New trends in optimal filtering and control for polynomial and time-delay systems

0. 1 Introduction Although the general optimal solution of the ?ltering problem for nonlinear state and observation equations confused with white Gaussian noises is given by the Kushner equation for the conditional density of an unobserved state with respect to obser- tions (see [48] or [41], Theorem 6. 5, formula (6. 79) or [70], Subsection 5. 10. 5, formula (5. 10. 23)), there are a very few known examples of nonlinear systems where the Ku- ner equation can be reduced to a ?nite-dimensional closed system of ?ltering eq- tions for a certain number of lower conditional moments. The most famous result, the Kalman-Bucy ?lter [42], is related to the case of linear state and observation equations, where only two moments, the estimate itself and its variance, form a closed system of ?ltering equations. However, the optimal nonlinear ?nite-dimensional ?lter can be - tained in some other cases, if, for example, the state vector can take only a ?nite number of admissible states [91] or if the observation equation is linear and the drift term in the 2 2 state equation satis?es the Riccati equation df /dx + f = x (see [15]). The complete classi?cation of the "general situation" cases (this means that there are no special - sumptions on the structure of state and observation equations and the initial conditions), where the optimal nonlinear ?nite-dimensional ?lter exists, is given in [95].

1 Optimal Filtering for Polynomial Systems	p. 1
1.1 Filtering Problem for Polynomial State over Linear Observations	p. 1
1.1.1 Problem Statement	p. 1
1.1.2 Optimal Filter for Polynomial State over Linear Observations	p. 2
1.1.3 Optimal Third-Order State Filter for Automotive System	p. 8
1.1.4 State Estimation of Bilinear Terpolymerization Process	p. 10
1.2 Filtering Problem for Polynomial State with Partially Measured Linear Part	p. 17
1.2.1 Problem Statement	p. 17
1.2.2 Optimal Filter for Polynomial State with Partially Measured Linear Part over Linear Observations	p. 18
1.2.3 Example	p. 21
1.3 Filtering Problem for Polynomial State with Multiplicative Noise	p. 24
1.3.1 Problem Statement	p. 24
1.3.2 Optimal Filter for Polynomial State with Multiplicative Noise over Linear Observations	p. 25
1.3.3 Example	p. 29
1.4 Filtering Problem for Polynomial State with Partially Measured Linear Part and Multiplicative Noise	p. 33
1.4.1 Problem Statement	p. 33
1.4.2 Optimal Filter for Polynomial State with Partially Measured Linear Part and Polynomial Multiplicative Noise over Linear Observations	p. 35
1.4.3 Cubic Sensor Optimal Filtering Problem	p. 38
1.5 Filtering Problem for Linear State over Polynomial Observations	p. 40
1.5.1 Problem Statement	p. 40
1.5.2 Optimal Filter for Linear State over Polynomial Observations	p. 41
1.5.3 Example: Third-Order Sensor Filtering Problem	p. 43
2 Further Results: Optimal Identification and Control Problems	p. 47
2.1 Optimal Joint State and Parameter Identification Problem for Linear Systems	p. 47
2.1.1 Problem Statement	p. 47
2.1.2 Optimal State Filter and Parameter Identifier for Linear Systems	p. 48
2.1.3 Example	p. 50
2.2 Dual Optimal Control Problems for Polynomial Systems	p. 53
2.2.1 Optimal Control Problem for Bilinear State with Linear Input	p. 53
2.2.2 Optimal Regulator for Terpolymerization Reactor	p. 56
2.2.3 Optimal Control for Third-Order Polynomial State with Linear Input	p. 61
2.2.4 Optimal Third-Order Polynomial Regulator for Automotive System	p. 62
2.3 Optimal Controller Problem for Third-Order Polynomial Systems	p. 65
2.3.1 Problem Statement	p. 65
2.3.2 Separation Principle for Polynomial Systems	p. 66
2.3.3 Optimal Controller Problem Solution	p. 68
2.3.4 Optimal Third-Order Polynomial Controller for Automotive System	p. 69
3 Optimal Filtering Problems for Time-Delay Systems	p. 75
3.1 Filtering Problem over Observations with Multiple Delays	p. 75
3.1.1 Problem Statement	p. 75
3.1.2 Optimal Filter over Observations with Multiple Delays	p. 76
3.1.3 Example	p. 80
3.2 Filtering Problem for Linear Systems with State Delay	p. 84
3.2.1 Problem Statement	p. 84
3.2.2 Optimal Filter for Linear Systems with State Delay	p. 85
3.2.3 Example	p. 89
3.3 Filtering Problem for Linear Systems with State and Observation Delays	p. 94
3.3.1 Problem Statement	p. 94
3.3.2 Optimal Filter for Linear Systems with State and Observation Delays	p. 95
3.3.3 Optimal Filter for Linear Systems with Commensurable State and Observation Delays	p. 98
3.3.4 Example	p. 99
3.3.5 Discussion	p. 103
3.4 Filtering Problem for Linear Systems with State and Multiple Observation Delays	p. 103
3.4.1 Problem Statement	p. 103
3.4.2 Optimal Filter for Linear Systems with State and Multiple Observation Delays	p. 105
3.4.3 Optimal Filter for Linear Systems with Commensurable State and Observation Delays	p. 108
3.4.4 Example	p. 110
3.5 Filtering Problem for Linear Systems with Multiple State and Observation Delays	p. 113
3.5.1 Problem Statement	p. 113
3.5.2 Optimal Filter for Linear Systems with Multiple State and Observation Delays	p. 114
3.6 Alternative Optimal Filter for Linear State Delay Systems	p. 118
3.6.1 Example	p. 120
3.7 Filtering Problem for Nonlinear State over Delayed Observations	p. 122
3.7.1 Problem Statement	p. 122
3.7.2 Optimal Filter for Nonlinear State over Delayed Observations	p. 123
3.7.3 Example	p. 127
4 Optimal Control Problems for Time-Delay Systems	p. 131
4.1 Optimal Control Problem for Linear Systems with Multiple Input Delays	p. 131
4.1.1 Problem Statement	p. 131
4.1.2 Optimal Control Problem Solution	p. 132
4.1.3 Example	p. 132
4.1.4 Proof of Optimal Control Problem Solution	p. 134
4.1.5 Duality between Filtering and Control Problems for Time-Delay Systems	p. 138
4.2 Optimal Control Problem for Linear Systems with Equal State and Input Delays	p. 141
4.2.1 Problem Statement	p. 141
4.2.2 Optimal Control Problem Solution	p. 141
4.2.3 Example	p. 142
4.2.4 Proof of Optimal Control Problem Solution	p. 146
4.3 Optimal Control Problem for Linear Systems with Multiple State Delays	p. 148
4.3.1 Problem Statement	p. 148
4.3.2 Optimal Control Problem Solution	p. 148
4.3.3 Example	p. 150
4.3.4 Proof of Optimal Control Problem Solution	p. 153
4.4 Optimal Control Problem for Linear Systems with Multiple State and Input Delays	p. 156
4.4.1 Problem Statement	p. 156
4.4.2 Optimal Control Problem Solution	p. 157
4.4.3 Example	p. 159
4.4.4 Proof of Optimal Control Problem Solution	p. 163
4.5 Optimal Controller Problem for Linear Systems with Input and Observation Delays	p. 166
4.5.1 Problem Statement	p. 166
4.5.2 Separation Principle for Time-Delay Systems	p. 167
4.5.3 Optimal Control Problem Solution	p. 168
4.5.4 Example	p. 168
5 Sliding Mode Applications to Optimal Filtering and Control	p. 175
5.1 Optimal Robust Sliding Mode Controller for Linear Systems with Input and Observation Delays	p. 175
5.1.1 Problem Statement	p. 175
5.1.2 Design Principles for State Disturbance Compensator	p. 176
5.1.3 Design Principles for Observation Disturbance Compensator	p. 177
5.1.4 Robust Sliding Mode Controller Design for Linear System with Input and Observation Delays	p. 179
5.1.5 Example	p. 181
5.2 Optimal and Robust Control for Linear State Delay Systems	p. 185
5.2.1 Optimal Control Problem	p. 185
5.2.2 Optimal Control Problem Solution	p. 185
5.2.3 Robust Control Problem	p. 186
5.2.4 Design Principles	p. 187
5.2.5 Robust Sliding Mode Control Design for Linear State Delay Systems	p. 189
5.2.6 Example	p. 189
5.2.7 Proof of Optimal Control Problem Solution	p. 192
References	p. 199
Index	p. 205

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