Stability analysis of a Lurie system with scalar retarded control and switching
Keywords:
Lurie systems, delay, synchronous and asynchronous switching, asymptotic stability, dwell-time approachAbstract
ntroduction: To ensure the stability of switched linear systems, M. S. Branicky proposed a method for finding conditions on the switching law. It is also known that the presence of delays can disrupt stability. It is advisable to extend this method to a nonlinear control system, which makes it possible to set arbitrary concentrated delays during control or switching. Purpose: To investigate a nonlinear system with subsystems composed of linear parts and control assumed to be a scalar nonlinearity with a greater than one rational degree. Also, for an arbitrary delay in control, to obtain conditions for the switching law that would guarantee the stability of the solution. Results: Cases of both synchronous (simultaneous for all parameters of the system) and asynchronous switchings have been studied in connection with the occurrence of delays when developing a controlling action, as well as receiving information about active subsystem changing. Multiple Lyapunov – Krasovsky functionals were constructed for each system under discussion, which made it possible to find conditions on the switching law under which the solutions will be locally asymptotically stable. As we have found out it is sufficient for this to choose the switching moments so that the duration of the subsystem should unboundedly tend to infinity over time. We also demonstrate that in the discrete case, similar conditions also ensure stability at a sufficiently small discretization step. We have carried out numerical modeling for both synchronous and asynchronous switching between continuous and discrete subsystems. The graphs presented in the paper are consistent with the theoretical conclusions.