Stabilization of programmed motions by the virtual output method

Abstract

with a scalar control, we consider the problem on the stabilization of a given programmed motion (x∗(t), u∗(t)), t ≥ 0, for a completely known state vector. For nonlinear systems, one known approach is based on the reduction of system (1) in R to a regular canonical [1, 2] or quasicanonical [3] form. To reduce the system to these forms, one should find a function φ(x) whose derivatives along the trajectories of system (1) up to a given order do not contain the control and verify a number of additional conditions. It is convenient to treat the function φ as a virtual output of system (1) and use the terminology of the theory of a normal form of an affine system with respect to a given output [4, p. 143]. An equilibrium stabilization method is known [4, p. 172] for stationary affine systems with given output reducible to a normal form with asymptotically stable zero dynamics. If the zero dynamics is unstable, then for stationary systems, one can use the method suggested in [5, 6] for finding a virtual output, which is then used for the design of a normal form with asymptotically stable zero dynamics. The problem on the stabilization of a given change of the output of an affine system was considered in [4, p. 180]. The results obtained for the problem on the stabilization of a programmed motion permit one to indicate conditions under which the programmed motion is stable in the variables of the normal form; however, the asymptotic stability problem remains open. The problem of the uniform asymptotic stabilization of a programmed motion can be reduced to the problem of uniform asymptotic stabilization for the zero equilibrium of a nonstationary affine system, which is obtained from the original system by passage to deviations from the programmed motion. The normal form of a nonstationary affine system was introduced in [7], and conditions providing the uniform asymptotic stabilization of the zero equilibrium in the variables of the normal form were presented there. The uniform asymptotic stability of the nonstationary zero dynamics is an important requirement. If the zero dynamics is unstable, then the problem of designing a stabilizing feedback requires a separate study. In the present paper, we consider stationary affine systems reducible to a quasicanonical form of special structure and give a method for designing a nonstationary feedback providing the asymptotic stabilization of a given programmed motion. The method is based on the design of a virtual output of the nonstationary system in terms of the variables describing the deviation from the programmed trajectory such that the latter system can be reduced, with respect to this output, to a normal form with uniformly asymptotically stable zero dynamics. A related example is given.