated Predictor State Feedback Design
6.2.1 Preliminaries
6.2.2 An Admissible Delay Bound
6.2.3 Numerical Examples
6.3 Delay Independent Truncated Predictor Output Feedback Design
6.3.1 Feedback Design
6.3.2 Stability Analysis
6.3.3 Numerical Examples
6.4 Conclusions
6.5 Notes and References
7 Regulation of Continuous-Time Linear Input Delayed Systems Without Delay Knowledge
7.1 Introduction
7.2 A Feedback Law with a Time-Varying Parameter
7.3 An Update Algorithm for the Feedback Parameter
7.4 Proof of the Properties of the Closed-Loop Signals
7.5 The PDE Description of the Closed-Loop System
7.6 Regulation Under the Update Algorithm
7.7 A Numerical Example
7.8 Conclusions
7.9 Notes and References
8 Regulation of Discrete-Time Linear Input Delayed Systems Without Delay Knowledge
8.1 Introduction
8.2 An Adaptive Feedback Law
8.3 Closed-Loop Analysis
8.3.1 The Boundedness of V(x(k),γ(k)) and l=2R∞γ(l)V(x(l),γ(l))
8.3.2 The Boundedness of γ(k) Away from Zero
8.3.3 The Regulation of the State and the Input Given a Sufficiently Small γ(0)
8.3.4 The Regulation of the State and the Input Given Any γ(0)
8.4 A Numerical Example
8.5 Conclusions
8.6 Notes and References
References
Index
