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On several composite quadratic Lyapunov functions for switched systems

Tingshu Hu, Umass LowellLiqiang Ma, Umass Lowell Zongli Lin, Univ. of Virginia

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Outline

Background on switched systems and sliding motion Approach of this work, main issues

Use three Lyapunov functions to construct switching laws How to ensure stability in the presence of sliding motion?

Main Results Switching law based on directional derivatives Directional derivatives along sliding surface Stabilization via min function Dual result for max function and convex hull function

Conclusions

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Switched systems and LDIs

Given a family of linear systems

NixAx i ,,2,1,

A linear differential inclusion (LDI):

NixAx i ,,2,1: Switching controlled by unknown forceExpect the worst case

ni

ix

R

xixxAx

where

if )(,)( Switching orchestrated by controller,Can be optimized for best performance

This work considers the second type, the switched systems

A switched system

Two approaches:• Analytical methods for lower-order sys [Antsaklis, Michel, Hu, Xu, Zhai]• Using Lyapunov functions for switching laws construction and stability analysis

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Earlier constructive approaches

Find P > 0 and [0,1] such that [Wicks, Peletics & Decarlo, 1998]

0))1(())1(( 2121 AAPPAA T

A switching law can be constructed for quadratic stability

Find P1, P2>0, ≥ or ≤ such that [Wicks & Decarlo, 1997]

)(

)(

2122222

1211111

PPAPPA

PPAPPAT

T

A switching law can be constructed based on

V(x) = max{xTP1 x, xTP2x}, or V(x) = min{xTP1 x, xTP2x}.

Stability ensured only if no sliding motion occurs.

Both methods involving BMIs, harder than LMIs but numerically possible More recent development based on BMIs: [Decarlo, Branicky, Pettesson,

Lennartson, Zhai, etc].

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Effect of sliding motion

)(),( 21222221211111 PPAPPAPPAPPA TT

Assume the matrix condition is satisfied [Wicks & Decarlo, 1997]

When sliding motion occurs, the system can be stable or unstable

Sliding motion not unusual in switched systems; It may be a result of optimization Not realizable but can be approximated via hysteresis, delay, fast sampling, etc.

Based on Vmin

Based on Vmax 0, 21 0, 21

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Approach of this work, main issues

Approach: Use three types of Lyapunov functions to construct switching

laws Functions composed from a family of quadratic functions; Lead to semi-definite matrix conditions, numerically possible; Two types of functions are not everywhere differentiable

Main issues: How to dealing with non-differentiable Lyapunov functions?

Use directional derivatives

How to address stability in the presence of possible sliding motion? Exclude the existence of sliding motion Characterize directional derivatives along sliding direction

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Three Lyapunov functions

0,,, 21 JPPP

0,1:1

jj

J

j

JR

xPPxxV JJc1

11

min2

1:)(

Given matrices:

3) The convex hull function:

2) The max function:

max ,,2,1:max 2

1:)( JjxPxxV j

T

1) The min function:

min ,,2,1:min 2

1:)( JjxPxxV j

T

Level set =

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Only the convex hull function is everywhere differentiable Vc and Vmax are convex conjugate pairs, they have been successfully applied to LDIs and saturated systems [Goebel, Hu, Lin, Teel, Zaccarian]

. Let

Level set =

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

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Level set =

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

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

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Switching law based on directional derivatives

Consider a function V(x). The one-sided directional derivative at x along the direction is,

t

xVtxVxV

t

)()(lim:);(

0

)(:max);( maxmax xVxPxPxxV jT

jT

)(:min);( minmin xVxPxPxxV jT

jT

NixAx i ,,2,1,

Then,

For the family of linear systems

Let the switching law be constructed as

),(minarg)(],1[

xAxVx iNi

xAx x)(

V(x) can be Vmax(x), Vmin(x), or Vc(x)

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

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

)(max21 xVxPxxPx TT

) ,(Typically x

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How sliding motion complicates the analysis?

Along sliding direction:

(0,1) ,)1( 21

surface switching the to tangential is that such xxAxAx

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

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

xA1

xA2

x

xxA1

xA2

Sliding along the set of differentiable points is easy to deal with

);()1();())1(;();( 2121 xAxVxAxVxAxAxVxxV

Sliding along the set of non-differentiable points is more complicated

);()1();())1(;();( 2121 xAxVxAxVxAxAxVxxV

ensured notstability set, level the of outward points increases, If

))(( ,0);(

x txVxxV

set. level the of inward points If

))(( ,0);(

xtxVxxV

decreases,

, 21 xAxxAx

What really matters is :);( xxV

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Directional derivatives along sliding surfaces

A1xA2x

A1x+(1-)A2x

Different situations w.r.t Vmax and Vmin

);()1();())1(;( 2121 xAxVxAxVxAxAxV

A2x

A1xA1x+(1-)A2x

0);( ,0);( 2max1max xAxVxAxV

somefor 0))1(;(But 21 xAxAxV

0);( ,0);( 2min1min xAxVxAxV

somefor 0))1(;(But 21 xAxAxV

Not necessary to require

0);( ,0);( 2max1max xAxVxAxV

Not sufficient to require

0);( ,0);( 2min1min xAxVxAxV

How can we ensure along sliding direction?0);( xxV

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Some key points in this work

A1xA2x

A1x+(1-)A2x

For Vmax, If there exists a s.t.

0))1(;( 21max xAxAxV

at the non-differentiable point, then

0))1(;( 21max xAxAxV

along the switching surface.

For Vmin, no sliding motion exist in the set of non-differentiable points

),(minarg)(],1[

xAxVx iNi

Note:

A1x and A2x points away from switching surface

Only need to consider the set of differentiable points

A2x

A1xA1x+(1-)A2x

0))1(;( 21max

es.inequalitimatrix with xAxAxV :condition this Interprete

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Stabilization via min function

),(minarg)( min],1[

xAxVx iNi

The switching law:

Proposition 1: There exist no sliding motion in the set of x where Vmin(x) is not differentiable.

xAx x)(

Matrix condition: Consider Vmin= min{xTPj x: j=1,2,…,J}. If there exist ij≥0, ij≥0, i=1

Nij=1, such that

JjPPPAPPA jkj

J

kjk

N

iiijjj

TN

iiij ,,2,1,)(

111

Then for every solution, texVtxV ))0(())(( minmin

Stability ensured by matrix condition even if sliding motion occurs.

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Stabilization via min function

Special case with two Ai’s: The matrix inequalities [Wicks & Decarlo, 1997]

)(),( 12222222111111 PPAPPAPPAPPA TT

ensures stabilization regardless of sliding motion.

The number of matrices Pj’s (J) does not need to be equal to the number of Ai’s (N) : J≥N, or J<N. As J increases, the convergence rate increases.

Example: ,

200

331

331

,

201

322

363

21

AA

System cannot be stabilized via quadratic Lyapunov functions: No P>0 and satisfy

0))1(())1(( 2121 AAPPAA T

both neutrally stable

With J=2, maximal 0.3375;With J=3, maximal 0.3836;With J=4, maximal 0.4656;

texVtxV ))0(())(( minmin

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A dual result for Vmax and Vc

Recall Vmax and Vc are conjugate functions

Consider the dual switched systems:

))(;(minarg)( , max],1[

1)(1xAxVxxAx i

Nix

))(;(minarg)( ,],1[

2)(2

Tic

Ni

T AVA

Proposition: Suppose N=2. Sys 1 is stable iff Sys 2 is stable.

Sys 1:

Sys 2:

Remarks: ₋ Results also obtained for the case N >2. ₋ It is easier to obtain matrix conditions via Sys. 2 since Vmax is piecewise quadratic.

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Example: A pair of dual systems

))(;(minarg)( , max],1[

1)(1xAxVxxAx i

Nix

))(;(minarg)( ,],1[

2)(2

Tic

Ni

T AVA

Sys 1:

Sys 2:

9.07.0

7.07.0,

1.16.0

6.045.0,

11

21,

15.0

002121 PPAA

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

x1

x 2

A1x

A2x

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5

1

2

Sliding motion occurs for both systems. They are stable with the same convergence rate w.r.t correspoding Lyapunov function.

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Conclusions

Switching laws constructed via three Lyapunov functions The min function The max function The convex hull function

Sliding motion carefully considered by using the directional derivatives

When min function is used, sliding motion does not exist in the set of non-differentiable points When max function is used, Vmax decreases along the sliding surface iff it decreases along a certain convex combination of A1x and A2x.

A dual result obtained via max function and convex hull function Condition for stabilization characterized by BMIs.