Weighted stochastic Riccati equations for generalization of linear optimal control

Abstract

This paper presents weighted stochastic Riccati (WSR) equations for designing multiple types of controllers for linear stochastic systems. The system matrices are independent and identically distributed (i.i.d.) to represent noise in the systems. While the stochasticity can invoke unpredictable control results, it is essentially difficult to design controllers for systems with i.i.d. matrices because the controllers can be solutions to non-algebraic equations. Although an existing method has tackled this difficulty, the method has not realized the generality because it relies on the special form of cost functions for risk-sensitive linear (RSL) control. Furthermore, designing controllers over an infinite-horizon remains challenging because many iterations of solving nonlinear optimization is needed. To overcome these problems, the proposed WSR equations employ a weighted expectation of stochastic equations. Solutions to the WSR equations provide multiple types of controllers characterized by the weight, which contain stochastic optimal and RSL controllers. Two approaches calculating simple recursive formulas are proposed to solve the WSR equations without solving the nonlinear optimization. Moreover, designing the weight yields a novel controller termed the robust RSL controller that has both a risk-sensitive policy and robustness to randomness occurring in stochastic controller design.