Control and estimation of systems with input/output delays

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].

1 Krein Space	p. 1
1.1 Definition of Krein Spaces	p. 1
1.2 Projections in Krein Spaces	p. 3
1.3 Kalman Filtering Formulation in Krein Spaces	p. 4
1.4 Two Basic Problems of Quadratic Forms in Krein Spaces	p. 5
1.4.1 Problem 1	p. 5
1.4.2 Problem 2	p. 6
1.5 Conclusion	p. 6
2 Optimal Estimation for Systems with Measurement Delays	p. 7
2.1 Introduction	p. 7
2.2 Single Measurement Delay Case	p. 7
2.2.1 Re-organized Measurements	p. 9
2.2.2 Re-organized Innovation Sequence	p. 11
2.2.3 Riccati Difference Equation	p. 12
2.2.4 Optimal Estimate x(t \| t)	p. 13
2.2.5 Computational Cost	p. 15
2.3 Multiple Measurement Delays Case	p. 17
2.3.1 Re-organized Measurements	p. 18
2.3.2 Re-organized Innovation Sequence	p. 19
2.3.3 Riccati Equation	p. 20
2.3.4 Optimal Estimate x(t \| t)	p. 22
2.3.5 Numerical Example	p. 24
2.4 Conclusion	p. 26
3 Optimal Control for Systems with Input/Output Delays	p. 27
3.1 Introduction	p. 27
3.2 Linear Quadratic Regulation	p. 28
3.2.1 Duality Between Linear Quadratic Regulation and Smoothing Estimation	p. 29
3.2.2 Solution to Linear Quadratic Regulation	p. 34
3.3 Output Feedback Control	p. 41
3.4 Examples	p. 44
3.5 Conclusion	p. 50
4 H[infinity] Estimation for Discrete-Time Systems with Measurement Delays	p. 53
4.1 Introduction	p. 53
4.2 H[infinity] Fixed-Lag Smoothing	p. 54
4.2.1 An Equivalent H[subscript 2] Estimation Problem in Krein Space	p. 55
4.2.2 Re-organized Innovation Sequence	p. 58
4.2.3 Calculation of the Innovation Covariance	p. 59
4.2.4 H[infinity] Fixed-Lag Smoother	p. 63
4.2.5 Computational Cost Comparison and Example	p. 67
4.2.6 Simulation Example	p. 69
4.3 H[infinity] d-Step Prediction	p. 69
4.3.1 An Equivalent H[subscript 2] Problem in Krein Space	p. 70
4.3.2 Re-organized Innovation	p. 73
4.3.3 Calculation of the Innovation Covariance	p. 74
4.3.4 H[infinity] d-Step Predictor	p. 76
4.4 H[infinity] Filtering for Systems with Measurement Delay	p. 77
4.4.1 Problem Statement	p. 77
4.4.2 An Equivalent Problem in Krein Space	p. 78
4.4.3 Re-organized Innovation Sequence	p. 80
4.4.4 Calculation of the Innovation Covariance Q[subscript w] (t)	p. 82
4.4.5 H[infinity] Filtering	p. 84
4.5 Conclusion	p. 85
5 H[infinity] Control for Discrete-Time Systems with Multiple Input Delays	p. 87
5.1 Introduction	p. 87
5.2 H[infinity] Full-Information Control Problem	p. 88
5.2.1 Preliminaries	p. 89
5.2.2 Calculation of v*	p. 91
5.2.3 Maximizing Solution of J[subscript N] with Respect to Exogenous Inputs	p. 96
5.2.4 Main Results	p. 104
5.3 H[infinity] Control for Systems with Preview and Single Input Delay	p. 106
5.3.1 H[infinity] Control with Single Input Delay	p. 106
5.3.2 H[infinity] Control with Preview	p. 108
5.4 An Example	p. 111
5.5 Conclusion	p. 113
6 Linear Estimation for Continuous-Time Systems with Measurement Delays	p. 115
6.1 Introduction	p. 115
6.2 Linear Minimum Mean Square Error Estimation for Measurement Delayed Systems	p. 116
6.2.1 Problem Statement	p. 116
6.2.2 Re-organized Measurement Sequence	p. 117
6.2.3 Re-organized Innovation Sequence	p. 118
6.2.4 Riccati Equation	p. 119
6.2.5 Optimal Estimate x(t \| t)	p. 120
6.2.6 Numerical Example	p. 121
6.3 H[infinity] Filtering for Systems with Multiple Delayed Measurements	p. 122
6.3.1 Problem Statement	p. 123
6.3.2 An Equivalent Problem in Krein Space	p. 124
6.3.3 Re-organized Innovation Sequence	p. 126
6.3.4 Riccati Equation	p. 127
6.3.5 Main Results	p. 129
6.3.6 Numerical Example	p. 130
6.4 H[infinity] Fixed-Lag Smoothing for Continuous-Time Systems	p. 132
6.4.1 Problem Statement	p. 132
6.4.2 An Equivalent H[subscript 2] Problem in Krein Space	p. 133
6.4.3 Re-organized Innovation Sequence	p. 136
6.4.4 Main Results	p. 137
6.4.5 Examples	p. 140
6.5 Conclusion	p. 141
7 H[infinity] Estimation for Systems with Multiple State and Measurement Delays	p. 143
7.1 Introduction	p. 143
7.2 Problem Statements	p. 144
7.3 H[infinity] Smoothing	p. 145
7.3.1 Stochastic System in Krein Space	p. 146
7.3.2 Sufficient and Necessary Condition for the Existence of an H[infinity] Smoother	p. 148
7.3.3 The Calculation of an H[infinity] Estimator z(t, d)	p. 149
7.4 H[infinity] Prediction	p. 157
7.5 Conclusion	p. 162
8 Optimal and H[infinity] Control of Continuous-Time Systems with Input/Output Delays	p. 163
8.1 Introduction	p. 163
8.2 Linear Quadratic Regulation	p. 164
8.2.1 Problem Statements	p. 164
8.2.2 Preliminaries	p. 165
8.2.3 Solution to the LQR Problem	p. 169
8.2.4 An Example	p. 175
8.3 Measurement Feedback Control	p. 178
8.3.1 Problem Statement	p. 179
8.3.2 Solution	p. 180
8.4 H[infinity] Full-Information Control	p. 185
8.4.1 Problem Statement	p. 185
8.4.2 Preliminaries	p. 187
8.4.3 Calculation of v*	p. 190
8.4.4 H[infinity] Control	p. 195
8.4.5 Special Cases	p. 199
8.5 Conclusion	p. 203
References	p. 205
Index	p. 211

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