### Docent Seminar, Optimization and Systems Theory

Friday, April 12, 2002, 11.00-12.00, Room 3721, Lindstedtsv. 25

**Ulf Jönsson**

Optimization and Systems Theory

KTH

E-mail: ulf.jonsson@math.kth.se

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On reachability analysis of uncertain systems

Reachability analysis is an important tool in verification and
synthesis of control systems. It refers to the problem of computing
bounds on the set of states that can be reached by a dynamical
system. Reachability analysis has received a lot of attention in recent
work on hybrid and switched dynamical systems where the aim has been
to extend existing verification procedures for discrete systems to
systems that involve continuous dynamics. The reachability tools that
have been been proposed use coarse uncertainty
descriptions such as differential inclusions, set disturbances, and
ellipsoidal approximations. In this lecture we consider reachability
analysis of systems where the disturbances and the model discrepancies
are characterized by integral quadratic constraints. This gives
improved approximation of many types of unmodeled dynamics.
Two specific problems of reachability analysis can be identified:

1. Reach set computation, which is the problem of computing bounds
on the reach set for trajectories of finite time extent.

2. Transition analysis, which is the problem of estimating the mapping
from one switching surface to another.

Two examples will illustrate the motivation for these two
problems. Reach set computation will be used to prove that a robot
stays within a close neighborhood of its desired path, which is
designed based on some nominal dynamics. Transition analysis on the
other hand will be used to prove a certain type of
robustness for limit cycles in piecewise linear systems.

Our analysis results in a nonconvex optimal control problem, which can be
addressed using Lagrange relaxation. We discuss how the dual
optimization can be performed and review some special conditions under which
there is no duality gap.

Calendar of seminars

*Last update: March 27, 2002 by
Anders Forsgren,
anders.forsgren@math.kth.se.
*