An LQR theory for systems with asymmetric saturating actuators

This paper presents an LQR-based method, referred to as A-SLQR, for systems with asymmetric (i.e., non-odd) saturating actuators. The development is based on a technique referred to as stochastic linearization, which approximates the saturation function by an equivalent gain and an equivalent bias. The equations for computing the gain and bias are presented, and their properties are explored. Using the stochastically linearized system, the A-SLQR problem is formulated as an optimization problem, and is solved using the Lagrange multiplier method. The solution is shown to consist of the usual Lyapunov and Riccati equations, coupled with additional equations that account for the equivalent gain and bias. The efficacy of the developed technique is illustrated through a detailed numerical example, where it is shown that the technique outperforms the standard LQR because saturation is taken into account directly at the design stage.