An iterative learning control theory for a class of nonlinear dynamic systems

-An iterative learning control scheme is presented for a class of nonlinear dynamic systems which includes holonomic systems as its subset. The control scheme is composed of two types of control methodology: a linear feedback mechanism and a feedforward learning strategy. At each iteration, the linear feedback provides stability of the system and keeps its state errors within uniform bounds. The iterative learning rule, on the other hand, tracks the entire span of a reference input over a sequence of iterations. The proposed learning control scheme takes into account the dominant system dynamics in its update algorithm in the form of scaled feedback errors. In contrast to many other learning control techniques, the proposed learning algorithm neither uses derivative terms of feedback errors nor assumes external input perturbations as a prerequisite. The convergence proof of the proposed learning scheme is given under minor conditions on the system parameters. 1. In troduct ion THE VAST MAJORITY of conventional control techniques have been devised for linear time-invariant systems which are assumed to be completely known and well understood. In most practical instances, however, the systems to be controlled are nonlinear and the basic physical processes in it are not completely known a priori. T h e situation becomes even more difficult when the unknown parameters of the system model change continuously or on a number of occasions. These types of parameter changes as well as model uncertainties are extremely difficult to manage even with the adaptive control techniques. The adaptive control techniques have been verified to work well for many dynamic systems with unknown but fixed parameters. However, their applications may not be easily extended to systems with unknown time-varying parameters. As a way to overcome these difficulties, a number of iterative learning control techniques have been proposed, which improve tracking performance through a number of iterative operations (Arimoto et aL (1984); Bondi et al. (1988); Casalino and Bartolini (1984); Craig (1984); Kawamura et aL (1988); Kuc and Nam (1989); Messner et al. (1990); Miller et al. (1987); Miyamoto et al. (1988); Oh and Suh (1988)). In contrast to the adaptive control schemes which achieves asymptotic tracking in the time domain, the leaning control schemes are capable of tracking the entire * Received 14 October 1991; revised 10 February 1992; received in final form 3 April 1992. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor T. J. McAvoy under the direction of Editor P. C. Parks. t Department of Electrical Engineering, Pohang Institute of Science and Technology, Pohang, P.O. Box 125,790-600, Korea. 1215 profile of the reference input because they are executed in an iterative manner. In other words, with the learning control schemes, the tracking errors in any phase of the time domain, either in the transient phase or in the steady state, can be made to be within the specified error bound. Moreover, the iterative learning control schemes work equally well for unknown dynamic systems with or without parameter changes. Among the publications in the area of learning control, Arimoto et al. (1984) proposed a general learning method for a class of nonlinear systems whose input and output gain matrices are of linear time-invariant form. In their scheme, the time derivative of the system output error is used to modify the control input for the next iteration. Their learning algorithm converges if the learning gain matrix satisfies certain conditions and has been applied to a robot system via a simple nonlinear transformation. Bondi et al. (1988) developed a learning algorithm for robot systems which uses position, velocity and acceleration signals in updating the control input at each iteration. Their result is based on the high-gain feedback concept which sets up uniform upper bounds on the trajectory errors. This bound has been used to prove the convergence of their learning controller. Miller et al. (1987) applied the idea of CMAC memory (Albus (1981)) to learning rule and developed a general learning controller for robot manipulators. They also used a high-gain feedback control approach in providing stability of the closed-loop system at each iteration. In this paper, we present a simple iterative learning control scheme which can be applied to a broad class of nonlinear systems---specifically to holonomic mechanical systems including robot systems. Its learning rule is based on feedback error signals from a linear PD controller, which provides several unique features. First, in contrast to many other learning schemes (Arimoto et al. (1984); Bondi et al. (1988); Miller et al. (1987)), the proposed learning controller does not use any derivative terms of system state/output errors, which implies that the measurement or estimation of the acceleration signal is not necessary when the learning control is applied to robot systems. Second, the learning controller does not require any external input perturbations. This feature enhances the robustness of the control scheme since the controller is free from any input perturbations which, when selected poorly, may jeopardize stable operation of the system. Third, the learning controller essentially takes into account the dominant system dynamics in the sense that it uses feedback information in updating the feedforward learning input. Fourth, its convergence is derived under simple conditions which normally hold even when the system has some bounded uncertainties. Finally, bounded input disturbances have been accommodated in the learning controller and eventually eliminated at the final stage of learning. This paper is organized as follows: Section 2 formulates the problem for a class of nonlinear systems. Section 3 derives the existence of a feedback law which keeps the tracking errors within the specified error bounds. Under stable closed loop systems, Section 4 develops the learning rules which drive the tracking error to zero.