Robust output feedback control of nonlinear singularly perturbed systems
Nonlinear singularly perturbed systems with time-varying uncertain variables are considered. The system under consideration is: \[ \begin{aligned} \dot x & = f_1(x,\theta(t))+ Q_1(x,\theta(t))z+ g_1(x,\theta(t))u,\\ \varepsilon\dot z & = f_2(x,\theta(t))+ Q_2(x,\theta(t))z+ g_2(x,\theta(t))u,\tag{1}\\ y & = h(x),\end{aligned} \] where \(x\in \mathbb{R}^n\), \(z\in\mathbb{R}^p\) are the states, \(u\in\mathbb{R}\) denotes the manipulated input, \(\theta\in\mathbb{R}^q\) denotes the vector of the uncertain time-varying variables, \(y\in\mathbb{R}\) denotes the controlled output, \(\varepsilon\) is a small positive parameter, \(f_j(x,\theta(t))\), \(g_j(x,\theta(t))\), \(j= 1,2\) are sufficiently smooth vector functions, \(Q_j(x,\theta(t))\), \(j= 1,2\) are sufficiently smooth matrices and \(h(x)\) is a sufficiently smooth scalar function. It is assumed that the fast subsystem is asymptotically stable and the slow subsystem is input/output linearizable and possesses an input-to-state stable (ISS) inverse dynamics. For these systems, a robust output feedback controller is synthesized in order to ensure boundedness of the state and enforces robust asymptotic output tracking with attenuation of the effect of the uncertain variables on the output of the closed-loop system. The controller is constructed through combination of a high-gain observer with a robust state feedback controller synthesized via Lyapunov's direct method. The proposed controller enforces the aforementioned properties in the closed-loop system, for initial conditions, uncertainty and rate of change of uncertainty in arbitrarily large compact sets, provided that the singular perturbation parameter is sufficiently small and the observer gain is sufficiently large.
scientific article; zbMATH DE number 1389424
scientific article