Stabilization of nonstationary affine systems by the virtual output method

Abstract

with a scalar control, we consider the problem on the stabilization of the equilibrium x = 0 for a completely known state vector. One of approaches is based on the reduction of the system to a special form [1; 2, p. 286] referred to as canonical for which the solution of the stabilization problem is known. However, conditions under which system (1) can be reduced in R × [0,+∞) to a canonical form are quite cumbersome, and such a transformation does not necessarily exist. Conditions under which system (1) can be reduced to a quasicanonical form [1] are less cumbersome, but methods for stabilization of the equilibrium for such systems are developed only in special cases and mainly for stationary systems. One method that permits one to solve the stabilization problem for equilibria of stationary affine systems was given in [3, 4]. It is based on finding a special function of state variables of the system, which is referred to as a virtual output. This output should define a transformation of the system to a normal form [5, p. 139; 6, p. 113] with asymptotically stable zero dynamics. For such a normal form, there is a method for constructing a stabilizing state feedback, and this feedback written out via the original variables solves the stabilization problem. In the present paper, we generalize the virtual output method for the construction of a stabilizing feedback to the case of nonstationary affine systems with scalar control reducible to a quasicanonical form and develop new methods of uniform stabilization of equilibria of nonstationary affine systems. We give related information on the reduction to quasicanonical and normal forms for nonstationary affine systems, justify the virtual output method for the construction of a feedback uniformly stabilizing the zero equilibrium of a nonstationary system that can be reduced by a virtual output to a normal form with relative degree 2, and consider an example. We consider a special case, and on its basis, we construct an example of the solution of the stabilization problem for a programmed trajectory.