On the efficiency of a linear controller under additional geometric constraints

Abstract

One heuristic approach to taking into account geometric constraints on the controls in stabilization problems is to use controls obtained by truncation (at the constraint values) of a control signal linear in the phase variables. With the introduction of the truncated control, the originally linear system becomes substantially nonlinear, which complicates the analysis. In numerous papers, the phase plane method was used to analyze the control defined as the sign of a control signal linear in the phase variables. In [1, 2], the asymptotic stability of linear dynamical systems with nonlinear controls of special type different from that considered below was studied. The problem of stabilization of a mechanical system by a geometrically constrained control was considered in [3]. The asymptotic stability of an arbitrary linear system with a truncated control was studied in [4], where some estimates for the attraction domain of the trivial solution of the system were obtained and necessary and sufficient conditions under which this domain can be made arbitrarily large were given. In the present paper, we solve the problem of ensuring the asymptotic stability of amechanical system with arbitrarily many degrees of freedom and with componentwise geometric constraints on the control.