×

zbMATH — the first resource for mathematics

Separation of motions in multirate discontinuous systems with time delay. (English. Russian original) Zbl 0932.93040
Autom. Remote Control 58, No. 7, Pt. 2, 1263-1275 (1997); translation from Autom. Telemekh. 1997, No. 7, 240-254 (1997).
The author considers a system of the form \[ \mu dz/dt = g(z,s,x,u), \quad ds/dt = h_1(z,s,x,u), \quad dx/dt = h_2(z,s,x,u) \] where \(z \in R^m\), \(s \in R\), \(x \in R^n\), \(u(s) = - \text{sign}[s(t-1)]\), \(g\),\(h_1\) and \(h_2\) are smooth functions and \(\mu\) is a small parameter. Sufficient conditions are given under which for sufficiently small \(\mu\) there exists an orbitally asymptotically stable periodic solution. The algorithm of the construction of an asymptotics of a periodic solution is suggested. An example illustrating how the obtained results can be used for the separation of motions in singularly perturbed discontinuous systems with delay is given.

MSC:
93C23 Control/observation systems governed by functional-differential equations
34C25 Periodic solutions to ordinary differential equations
34K13 Periodic solutions to functional-differential equations
34D15 Singular perturbations of ordinary differential equations
93D20 Asymptotic stability in control theory
93D15 Stabilization of systems by feedback
PDF BibTeX XML Cite