A numerical algebraic geometry approach to nonlinear constrained optimal control

Abstract

A new method for nonlinear constrained optimal control based on numerical algebraic geometry is presented. First, the optimal control problem is formulated as a parametric optimization program. Then, certain structural information related to the optimization problem is computed off-line. Afterwards, given this information, numerical algebraic geometry techniques are used to efficiently obtain the optimal control input (i.e. optimal solution) of the original control problem in real time. By using homotopy continuation over the field of complex numbers, this approach has a probability-one guarantee of finding the global optimal solution to the problem at hand.

A numerical algebraic geometry approach to regional stability analysis of polynomial systems

“…Numerical algebraic geometry has found application in optimal control, for example [2]. Numerical algebraic geometry is also a powerful tool for solving problems in kinematics [16].…”

“…The optimal sublevel set Ω V * of Theorem 2 can be found if one can solve optimization problem (2). Since V (x) has compact sub-level sets and is continuous, it is easy to show that the optimal value of (2) is achieved if (2) is feasible.…”

Polytopic Approximation of Explicit Model Predictive Controllers

“…The first is that only a restricted class of systems give rise to problems that can be reasonably solved parametrically. For example, linear systems with piecewise linear [5], [30] or quadratic cost functions [8], piecewise affine systems (PWA) with linear [4] and quadratic [7] cost functions and some classes of polynomial systems [3]. If the control law can be computed explicitly, then it is often the case that the complexity of the control law (i.e.…”

An optimal control application in power electronics using numerical algebraic geometry

“…In contrast to other parametric optimization approaches based on continuation suggested mainly in the nineties (see [6] for a more recent survey), the approach followed here [4] is based on polynomial homotopy continuation over the field of complex numbers. This bears the advantage of having a deep and rich supporting theory which can be used, among other things, to avoid singularities.…”

“…Motivated by this framework, in the present work we apply the approach recently proposed in [4] to the optimal control † Institute for Mathematics and Its Applications, University of Minnesota, MN 55455-0436, USA * Automatic Control Laboratory, ETH Zurich, 8092 Zurich, Switzerland…”