Control of impulsive renewal systems: Application to direct design in networked control

We consider the control of impulsive systems with independent and identically distributed intervals between jumps. The control action and output measurement are assumed to take place only at jump times. We give necessary and sufficient conditions, in the form of LMIs, for mean square stabilizability and detectability and solve an infinite horizon quadratic optimal control problem, under appropriate stabilizability and detectability properties of the system. The class of systems considered is especially suited to model networked control systems utilizing CSMA-type protocols, with stochastic intervals between transmissions and packet drops. In this setting, the analysis and synthesis tools mentioned above are used to (i) prove that for an emulation-based design, stability of the closed-loop is preserved if the distribution of the intervals between transmissions assigns high probability to fast sampling (ii) illustrate through a benchmark example the potential advantages of controller direct-design over an emulation-based design.