Pariz-Karimpour Feb 2011 1 Chapter 3 Reference: Switched linear systems control and design Zhendong Sun, Shuzhi S. Ge.

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Presentation transcript:

Pariz-Karimpour Feb Chapter 3 Reference: Switched linear systems control and design Zhendong Sun, Shuzhi S. Ge

Pariz-Karimpour Feb Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

Pariz-Karimpour Feb Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

Pariz-Karimpour Feb

5 Introduction

6 Introduction

7 ?Introduction

8 Introduction Example

Pariz-Karimpour Feb LetIntroduction

Pariz-Karimpour Feb Introduction

Pariz-Karimpour Feb This lecture provide: Basic observation on the ability and limitation of switching design Analyze and design of some switching for Stability and robustness Introduction

Pariz-Karimpour Feb Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

Pariz-Karimpour Feb

Pariz-Karimpour Feb Algebraic Criteria Equivalence of the Stabilization Notions Periodic and Synchronous Switchings Special Systems Robustness Issues General Results

Pariz-Karimpour Feb Algebraic Criteria

Pariz-Karimpour Feb Algebraic Criteria

Pariz-Karimpour Feb Algebraic Criteria

Pariz-Karimpour Feb Example Algebraic Criteria

Pariz-Karimpour Feb Algebraic Criteria

Pariz-Karimpour Feb Example Algebraic Criteria

Pariz-Karimpour Feb Algebraic Criteria

Pariz-Karimpour Feb Algebraic Criteria Equivalence of the Stabilization Notions Periodic and Synchronous Switchings Special Systems Robustness Issues General Results

Pariz-Karimpour Feb Does this equivalence still hold for switched linear systems To establish the equivalence, we need the concept of switched convergence Equivalence of the Stabilization Notions

Pariz-Karimpour Feb Equivalence of the Stabilization Notions

Pariz-Karimpour Feb Equivalence of the Stabilization Notions

Pariz-Karimpour Feb Equivalence of the Stabilization Notions

Pariz-Karimpour Feb Equivalence of the Stabilization Notions

Pariz-Karimpour Feb R2R2... RlRl R1R1 RiRi Equivalence of the Stabilization Notions

Pariz-Karimpour Feb R2R2... RlRl R1R1 RiRi Equivalence of the Stabilization Notions Since

Pariz-Karimpour Feb Equivalence of the Stabilization Notions

Pariz-Karimpour Feb Equivalence of the Stabilization Notions

Pariz-Karimpour Feb Equivalence of the Stabilization Notions

Pariz-Karimpour Feb Equivalence of the Stabilization Notions

Pariz-Karimpour Feb Algebraic Criteria Equivalence of the Stabilization Notions Periodic and Synchronous Switchings Special Systems Robustness Issues General Results

Pariz-Karimpour Feb Periodic and Synchronous Switchings

Pariz-Karimpour Feb Periodic and Synchronous Switchings

Pariz-Karimpour Feb Periodic and Synchronous Switchings

Pariz-Karimpour Feb Periodic and Synchronous Switchings

Pariz-Karimpour Feb Algebraic Criteria Equivalence of the Stabilization Notions Periodic and Synchronous Switchings Special Systems Robustness Issues General Results

Pariz-Karimpour Feb Special Systems

Pariz-Karimpour Feb Special Systems

Pariz-Karimpour Feb Special Systems

Pariz-Karimpour Feb Algebraic Criteria Equivalence of the Stabilization Notions Periodic and Synchronous Switchings Special Systems Robustness Issues General Results

Pariz-Karimpour Feb Robustness Issues

Pariz-Karimpour Feb Robustness Issues

Pariz-Karimpour Feb Robustness Issues Proof of theorem 3.15 (continue)

Pariz-Karimpour Feb Robustness Issues Proof of theorem 3.15 (continue)

Pariz-Karimpour Feb Robustness Issues

Pariz-Karimpour Feb Robustness Issues

Pariz-Karimpour Feb Robustness Issues

Pariz-Karimpour Feb Proof of theorem 3.19 (continue) Robustness Issues

Pariz-Karimpour Feb Proof: By theorem 3.19 we have Robustness Issues

Pariz-Karimpour Feb Robustness Issues

Pariz-Karimpour Feb Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

Pariz-Karimpour Feb

Pariz-Karimpour Feb Periodic Switching

Pariz-Karimpour Feb m …… 12m 12m Periodic Switching

Pariz-Karimpour Feb Define the fundamental matrix as: Periodic Switching

Pariz-Karimpour Feb Periodic Switching

Pariz-Karimpour Feb m …… 12m 12m = 1 = 2 Periodic Switching

Pariz-Karimpour Feb m …… 12m 12m = 1 = 2 Periodic Switching

Pariz-Karimpour Feb Periodic Switching

Pariz-Karimpour Feb i) The system state is bounded if the perturbation is bounded ii) The system state is bounded and convergent if the perturbation is bounded and convergent iii) The system state is exponentially convergent if the perturbation is exponentially convergent Periodic Switching

Pariz-Karimpour Feb i) Periodic Switching

Pariz-Karimpour Feb ii) Periodic Switching

Pariz-Karimpour Feb iii) Periodic Switching

Pariz-Karimpour Feb Periodic Switching

Pariz-Karimpour Feb Periodic Switching

Pariz-Karimpour Feb Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

Pariz-Karimpour Feb

Pariz-Karimpour Feb State-space-partition-based Switching A Modified Switching Law Observer-based Switching State-feedback Switching

Pariz-Karimpour Feb State-space-partition-based Switching

Pariz-Karimpour Feb Switching strategy State-space-partition-based Switching

Pariz-Karimpour Feb State-space-partition-based Switching

Pariz-Karimpour Feb State-space-partition-based Switching

Pariz-Karimpour Feb State-space-partition-based Switching

Pariz-Karimpour Feb State-space-partition-based Switching

Pariz-Karimpour Feb State-space-partition-based Switching

Pariz-Karimpour Feb State-space-partition-based Switching

Pariz-Karimpour Feb State-space-partition-based Switching

Pariz-Karimpour Feb State-space-partition-based Switching

Pariz-Karimpour Feb

Pariz-Karimpour Feb function y=myfun2(x) if x(1)~=x(2);y=1; else y=0; end function y=myfun1(w) if w==1; y=[1;0];end if w==2; y=[0;1];end end function y=myfun(w) x=w(1:2);sigk=w(3); A1=[-2 0;0 1];A2=[1 0;0 -2];x0=[1;-1]; P=0.5*eye(2); Q(1).s=A1'*P+P*A1;Q(2).s=A2'*P+P*A2;r(1)=0.4;r(2)=0.4; if (x'*Q(sigk).s*x) > (-r(sigk)*x'*x) [c,y]=min([x'*Q(1).s*x, x'*Q(2).s*x]); else y=sigk; end State-space-partition-based Switching

Pariz-Karimpour Feb State-space-partition-based Switching

Pariz-Karimpour Feb State-space-partition-based Switching

Pariz-Karimpour Feb State-space-partition-based Switching

Pariz-Karimpour Feb State-space-partition-based Switching

Pariz-Karimpour Feb State-space-partition-based Switching A Modified Switching Law Observer-based Switching State-feedback Switching

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb Modified Switching strategy A Modified Switching Law

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb A Modified Switching Law

Pariz-Karimpour Feb State-space-partition-based Switching A Modified Switching Law Observer-based Switching State-feedback Switching

Pariz-Karimpour Feb Observer-based Switching

Pariz-Karimpour Feb Observer-based Switching

Pariz-Karimpour Feb Observer-based Switching

Pariz-Karimpour Feb Observer-based Switching

Pariz-Karimpour Feb Observer-based Switching

Pariz-Karimpour Feb

Pariz-Karimpour Feb

Pariz-Karimpour Feb Observer-based Switching

Pariz-Karimpour Feb

Pariz-Karimpour Feb Observer-based Switching

Pariz-Karimpour Feb Observer-based Switching

Pariz-Karimpour Feb Check the assumption 3.2 for the system 2- Repeat the system simulatrion by 3- Choose suitable L 1 and L 2 and repeat the simulation. 4- Examine the system for y=x 1 for the first system and y=x 2 for the second one. Exercises: 5- According to exercise 4 derive another condition for observer base switching.

Pariz-Karimpour Feb Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

Pariz-Karimpour Feb

Pariz-Karimpour Feb Combined Switching

Pariz-Karimpour Feb Periodic switching 0 12m …… 12m 12m Combined Switching

Pariz-Karimpour Feb State feedback switching Combined Switching

Pariz-Karimpour Feb Combined Switching

Pariz-Karimpour Feb Switching Strategy Description Robustness Properties Observer-based Switching Extensions Combined Switching

Pariz-Karimpour Feb Switching Strategy Description

Pariz-Karimpour Feb Switching Strategy Description

Pariz-Karimpour Feb tktk t k+2 t k+1 Switching Strategy Description

Pariz-Karimpour Feb Proof: Switching Strategy Description

Pariz-Karimpour Feb Switching Strategy Description

Pariz-Karimpour Feb Switching Strategy Description

Pariz-Karimpour Feb Switching Strategy Description

Pariz-Karimpour Feb Switching Strategy Description

Pariz-Karimpour Feb Switching Strategy Description

Pariz-Karimpour Feb Switching Strategy Description

Pariz-Karimpour Feb Switching Strategy Description

Pariz-Karimpour Feb

Pariz-Karimpour Feb By student (#2) Switching Strategy Description

Pariz-Karimpour Feb Switching Strategy Description Robustness Properties Observer-based Switching Extensions Combined Switching

Pariz-Karimpour Feb Proof: By one of the student (#3) Robustness Properties

Pariz-Karimpour Feb By student (#3) Robustness Properties

Pariz-Karimpour Feb Switching Strategy Description Robustness Properties Observer-based Switching Extensions Combined Switching

Pariz-Karimpour Feb Observer-based Switching

Pariz-Karimpour Feb Observer-based Switching

Pariz-Karimpour Feb Proof: By one of the student (#4) Observer-based Switching

Pariz-Karimpour Feb Switching Strategy Description Robustness Properties Observer-based Switching Extensions Combined Switching

Pariz-Karimpour Feb Extensions

Pariz-Karimpour Feb Extensions

Pariz-Karimpour Feb Extensions

Pariz-Karimpour Feb Extensions

Pariz-Karimpour Feb Let

Pariz-Karimpour Feb Let t0t0 t4t4 t1t1 t2t2 t3t3 Extensions

Pariz-Karimpour Feb Extensions

Pariz-Karimpour Feb Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems

Pariz-Karimpour Feb

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Numerical Examples

Pariz-Karimpour Feb Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching 3.6. Numerical Examples Stabilizing Switching for Autonomous Systems Summary

Pariz-Karimpour Feb Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching Stabilizing Switching for Autonomous Systems Summary

Pariz-Karimpour Feb Stabilizing Switching for Autonomous Systems Summary 3.1. Introduction

Pariz-Karimpour Feb Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching Stabilizing Switching for Autonomous Systems Summary

Pariz-Karimpour Feb Stabilizing Switching for Autonomous Systems Summary 3.2. General Results Algebraic Criteria

Pariz-Karimpour Feb Stabilizing Switching for Autonomous Systems Summary 3.2. General Results Algebraic Criteria Equivalence of the Stabilization Notions

Pariz-Karimpour Feb Stabilizing Switching for Autonomous Systems Summary 3.2. General Results Algebraic Criteria Equivalence of the Stabilization Notions Periodic and Synchronous Switching

Pariz-Karimpour Feb Stabilizing Switching for Autonomous Systems Summary 3.2. General Results Algebraic Criteria Equivalence of the Stabilization Notions Periodic and Synchronous Switching Special Systems Robustness Issues

Pariz-Karimpour Feb Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching Stabilizing Switching for Autonomous Systems Summary

Pariz-Karimpour Feb Stabilizing Switching for Autonomous Systems Summary 3.3. Periodic Switching 0 12m …… 12m 12m

Pariz-Karimpour Feb Stabilizing Switching for Autonomous Systems Summary 3.3. Periodic Switching

Pariz-Karimpour Feb Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching Stabilizing Switching for Autonomous Systems Summary

Pariz-Karimpour Feb Stabilizing Switching for Autonomous Systems Summary 3.4. State Feedback Switching Switching strategy

Pariz-Karimpour Feb Stabilizing Switching for Autonomous Systems Summary 3.4. State Feedback Switching Modified Switching strategy

Pariz-Karimpour Feb Stabilizing Switching for Autonomous Systems Summary 3.4. State Feedback Switching Observer Based Switching strategy

Pariz-Karimpour Feb Introduction 3.2. General Results 3.3. Periodic Switching 3.4. State-feedback Switching 3.5. Combined Switching Stabilizing Switching for Autonomous Systems Summary

Pariz-Karimpour Feb Stabilizing Switching for Autonomous Systems Summary 3.5. Combined Switching

Pariz-Karimpour Feb Stabilizing Switching for Autonomous Systems Summary 3.5. Combined Switching (Robustness Property)

Pariz-Karimpour Feb Stabilizing Switching for Autonomous Systems Summary 3.5. Combined Switching (Extension)

Pariz-Karimpour Feb