### Optimization and Systems Theory Seminar

Monday, May 18, 1998, 11.00-12.00, Room 3733,
Lindstedtsvägen 25

**Professor
Jan C. Willems**

Mathematics Institute

University of Groningen

####
A behavioral approach to robust control

The control configuration that we will discuss can be described as
follows. Consider a given plant that relates three (vector-valued)
time-signals: disturbances $d$, to-be-controlled variables $z$, and
control variables $c$. Control is viewed as interconnecting a
controller to the plant, with the controller acting through the
control variables only. Before the controller is attached to the
plant, the variables $w=(d,z)$ define a set $\mathcal{P}$ of
realizable time-signals; $\mathcal{P}$ is, for example, a linear
shift-invariant subspace of a suitable function space.\\
When the controller is attached to the plant, it restricts
$\mathcal{P}$ to $\mathcal{K} \subset \mathcal{P}$: $\mathcal{K}$ is
called the {\em controlled behavior}. The fact that the controller can
act on the control variables only, translates into the requirement
that $\mathcal{K}$ must contain a given subspace $\mathcal{N}$ of
$\mathcal{P}$. The control problem is thus to find, for a given
$\mathcal{N}$ and $\mathcal{P}$, a controlled behavior $\mathcal{K}$
with $\mathcal{N} \subset \mathcal{K} \subset \mathcal{P}$ such that
$\mathcal{K}$ meets certain specifications. In robust control,
$\mathcal{K}$ must leave the exogenous disturbances $d$ free, it must
be stable (appropriately defined), and it must be attenuate the
disturbances ($\mid \mid z \mid \mid \ \leq \ \mid \mid d \mid \mid$
in suitable norms).

The purpose of this talk is to explain this formulation of the control
problem carefully, and sketch its solution. This solution involves the
theory of dissipative systems, and centers around a very subtle
coupling condition involving storage functions for the dissipative
systems $\mathcal{N}$ and $\mathcal{P}^\bot$.

Calendar of seminars

*Last update: May 6, 1998 by
Anders Forsgren,
andersf@math.kth.se.
*