Robust Stabilization Design for Stochastic Partial Differential Systems Under Spatio-temporal Disturbances and Sensor Measurement Noises

We investigate an ${H_{\infty } }$ robust observer-based stabilization design problem for linear stochastic partial differential systems (LSPDSs) under spatio-temporal disturbances and sensor measurement noises. A general theoretical ${H_{\infty } }$ robust observer-based stabilization method is introduced at the beginning for LSPDSs under intrinsic fluctuation, external disturbance and sensor measurement noise in the spatio-temporal domain. A complex Hamilton Jacobi integral inequality (HJII) needs to be solved when designing the robust ${H_{\infty } }$ observer-based stabilization for LSPDSs. For simplifying the design procedure, a stochastic state space model is first developed via the semi-discretization finite difference scheme to represent the stochastic partial differential system at each grid node. Then the stochastic state space models at all grid nodes are merged together into a stochastic spatial state space model. Based on this stochastic spatial state space model, an implementable ${H_{\infty } }$ robust stabilization design is proposed for LSPDSs via an iterative linear matrix inequality (ILMI) method. The proposed robust ${H_{\infty } }$ stabilization design can efficiently attenuate the effect of spatio-temporal external disturbances and measurement noises upon LSPDSs from the area energy point of view. Finally, a robust ${H_{\infty } }$ stabilization example simulation is shown to illustrate the design procedure and to confirm the performance of the proposed robust stabilization design method.