A Higher-Order Method for Dynamic Optimization of a Class of Linear Systems

Abstract

This paper deals with optimization of a class of linear dynamic systems with n states and m control inputs, commanded to move between two fixed states in a prescribed final time. This problem is solved conventionally using Lagrange’s multipliers and it is well known that the optimal solution satisfies 2n first-order linear differential equations in the state and Lagrange multiplier variables. In this paper, a new procedure for dynamic optimization is presented that does not use Lagrange multipliers. In this new procedure applied to a class of linear systems with controllability index p = (n/m), optimal solution satisfies m differential equations of order 2p. The boundary conditions on these m variables are computed in terms of higher derivatives (up to p − 1) at the initial and final time. These higher-order differential equations are solved using classical weighted residual methods, methods relatively unknown to controls community but extremely popular with researchers in mechanics. This new procedure for dynamic optimization, higher order necessary condition solved by weighted residual method, is computationally more efficient compared to other conventional procedures, offering benefits for real-time applications.

Optimal planning of an under-actuated planar body using higher-order method

“…This higherorder form (also, the flat form) has been investigated for the solution of the optimal trajectory generation problem by Agrawal and coworkers [ 11, [2], We have traded the niiniber of states in the problem for higher derivatives of a smaller set of states.…”

“…These include linear time invariant systems [l], classes of linear. time varying systems [2], arid classes of nonlinear systems, such as feedback liriearizable systems [3] arid chairled systems [SI. 111 the higher-order method, the state equations are explicitly embedded into the 0-7803-4300-X-5/98 $10.00 0 1998 IEEE states and inputs, changing the constrained optimization problem into an unconstrained optimization problem.…”