Linear quadratic regulation for linear time-varying systems with multiple input delays part I: discrete-time case

Although the problem of linear quadratic regulation has been well studied for discrete linear systems without delays or with only a single input delay in the past decades, the same problem for systems with multiple input delays remains to be solved. This paper aims to present a complete solution to the LQR problem for systems with multiple input delays. It shall be shown that the problem is dual to that of fixed-lag smoothing for a backward system without involving delays. Thus, the optimal LQR controller can be obtained by computing the gain matrices of the standard fixed-lag smoothing. An explicit solution for the LQR problem is given in terms of the solution of a Riccati difference equation.