Automating Verification of Cooperation, Control, and Design in Traffic Applications - Formal Methods and Hybrid Real-Time Systems - page 151

α ∗

P I

μ i m Q i m + I

: A m P + PA m −

∀ ∈ M

m 0

i

The matrices Q i m ∈ R -procedure [55]. They are

computed a priori from the invariants I m such that I m ⇒

n

are the result of the so-called S

0 for all i .
The details of this computation, which only involves basic algebra in the case of
polytopic invariants, can be found in [42].

x T Q i m x ≥

f(x)

x

(a) Convex function (b) Convex set

Fig. 15. Convex set and function

Intuitively, this linear matrix inequality can be visualized as follows.
Figure 15(b) shows an illustration of the parameter space of the Lyapunov func-
tion candidate. Note that the parameter space will generally be high-dimensional
(for example 10 dimensions in case of 4 continuous variables, plus the

-procedure
variables μ i m ), so the parameter space for an actual system can not be represented
visually in a meaningful way. Each linear matrix inequality constraint bounds the
set of feasible Lyapunov functions with a convex (that is, “curving inward”, see
Fig. 15(a)) constraint, resulting in a convex solution set. Each point in this so-
lution set corresponds to one admissible Lyapunov function for the system, and
identifying one is a convex feasibility problem , which can be solved with standard
nonlinear optimization software [7]. Additionally, it is possible to identify an op-
timal feasible point, with respect to a convex constraint. This is for instance used
to maximize the volume of the ellipsoid or the value of k in Section 4. One can also
use this to obtain an estimate on the convergence rate of an asymptotically stable
system [42]. As opposed to linear optimization, the optimum will not generally lie
on the edge of the feasible set - therefore interior point algorithms [40] are used.
Here the convexity of the solution set can be exploited.

S

6.3 Stability of the Drive Train with Continuous-Time Controller

For the drive train with continuous controller, as described above, the solver
CSDP [7] gives the following solution