The Approximation of Reachable Sets of Nonlinear Differential Controlled Dynamical Systems

Abstract

This paper considers the problem of reachable sets computation and approximation for nonlinear differential controlled dynamical system. The object of study is the class of controlled dynamical systems described by vector nonlinear differential equations. Firstly, the system under study is sequentially transformed using linearization along reference trajectory and discretization of this linearized system. Thus, the initial nonlinear model is associated with its linear discrete-time approximation. In this paper it is assumed that the state vector of a system and the control action are constrained by geometrical sets, i.e. by convex, closed and limited polyhedra with finite number of vertices because of natural causes. The reachable set computation is implemented using general recursion algebraic method and its modification. The final section of the work provides the simulation results on few examples of relative motion of two spacecrafts (Clochessy-Wiltshire system of equations) and predator-prey system (Lotka-Volterra model). For each example the computer simulation results and comparative analysis of the reachable sets approximation accuracy for a specific nonlinear differential dynamical system using the reachable sets of corresponding linear discrete-time dynamical systems, which was constructed with the general recursion algebraic method of reachable sets computation and its modification.