Linear Quadratic Regulation for Discrete-time Systems with Single Input Delay

The optimal control problem of linear quadratic regulation (the LQR problem) for linear discrete-time systems with single input delay is studied in this paper. A new and simple approach is applied to derive the optimal control input sequences. With the established duality, we first show that the LQR problem is equivalent to an optimization problem in Krein space. The latter problem is finally converted to a strictly convex quadratic programming problem. Thus we convert the delayed control input into control input delay-free, and convert a dynamic LQR optimal control problem for the linear discrete-time systems into a static mathematical programming model. The optimal control input sequences are successfully derived by solving this strictly convex quadratic programming problem. Our approach is simple and yet very effective in dealing with the LQR problem for the linear discrete-time systems with single input delay.