An optimal control application in power electronics using numerical algebraic geometry

Abstract

Abstract-We present an optimal control application in power electronics using the homotopy continuation method for solving systems of polynomial equations. The proposed approach breaks the computations associated with the optimal control problem into two parts, an off-line and an on-line. In the off-line part, the approach solves a generic polynomial system by means of a linear homotopy and stores its solution. Then, the on-line part uses this solution and, given the initial state value, it calculates by means of a coefficient parameter homotopy the optimal control input of the problem. The approach exhibits a probability-one guarantee of finding the global optimal solution to the problem at hand.

A numerical algebraic geometry approach to nonlinear constrained optimal control

“…Becauase the efficiency of tracking depends on the structure of the polynomial system, there is no nice factor by which computational cost increases compared to size. However, the algorithm of this paper has been successfully applied to a larger, more realistic problem from power electronics in [27]. In Figure 1 we can observe the dynamical response of the system for N =1 , 2, and 3.…”

Numerical Algebraic Geometry for Optimal Control Applications

“…We consider the Duffing oscillator, given bÿ y(t) + 2ζẏ(t) + y(t) + y(t) 3 = u(t), (3.6) where t ∈ R + is time, y ∈ R is the continuous state variable and u ∈ R the control input, see [13].…”

“…Adequate provisions however can be made in terms of delay compensation and state estimation techniques: the interest herein rather lies, all other things being equal, in analyzing the benefit derived from the explicit inclusion of the nonlinear dynamics. Additionally, the considered system along with its associated control problem description, nonlinear model derivation and optimal control formulation have been extensively described in [3], whereto the interested reader is referred for full details: in the following a summary of the obtained results will be given. In particular, a control model of the form…”

Comparison of Computational Methods for Reaction Equilibrium

“…Nitrogen and hydrogen atom balances are given in Eqs. 8 and 9 (7) (7) (9) where is the total concentration given by Eq. 10 (10) The Gröbner bases can be found using Maple [8]; Figure 2 shows the steps in Maple when and are denoted and , respectively.…”

“…In [9], it is recommended to instead use ideas from continuation/homotopy to find all solutions; continuation has better numerical properties than current implementations of the Gröbner basis method.…”