(204d) Moving Horizon Estimation Via Carleman Linearization and Stability Analysis

In order to control dynamic systems, knowledge of the status of the system e.g. state variables is critical. However, the online measurement of state variables is not always feasible. Consequently, there are many different approaches in the literature to estimate these unmeasurable state variables of a dynamic system. However, the estimation problem would become harder when the system is corrupted with noise signals. Moving Horizon Estimation (MHE) is one of the most successful estimation approaches in dealing with such random uncertainties [1]. The idea behind MHE design is very similar to MPC design. To handle the constraints on input and state variables, these two methods formulate and solve a constrained dynamic nonlinear optimization problem. However, solving this optimization problem is usually time-consuming and online convergence of the scheme to the optimal decision variables is sometimes infeasible. In this work, we propose a new approach that accelerates computations in MHE by exploiting Carleman linearization and prove the stability of this new design.

Extended Kalman Filter (EKF) is the alternative method for state estimation in nonlinear systems in presence of noise. In fact, EKF is a simplified version of MHE which has an analytical closed-form solution; it uses only the last measurement data and linearizes the nonlinear system around the last estimate of states. In contrast, MHE considers the entire output sequence and also takes constraints and nonlinearities into account. This work focuses on design of a new estimator using MHE's structure with a constant estimation horizon. The estimator further simplifies the associated dynamic optimization problem through use of Carleman linearization, which is more accurate than first-order Taylor expansions. Therefore, the new design of MHE gives a better balance between accuracy and simplicity. In a previous work, we employed this design method and also performed a sensitivity analysis for a system with unknown disturbance [2, 3]. In the current work, in contrast, we build on our previous work by considering system noise and also studying MHE's stability.

The stability of MHE is an important issue. This issue is particularly very thoroughly addressed for constrained linear systems. For constrained nonlinear problems, Rao et al. presented an algorithm to design a stable MHE by assuming boundedness of noise signals [4]. Also, they assumed the estimator is able to solve the optimization problem ideally and provide the global optimal solution. Later, some papers considered the optimization error in their MHE design [5]. Our work falls into the second category. We consider the maximum error for Carleman linearization at every time step and prove the estimation error is still bounded. Also it is shown that this error bound can be reduced to any desirable level by choosing a proper order of Carleman linearization. Finally, we employ the proposed MHE in a practical chemical engineering example. The simulation running time is reduced remarkably by employing the proposed MHE.

 Reference:

[1] Haseltine, E. L., & Rawlings, J. B. (2005). Critical evaluation of extended Kalman filtering and moving-horizon estimation. Industrial and Engineering Chemistry Research, 44, 2451-2460. 

[2] Hashemian, N., & Armaou, A. (2015). Fast moving horizon estimation of nonlinear processes

via Carleman linearization. American Control Conference, Chicago, IL.

[3] Hashemian, N., & Armaou, A. (2014). Moving horizon estimation using Carleman linearization and sensitivity analysis. Annual Meeting AIChE, Atlanta, GA.

[4] Rao, C. V., Rawlings, J. B., & Mayne, D. Q. (2003). Constrained state estimation for nonlinear discrete-time systems: stability and moving horizon approximations. IEEE Transactions on Automatic Control, 48, 246-258.

[5] Alessandri, A., Baglietto, M., & Battistelli, G. (2008). Moving-horizon state estimation for nonlinear discrete-time systems: New stability results and approximation schemes. Automatica, 44, 1753-1765.