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, 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 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 systems
• Branicky considers the following class of system:
x (t )  f i ( x(t ))
• 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.
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:
 3.5  4.5 
f1 ( x)  A1 x, A1  

13
.
5

14
.
5


4.5 
 3.5
f 2 ( x)  A2 x, A2  


13
.
5

14
.
5


Instability
Main result in the area
• Branicky adopts the following notation:
S  x0 : (i0 , t0 ), (i1 , t1 ),....(iN , t N ),....
• The rules are interpreted as follows:
(ik , t k ) : x  f ik ( x(t ), t ), t k  t  t k 1
• This trajectory is denoted xs(t).
Main result in the area
• The switching is assumed to be minimal. This means
that
i j  i j 1
• The endpoints at which the i’th system is active is
denoted:
S |i
• Let T be a strictly increasing sequence of times:
T  t0 , t1 ,..., t N ,...
Main result in the area
• Let T be a strictly increasing sequence of times:
T  t0 , t1 ,..., t N ,...
• The even sequence of T is given by:
 (T ) : t0 , t2 ,....
• The interval completion I(T) of a strictly increasing of
times T is the set:
 [t 2 j , t 2 j 1 ]
jZ 
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:
1.
V ( x (t ))  0 t  I (T )
2.
V is monotonica lly nonincreas ing on  (T)
Definition: Lyapunov-like functions
V (x)
t0 t1 t 2 t3
t 2 j t 2 j 1
Main result
Suppose that we have candidate Lyapunov
functions Vi : i  1,2,.., N , and vector fields
x  f i (x ) with 0  f i (0) for all i. Let S be the set of
all switching sequences associated with the system.
If for every possible switching sequence we have
that for all i, Vi is Lyapunov like for fi and xs(t) over
S|I, then the system is stable in the sense of
Lyapunov.
Main result
V1 ( x)
x  f i ( x), i  [1,2]
V2 ( x)
t 0 t1 t 2 t3 t 4 t5 t6 t7
A slight variation of the previous result
Suppose that we have a finite number of Lyapunov
functions Vi : i  1,2,.., N corresponding to the
continuous-time vector fields x  f i (x ) . Let Sk
be the switching times of the system. If, whenever we
switch into mode i, with corresponding Lyapunov
function Vi, we have that Vi ( sk )  Vi ( s j ), where
s j  sk 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 result
V1 ( x)
x  f i ( x), i  [1,2]
V2 ( x)
t 0 t1 t 2 t3 t 4 t5 t6 t7
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.