Multiple-Lyapunov Functions for Guaranteeing the Stability of

a class of Hybrid Systems

Some background informationSome background information• The original idea has been around for a number of years. See the following recent papers for details:

‘Asymptotic stability of m-switched systems using Lyapunov-like functions’ Peleties & DeCarlo, ACC91.

‘Stability of Switched and Hybrid Systems’, Branicky,LIDS Tech. Report, 2214, 1993.

‘Stability of Switched and Hybrid Systems’, Branicky, CDC94.

‘Multiple Lyapunov Functions and other Analysis Tools for Switched and Hybrid Systems’, IEEE Trans. AC, 1998.

Some background informationSome background information

• I am going to focus on work described in the following papers

‘Stability of Switched and Hybrid Systems’, Branicky,LIDS Tech. Report, 2214, 1993.

‘Stability of Switched and Hybrid Systems’, Branicky, CDC94.

‘Multiple Lyapunov Functions and other Analysis Tools for Switched and Hybrid Systems’, IEEE Trans. AC, 1998.

This work is related to the piecewise quadratic Lyapunov

functions studied by Rantzer and Johanson.

A stability result for non-linear systemsA stability result for non-linear systems

• Branicky considers the following class of system:

• The following assumption is made:

– Each fi is assumed to be globally Lipschitz continuous

– Each fi is assumed to be exponentially stable

– The i’s are picked in such a way that there are finite switches in finite time.

))(()( txftx i

A stability result for non-linear systemsA stability result for non-linear systems

• It is well known that switching between exponentially stable vector fields will not necessarily result in a stable system.

• Example:

5.145.13

5.45.3,)(

5.145.13

5.45.3,)(

222

111

AxAxf

AxAxf

InstabilityInstability

Main result in the areaMain result in the area

• Branicky adopts the following notation:

• The rules are interpreted as follows:

• This trajectory is denoted xs(t).

),....,),....(,(),,(: 11000 NN tititixS

1),),((:),( kkikk tttttxfxtik

Main result in the areaMain result in the area

• The switching is assumed to be minimal. This means that

• The endpoints at which the i’th system is active is denoted:

• Let T be a strictly increasing sequence of times:

1 jj ii

iS |

,...,...,, 10 NtttT

Main result in the areaMain result in the area

• Let T be a strictly increasing sequence of times:

• The even sequence of T is given by:

• The interval completion I(T) of a strictly increasing of times T is the set:

,...,...,, 10 NtttT

,....,:)( 20 ttT

Zj

jj tt ],[ 122

Definition: Lyapunov-like functionsDefinition: Lyapunov-like functions

• Let V be a function that is a continuous positive definite function about the origin, with continuous partial derivatives. Given a strictly increasing sequence of times T in R, we say that V is Lyapunov-like for the function f and the trajectory x(t) if:

(T)on ingnonincreaslly monotonica is .2

)(0))((.1

V

TIttxV

Definition: Lyapunov-like functionsDefinition: Lyapunov-like functions

)(xV

0t 1t 2t 3t jt2 12 jt

Main resultMain result

S u p p o s e th a t w e h a v e c a n d id a te L y a p u n o vfu n c t io n s NiV i ,..,2,1: , a n d v e c to r f ie ld s

)( xfx i w i th )0(0 if fo r a l l i . L e t S b e th e s e t o fa l l s w i tc h in g s e q u e n c e s a s s o c ia te d w i th t h e s y s te m .

I f fo r e v e r y p o s s ib le s w i tc h in g s e q u e n c e w e h a v eth a t fo r a l l i , V i i s L y a p u n o v l ik e fo r f i a n d x s( t ) o v e rS | I , th e n th e s y s te m i s s ta b le in th e s e n s e o fL y a p u n o v .

Main resultMain result

)(1 xV

0t 1t 2t 3t 4t 5t 6t 7t

)(2 xV

]2,1[),( ixfx i

A slight variation of the previous resultA slight variation of the previous result

Suppose that we have a finite number of Lyapunovfunctions NiVi ,..,2,1: corresponding to thecontinuous-time vector fields )(xfx i . Let Sk

be the switching times of the system. If, whenever weswitch into mode i, with corresponding Lyapunovfunction Vi, we have that ),()( jiki sVsV where

kj ss is the last time that we switched out of mode i,

then the system is stable in the sense of Lyapunov.

A slight variation of the previous resultA slight variation of the previous result

)(1 xV

0t 1t 2t 3t 4t 5t 6t 7t

)(2 xV

]2,1[),( ixfx i

Advantages of Branicky’s approachAdvantages of Branicky’s approach

• Easy to understand and will therefore be used in industry.

• Can be used for heterogeneous systems (truly varying structure).

• Any type of Lyapunov function can be used.

Branicky’s approach: DisadvantagesBranicky’s approach: Disadvantages

• No constructive procedure for choosing the Lyapunov functions. The choice of these functions is critical to the performance of the system.

• Cannot in present form cope with unstable sub-systems.

• Requires N-candidate Lyapunov functions and places conditions on all of the candidate Lyapunov functions.

• This is probably not necessary.

Branicky’s approach:Branicky’s approach:

• Another way of saying:

‘Wait long enough before you switch and the system will be stable’

• Does not really tell you how long to wait.

• Result is related to all the other slowly varying system results in the literature.

• Computationally intensive - need to store lots of values.

• Very conservative!

Open questionsOpen questions

• Linear systems - How do you pick the Lyapunov functions to minimise the dwell time.

• Can the conditions be relaxed. Do we need N-candidate Lyapunov functions.

• What about forced switching. One basic assumption is the longer you wait the more stable you are. This is not always the case in practice.

• No need to restrict the results to switching instants.