Control problems on a finite time horizon and uncertain systems

The paper examines three control problems on a finite time horizon for an uncertain linear time-invariant finite-dimensional system. For the unperturbed system we provide solutions to the following problems: (I) steering the output at time r to a point inside a given ball, i.e. ||y(τ) - y₁||²_Rq ≤ α (II) tracking a given reference signal on a finite time interval with a prescribed accuracy, i.e. ||y(·)- y₁(·)||²_L2(0,τRq) ≤ α (III) recovering an initial state from an output observation made on a finite time interval. In problems (I) and (II) the control which does the job is to minimize energy and is expressed in feedback form. If we now apply these solutions to an uncertain system then, in general yΔ(τ) ≠ y(T), yΔ(·) ≠y(·) and xΔ ≠ x₀, where the subscript Δ denotes the perturbed system. We present an approach to computing estimates for the deviations of the output or recovered initial state of the perturbed system from the output or initial state of the unperturbed one. Uncertainty in the system description is modelled by unknown (norm bounded) additive perturbations of the system matrix as well as input and output matrices.