Estimation problems for uncertain nonlinear dynamical systems with ellipsoidal state constraints

Abstract

The paper deals with the estimation problems for uncertain nonlinear dynamical systems in the case when a probabilistic description of noise and errors is not available, but only bounds on them known. Such models are found in many applied areas ranged from engineering problems from physics to economics as well as to biological and ecological modeling when it occurs that a stochastic nature of the errors is questionable because of limited data or because of complexity and nonlinearity of the model. As alternative to a stochastic characterization, a so-called set-membership approach has been proposed and intensively developed in the last decades. The solution of many control and estimation problems under uncertainty involves constructing reachable sets and their analogs. For models with linear dynamics under such set-membership uncertainty, there are several constructive approaches, which allow finding effective estimates of reachable sets. Certainly, concrete problems are mostly nonlinear in their parameters and the set of feasible system states is usually non-convex or even non-connected. The key issue in nonlinear set-membership estimation is to find suitable techniques, which produce related bounds for the set of unknown system states without being too computationally demanding. Here the problem of estimating reachable sets of nonlinear dynamical control systems with combined bilinear and quadratic nonlinearity and with uncertainty in initial states is studied. Applying results of the theory of trajectory tubes of control systems and related techniques of differential inclusions theory we present approaches that allow finding the upper ellipsoidal estimates of reachable sets. The main new result consists in obtaining ellipsoidal estimates for reachable sets of nonlinear dynamical system with additional state constraint of ellipsoidal type. Related numerical algorithms, examples and results of computer simulations are included.