By Eli Gershon

Complicated issues up to speed and Estimation of State-Multiplicative Noisy structures starts with an creation and large literature survey. The textual content proceeds to hide the sphere of H∞ time-delay linear structures the place the problems of balance and L2−gain are provided and solved for nominal and unsure stochastic platforms, through the input-output process. 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 regulate also are offered and solved. the second one a part of the monograph matters non-linear stochastic nation- multiplicative platforms and covers the problems of balance, regulate and estimation of the structures within the H∞ experience, for either continuous-time and discrete-time situations. The e-book additionally describes distinctive issues reminiscent of stochastic switched platforms with live time and peak-to-peak filtering of nonlinear stochastic platforms. 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 ebook is rounded out via a three-part appendix containing stochastic instruments beneficial for a formal appreciation of the textual content: a easy advent to stochastic keep watch over techniques, points 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. complex themes up to speed and Estimation of State-Multiplicative Noisy structures might be of curiosity to engineers engaged on top of things platforms examine and improvement, to graduate scholars focusing on stochastic keep an eye on idea, and to utilized mathematicians drawn to keep an eye on difficulties. The reader is predicted to have a few acquaintance with stochastic keep watch over conception and state-space-based optimum keep watch over thought and techniques for linear and nonlinear systems.

Table of Contents

Cover

Advanced issues up to speed 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 process for behind schedule Systems

1.2.1 Continuous-Time Case

1.2.2 Discrete-Time Case

1.3 Non Linear regulate 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 approaches - 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 strong balance - The Norm-Bounded Case

2.3.3 powerful balance - The Polytopic Case

2.4 Bounded genuine 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 an eye on - The Nominal Case

2.5.2 powerful State-Feedback regulate - The Norm-Bounded Case

2.5.3 strong Polytopic State-Feedback Control

2.5.4 instance - State-Feedback Control

2.6 Stochastic Filtering for not on time Systems

2.6.1 Stochastic Filtering - The Nominal Case

2.6.2 powerful Filtering - The Norm-Bounded Case

2.6.3 powerful Polytopic Stochastic Filtering

2.6.4 instance - Filtering

2.7 Stochastic Output-Feedback keep watch over 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 powerful 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 regulate 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 - strong State-Feedback

5.6 not on time Filtering

5.6.1 instance - Filtering

5.7 Conclusions

6 H-infinity-Like keep an eye on 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 structures - 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 strong SF regulate 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 watch over 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 platforms with stay 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 an eye on and Filtering

12.1 Introduction

12.2 challenge formula and Preliminaries

12.3 balance and P2P Norm certain 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 regulate of Switched Systems

13.4 Non Linear structures: dimension Output-Feedback Control

13.5 Discrete-Time Non Linear structures: l2-Gain

13.6 L-infinity regulate and Estimation

A Appendix: Stochastic regulate methods - easy Concepts

B The LMI Optimization Method

C Stochastic Switching with live Time - Matlab Scripts

References

Index

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**Example text**

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. 16), τ (t) is an unknown time-delay which satisﬁes: 0 ≤ τ (t) ≤ h, τ˙ (t) ≤ d < 1. 17) In order to solve the above problem, we introduce the following operators: Δ (Δ1 g)(t) = g(t − τ (t)), Δ (Δ2 g)(t) = t t−τ (t) g(s)ds.

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 ]. 54) where ˜ Aˆ0 + m) + (Aˆ0 + m)T Q ˜+ Υ11 = Q( 1 R1 .

1. 16) has been done in [55] for the nominal case (with no deterministic norm-bounded uncertainties). 24). 10). 2. 25) ⎢ ⎥<0 ⎢ ∗ ∗ ⎥ ∗ − Q h QE h QE f f 0 f 1 ⎢ ⎥ ⎣ ∗ ∗ ⎦ ∗ ∗ − 1 In 0 ∗ ∗ ∗ ∗ ∗ − 2 In where Ψˆ11 Ψˆ12 Ψˆ14 Ψˆ22 Ψˆ24 = QA0 + Qm + AT0 Q + QTm + = QA1 − Qm + αG ¯ T QH, = h f AT0 Q + h f QTm , ¯ 1T H ¯1 = −R1 + H T QH + 2 H T T = h f A1 Q − h f Qm . 2. The above result provides a delay dependent stability condition. A corresponding delay independent (but rate dependent) result is readily obtained by choosing m = 0 and f → 0.