Mode-estimator-free quadratic control of jump linear systems with mode-detection random delay

Linear quadratic regulator algorithms are developed via Dynamic Programming for discrete-time dynamical systems with a jumping parameter ("the mode") modeled as the state of a finite homogenous Markov chain and with additive zero-mean white process noise. The continuous state is assumed to be known while the mode is detected after some random delay, which probabilistic model is given. The optimal solution is presented, the conflict between optimality and practicality is analyzed, and a computationally efficient suboptimal algorithm is suggested. Specific algorithms for the cases of a known constant delay and of no-delay are derived as by-products. The proposed algorithms have some of the classical LQG features: recursion, linear state feedback, state quadratic optimal cost-to-go. which are direct consequences of the nested property of the information patterns and of the full state information. The gain computation, however, depends on the specific information structure. The proposed suboptimal controller is a finite memory controller and a look-up table for the gains can be computed off-line. Comparative results of an extensive Monte-Carlo simulation illustrate the efficiency of the proposed algorithm that mitigates the destabilizing effect of an ignored random mode-detection delay. Using a formal limiting procedure the algorithm for continuous time and continuous delay is also provided. It includes as a special case a previous algorithm developed under the assumption of instantaneous mode detection.