Brief paperEnlarging the terminal region of nonlinear model predictive control using the support vector machine method☆
Introduction
We consider the receding horizon model predictive control (RHMPC) approach for a nonlinear discrete-time constrained dynamic system:where and are the state and control variables and and are the corresponding constraint sets. Many approaches to RHMPC for such a system have been proposed (see for example, Allgöwer & Zheng, 2000; Mayne, Rawlings, Rao, & Scokaert, 2000; and others). Set-theoretic approaches (Chen & Allgöwer, 1998; Magni, Nicolao, Magnani, & Scattolini, 2001; Mayne and Michalska, 1990, Michalska and Mayne, 1993 and others) use a terminal constraint set and a terminal cost F to increase the domain of attraction and minimize online computational effort. These methods typically rely on properties of the linearized system for the characterizations of and F and prove, under reasonable assumptions, the asymptotic stability of the origin under RHMPC. With limited assumptions on the properties of f, the domains of attraction for such methods are naturally small. Unlike the set-theoretic approach, this paper shows the use of approximating functions in RHMPC by characterizing and F using support vector machine (SVM) learning. The resulting and hence the domains of attraction are much larger. When X is compact, this approach also results in lower online computational effort as shorter horizons can be used. The use of approximating function in model predictive control (MPC) is not new. Parisini and Zoppoli (1995) apply neural networks to directly approximate the closed-loop MPC control law, without the use of and F. Such an approach requires accurate approximation to ensure closed-loop stability. Our approach exploits the flexibility in the choices of and F and is less demanding in terms of the approximating accuracy. We demonstrate the effectiveness of the approach on two low-dimensional systems (). For higher-dimensional systems, the online computations are expected to increase reasonably although the off-line computations become increasingly demanding with n.
Standard notation is adopted: is the interior of , is the inner product of , and is the norm on the combined vector for .
Section snippets
Preliminaries
The RHMPC of (1) is based on the solution, at each time t, of the following finite horizon (FH) optimal control problem over where is the stage cost, is the terminal set and F is the terminal cost. Let (3)–(6) are satisfied.}, the set of states steerable to in N steps or less. Suppose the system satisfy the following
Characterization of
Our characterization of follows the approach described in Ong, Keerthi, Gilbert, and Zhang (2004). It considers system (7) with where K is determined from some systematic procedure. The characterization of takes the form of a scalar function, such that closely approximates . To do so, (7) is solved numerically for many initial points and for each , the condition of is determined. Two point sets, one containing points in and the other
Characterization of F
Like SVC, support vector regression (SVR) (Scholköpf and Smola, 2002, Shevade, Keerthi, Bhattacharyya & Murthy, 2000) finds a regression function that approximates represented by a group of points . With (A2), the computation of via (10) can be achieved over a finite number of terms if is the standard linear quadratic (LQ) cost. Let of (8) be the Lyapunov set of the linearized system and be the shortest t such that for under (7).
Enforcing feasibility
While can tolerate reasonable inaccuracy to , the approximation of by introduces a possible difficulty: FH can become infeasible because is not positively invariant. When is a reasonable approximation of , such a problem rarely occurs. To guarantee feasibility, we exploit the availability of (8) for the case where is the standard LQ cost using a modified FH, FHM. FHM optimizes over the same space with the following changes: F is replaced with in (2), (6) is
Numerical examples
We illustrate our approach with two examples using a P4 machine with a 1.8 GHz processor. The nonlinear optimization routine used for the solutions of FH and FHM is based on the algorithm provided for in the Matlab optimization toolbox. The choice of in is 0.05. Example 6.1 (Chen & Allgöwer, 1998) is the discrete-time implementation of the continuous-time system with sampling period of 0.1 s. The equilibrium point is unstable when and the linearized system
Conclusions
This paper shows the application of SVM learning to the implementation of RHMPC for nonlinear systems. The resulting terminal set, learned by support vector classification, is much larger than those seen in the literature. Consequently, the domain of attraction of the origin under RHMPC is greatly enlarged. The large terminal set also leads to a lower online computational effort via the use of a shorter horizon. The offline computation grows rapidly with the order of the system but is expected
Chong-Jin Ong received his B.Eng (Hons) and M.Eng. degrees in mechanical engineering from the National University of Singapore in 1986 and 1988, respectively, and the M.S.E. and Ph.D. degrees in mechanical and applied mechanics from University of Michigan, Ann Arbor, in 1992 and 1993, respectively. He joined the National University of Singapore in 1993 and is now an Associate Professor with the Department of Mechanical Engineering. His research interests are in robotics, control theories and
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Chong-Jin Ong received his B.Eng (Hons) and M.Eng. degrees in mechanical engineering from the National University of Singapore in 1986 and 1988, respectively, and the M.S.E. and Ph.D. degrees in mechanical and applied mechanics from University of Michigan, Ann Arbor, in 1992 and 1993, respectively. He joined the National University of Singapore in 1993 and is now an Associate Professor with the Department of Mechanical Engineering. His research interests are in robotics, control theories and machine learning.
D. Sui received the B.Eng. and M.Eng. degrees in electronic engineering department from the Northwestern Polytechnical University, PR China in 2000 and 2002, respectively. She is currently a Ph.D. candidate at the Department of Mechanical Engineering, National University of Singapore. Her research mainly focuses on model predictive control and robust control.
Elmer Gilbert received the degree of Ph.D from the University of Michigan in 1957. Since then, he has been in the Department of Aerospace Engineering at the University of Michigan and is now Professor Emeritus. His current interests are in optimal control, nonlinear systems, robotics and machine learning. He has published numerous papers and holds eight patents. He received IEEE Control Systems Field Award in 1994 and the Bellman Control Heritage Award of the American Automatic Control Council in 1996. He is a member of the Johns Hopkins Society of Scholars, a Fellow of the American Association for the Advancement of Science, a Fellow of the Institute of Electrical and Electronics Engineers and a member of the National Academy of Engineering (USA).
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Martin Guay under the direction of Editor Frank Allgower.