Disturbance rejection in control systems with saturating actuators (Q1570927)

The authors consider a control system such that plant \(P(s)\) and controller \(C(s)\) have the following state-space representations \[ \dot x_p= A_0 x_p + B_1 w + B_2 v, \qquad z= C_1 x_p,\qquad y= C_2 x_p, \] and \[ \dot x_c= M x_c + L y, \qquad u= K x_c, \] where \(x_p\) and \(x_c\) are the states of the plant and the controller respectively and \(u, v, w, y, z \in \mathbb{R}.\) The disturbance \(w\) is a standard white noise and the actuator characteristics is \(v =\phi(u)\) where \(\phi(u)\) is standard saturation function \[ \phi(u) = \begin{cases} +1, \quad u > +1 \\ u, \quad -1 \leq u \leq +1 \\ -1,\quad u < -1.\end{cases} \] Let \(\sigma_z^2\) and \(\sigma_u^2\) be (steady-state) variances of \(z\) and \(u\). The authors consider the following problem. Analysis problem: given \(P(s)\) and \(C(s)\), assuming that the controller \(C(z)\) stabilize the system when \(w \equiv 0\), determine the variance \(\sigma_z^2.\) Synthesis problem: given \(P(s)\) and \(\rho >0\), find a controller \(C(s)\) that stabilizes the system when \(w \equiv 0\) and minimizes \(J = \sigma_z^2 + \rho \sigma_u^2.\) The authors use the method of stochastic linearization. The solution involves nonlinear equations and simple bisection algorithms that can be used for the solution of these equations are provided.