A new technique for solving polynomial nonlinear constrained optimal control problems is presented. The problem is reformulated into a parametric optimization problem, which in turn is solved in a two step procedure. First, in a pre-computation step, the equation part of the corresponding first order optimality conditions is solved for a generic value of the parameter. Relying on the underlying algebraic geometry, this first solution makes it possible to solve efficiently and in real time the corresponding optimal control problem at the measured parameter value for each subsequent time step. This approach has a probability one guarantee of finding the global optimal solution at each step. Controller synthesis for two applications from the area of power electronics featuring a dc-ac converter and a dc-dc converter are discussed to motivate the proposed approach.
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.
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.
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