Volume-Optimal Inner and Outer Ellipsoids

Approximating a complex set K by a simple geometrical form (such as a polytope, an orthotope, a sphere or an ellipsoid) is often of practical interest. Consider for instance the situation where a vector u has to be chosen so as to satisfy the property (8.1) % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGWbWaaeWaa8aabaWdbiaadwhacaGGSaGaamiEaaGaayjkaiaa % wMcaaiabgIGiolaadsfacaGGSaGaeyiaIiIaamiEaiabgIGiolaado % facaGGSaaaaa!433E! $$ p\left( {u,x} \right) \in T,\forall x \in S, $$ where x and p(.,.) are vector-valued and where T and S are given sets. This can be of interest for instance in robust control, where the controller characterized by u must be designed in order to guarantee some given performances—at least stability—corresponding to a target set T for the process under study, given the information that the model parameters x lie in some specified feasible domain S. The information about S can be derived using the parameter bounding methodology, where one assumes that observations with bounded errors are performed on the process.(1)