An LQR/LQG theory for systems with saturating actuators

An extension of the LQR/LQG methodology to systems with saturating actuators, referred to as SLQR/SLQG, where S stands for saturating, is obtained. The development is based on the method of stochastic linearization, whereby the saturation is replaced by a gain, calculated as a function of the variance of the signal at its input. Using the stochastically linearized system and the Lagrange multiplier technique, solutions of the SLQR/SLQG problems are derived. These solutions are given by standard Riccati and Lyapunov equations coupled with two transcendental equations, which characterize both the variance of the signal at the saturation input and the Lagrange multiplier associated with the constrained minimization problem. It is shown that, under standard stabilizability and detectability conditions, these equations have a unique solution, which can be found by a simple bisection algorithm. When the level of saturation tends to infinity, these equations reduce to their standard LQR/LQG counterparts. In addition, the paper investigates the properties of closed-loop systems with SLQR/SLQG controllers and saturating actuators. In this regard, it is shown that SLQR/SLQG controllers ensure semi-global stability by appropriate choice of a parameter in the performance criterion. Finally, the paper illustrates the techniques developed by a ship roll damping problem.