By Eli Gershon

Complicated subject matters on top of things and Estimation of State-Multiplicative Noisy platforms starts with an advent and large literature survey. The textual content proceeds to hide the sphere of H∞ time-delay linear platforms the place the problems of balance and L2−gain are awarded and solved for nominal and unsure stochastic structures, through the input-output procedure. It provides strategies to the issues of state-feedback, filtering, and measurement-feedback keep watch over for those structures, for either the continual- and the discrete-time settings. within the continuous-time area, the issues of reduced-order and preview monitoring keep watch over also are awarded and solved. the second one a part of the monograph issues non-linear stochastic nation- multiplicative structures and covers the problems of balance, regulate and estimation of the platforms within the H∞ feel, for either continuous-time and discrete-time situations. The ebook additionally describes specific themes comparable to stochastic switched structures with stay time and peak-to-peak filtering of nonlinear stochastic structures. The reader is brought to 6 sensible engineering- orientated examples of noisy state-multiplicative keep watch over and filtering difficulties for linear and nonlinear platforms. The booklet is rounded out by way of a three-part appendix containing stochastic instruments worthy for a formal appreciation of the textual content: a easy creation to stochastic regulate tactics, features of linear matrix inequality optimization, and MATLAB codes for fixing the L2-gain and state-feedback keep watch over difficulties of stochastic switched structures with dwell-time. complicated issues up to the mark and Estimation of State-Multiplicative Noisy structures should be of curiosity to engineers engaged up to speed platforms study and improvement, to graduate scholars focusing on stochastic keep watch over concept, and to utilized mathematicians drawn to keep watch over difficulties. The reader is predicted to have a few acquaintance with stochastic keep watch over conception and state-space-based optimum keep an eye on conception and strategies for linear and nonlinear systems.

Table of Contents

Cover

Advanced issues on top of things and Estimation of State-Multiplicative Noisy Systems

ISBN 9781447150695 ISBN 9781447150701

Preface

Contents

1 Introduction

1.1 Stochastic State-Multiplicative Time hold up Systems

1.2 The Input-Output method for behind schedule Systems

1.2.1 Continuous-Time Case

1.2.2 Discrete-Time Case

1.3 Non Linear keep an eye on of Stochastic State-Multiplicative Systems

1.3.1 The Continuous-Time Case

1.3.2 Stability

1.3.3 Dissipative Stochastic Systems

1.3.4 The Discrete-Time-Time Case

1.3.5 Stability

1.3.6 Dissipative Discrete-Time Nonlinear Stochastic Systems

1.4 Stochastic tactics - brief Survey

1.5 suggest sq. Calculus

1.6 White Noise Sequences and Wiener Process

1.6.1 Wiener Process

1.6.2 White Noise Sequences

1.7 Stochastic Differential Equations

1.8 Ito Lemma

1.9 Nomenclature

1.10 Abbreviations

2 Time hold up structures - H-infinity regulate and General-Type Filtering

2.1 Introduction

2.2 challenge formula and Preliminaries

2.2.1 The Nominal Case

2.2.2 The strong Case - Norm-Bounded doubtful Systems

2.2.3 The powerful Case - Polytopic doubtful Systems

2.3 balance Criterion

2.3.1 The Nominal Case - Stability

2.3.2 powerful balance - The Norm-Bounded Case

2.3.3 strong balance - The Polytopic Case

2.4 Bounded actual Lemma

2.4.1 BRL for not on time State-Multiplicative platforms - The Norm-Bounded Case

2.4.2 BRL - The Polytopic Case

2.5 Stochastic State-Feedback Control

2.5.1 State-Feedback keep watch over - The Nominal Case

2.5.2 strong State-Feedback keep watch over - The Norm-Bounded Case

2.5.3 powerful Polytopic State-Feedback Control

2.5.4 instance - State-Feedback Control

2.6 Stochastic Filtering for behind schedule Systems

2.6.1 Stochastic Filtering - The Nominal Case

2.6.2 strong Filtering - The Norm-Bounded Case

2.6.3 strong Polytopic Stochastic Filtering

2.6.4 instance - Filtering

2.7 Stochastic Output-Feedback keep an eye on for not on time Systems

2.7.1 Stochastic Output-Feedback keep an eye on - The Nominal Case

2.7.2 instance - Output-Feedback Control

2.7.3 strong Stochastic Output-Feedback regulate - The Norm-Bounded Case

2.7.4 powerful Polytopic Stochastic Output-Feedback Control

2.8 Static Output-Feedback Control

2.9 strong Polytopic Static Output-Feedback Control

2.10 Conclusions

3 Reduced-Order H-infinity Output-Feedback Control

3.1 Introduction

3.2 challenge Formulation

3.3 The not on time Stochastic Reduced-Order H regulate 8

3.4 Conclusions

4 monitoring keep an eye on with Preview

4.1 Introduction

4.2 challenge Formulation

4.3 balance of the not on time monitoring System

4.4 The State-Feedback Tracking

4.5 Example

4.6 Conclusions

4.7 Appendix

5 H-infinity keep watch over and Estimation of Retarded Linear Discrete-Time Systems

5.1 Introduction

5.2 challenge Formulation

5.3 Mean-Square Exponential Stability

5.3.1 instance - Stability

5.4 The Bounded genuine Lemma

5.4.1 instance - BRL

5.5 State-Feedback Control

5.5.1 instance - powerful State-Feedback

5.6 behind schedule Filtering

5.6.1 instance - Filtering

5.7 Conclusions

6 H-infinity-Like regulate for Nonlinear Stochastic Syste8 ms

6.1 Introduction

6.2 Stochastic H-infinity SF Control

6.3 The In.nite-Time Horizon Case: A Stabilizing Controller

6.3.1 Example

6.4 Norm-Bounded Uncertainty within the desk bound Case

6.4.1 Example

6.5 Conclusions

7 Non Linear platforms - H-infinity-Type Estimation

7.1 Introduction

7.2 Stochastic H-infinity Estimation

7.2.1 Stability

7.3 Norm-Bounded Uncertainty

7.3.1 Example

7.4 Conclusions

8 Non Linear platforms - dimension Output-Feedback Control

8.1 advent and challenge Formulation

8.2 Stochastic H-infinity OF Control

8.2.1 Example

8.2.2 The Case of Nonzero G2

8.3 Norm-Bounded Uncertainty

8.4 In.nite-Time Horizon Case: A Stabilizing H Controller 8

8.5 Conclusions

9 l2-Gain and powerful SF keep an eye on of Discrete-Time NL Stochastic Systems

9.1 Introduction

9.2 Su.cient stipulations for l2-Gain= .:ASpecial Case

9.3 Norm-Bounded Uncertainty

9.4 Conclusions

10 H-infinity Output-Feedback keep an eye on of Discrete-Time Systems

10.1 Su.cient stipulations for l2-Gain= .:ASpecial Case

10.1.1 Example

10.2 The OF Case

10.2.1 Example

10.3 Conclusions

11 H-infinity keep an eye on of Stochastic Switched structures with reside Time

11.1 Introduction

11.2 challenge Formulation

11.3 Stochastic Stability

11.4 Stochastic L2-Gain

11.5 H-infinity State-Feedback Control

11.6 instance - Stochastic L2-Gain Bound

11.7 Conclusions

12 strong L-infinity-Induced keep watch over and Filtering

12.1 Introduction

12.2 challenge formula and Preliminaries

12.3 balance and P2P Norm sure of Multiplicative Noisy Systems

12.4 P2P State-Feedback Control

12.5 P2P Filtering

12.6 Conclusions

13 Applications

13.1 Reduced-Order Control

13.2 Terrain Following Control

13.3 State-Feedback keep an eye on of Switched Systems

13.4 Non Linear structures: size Output-Feedback Control

13.5 Discrete-Time Non Linear structures: l2-Gain

13.6 L-infinity keep watch over and Estimation

A Appendix: Stochastic regulate methods - simple Concepts

B The LMI Optimization Method

C Stochastic Switching with reside Time - Matlab Scripts

References

Index

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**Example text**

009⎦ , Cc = 1 1 0 . 22 Case No. 22 and various values of d. 7 Stochastic Output-Feedback Control for Delayed Systems 49 achieved for the non-delayed case is, obviously, the lowest one. In cases 2– 5, it is seen that the attenuations levels increase as the bound on the delay derivative (d) increases. Similar behavior appears for diﬀerent delay lengths (not shown). Also, the maximum value of the derivative bound is reduced as the delay length is increased. 10). 14. 10). 56) which includes the adˆ T 1H ˆ T 2H ˆ 0 in Υ¯11 and the additional term of H ˆ 1 in the ditional term of H 0 1 (2, 2) block, where ˆ0 = E0 , E 0 ˆ1 = E1 , E 0 ¯0 H ¯ 2 Cc , ˆ0 = H H ¯1 0 .

3 Stability Criterion We ﬁrst consider the issue of stability of the stochastic delayed nominal autonomous system and then the stability of the norm-bounded and polytopic uncertain systems. Once the criteria for stability is found, we formulate and obtain the nominal and robust H∞ BRL for the latter system. We use the resulting BRL to solve the state-feedback, ﬁltering, dynamic output-feedback, and static output-feedback control problems that were deﬁned above. 16) 26 2 Time Delay Systems – H∞ Control and General-Type Filtering ˜ 2 ([0, ∞); Rq ) is an exogenous where x(t) ∈ Rn is the state vector, w(t) ∈ L Ft disturbance, and A0 , A1 , B1 and G, H are time invariant matrices and where β(t), ν(t) are zero-mean real scalar Wiener processes satisfying: E{β(t)β(s)} = min(t, s), E{ν(t)ν(s)} = min(t, s), E{β(t)ν(s)} = α ¯ · min(t, s), |¯ α| ≤ 1.

49) ⎡ and where ¯ Y ] 0 0 0 h f E T [X ¯ Y] 0 0 0 0 Υ2T = E0T [X 0 ¯ Y ] 0 0 0 h f E T [X ¯ 0] 0 0 0 0 Υ3T = E0T [X 0 ¯ Y ] 0 0 0 h f E T [X ¯ Y] 0 0 0 0 Υ4T = E1T [X 1 T T T , , . 45). 12). 12. 12). 50). 51) ˜ f hΥi,14 . 47). 5 1 , d = 0. e α ¯ = 0). 11 . 1 Stochastic Output-Feedback Control – The Nominal Case In this section we address the dynamic output-feedback control problem of the delayed state-multiplicative uncertain noisy system [59]. 7). 52) Gξ(t)dβ(t) + F˜ ξ(t)dζ(t), ξ(θ) = 0, over[−h 0], ˜ z˜(t) = Cξ(t), with the following matrices: Aˆ0 = ˜ = H H0 0 0 A0 B2 Cc Bc C2 Ac ˜= , G G0 0 0 , Aˆ1 = , F˜ = A1 0 Bc C¯2 0 0 0 Bc F 0 ˜= , B B1 0 0 Bc D21 , C˜ = [C1 D12 Cc ].