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Without loss of generality we assume that the origin of the continuous state
space
n
is the equilibrium point all trajectories converge to. If one wants to
show asymptotic stability with respect to a different equilibrium - as is the case
in the drive train example - the state space of the hybrid system can simply be
“shifted” to move this point into the equilibrium.
Intuitively, the stability property guarantees that there is an upper bound on
how far the system can stray from the equilibrium point, depending on its initial
state. Moreover, the global attractivity property tells us that the system will
eventually converge to the equilibrium point. Together, this implies that there
is an upper bound on the temporary change of state a disturbance can cause,
relative to the size of the disturbance, and that eventually the system will have
compensated for the disturbance.
We will consider hybrid systems with a finite number of discrete modes. With
each mode
m
, we associate an ane differential equation
x
•
=
A
m
x
+
b
m
R
with
n
for describing the continuous evolution of the system's
state variables. A possible transition between a pair of modes
m
1
and
m
2
is given
as a quantifier-free first-order predicate over the continuous variables. No discrete
updates of continuous variables are allowed. We also allow for an invariant in
each mode, given by a quantifier-free first-order predicate
7
on the continuous
variables. The system may only stay in a mode while its invariant is satisfied.
We assume that the system does not exhibit Zeno or blocking behavior, so that
all trajectories are continuous and unbounded in time (cf. Appendix 8).
Since the state space of such a hybrid system is
n
×
n
,
b
n
A
m
∈
R
∈
R
n
, the cartesian product
of the continuous and discrete state (mode) space, one is usually interested in
local stability. The invariants specify which continuous states can be active with
which modes - combinations violating the invariants need not be considered.
Therefore the stability property is local as defined by the invariants. Further-
more, we only expect the continuous variables to converge, but for all permissible
initial hybrid states (
x
(0)
,
m
(0)).
For systems of this kind, local asymptotic stability can be shown with the
help of a
common Lyapunov function
.Itisdefinedasfollows(see[29,9]).
R
×
M
n
Definition 5.
Consider a hybrid system with state vector x
∈
R
and mode
m
is the finite set of modes. Assume that the dynamics in mode
m are given as x
•
=
f
m
(
x
)
and that the invariant belonging to mode m is the
predicate I
m
.A
(common) Lyapunov function
for this system is then a function
V
:
∈
M
,where
M
n
R
→
R
such that:
a) V
(
x
)=0
if x
=0
and V
(
x
)
>
0
otherwise
b) for all m: V
m
(
x
):=
d
dx
(
x
)
f
m
(
x
)
<
0
if
0
=
x
I
m
V
m
(0) = 0
c)
0
I
m
⇒
d) V
(
x
)
→∞
when
||
x
||→ ∞
7
In principle, any quantifier-free predicate over the continuous variables is admissible
for mode transitions or invariants. If the resulting invariant set is not a convex poly-
hedron, it will need to be over-approximated for the actual computation, increasing
conservativeness.
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