### 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.