Time-variant receding-horizon control of nonlinear systems

Abstract

Nonlinear receding-horizon state control problem is considered including time-variant parameters explicitly in the formulation. The present problem can be applied not only to control of nonlinear time-variant systems but also to nonlinear tracking control problems. The conventional real-time optimization algorithm based on the stabilized continuation method is extended for the present problem with time-variant parameters. A simplified model of a space vehicle is employed as a numerical example, and a receding-horizon tracking control law is derived for the modsl. The obtained tracking control law achieves the best possible performance even if a given reference trajectory cannot be tracked perfectly with any control input. Introduction Stabilized continuation method is shown to be suitable for real-time optimization in nonlinear recedinghorizon state feedback and nonlinear receding-horizon state estimation,which can be applied to various problems in the field of engineering. The conventional formulation of the receding-horizon state problem does not include time-variant parameters explicitly in contrast to the state estimation problem in which timevariant parameters appear necessarily as measured output and known input. Though a time-variant receding-horizon state problem can be expressed as a time-invariant problem by regarding the time as a state, it is preferable in practical design to formulate the problem including timevariant parameters explicitly. Especially, explicit treatment of time-variant parameters is convenient if they are not given a priori as time-dependent functions but are given on line as external inputs, as is often a case in tracking control problems. The main objective of this_paper is to extend-theconventional real-time optimization algorithm to the receding-horizon state problem with explicit timevariant parameters. Slight modification of the conventional algorithm yields an algorithm that requires the derivative of the time-variant parameter with respect to time over a finite time interval in the future. Another algorithm is also derived by further modification so that the derivative of the time-variant parameter appears only in a boundary condition, which results in less computation and data storage than the former algorithm. A tracking control problem of a nonlinear space vehicle is employed as a numerical example in this paper. The state equation of the model is nonlinear and nonholonomic in spite of its simplicity. The given reference trajectory cannot be tracked perfectly by the model, and tracking error does not converge to zero. Therefore an infinite-horizon performance index cannot be used to design a tracking control law. However, minimization of a receding-horizon performance index yields a tracking control law that achieves the best possible performance. Simulation results demonstrate closed-loop characteristics of the designed tracking control law. Problem Formulation The dynamical system treated here is expressed in the following differential equation: Assistant Professor, Institute of Engineering Mechanics, Member AIAA. Copyright © 1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. where x(t) dsnotes the state, u(t) the input to the system, and p(f) the vector of time-variant parameters. The timevariant parameter p(t) is assumed to be known. In this paper, an optimal state law is designed so as to minimize a receding horizon performance index: (2) J = (p[x(t + T), P(t + T)] + J LJXT), u(r), p(r)]d Since the time-variant parameter p(t) is included both in the state equation and in the performance index, the present problem can deal with not only control of time-variant systems but also tracking control problems. A command input is regarded as a time-variant parameter in a tracking control problem. The performance index evaluates the 1 American Institute of Aeronautics and Astronautics performance from the present time t to the finite future t+T. Since the time interval in the performance index is finite, the integrant in the performance index does not have to converge to zero as time increases. Therefore the recedinghorizon tracking control law can be determined even if the tracking error does not converge to zero with any control input. In contrast to the receding horizon control, a performance index may not be bounded and there may not be any optimal solutions in an infinite-horizon problem. The present receding horizon control problem can be converted to a family of finite-horizon optimal control problems parameterized by time t as follows: