TimeOptimal Control Problem for a Special Class o

Authors: Grigory M Sklyar Svetlana Yu Ignatovich Sergey E Shugaryov

Publish Date: 2014/07/19

Volume: 165, Issue: 1, Pages: 62-77

PDF Link

Abstract

We consider the timeoptimal control problem to the origin for a class of nonlinear systems called dualtolinear systems We obtain the general description of possible optimal controls In particular we show that optimal controls take the values 1 0 and +1 only and have a finite number of points of discontinuity We describe a class of nonlinear affine control systems which can be approximated by dualtolinear systems in the sense of time optimalityOne of the most powerful and wellinvestigated tools of the nonlinear control theory is the method of linearization that is finding the precise mapping that transforms given initial system to a linear one A pioneer result in this direction was obtained in 1973 by Korobov 1 who introduced and studied the socalled class of triangular systems in connection with the controllability and stabilizability problems for nonlinear systems Later on the class of triangular systems was considered in many works in particular in connection with the problem of linearizability ie the possibility to transform a nonlinear system to a linear one A way using “Lie brackets technique” was proposed in 1973 by Krener 2 and developed in numerous works However the class of nonlinear linearizable systems is rather small So the next step was to develop methods of approximation in some sense of a given nonlinear system by a linear one In 3 the approximation was considered for nonlinear affine systems with real analytic righthand side and a concept of approximation in the sense of time optimality was introduced Moreover necessary and sufficient conditions were obtained under which the system is approximated by a certain linear systemIf these conditions are not satisfied the question arises how to approximate the original system by another nonlinear affine system of a simpler form Further progress was achieved by developing the algebraic approach 4 5 6 7 8 9 10 As a result it was shown that the approximating system can be constructed with the use of some special structures in the algebra of nonlinear power momentsIn the present paper we consider the timeoptimal control problem for affine systems with real analytic righthand side including control and the first coordinate only It turns out that optimal controls take values 1 0 +1 and have a finite number of points of discontinuity We study the question of approximation in the sense of time optimality following the approach proposed in 3 5 We find conditions under which a system of the considered class approximates an affine control system These conditions are “dual” to the corresponding conditions for systems approximated by linear ones 3 That gives a certain reason to interpret such systems as dualtolinear systemsThe paper is organized as follows In Sect 2 we consider the timeoptimal control problem for dualtolinear systems and show that optimal controls take values 1 0 and +1 only and have a finite number of points of discontinuity and describe possible optimal controls Section 3 contains the threedimensional example Finally in Sect 4 we describe the structure of right ideals induced by dualtolinear systems in the algebra of nonlinear power moments and consider the question of approximation in the sense of time optimalityThe equality 7 implies that widehatx 1t is a root of the function Pz for any tin t 2t 1 Since widehatx 1t is continuous it equals one of these roots identically on t 2t 1 ie widehatx 1t=mathrm const Therefore widehatut=dotwidehatx 1t=0 for all tin t 2t 1 square Let psi 1t 1=psi 1t 2=0 and psi 1tnot =0 for all tin t 2t 1 where 0t 2t 1le widehattheta Suppose there exists a strongly increasing sequence tau k k=1infty such that tau krightarrow t 2 as krightarrow infty and psi 1tau k=0 kge 1 Then Pz has at least two different roots and t 1t 2d 0

Keywords: