Disturbance rejection in control systems with saturating actuators
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.
scientific article; zbMATH DE number 1472215
scientific article