Optimal control by least squares support vector machines

Support vector machines have been very successful in pattern recognition an d function estimation problems. ln this paper we introduce the use of least squares support vector machines (LS-SVM's) for the optimal control of nonl inear systems. Linear and neural full static state feedback controllers are considered. The problem is formulated in such a way that it incorporates t he N-stage optimal control problem as well as a least squares support vecto r machine approach for mapping the state space into the action space. The s olution is characterized by a set of nonlinear equations. An alternative fo rmulation as a constrained nonlinear optimization problem in less unknowns is given, together with a method for imposing local stability in the LS-SVM control scheme. The results are discussed for support vector machines with radial basis function kernel. Advantages of LS-SVM control are that no num ber of hidden units has to be determined for the controller and that no cen ters have to be specified for the Gaussian kernels when applying Mercer's c ondition. The curse of dimensionality is avoided in comparison with definin g a regular grid for the centers in classical radial basis function network s. This is at the expense of taking the trajectory of state variables as ad ditional unknowns in the optimization problem, while classical neural netwo rk approaches typically lead to parametric optimization problems. In the SV M methodology the number of unknowns equals the number of training data, wh ile in the primal space the number of unknowns can be infinite dimensional. The method is illustrated both on stabilization and tracking problems incl uding examples on swinging up an inverted pendulum with local stabilization at the endpoint and a tracking problem for a ball and beam system. (C) 200 1 Elsevier Science Ltd. All rights reserved.