Methods of constructing guaranteed estimates of parameters of linear systems and their statistical properties

This paper is concerned with the properties of guaranteed estimates of unknown parameters of linear systems. In the theory of guaranteed or minimax estimation [1,2] as opposed to mathematical statistics, the nature of the perturbations is assumed to be uncertain and we consider either the problem of finding estimates that minimize the estimation error under the worst (from the viewpoint of an observer) possible perturbations from some a priori known set, or the problem of finding the whole set of parameters compatible with the observed signal. When there is no sufficiently complete description of the random perturbations the guaranteed approach can also be used for stochastic systems. In this case the assumption of random noise implies that the guaranteed estimates have additional properties. In this paper (see also [3-6]) we obtain sufficient conditions for the convergence of these estimates to the actual values of the unknown parameters. We consider the case when we have geometrical constraints, implying that the perturbations are bounded at each instant of time. In this case to develop an exact description of the information sets mentioned above is a very cumbersome nonlinear programming problem. A method of approximating guaranteed estimates which only requires the solution of a linear programming problem is suggested, and examples are given.