Classification of linear dynamical systems in the plane in admitting a stabilizing hybrid feedback control
The authors consider a stabilization problem for the following controlled system \[ \dot x = A x + B u,\qquad y = c x \tag{1} \] where \( x \in \mathbb{R}^2\), \(u \in \mathbb{R}^1\), \(y \in \mathbb{R}^1 \). For stabilization a linear output feedback control \( u = G y \) is used. A simple example, considered by Z. Artstein, \[ \dot x_1 = x_2, \quad \dot x_2 = -x_1 + u, \quad y = x_1, \quad u = u (y) \tag{2} \] shows that although system (2) is both controllable and observable, it cannot be stabilized by such feedback (even nonlinear and discontinuous). It was, however, shown by Z. Artstein that there exists a hybrid feedback control, under which system (2) becomes asymptotically stable. A hybrid feedback control procedure attaches a discrete time controller (an automaton) to the underlying continuous dynamical system and prescribes when and how a control \(u\) should be switched on and off. As Artstein's example shows, such a hybrid feedback may help when the ordinary feedback fails to stabilize the system. In the paper, a classification of linear controlled planar systems is suggested and all cases where the system admits a linear hybrid stabilization are described. In connection with the results obtained, the authors point at a preprint devoted to the same problem: \textit{C. Benassi} and \textit{A. Gavioli}, Hybrid stabilization of planar linear systems with one-dimensional outputs. Preprint, Dipartamento di Matematica Pura e Applicata, Università di Modena, Italy, August, 1999.
scientific article; zbMATH DE number 1545849