Linear quadratic regulation for discrete-time systems with multiplicative noises and input delays

This paper considers the stochastic LQR problem for systems with multiple input delays and stochastic parameter uncertainties in the state and input matrices. The problem is known to be difficult due to the presence of interactions between the delayed input channels and the stochastic parameter uncertainties in the channels. The key to our approach is an innovation analysis which allows us to establish a duality between the stochastic LQR problem and a smoothing problem for an associated delay free system with stochastic parameter uncertainties. Our solution is given in terms or two Riccati difference equations (RDEs) of the same dimension as that of the plant (ignoring the delays) and hence is very simple in computation.