The H∞‐optimal control problem of CSVIU systems

Abstract

The article devises an ‐norm theory for the CSVIU (control and state variations increase uncertainty) class of stochastic systems. This system model appeals to stochastic control problems to express the state evolution of a possibly nonlinear dynamic system restraint to poor modeling. It introduces the control with infinity energy disturbance signals, contrary to usual stochastic formulations, which mimic deterministic models dealing with finite energy disturbances. The reason is to portray the persistent perturbations due to the environment more naturally. In this regard, it develops a refined connection between a notion of stability and the system's power finiteness. It delves into the control solution employing the relations between optimization and differential games, connecting the worst‐case stability analysis of CSVIU systems with a perturbed Lyapunov type of equation. The norm characterization relies on the optimal cost induced by the min–max control strategy. The solvability of a generalized game‐type Riccati equation couples to the rise of a pure saddle point, yielding the global solution of the CSVIU dynamic game. In the way to the optimal stabilizing compensator, the article finds it based on linear feedback stabilizing gain from solving the Riccati equation together with a spectral radius test of an ensuing matrix. The optimal disturbance compensator produces inaction regions in the sense that, for sufficiently minor deviations from the model, the optimal action is constant or null in the face of the uncertainty involved. Finally, an explicit solution approximation developed by a Monte Carlo method appears, and a numerical example illustrates the synthesis.