pariz-karimpour feb 2011 1 chapter 3 reference: switched linear systems control and design zhendong...

Pariz-Karimpour Feb 2011

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Chapter 3

Reference: Switched linear systems control and design

Zhendong Sun, Shuzhi S. Ge


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3.1. Introduction3.1. Introduction

3.2. General Results3.2. General Results

3.3. Periodic Switching3.3. Periodic Switching

3.4. State-feedback Switching3.4. State-feedback Switching

3.5. Combined Switching3.5. Combined Switching

3.6. Numerical Examples 3.6. Numerical Examples

Stabilizing Switching for Autonomous SystemsStabilizing Switching for Autonomous Systems


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IntroductionIntroduction


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7

?



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j

j

21

21

2

1

4142.0

4142.2

2

1


Example


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Let



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This lecture provide:

• Basic observation on the ability and limitation of switching design

• Analyze and design of some switching for Stability and robustness



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3.2.1. Algebraic Criteria3.2.1. Algebraic Criteria

3.2.2. Equivalence of the Stabilization Notions3.2.2. Equivalence of the Stabilization Notions

3.2.3. Periodic and Synchronous Switchings3.2.3. Periodic and Synchronous Switchings

3.2.4. Special Systems3.2.4. Special Systems

3.2.5. Robustness Issues3.2.5. Robustness Issues

General ResultsGeneral Results


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Algebraic CriteriaAlgebraic Criteria


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j

j

21

21

2

1

4142.0

4142.2

2

1

Example



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j

j

21

21

2

1

4142.0

4142.2

2

1

Example


2,2)( 11 TAA

1.6,1.2)( 22 TAA


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Does this equivalence still hold for switched linear systems

To establish the equivalence, we need the concept of switched convergence

Equivalence of the Stabilization NotionsEquivalence of the Stabilization Notions


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kjttt

eeetΦ

jj

ttAttAttA

xijjjijji

,...,1,0],[

...),0,(

1

)()()(111



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R2. . .

Rl

. . .

. . .

R1

Ri



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R2. . .

Rl

. . .. .

.

R1

Ri

TttttttttttAttAttA

ttAttAttAttAttAttA

x

eeeeeeee

eeeeeetΦ

jjjijjjijji

ijjjijjiijjjijji

)()()()()()()(

)()()()()()(

11111

111111

......

.......),0,(


SinceSince


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0

000,0

22,,0;0(,,0;( 0

00

z

zT

xT

x ttx

exext

1

1

0

0

0

00

0

00,

42,,0;(,,0;( 0

00

z

z

z

z

z

zT

x

z

zT

x ttttx

extext

2

2

1

1

0

0

1

1

0

0

0

1

1

0

00

,

82,,0;(,,0;( 0

00

z

z

z

z

z

z

z

z

z

z

Tx

z

z

z

zT

x

tttttt

xexttext



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02 xe T

4

)(txtt

TT exexe 0

2ln

02


22 0xe T

2

42 0xe T

3

82 0xe T


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PeriodicPeriodic andand SynchronousSynchronous SwitchingsSwitchings


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Special SystemsSpecial Systems


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Robustness IssuesRobustness Issues


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Proof of theorem 3.15 (continue)


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Proof: By theorem 3.19 we have



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Periodic SwitchingPeriodic Switching


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0

1 2 m…… 1 2 m…… 1 2 m……



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Define the fundamental matrix as:



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0

1 2 m…… 1 2 m…… 1 2 m…… = 1 = 2

)1()1(

stisfy,let , any For

222111

2112

lsllsl

llss



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0

1 2 m…… 1 2 m…… 1 2 m…… = 1 = 2



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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



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i)



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ii)



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iii)



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3.4.1. State-space-partition-based Switching3.4.1. State-space-partition-based Switching

3.4.2. A Modified Switching Law3.4.2. A Modified Switching Law

3.4.3. Observer-based Switching3.4.3. Observer-based Switching

State-feedback SwitchingState-feedback Switching


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State-space-partition-based SwitchingState-space-partition-based Switching


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Switching strategy



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function y=myfun2(x) if x(1)~=x(2);y=1;else y=0;endend

function y=myfun1(w)if w==1; y=[1;0];endif w==2; y=[0;1];endend

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;endend



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A Modified Switching LawA Modified Switching Law


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Modified Switching strategy



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Observer-basedObserver-based SwitchingSwitching


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1- Check the assumption 3.2 for the system

2- Repeat the system simulatrion by

0

2

5.0

021 LL

3- Choose suitable L1 and L2 and repeat the simulation.

4- Examine the system for y=x1 for the first system and y=x2 for the second one.

Exercises:

5- According to exercise 4 derive another condition for observer base switching.


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Combined SwitchingCombined Switching


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Periodic switchingPeriodic switching

0

1 2 m…… 1 2 m…… 1 2 m……



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State feedback switchingState feedback switching



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3.5.1. Switching Strategy Description3.5.1. Switching Strategy Description

3.5.2. Robustness Properties3.5.2. Robustness Properties


3.5.4. Extensions3.5.4. Extensions



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Switching Strategy DescriptionSwitching Strategy Description


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tktk+2tk+1



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Proof:



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136By student (#2)



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138Proof: By one of the student (#3)

Robustness PropertiesRobustness Properties


139By student (#3)

Robustness PropertiesRobustness Properties


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Observer-based SwitchingObserver-based Switching


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Proof: By one of the student (#4)



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Extensions Extensions


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Let 4321 ,,, jjjj 4321 ,,,

Let 1)( 10 jt


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Let 1)( 10 jt

t0

t4t11

1j 2j

2 t2 t3

3j 4j

3 4



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3.6. Numerical Examples3.6. Numerical Examples



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Numerical ExamplesNumerical Examples


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Stabilizing Switching for Autonomous SystemsStabilizing Switching for Autonomous SystemsSummarySummary


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3.2.3. Periodic and Synchronous Switching3.2.3. Periodic and Synchronous Switching


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3.2.3. Periodic and Synchronous Switching3.2.3. Periodic and Synchronous Switching




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0

1 2 m…… 1 2 m…… 1 2 m……


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3.4. State Feedback Switching3.4. State Feedback Switching

Switching strategy


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Modified Switching strategy


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Observer Based Switching strategy


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3.5. Combined Switching (3.5. Combined Switching (Robustness PropertyRobustness Property))


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3.5. Combined Switching (3.5. Combined Switching (ExtensionExtension))


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pariz-karimpour feb 2011 1 chapter 3 reference: switched linear systems control and design zhendong...

Documents

ge slide

letintroduction slide

robustness introduction

stabilization notions

periodic switching

combined switching

example algebraic criteria

introduction example