Optimal control of linear multivariable systems with quadratic performance criteria

Abstract

The optimal control of continuous-time linear multivariable systems with finite- and infinite-time quadratic performance is investigated using the maximum principle. The time-varying optimal-control law is shown to be related to the matrix Riccati differential equation, and a solution is obtained in terms of the components of the transition matrix associated with the linear system of 2n differential equations in the n state variables and the n adjoint variables. The time-varying transition-matrix formulation is also used to develop a solution of the steady-state matrix Riccati and associated Lyapunov equations. The infinite-time optimal-control solution is derived in terms of partitioned eigenvector components related to the stable modes of the augmented system, and the technique is shown to be similar in principle to the method previously developed for reducing the order of a set of matrix differential equations by neglecting high-order modes. The results are also shown to be related to the solution of the algebraic Riccati equation and are used to develop an eigenvector solution for the Lyapunov equation. The techniques are extended to consider the optimal control of the linear system containing input derivatives. Results are obtained for the optimal control of a 1st- and 2nd-order linear system and for a 6th-order model representing the dynamics of a superheater system. The various techniques available for obtaining a solution of the linear optimal-control problem, namely the recurrence-type control algorithm based on dynamic programming, the eigenvector solution of the steady-state Riccati equation and the solution obtained by numerical integration of the time-varying Riccati equation, are compared.