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Modal Model Predictive Control of Kuramoto-Sivashinsky Equation with State and Input Constraints

Stevan Dubljevic, Cardiology Department, Cardiovascular Research Laboratories, David Geffen School of Medicine, University of California, Los Angeles, 675 Charles E. Young Drive South, Box 951760 Room#3645 MRL, Los Angeles, CA 90095-1760

This paper focuses on model predictive control of linear dissipative PDEs with input and state constraints, and in particular, on predictive control of a dissipative fourth-order equation given by the Kuramoto-Sivashinsky equation, which describes a variety of physicochemical phenomena like long-wave motions of the liquid thin film over a vertical plate or evolution of laminar fronts [1]. The problem of stabilization of Kuramoto-Sivashinsky equation has been addressed within the output feedback formulation by Christofides and Armaou ([2],[3]) and within optimal control setting by Lee and Tran [4]. However, the important aspect of inclusion of state and input constraints in the controller synthesis has not be addressed.

In this work modal model predictive control synthesis for control of Kuramoto-Sivashinsky equation with state and input constraints has been developed. The evolution of a linear dissipative PDE is initially given by an abstract evolution equation in an appropriate Hilbert space. Modal decomposition technique is used to decompose the infinite dimensional system into an interconnection of a finite-dimensional (slow) subsystem with an infinite-dimensional (fast) subsystem. The important consequence of such structured dynamics is the notion of decoupled modes which is of paramount importance for the synthesis of a low-order modal model predictive controller (MMPC). The MMPC synthesis is then formulated in a way that the construction of the cost functional accounts only for the weighted evolution of the slow (finite-dimensional) states, while in the state constraints a high-order (finite-dimensional) approximation of the fast states is utilized ([5],[6]). As an example of the proposed controller synthesis methodology, the optimal stabilization under the presence of input and state constraints of spatially-uniform unstable steady state of Kuramoto-Sivashinsky equation subject to periodic boundary conditions is considered. Simulation results demonstrate successful application of the proposed predictive control technique with infinite-dimensional closed-loop system stability and the state constraint being enforced at a point within the spatial domain.

[1] L. H. Chen and H. C. Chang, Nonlinear waves on liquid film surface-II. Bifurcation analyses of the long-wave equation, Chemical Engineering Science, vol. 41, pp. 2477-2486, 1986.

[2] A. Armaou and P. D. Christofides, Feedback control of the Kuramoto-Sivashinsky equation, Physica D., vol. 137, pp. 49-61, 1999.

[3] P. D. Christofides and A. Armaou, Global stabilization of the Kuramoto-Sivashinsky equation via distributed output feedback control, Syst. & Contr. Lett., vol. 39, pp. 283-294, 2000.

[4] C. H. Lee and H. T. Tran, Reduced-order-based feedback control of the Kuramoto-Sivashinsky equation, Journal of Computational and Applied Mathematics, vol. 173, pp. 1-19, 2005.

[5] Dubljevic, S., N. H. El-Farra, P. Mhaskar and P. D. Christofides, Predictive Control of Parabolic PDEs with State and Control Constraints, Inter. J. Rob. & Non. Contr., in press.

[6] Dubljevic, S., P. Mhaskar, N. H. El-Farra and P. D. Christofides, Predictive Control of Transport-Reaction Processes, Comp. & Chem. Eng., 29, 2335-2345, 2005.