(110h) Discrete Output Regulation of Kuramoto-Sivashinsky Equation | AIChE

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(110h) Discrete Output Regulation of Kuramoto-Sivashinsky Equation

Authors 

Xie, J. - Presenter, University of Alberta
Dubljevic, S., University of Alberta
Kuramoto-Sivashinsky equation (KSE) as a fourth-order nonlinear partial differential equation (PDE) has been extensively utilized for mathematical modeling of falling film fronts, unstable flame fronts, phase turbulence in Belousov-Zhabotinsky reaction- diffusion and interfacial instabilities between multiple viscous phases [1-2]. Considering the fourth-order spatial derivative term and nonlinear multiplication term of the state with its first-order spatial derivative term, solving this high order parabolic KSE is not straightforward, posing a challenge for the associated regulator design [3-5]. On the other hand, it is difficult to install spatially distributed sensors for condition monitoring of the front dynamics either due to physical constraints or prohibitive costs. Thus, it is of significance to design state observers for state reconstruction based on boundary measurements. Considering the discrete nature of measurement and controller realization, a discrete output feedback regulator design technique is proposed for output reference tracking and disturbance rejection.

This work proposes a discrete-time output feedback regulator of this infinite- dimensional Kuramoto-Sivashinsky partial differential equation (PDE) by using discrete-time Sylvester regulation equations. For simplicity, a linear KSE system is obtained by linearization of KSE at equilibrium working point of interest. To discretize the resulting continuous linear KSE model, the state-of-the-art Cayley-Tustin time discretization method is applied without any spatial approximation or spatial order reduction, leading to an energy and structure preserving discretization configuration[6-7]. Additionally, the four-by-four-matrix-form resolvent operator is solved in order to realize this discrete distributed parameter system setting. Considering the difficulties and/or prohibitive cost for spatially distributed sensor installation, a discrete-time infinite-dimensional Kalman filter is designed for the discrete stochastic KSE system to take measurement and process noises into account[8]. Based on the estimated state by pre-designed Kalman filter, a discrete output feedback regulator is designed for output reference tracking and disturbance rejection. Based on Internal Model Design theory[9], a discrete-time Sylvester regulation framework is proposed along with the well-known continuous Sylvester regulation equations[10]. Finally, different types of signals (polynomial and sinusoidal signals) are investigated in simulations to verify the effectiveness of the proposed method.

References:

[1] Kuramoto Y., Tsuzuki T. 1975. On the formation of dissipative structures in reaction–diffusion systems, Progress of Theoretical Physics, 54: 687-699.

[2] Michelson D.M., Sivashinsky G.I. 1977. Nonlinear analysis of hydrodynamic instability in laminar flames. II. Numerical experiments, Acta Astronautica, 4: 1207-1221.

[3] Dubljevic S. 2010b. Model predictive control of Kuramoto–Sivashinsky equation with state and input constraints, Chemical Engineering Science, 65(15): 4388-4396.

[4] Dubljevic, S. 2010a. Boundary model predictive control of kuramoto-sivashinsky equation with input and state constraints. Computers & Chemical Engineering, 34(10): 1655-1661.

[5] Yang Y, Dubljevic S. Boundary model predictive control of thin film thickness modelled by the Kuramoto-Sivashinsky equation with input and state constraints, Journal of Process Control, 2013.http://dx.doi.org/10.1016/j.jprocont.2013.03.009.

[6] Xu, Q., Dubljevic, S., 2017. Linear model predictive control for transport-reaction processes. AIChE Journal 63 (7), 2644-2659.

[7] Havu, V., Malinen, J., 2007. The cayley transform as a time discretization scheme. Numerical Functional Analysis and Optimization 28 (7-8), 825-851.

[8] Xie J, Xu Q, Ni D, and Dubljevic S, Observer and filter design for linear transport- reaction systems, European Journal of Control, 2019. DOI: https://doi.org/10.1016/ j.ejcon.2019.01.005.

[9] Francis B. A. and Wonham W. M., The internal model principle of control theory, Automatica, vol. 12, no. 5, pp. 457–465, 1976.

[10] Xie J, Zhang L, and Dubljevic S, Discrete Output Feedback Regulator Design for Heterodirectional Hyperbolic Pipeline Systems, IEEE Transactions on Control Systems Technology, submitted.

Topics 

Process Automation & Control
Fluid Mechanics
Regulations

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