An algebraic geometry approach to nonlinear parametric optimization in control

Abstract

We present a method for nonlinear parametric optimization based on algebraic geometry. The problem to be studied, which arises in optimal control, is to minimize a polynomial function with parameters subject to semialgebraic constraints. The method uses Gröbner bases computation in conjunction with the eigenvalue method for solving systems of polynomial equations. In this way, certain companion matrices are constructed off-line. Then, given the parameter value, an on-line algorithm is used to efficiently obtain the optimizer of the original optimization problem in real time.

An optimal control application in power electronics using numerical algebraic geometry

“…One important feature of the presented approach lies in the fact that it scales well with the dimension of the parameter vector x 0 , in contrast to similar existing methods [9]. This is due to the fact that parameter x 0 is not treated symbolically -therefore its dimension is relatively irrelevant regarding the computational complexity of the approach -because a specific value (x 0 orx 0 ) is always assigned to it.…”

A numerical algebraic geometry approach to nonlinear constrained optimal control

“…The combined algebraic and numeric nature of the approach renders it capable of handling problems with an increased number of parameters, compared e.g. to the one developed in [10].…”

“…Symbolic methods based for instance on comprehensive Gröbner bases [21] or resultants, can deal with parametric problems arising in parametric optimal control, but tend to be quite expensive when it comes to computations [9], [10], [11]. Specifically, these techniques suffer from what is known as coefficient blowup and their precomputation phase can take a considerable amount of time.…”

“…In this paper, we focus on the important subclass of polynomial systems with a general polynomial performance index, following [9], [10], [11] and show how insight from numerical algebraic geometry, based on homotopy continuation for polynomial systems over the complex numbers, can be used to exploit the structure of the problem and overcome certain shortcomings of general nonlinear optimal control methods.…”

Explicit Predictive Control Laws with a Nonlinear Constraints Handling Mechanism

“…The main assumption consists in the convexity of the optimisation problem. In [6] a linearization is done over a specific class of functions (polynomial), in [7] a sampling of the parameters space is performed or one can replace directly the nonlinear bounds of the feasible domain by an approximated linearized one [8].…”