Objective
The purpose of this project is to open up a new area in linear-quadratic optimisation theory by combining ideas and methods that have been developed over the years in the co-operating research groups. Some of the pioneering work in this area has been developed by members of the research consortium, ensuring the feasibility of the project.
The project is devoted to several closely connected, but different, new problems of optimal control of linear discrete-time systems with uncertainties. The corresponding mathematical problems formalise several important engineering problems such as vibration damping. The novelty of the problems under consideration is in the presence of different kinds of uncertainties so that the information known about the systems is incomplete. Methods will be developed for constructing physically realisable regulators which are optimal and which do not depend on the unknown parameters, despite the fact that the optimal processes do. Moreover, the regulators should be robust in the sense that the minimal cost is continuous with respect to all parameters.
These problems also differ from more well-known versions of linear-quadratic optimisation problems in that there are quadratic constraints which may be non-convex, making the problem new and difficult. Nevertheless, some special properties of these, in general, non-convex global minimisation problems enable efficient solutions to be obtained. In fact, it is a special case of a class of optimisation problems in Hilbert space in which the feasible set is the intersection of a subspace with a set defined by the quadratic constraints. A general method for the solution of such problems has already been presented. The method is based on the generalisation of the Hausdorff theorem stating that the image of the sphere in Rn is convex under a quadratic mapping into R2. Earlier generalisations were obtained for mappings into Rm for m > 2. Appropriate generalisation of these results will concern the quadratic nature of the mapping and allow this method to be applied to some new interesting cases which are important in applications.
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10044 Stockholm
Sweden