Hybrid feedback stabilization of quasilinear systems in the plane (Q1771405)

The Artstein hybrid feedback algorithm [\textit{Z.~Artstein}, Example of stabilization with hybrid feedback, in Proceedings of the DIMACS/SYCON workshop on Hybrid systems III: verification and control: verification and control, New Brunswick, New Jersey, USA, 1996, Lect. Notes Comput. Sci. 1066, 171--185 (1996)] is applied to stabilize the following quasilinear dynamical system in the plane \[ \begin{aligned} \dot\xi&=a\xi+\eta+f_1(\xi ,\eta),\\ \dot\eta&=-\xi+a\eta+f_2(\xi ,\eta)+u,\tag{1}\\ y&=\xi , \end{aligned} \] where \(t\in [t_0,\infty )\), \(a\) is a real parameter, and \(f_i(\xi ,\eta)=o(| \xi | +| \eta | )\). The paper studies the case of incomplete observation where the control \(u\) is assumed to depend on the available observation \(y\), only. The challenge is to design a feedback control \(u\) which stabilize the zero solution of the system (1). It is known that in this case ordinary (linear or nonlinear) feedback controls do not work and one can, e.g., use special feedback controls called hybrid feedback controls (HFC). In general, HFC are not elementary, i.e. they are governed by automata with infinitely many states. In the paper, it is shown that in the case of the system (1) one can utilize Artstein's procedure based on an automaton with finitely many states. In addition, in contrast to many other works, where automata trace the plant continuously, the automata designed in the paper can only operate at some discrete moments of time, i.e. covers cases when a continuous tracing is unrealistic. The main results of the paper state that Artstein's procedure for the system (1) provides an arbitrary rate of asymptotic convergence/divergence of solutions, i.e. the complete controllability from below of the upper Lyapunov exponent and the uniform upper Lyapunov exponent is proved.