Feedback Implementation of Zermelo’s Optimal Control by Sugeno Approximation

This paper proposes an approach to implement optimal control laws of nonlinear systems in real time. Our methodology does not require solving two-point boundmy value problems online and may not require it off-line either. The optimal control law is learned using the original Sugeno controller (OSC) from a family of optimal trajectories. Wb compare the trajectories generated by the OSC and the trajectories yielded by the optimal feedback control law when applied to Zermelo’s ship steering problem. L Introduction Optimal control PIYSOIL 1996; Kirk, 1970] is one of oldest approaches to control engineering. It has many advantages: (1) State and control constraints can be include explicitly. (2) The cost fimction to be minimized can be often given a simple intuitively appealing inteqxetation. (3) Optimal control is a very geneml methodology applicable to multiinput-mtdti-output, nordina, stochastic, or infinite-dimensional systems. Hence, optimal control theory provides a unified approach to stating and solving very general control problems that are at the same time physically intuitive. Unfofiunately, optimal control theory suffers from a major disadvantage; namely, solving optimal control problems is in geneml compuationally difficult except in very special cases where a closed-form expmsion of the control law can be obtained. These cases include many nonlinear second-order systems and the celebrated linear quadratic regulato~ In general however the neceswuy conditions have no closed-form solution and are at least as difficult to obtain as to solve a nonlinear two-instant boundmy value problem (for the control of a system described by deterministic nonlinear ordimty differential equations. When the plant is stochastic or infinite-dimensional, the numerical difficulties are compounded.) The absence of simple closed-form solutions and online numerical solutions of the general open-loop control problem means that there is no general feedback implementation (except in the neighborhood of an optimal reference trajecto~ using the well-known neighboring optimal control Pryson and Ho; 1975].) The lack of feedback implementation is in our opinion the main reason why interest and research conducted in optimal control has greatly diminished. On the other band, fuzzy-logic controllers (FLCs) are essentially feedback control laws. While theses controllers crmbe easily made to incorpomte the heuristic knowledge of the control enginee~ and this can be an advantage in cases where this is about the only knowledge available, designing a FLC using detailed, mathematical, and exact descriptions of the plant is not very well-undemtood or pmcticed. Clearly, using alt available knowledge about the system should in principle yield control laws with superior performance. Hence, we investigate in this paper the possibility of designing fuzzy logic controllers that approximate optimal control laws; from another point of view; we investigate feedback implementation of optimal control laws using f~-logic controlled To illustrate this approach we consider the Zermelo’s problem; that is, the problem of docking a ship going at constant water speed in minimum time in a region of strong water currents using the heading angle as the control i nput. W obtain a family of open-loop solutions of this problem and use it to train the OSC. The resulting trained engine will then be a feedback implementation of (a least-squares approximation of) Zermelo’s optimal control. The Sugeno controllers ~uckley, 1993] are capable of approximating any continuous map within an arbitrmy accuracy. This paper is oqynized as follows. Section 2 provides the necessa~ background information on the optimal control of the ship steering problem. Section 3 discusses the training procedure used in designing the Sugeno-type controller from the data obtained from the optimal trajectories. Section 4 discusses the generation of training data and the elimination of angle discontinuity. Finally, section 5 summarizes the used procedure and shows simulation results. 2. Zermelo’s Optimal Control Problem The objective of Zermelo’s problem is to find a minimum-time path through a region of position-dependent vector 169 https://ntrs.nasa.gov/search.jsp?R=20010000388 2017-12-04T23:23:23+00:00Z