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Lecture Notes in Control and Information Sciences

By: Huanshui Zhang, Lihua Xie

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Paperback | 5 April 2007

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Time delay systems exist in many engineering ?elds such as transportation, communication, process engineering and more recently networked control s- tems. In recent years,time delaysystems haveattracted recurring interests from research community. Much of the research work has been focused on stability analysis and stabilization of time delay systems using the so-called Lyapunov- Krasovskii functionals and linear matrix inequality (LMI) approach. While the LMI approach does provide an e?cient tool for handling systems with delays in state and/or inputs, the LMI based results are mostly only su?cient and only numerical solutions are available. For systems with knownsingle input delay, there have been rather elegant- alytical solutions to various problems such as optimal tracking, linear quadratic regulation and H control. We note that discrete-time systems with delays can ? usually be converted into delay free systems via system augmentation, however, theaugmentationapproachleadsto muchhigher computationalcosts,especially for systems of higher state dimension and large delays. For continuous-time s- tems,time delayproblemscaninprinciple betreatedby thein?nite-dimensional system theory which, however,leads to solutions in terms of Riccati type partial di?erential equations or operator Riccati equations which are di?cult to und- stand and compute. Some attempts have been made in recent years to derive explicit and e?cient solutions for systems with input/output (i/o) delays. These include the study ontheH controlofsystemswith multiple input delaysbased ? on the stable eigenspace of a Hamlitonian matrix [46].

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Krein Space	p. 1
Definition of Krein Spaces	p. 1
Projections in Krein Spaces	p. 3
Kalman Filtering Formulation in Krein Spaces	p. 4
Two Basic Problems of Quadratic Forms in Krein Spaces	p. 5
Problem 1	p. 5
Problem 2	p. 6
Conclusion	p. 6
Optimal Estimation for Systems with Measurement Delays	p. 7
Introduction	p. 7
Single Measurement Delay Case	p. 7
Re-organized Measurements	p. 9
Re-organized Innovation Sequence	p. 11
Riccati Difference Equation	p. 12
Optimal Estimate x(t t)	p. 13
Computational Cost	p. 15
Multiple Measurement Delays Case	p. 17
Re-organized Measurements	p. 18
Re-organized Innovation Sequence	p. 19
Riccati Equation	p. 20
Optimal Estimate x(t t)	p. 22
Numerical Example	p. 24
Conclusion	p. 26
Optimal Control for Systems with Input/Output Delays	p. 27
Introduction	p. 27
Linear Quadratic Regulation	p. 28
Duality Between Linear Quadratic Regulation and Smoothing Estimation	p. 29
Solution to Linear Quadratic Regulation	p. 34
Output Feedback Control	p. 41
Examples	p. 44
Conclusion	p. 50
H[infinity] Estimation for Discrete-Time Systems with Measurement Delays	p. 53
Introduction	p. 53
H[infinity] Fixed-Lag Smoothing	p. 54
An Equivalent H[subscript 2] Estimation Problem in Krein Space	p. 55
Re-organized Innovation Sequence	p. 58
Calculation of the Innovation Covariance	p. 59
H[infinity] Fixed-Lag Smoother	p. 63
Computational Cost Comparison and Example	p. 67
Simulation Example	p. 69
H[infinity] d-Step Prediction	p. 69
An Equivalent H[subscript 2] Problem in Krein Space	p. 70
Re-organized Innovation	p. 73
Calculation of the Innovation Covariance	p. 74
H[infinity] d-Step Predictor	p. 76
H[infinity] Filtering for Systems with Measurement Delay	p. 77
Problem Statement	p. 77
An Equivalent Problem in Krein Space	p. 78
Re-organized Innovation Sequence	p. 80
Calculation of the Innovation Covariance Q[subscript w] (t)	p. 82
H[infinity] Filtering	p. 84
Conclusion	p. 85
H[infinity] Control for Discrete-Time Systems with Multiple Input Delays	p. 87
Introduction	p. 87
H[infinity] Full-Information Control Problem	p. 88
Preliminaries	p. 89
Calculation of v*	p. 91
Maximizing Solution of J[subscript N] with Respect to Exogenous Inputs	p. 96
Main Results	p. 104
H[infinity] Control for Systems with Preview and Single Input Delay	p. 106
H[infinity] Control with Single Input Delay	p. 106
H[infinity] Control with Preview	p. 108
An Example	p. 111
Conclusion	p. 113
Linear Estimation for Continuous-Time Systems with Measurement Delays	p. 115
Introduction	p. 115
Linear Minimum Mean Square Error Estimation for Measurement Delayed Systems	p. 116
Problem Statement	p. 116
Re-organized Measurement Sequence	p. 117
Re-organized Innovation Sequence	p. 118
Riccati Equation	p. 119
Optimal Estimate x(t t)	p. 120
Numerical Example	p. 121
H[infinity] Filtering for Systems with Multiple Delayed Measurements	p. 122
Problem Statement	p. 123
An Equivalent Problem in Krein Space	p. 124
Re-organized Innovation Sequence	p. 126
Riccati Equation	p. 127
Main Results	p. 129
Numerical Example	p. 130
H[infinity] Fixed-Lag Smoothing for Continuous-Time Systems	p. 132
Problem Statement	p. 132
An Equivalent H[subscript 2] Problem in Krein Space	p. 133
Re-organized Innovation Sequence	p. 136
Main Results	p. 137
Examples	p. 140
Conclusion	p. 141
H[infinity] Estimation for Systems with Multiple State and Measurement Delays	p. 143
Introduction	p. 143
Problem Statements	p. 144
H[infinity] Smoothing	p. 145
Stochastic System in Krein Space	p. 146
Sufficient and Necessary Condition for the Existence of an H[infinity] Smoother	p. 148
The Calculation of an H[infinity] Estimator z(t, d)	p. 149
H[infinity] Prediction	p. 157
Conclusion	p. 162
Optimal and H[infinity] Control of Continuous-Time Systems with Input/Output Delays	p. 163
Introduction	p. 163
Linear Quadratic Regulation	p. 164
Problem Statements	p. 164
Preliminaries	p. 165
Solution to the LQR Problem	p. 169
An Example	p. 175
Measurement Feedback Control	p. 178
Problem Statement	p. 179
Solution	p. 180
H[infinity] Full-Information Control	p. 185
Problem Statement	p. 185
Preliminaries	p. 187
Calculation of v*	p. 190
H[infinity] Control	p. 195
Special Cases	p. 199
Conclusion	p. 203
References	p. 205
Index	p. 211
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Lecture Notes in Control and Information Sciences

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