Multiple Lyapunov Functions
and Other Analysis Tools for
Switched and Hybrid Systems
Michael Branicky
Dept. of Electrical Eng. and Computer Sc.
Case Western Reserve University
Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems – p. 1/16
Contents
•
Systems under consideration
•
Lyapunov-like functions
•
Stability using multiple Lyapunov functions
•
Limit cycles in continuous switched systems
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Switched systems
Systems of the form
ẋ(t) = fi (x(t))
where x ∈ Rn is the state vector, t ∈ R+ ∪ {0} is
time and i ∈ Q = {1, 2, . . . , N } is an index.
•
fi is globally Lipschitz continuous for all i
•
i is chosen such that the overall system is
non-Zeno
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Discrete-time switched systems
Systems of the form
xk+1 = fi (xk )
where k ∈ Z+ ∪ {0} is the discrete-time sample
number and i ∈ {1, 2, . . . , N } and everything else
is as before.
•
fi is globally Lipschitz continuous for all i
Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems – p. 4/16
Continuous switched systems
The system
ẋ(t) = fi (x(t))
is continuous at the switching times. That is, if at
times tj , j = 1, 2, 3, . . . the vector field fij−1
switches to fij , then
fij−1 (x(tj−1 )) = fij (x(tj ))
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Hybrid systems
Systems of the form
ẋ(t) = f (x(t), q(t))
q(t) = ν(x(t), q(t− ))
where x ∈ Rn is the state vector, t ∈ R+ is time
and q ∈ Q = {1, 2, . . . , N } is the discrete state.
We can also imagine an external input u ∈ Rm ,
either switching or continuous; this would make
the system a controlled hybrid system.
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Switching sequences
S = x0 ; {(ij , tj )}N
j=0
where x0 ∈ Rn is the initial state and (ij , tj ) are
pairs specifying
ẋ(t) = fij (x(t)) for tj ≤ t < tj+1
In the following, S denotes the set of all switching
sequences associated with the system.
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Switching sequences
•
Projections of switching sequences:
π1 (S) = x0 ; {(ij )}N
j=0 ,
•
π2 (S) = x0 ; {(tj )}N
j=0
Endpoints of time intervals where system i is
"switched in":
S|i = x0 ; {(i, tj )}, j ∈ {k ∈ Z+ ∪ {0}|fi is active}
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Switching sequences
•
Interval completion of a strictly increasing
Nj
sequence T = {tj }j=0 :
I(T ) =
Nj
[
[t2j , t2j+1 ]
j=0
•
Even sequence of T :
E(T ) = t0 , t2 , t4 , . . .
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Lyapunov-like functions
Definition 2.2 Given a strictly increasing time
sequence T , a continuous pos. def. function
V (x(t)) with V (0) = 0 and continuous partial
derivatives is Lyapunov-like for a trajectory x(t)
over T if
•
V̇ (x(t)) ≤ 0 for ∀t ∈ I(T ), x 6= 0
•
V (x(t)) is monotonically non-increasing on
E(T )
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Lyapunov-like stability for
switched systems
Theorem 2.3 Suppose we have candidate
Lyapunov fcts. Vi for corresponding vector fields
ẋ = fi (x), fi (0) = 0 for i ∈ Q. If, for all S ∈ S and
all i ∈ Q, Vi is Lyapunov-like for xS (t) (the
trajectory associated with the switching seq. S)
over S|i, then the system is stable in the sense of
Lyapunov.
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Lyapunov-like stability for
switched systems
•
The theorem holds for time-varying fi as well
•
It does not hold for infinite numbers of vector
fields
•
For restricted classes of vector fields, one
can use this to design (feedback laws and)
switching sequences that guarantee stability
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A switched system requiring
MLF
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Lyapunov-like stability for
"smoothly" switched systems
Theorem 2.7 Suppose we have a continuous
candidate Lyapunov-like function
Vλ : Rn × K → R+ ∪ {0} for corresponding vector
fields ẋ = f (x, λ), f (0, λ) = 0, λ ∈ K (a compact
set). If, for all S ∈ S and all λ ∈ K, Vλ satisfies
Vλj+1 (x(tj+1 )) < Vλj (x(tj ))
∀tj , tj+1 ∈ T
for xS (t) over S|λ, then the system is stable in the
sense of Lyapunov.
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Limit cycles in continuous
switched systems
Theorem 4.1 Suppose D is a simply connected
domain in R2 and f : R2 → R2 is a Lipschitz cont.
∂f1
∂f2
+
vector field such that ∇f (x) = ∂x
∂x2 , which
1
exists almost everywhere, is not zero almost
everywhere over any subregion in D.
Then D contains no closed trajectories of
x = f (x).
Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems – p. 15/16
Related literature
• Johansson and Rantzer, "Computation of piecewise
quadratic Lyapunov functions for hybrid systems,"
IEEE TAC 1998
• Pogromsky, Jirstrand and Spangeus, "On stability and
passivity of a class of hybrid systems," CDC 1998
• Mignone, Ferrari-Trecate and Morari, "Stability and
stabilization of piecewise affine and hybrid systems:
an LMI approach," CDC 2000
• Rubensson and Lennartson, "Stability and robustness
of hybrid systems using
discrete-time Lyapunov
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