Multiple-Lyapunov Functions for Guaranteeing the Stability of a class of Hybrid Systems

Some 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, ‘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 information I am going to focus on work described in the following papers ‘Stability of Switched and Hybrid Systems’, Branicky,LIDS Tech. Report, 2214, ‘Stability of Switched and Hybrid Systems’, Branicky, CDC94. ‘Multiple Lyapunov Functions and other Analysis Tools for Switched and Hybrid Systems’, IEEE Trans. AC, This work is related to the piecewise quadratic Lyapunov functions studied by Rantzer and Johanson.

A stability result for non-linear systems Branicky considers the following class of system: The following assumption is made: –Each f i is assumed to be globally Lipschitz continuous –Each f i is assumed to be exponentially stable –The i’s are picked in such a way that there are finite switches in finite time.

A 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:

Instability

Main result in the area Branicky adopts the following notation: The rules are interpreted as follows: This trajectory is denoted x s (t).

Main 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:

Main 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:

Definition: 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:

Definition: Lyapunov-like functions

Main result

A slight variation of the previous result

Advantages 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: 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: 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 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.