Pseudospectral Expansion-based Model Reduction for Control and Boundary Observation of Unstable Parabolic Partial Differential Equations

C. Xu, Z-G. Ren, Y. Ou, E. Schuster and X. Yu

Kongzhi Lilun Yu Yingyong/Control Theory and Applications v 30, n 7, p 793-800, July 2013

The reduce-then-design approach is widely used for controller synthesis of infinite dimensional systems. A drawback of the reduce-then-design method is the inherent loss of information due to the truncation before control design. Moreover, the order of the model truncation is a trade-off between model accuracy and real time computation. The stabilization of an unstable linear parabolic Partial Differential Equation (PDE) system with both Neumann boundary control and interior control is considered in this work. Point output measurement is available at one end of the physical domain. A proportional state feedback is proposed for the interior control with a symmetric kernel function, and the pseudospectral method is used to solve the stability conditions governed by the Sturm-Liouville systems. In addition, an observer is designed using the point measurement at one end of the physical domain, and used to propose an observer-based feedback controller for the PDE system. Both controller and observer gains are designed numerically to make the eigenvalues of the associated Sturm-Liouville problems stable. Simulations show the effectiveness of the proposed controller.