(139d) Robust Moving Horizon Estimation Based Output-Feedback Economic Model Predictive Control

The economic model predictive control (EMPC) framework deals with a reformulation of the conventional MPC quadratic cost function in which an economic (not necessarily quadratic) cost function is used directly as the cost in MPC, and thus, it may, in general, lead to time-varying process operation policies (instead of steady-state operation), which directly optimize process economics. In our previous work [1], a Lyapunov-based EMPC was developed which includes two operation modes. The EMPC design was also extended to the case of output feedback based on a high-gain observer which may be sensitive to measurement noise [2]. In this work, we propose a robust moving horizon estimation (RMHE) based output feedback EMPC design to significantly reduce the effect of noise in the performance of the output feedback EMPC.

The RMHE was initially developed in [3] which is based on an auxiliary nonlinear observer. Specifically, the auxiliary deterministic nonlinear observer that asymptotically tracks the nominal system state is taken advantage of to calculate a confidence region that contains the actual system state taking into account bounded model uncertainties every sampling time. The region is then used to design a constraint on the state estimate in the RMHE. The RMHE brings together deterministic and optimization-based observer design techniques. It was proved to give bounded estimation error in the case of bounded model uncertainties. It was also shown to compensate for the error in the arrival cost approximation and could be used together with different arrival cost approximation techniques to further improve the state estimate.

In the present work, in order to achieve fast convergence of the state estimates to the actual system state (thus an effective separation principle between the observer and controller designs) and the robustness of the system to measurement noise and model uncertainties, a high-gain observer is first applied for a small time period with continuous output measurements to drive the estimation error to a small value; after this initial small time period, the high-gain observer is switched off and the RMHE is put online to provide more accurate and smooth state estimates based on (possibly sampled) output measurements. The EMPC is designed based on the state estimates provided by the high-gain observer and RMHE. In the design of the EMPC, bounded measurement noise is taken into account explicitly. Sufficient conditions that ensure the closed-loop stability are provided. Extensive simulations based on a chemical process example illustrate the effectiveness of the proposed approach.

[1] Heidarinejad, M., Liu, J. & Christofides, P. D. Economic Model Predictive Control of Nonlinear Process Systems Using Lyapunov Techniques,  AIChE Journal, 2012, 58,  855-870.

[2] Heidarinejad, M., Liu, J. & Christofides, P. D. State Estimation-Based Economic Model Predictive Control of Nonlinear Systems, Systems & Control Letters, 2012, 61, 926-935.

[3] Liu, J. Moving Horizon State Estimation for Nonlinear Systems with Bounded Uncertainties Chemical Engineering Science, 2013, 93, 376-386.