Application of non-linear state variable transformations for a class of non-linear system optimization problems

Abstract

This paper presents a simple non-linear state variable transformation method for solving optimization problems for a class of non-linear systems, a1(x,[xdot])(d2x/dt2) + a2(x,[xdot])(dx/dt) =ƒu, and (dx/dt) +cx2 = hu, where ƒ, c, and h are constants and u is an input, |u|≤1. Choosing an appropriate transformation function, the systems can be transformed into either a bilinear or a linear system. Optimal solutions for the bilinear and linear systems have been solved analytically by several authors. Using an optimal solution in a transformed space, the solution of an original system can be obtained by an inverse transform. In a class of the second-order non-linear systems, the conditions for the applicability of the transformation are given and are classified by its coefficient's relation and the type of the transform functions. An example, solved graphically by Athan and Falb (1966), illustrates that a class of the second-order non-linear optimal control problems can be dealt with using this technique. Fur...