(582e) Equation-Free Control of Distributed Parameter Systems Using Discrete Empirical Interpolation Method and Proper Orthogonal Decomposition

Diffusion, convection and reaction exist in many chemical and material industry processes. These phenomena lead to the spatial variation of the state variables. Their mathematical description can be derived using conservation equations and is in the form of distributed parameter system (DPS). Controlling these processes is of significant importance to achieve product quality, safety and economic profit. One of the challenges in the controller design for DPS is due to the infinite dimensional state space dynamics when the system is considered in a functional space. A standard way to deal with this issue is to use Galerkinâ??s method to construct a reduced order model. The prerequisite basis functions in Galerkinâ??s method can be generated by proper orthogonal decomposition (POD) [1]. After the reduced order model is constructed, established control methods, including feedback linearization, Lyapunov based ones and back-stepping can be used to design the controller. To implement these methods though, the information of the system state is required. The state of distributed parameter systems can be estimated by Luenberger-based observers, which relaxes the requirement on sensors. However, both the Luenberger-based observers and aforementioned controller design methods assume that a mathematical model of the system is available; furthermore unmodeled dynamics and model uncertainty always exits [2].

This issue motivates us to propose an approach to control systems when the knowledge of the chemical and physical law that describes the systems is unavailable or incomplete but the effect of actuators is known. First, POD is applied to generate two sets of basis functions that are used to estimate the system dynamics and state. Then discrete empirical interpolation method (DEIM) [3] is employed to determine the location of sensors. DEIM was proposed to reduce the computational cost associated with the nonlinear term in the reduced order model generated by Galerkin-POD methodology. Because DEIM has the property that the selection of the interpolation indices can limit the growth of the error of the approximation, it is modified to determine sensor location in our method. Compared with other sensor network design methods, this approach does not require the a priori knowledge of the governing equation. Having continuous measurements from these sensors, the state in the ROM is estimated by a static observer. Using a similar approach as the static observer, a mapping from the measurement of the velocity sensors onto the projection of the dynamics is generated. The estimation of the system dynamics and state make the model redundant in controller design.

Compared with other equation free methods, including divide-and-conquer techniques (such as memory based local modeling), subspace identification, neural network, and autoregressive moving average with exogenous inputs (ARMAX), the proposed method is computationally cheaper. The method performance is illustrated on a diffusion reaction process.

REFERENCES

[1] Lawrence Sirovich. Turbulence and the dynamics of coherent structures .1. coherent structures. Quarterly of Applied Mathematics, 45(3):561â??571, 1987.

[2] Zhong Sheng Hou and Zhuo Wang. From model-based control to data-driven control: Survey, classification and perspective. Information Sciences, 235:3â??35, 2013.

[3] Saifon Chaturantabut and DC Sorensen. Nonlinear model reduction via discrete empirical interpolation. SIAM Journal on Scientific Computing, 32(5):2737â??2764, 2010.