  This paper presents weighted stochastic Riccati (WSR) equations for designing
multiple types of optimal controllers for linear stochastic systems. The
stochastic system matrices are independent and identically distributed (i.i.d.)
to represent uncertainty and noise in the systems. However, it is difficult to
design multiple types of controllers for systems with i.i.d. matrices while the
stochasticity can invoke unpredictable control results. A critical limitation
of such i.i.d. systems is that Riccati-like algebraic equations cannot be
applied to complex controller design. To overcome this limitation, the proposed
WSR equations employ a weighted expectation of stochastic algebraic equations.
The weighted expectation is calculated using a weight function designed to
handle statistical properties of the control policy. Solutions to the WSR
equations provide multiple policies depending on the weight function, which
contain the deterministic optimal, stochastic optimal, and risk-sensitive
linear (RSL) control. This study presents two approaches to solve the WSR
equations efficiently: calculating WSR difference equations iteratively and
employing Newton's method. Moreover, designing the weight function yields a
novel controller termed the robust RSL controller that has both a
risk-sensitive policy and robustness to randomness occurring in stochastic
control design.
