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Absolute Stability with a Absolute Stability with a Generalized Sector ConditionGeneralized Sector Condition

Tingshu HuTingshu Hu

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Outline

Background, problems and tools Absolute stability with a conic sector, circle criterion, LMIs The generalized sector bounded by PL functions (PL sector) Composite quadratic Lyapunov functions

Main results: Estimation of DOA with invariant level sets Quadratics : Invariant ellipsoid LMIs Composite quadratics : Invariant convex hull of ellipsoids BMIs An example

Building up the main results ─ Foundation: Stability analysis of systems with saturation Main idea: Describing PL sector with saturation functions Absolute stability stability for a family of saturated systems

Summary

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System with a conic sector condition

0)),()(),(( 21 uktuuktu

The conic sector condition:

A system with a nonlinear and/or uncertain component:

),( tu BAsIF 1)( vu

v

uk1

u

uk2

Question: what is the condition of robust stability for all possible u,t) satisfying the sector condition?

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Stability for a nonlinear system

Consider a nonlinear system:

),( txfx

Stability is about the convergence of the state to the origin or an equilibrium point. Also, if it is initially close to the origin, it will stay close.

Stability region: the set of initial x0 such that the statetrajectory converges to the origin.

Global stability: the stability region is the whole state space.

0 allfor 0)( xtx

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Quadratic function and level sets

Given a nn real symmetric matrix P, P=PT. If xTPx>0 for all xRn\{0}, we call P a positive definite matrix, and denote P > 0. (Negative definite can be defined similarly)

With P > 0, define V(x)= xTPx. Then V is a positive definite function, i.e., V(x) > 0 for all xRn\{0}.

Level sets of a quadratic function: Ellipsoids. Given

PxxRxP Tn :),(

1 23 4

),(),(),(),( 4321 PPPP

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Quadratic stability

),( txfx The system:

Stability condition: If for all x (P,)\{0}, 0),(, txPfxxPx T

Then (P, ) is a contractively invariant set and a region quadratic stability.

(*)

Condition (*) means that along the boundary of (P,) for any 0 , the vector points inward of the boundary

),( txfx

In Lyapunov stability theory, the quadratic Lyapunov function is replaced with a more general positive-definite function

Px

x

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Quadratic stability for linear systems

For a linear system:

Stability condition: If for all x (P,)\{0},

0PAxxT

Then (P, ) is a contractively invariant set and a region of quadratic stability.

(*)

Axx

(*) is equivalent to definite)-(negative0 PAPAT

Lyapunov matrix inequality.

As long as there exists a P satisfying the matrix inequality, the linear system is stable

0)( xPAPAx TT

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Absolute stability with conic sector

0)),()(),(( 21 uktuuktu

The conic sector condition:

Consider again the system with a nonlinear component:

v

uk1

u

uk2

Absolute stability: the origin is globally stable for any satisfying the sector condition

u,t)vu

F(sI-A)-1B

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Conditions for absolute stability

0)()'(

0)()'(

22

11

BFkAPPBFkA

BFkAPPBFkA

Popov criterion Circle criterion LMI condition Quadratic stability

Description with linear differential inclusion (LDI):

],[:)( 21 kkkxkBFAx

Quadratic stability: exists P=P’ >0 such that

),( tu BAsIF 1)( vu

v

uk1

u

uk2

xBFkAxBFkA )(,)(co 21 Px

x

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Motivation for a generalized sector

uk1

u

uk2Limitations of the conic sector: • not flexible• could be too conservative

Note: Subclass of the conic sector has been considered, e.g., slope restricted,Monotone ( Dewey & Jury, Haddad & Kapila, Pearson & Gibson, Willems, Safonov et al,Zames & Falb, etc.)

Our new approach extend the linear boundary functions to nonlinear functions basic consideration: numerical tractability Our Choice: Piecewise linear convex/concave boundary functions

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A piecewise linear (PL) sector

Let and be

odd symmetric, piecewise linear convex or concave for u > 0

The generalized sector condition:

u

v),( tu

)(1 u

v

u

)(2 u

)(1 u

)(2 uMain feature: More flexible and still tractable

0))(),())((),(( 21 utuutu

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A tool: the composite quadratic function

0,,, 21 JQQQ

0,1: 1 jJJR

xQQxxV JJc1

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min2

1:)(

Given J positive definite matrices:

Denote

The composite quadratic function is defined as:

The level set of VC is the convex hull of ellipsoids Convex, differentiable

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Applying composite quadratics to conic sectors

Recall: A systems with conic sector condition can be describedwith a LDI: xAxAx 21 ,co

Theorem: Consider Vc composed from Qj’s. If there exist ijk ≥ 0,

i = 1,2 , j,k =1,2,…,J , such that

)(1

'jk

J

kijkijji QQAQQA

Then .00)( xxVc )0)(( xxV T

Example: A linear difference inclusion: x(k+1)co{A1x, A2(a)x}

1,4.04.0/4.04.0)(,4.04.0

4.04.021

aa

aaAA

With quadratics, the maximal a ensuring stability is a1=4.676;With composite quadratics (N=2), the maximal a is a2=7.546

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Main results: Invariant level sets

Quadratics : Invariant ellipsoid LMIs Composite quadratics : Invariant convex hull of ellipsoids BMIs An example

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Absolute stability analysis via absolutely invariant level sets

1)(:)1( xVRxL nV

Consider the system: ),( tFxBAxx

LV (1) is absolutely contractively invariant (ACI) if it is contractively invariant for all co {

For a Lyapunov candidate V ( x ), its 1-Level set is

The set LV (1) is contractively invariant (CI) if

RtLxtFxBAxxVV VT },0{\)1(0)),(()(

Quadratics : ACI ellipsoids, Composite quadratics: ACI convex hull of ellipsoids

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Result 1: contractive invariance of ellipsoid

NiY ni ,...2,1,R1

),( if,

],( if,

],0[ if,

)( 2111

10

NNN buucuk

bbuucuk

buuk

u

,)(FxBAxx

Consider the system,

Theorem: An ellipsoid ( Q ) is contractively invariant iff

0)()'( 00 QBFkABFkAQ

and there exist such that

NiQQFkY

FQkYc

BYBYAQQA

ii

iii

ii

,,2,1,0''

,0'''2

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Result 2: Quadratics → ACI ellipsoids

NiYY nii ,...2,1,R, 121

2,1,

),( if,

],( if,

],0[ if,

)(

,,,

2111

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q

bucuk

bbucuk

buuk

u

qNqNqN

qqqq

qq

q

R,

,)(),(co),(

,),(

21

tu

uutu

tFxBAxx

The system,

Theorem: An ellipsoid ( Q ) is ACI if and only if

2,1,0)()'( 00 qQBFkABFkAQ qq

and there exist such that

2,1,,,2,1,0''

,0'''

2

qNiQQFkY

FQkYc

BYBYAQQA

iqiq

iqiqiq

iqiq

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Result 3: ACI of convex hull of ellipsoids

],,1[],,1[,R, 121 JjNiYY n

jiji

2,1,)()()'(1

000

qQQQBFkABFkAQJ

kjkqjkjqqj

and there exist such that

],1[,2,1],,1[,0''

,)('''

2

1

JjqNiQFQkY

FQkYc

QQBYBYAQAQ

jjiqiqj

jiqiqjiq

J

kjkiqjkiqjiqjjj

Consider Vc composed from Qj’s. LVc (1) is the convex hull of (Qj

-1).

Theorem: LVc (1) is ACI if there exist iqjk ≥ 0, i 0,1,…,N , q=1,2, j,k =1,2,…,J , such that

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Optimizing ACI level sets

],1[,2,1,)()()'(

],1[],,1[,0

'1

,sup

1000

1

JjqQQQBFkABFkAQ

JjKpQγx

x

J

kjkqjkjqqj

J

jjpjp

p

],1[,2,1],,1[,0''

,)('''

2

1

JjqNiQFQkY

FQkYc

QQBYBYAQAQ

jjiqiqj

jiqiqjiq

J

kjkiqjkiqjiqjjj

Choose reference points x1,x2,…,xK . Determine ACI LVc (1) such

that xp’s are inside LVc (1) with maximized. 1x

2x

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8.84.0

,7.0

6.0,

3.01

01.03.0

,,co),(

F

BA

FxBAxx

-5 0 5-1.5

-1

-0.5

0

0.5

1

1.5

Example

A second order system:

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Reference point:

1

10x

Maximal

LVc(1):

(Q1-1):

(Q2-1):

21-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

-3

-2

-1

0

1

2

3

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Composite quadratics + PL sector max0.8718

Quadratics + PL sector max = 0.6401

Quadratics + conic sectormax 0.4724

A closed-trajectory under the “worst switching” w.r.t Vc

ACI convex hull

A diverging trajectory

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Building up the main results Stability analysis for systems with saturation Describing PL sector with saturation functions Stability for an array of saturated systems

Absolute stability

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Stability analysis for systems with saturation

1||),(0))(sat)()(sat( 11 fxkPxfxfxfxkfx

uk1

}1|,min{|)sgn()(sat,)(sat uuuxfbAxx The system

1':)( PxxRxP n

Problem: To characterize the (contractive) invariance of

Traditional approach: find k, 0 < k≤ 1, such that

u

then use the traditional absolute stability analysis tools

Note: The condition takes form of bilinear matrix inequalities

maxu

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New approach of dealing with saturation

vR1))(sat)()(sat( uuuvu

The basic idea: If |v| ≤ 1, then

u

-1

For any row vector h,

1||0))(sat)()(sat( hxfxfxhxfx

1||0))(sat)()(sat( 11 fxkfxfxfxkfx

Recall the traditional approach

Further more, the resulting condition for invariance of ellipsoid includes only LMIs is necessary and sufficient

We have full degree of freedom in choosing h as compared with

the one degree of freedom in choosing kin kf.

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Foundation: The necessary and sufficient condition for invariance of ellipsoid

,)(sat xfbAxx

Theorem: the ellipsoid ( Q) is contractively invariant for

if and only if there exists such that ,R1 ny

,0'

1

,0'''

0)()'(

Qy

y

bybyAQQA

QbfAbfAQ

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Building-up tool: description of PL functions with saturation functions

1b

u

c

kkcuku

1

1011 sat)(

1k

),( if,

],0[ if,)(

111

10

buucuk

buuku

Consider a PL function withonly one bend

0k

u

)(u

The necessary and sufficient condition for invariance of ellipsoid follows.

fxc

kkbcxfbkAx

xfbAxx

1

1011 sat)(

,)(

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Key step: description of PL functions with saturation functions

)(,co)()

0,,2,1:)(min)()

0 uukub

uNjuua

i

i

),( if,

],( if,

],0[ if,

)( 2111

10

NNN buucuk

bbuucuk

buuk

u

A PL function,

u

c

kkcuku

i

iiii

0sat)(Define

)(1 u )(2 u

u

)(u

)(3 u

uk0

Properties:

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Putting things together:Absolute stability via saturated systems

2,1],,1[

,sat)(: 0

qNi

Fxc

kkBcxFBkAxS

iq

iqqiqiqiq

,co,),(: tFxBAxxS

The original system and N systems with saturation,

ACI of a level set for S CI of the level set for all Siq

Stability analysis results contained in:

T. Hu, Z. Lin, B. M. Chen, Automatica, pp.351-359, 2002T. Hu and Z. Lin, IEEE Trans. AC-47, pp.164-169, 2002T. Hu, Z. Lin, R. Goebel and A. R. Teel, CDC04, to be presented.

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Summary • The systems: subject to PL sector condition • Tool: composite quadratic Lyapunov function• Problem: determine ACI sets (convex hull of ellipsoids)

• Key step: description of PL functions with saturations

• Main feature: more flexible as compared with

conic sector, and still tractable• Future topics: under PL sector condition,

characterize the nonlinear L2 gain

apply non-quadratics to study input-state,

input-output, state-output properties