Professor
Jan C. Willems
Mathematics Institute
University of Groningen
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 $\mathcal{P}^\bot$.