(711e) Modified Equation-Free Control of Distributed Parameter Systems with Model Mismatch

Many chemical and material industry processes exhibit spatial variation of the state variables because of diffusion, convection and reaction. These processes can be mathematically described by partial differential equations based on conservation equations. This kind of system is called distributed parameter system (DPS). To achieve economic profit and improve the product quality, controlling DPS is important. Since the infinite dimensional state space dynamics of DPS makes controller design nontrivial, a standard way to tackle the issue is to construct a reduced order model (ROM) using Galerkin’s method. Then controller and observer can be designed based on the ROM with less computational cost. However, most of the established methods require a mathematical model of the system.

To relax the requirement on the mathematical model, in [1] we proposed an equation-free method to control systems when the knowledge of the governing law is unavailable or incomplete and the actuator effect is known. This method was developed based on the feature of discrete empirical interpolation method (DEIM) [2] that the selection of the interpolation indices can limit the growth of the error of the approximation by DEIM. A set of snapshots (observation of the system) is assumed available. Estimation of the dynamics can be made using these previous observations and continuous measurement on limited number of locations.

In this presentation, we improve the accuracy of the estimation of the dynamics when the system dynamics no longer match previous observations. A correction term is added to reduce the error of the estimation of the system dynamics. The difference between the expected value of the state and the actual value is used to update the correction term. The modified version is applied to a diffusion reaction process.

REFERENCES

[1] Yang, M., Armaou, A. On the design of equation-free controllers for dissipative PDEs via DEIM, American Control Conference, 2017.

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