On determining the model structure of a nonlinear feedback control law for a model following nonlinear system

Abstract

GbstractThis paper presents a methodology to determine the model structure of a nonlinear feedback control law so that when applied to a model following nonlinear system with predominant nonlinearities, such as industrial robots, will cause the output of the model following system to accurately follow the output of the referenced model.The proposed algorithm will start with a control sequence for the nonlinear plant which is equal to the control sequence for the model. This control sequence will be updated iteratively, taking into consideration a library of nonlinear vector-valued functions associated with the dynamic equations of the nonlinear plant.The essential feature of the technique is that it recursively generates a sequence of projection matrices to be used as a mapping transformation operators so that the original vector-valued functions become conjugate directions. This leaves the original nonlinear dynamic equations intact and allows the n-term nonlinear control law to be assembled in n selections from the library.A fundamental problem in nonlinear control theory is to determine a feedback control law so that a set of boundary and dynamical constraints for a nonlinear system are satisfied and a certain performance index is minimized. In general the problem can be expressed as Min u E~ J(u(t)) = h(x(t),u(t),t) dt ( 1 )1 S t f for the dynamical systemTo design a feedback control law for nonlinear systems, several clever heuristics and algorithms have been developed in the past [l-31. The basic approach for such algorithms is to generate a series of control sequences hopefully' convergent to the optimal controller. Quite often, a linear control law is successful because the system does not have predominant nonlinearities, e.g. [41. In [41, Desrochers and AlJaar have used a simplified model to obtain a linear controller for a nonlinear F-8 aircraft fighter. In their findings they indicated that the linear controller obtained using the simplified model is compatible with the optimal linear and nonlinear controllers previously found by Gmard and Jordan [51.In an approach used by Nedeljkovic [l], the system is split into linear and nonlinear parts. Then an arbitrarily chosen control sequence, (1101, is updated iteratively, considering the linear and nonlinear variations.Another approach based on the conjugate gradient technique has been developed by Lasdon, Mitter, and Waren [2]. They applied the conjugate gradient method of Fletcher and Reeves to optimal control problems. Later, Haas modified the Lasdon-Mitter-Waren (LMW) algorithm by using the conjugate gradient method of Polak-Ribiere [31. Further, the well known DavidonFletcher-Powell (DFP) algorithm [6-71, uses the gradient information to recursively generate a set of matrices to approximate the inverse of the Hessian matrix. In a previous version of the proposed algorithm, a set f positive definite and symmetric projection matrices ME) (j=1,2, ...) are recursively generated to be substituted for the inverse of the Hessian matrix and to be used to genera...