Comments on “A closed-form optimal control for linear systems with equal state and input delays” [Automatica 41 (2005), 915–920]

Abstract

A method for solving the optimal control problems in linear systems with equal state and input delays has been recently presented by Basin and R-Gonzalez (2005, Automatica, 41, 915–920). In this note, based on a counterexample, it is shown that the method does not find the optimal solution.

Introduction

A recent paper by Basin and R-Gonzalez (2005) presents a method for solving the optimal control problems in linear systems with equal state and input delays. A brief review of the method is as follows.

Consider a linear system with equal time delays in state and input

$$\dot{x}\left(t\right)=A\left(t\right)x(t-h)+B\left(t\right)u(t-h)\text{,}$$

with the initial condition $x\left(s\right)=\varphi \left(s\right)$ for $s\in [t_0-h,t_0]$. The control problem is to find the control $u\left(t\right),t\in [t_0,T_1]$, that minimizes the criterion (2).

$$J=\frac{1}{2}{x}^T\left(T_1\right)\psi x\left(T_1\right)+\frac{1}{2}{\int }_{t_0}^{T_1}{u}^T\left(s\right)R\left(s\right)u\left(s\right)ds+\frac{1}{2}{\int }_{t_0}^{T_1}{x}^T\left(s\right)L\left(s\right)x\left(s\right)ds\text{.}$$

It is assumed that $u\left(t\right)=0$, for $t\in [t_0-h,t_0)$. According to Basin and R-Gonzalez (2005), the optimal control $u^{\ast }\left(t\right)$ is calculated as follows:

$$u^{\ast }\left(t\right)=R^{-1}\left(t\right)B^T\left(t\right)Q\left(t\right)x\left(t\right)\text{,}$$

where, $Q\left(t\right)$ satisfies the matrix equation (3) with the terminal condition $Q\left(T_1\right)=-\psi$.

$$\dot{Q}\left(t\right)=L\left(t\right)-Q\left(t\right)A\left(t\right)M_1^T\left(t\right)-M_1\left(t\right)A^T\left(t\right)Q\left(t\right)-Q\left(t\right)B\left(t\right)R^{-1}\left(t\right)B^T\left(t\right)Q\left(t\right)$$

where, $M_1\left(t\right)=0$ for $t$ in $[t_0,t_0+h)$, and $M_1\left(t\right)=I$ for all $t=t_0+h$.