(231d) Robust Moving Horizon Estimation for Nonlinear Systems With Bounded Uncertainties

State estimation plays an important role in feedback control, system monitoring, fault detection as well as system optimization because it is in general difficult to measure all the system state variables. In the case of linear systems, the standard solutions are the Kalman filter and the Luenberger observer. However, in the context of nonlinear systems, the observer design problem is much more challenging and has attracted significant attention.
Observer designs that explicitly account for nonlinear systems can track back to the earlier work by Thau [1]. In [2], the first systematic approach for the design of nonlinear observers was proposed in which a nonlinear state transformation is used to linearize the original nonlinear system to a form to which linear methods can be applied. Since then, numerous efforts have been made along this direction to develop nonlinear observers. Even though an effective systematic design method for general nonlinear systems is still not available, significant progress has been made in the design of observers for different specific classes of nonlinear systems with many successful applications to nonlinear chemical processes. Important features of these nonlinear observer designs are that the observers can be expressed analytically and have stability and convergence properties that can be proved rigorously. These observers can be classified as deterministic since, in the design of these observers, model uncertainties and measurement noise are not taken into account explicitly. In addition, only the current measurement is used in these observers which makes the observers sensitive to occasional spikes in process disturbances and measurement noise.

In another line of work, moving horizon estimation (MHE) based on batch least squares technique has become popular in recent years because of its ability to handle explicitly nonlinear systems and constraints on decision variables. In MHE, the state estimate is determined by solving online an optimization problem that minimizes the sum of squared errors. In order to have a finite dimensional optimization problem, the horizon (estimation window size into the past) of MHE is in general chosen to be finite. At a sampling time, when a new measurement is available, the oldest measurement in the estimation window is discarded, and the finite horizon optimization problem is solved again to get the new estimate of the state. The ability of MHE to handle constraints on process disturbances, measurement noise and states was shown to lead to improved performance. In order to account for the effect of historical data outside the estimation window, an arrival cost which summarizes the information of those data is included in the cost function of an MHE optimization problem. The arrival cost plays an important role in the performance and stability of an MHE scheme. The effect of a poorly approximated arrival cost in the observer performance in general can be reduced by using longer horizons. However, the use of longer horizons increases further the computational burden which may impede the applicability of MHE. Moreover, it is desirable to study the stability and robustness of MHE subject to bounded disturbances.

Motivated by the above observations and inspired by techniques developed in nonlinear model predictive control with Lyapunov-based stability constraints as well as the MHE scheme with contractive constraint, an MHE scheme augmented by an auxiliary nonlinear observer is proposed for nonlinear systems with bounded model uncertainties in this work. Specifically, a deterministic nonlinear observer that asymptotically tracks the nominal system state is used to calculate a confidence region that contains the actual system state taking into account the effects of bounded model uncertainties at every sampling time. This confidence region is then used to design a constraint on state estimates in the MHE scheme. The proposed MHE scheme brings together deterministic and optimization-based observer design techniques. First of all, the proposed MHE scheme is proved to give bounded estimation errors when certain conditions are satisfied. This property is highly desired from an application point of view but is not available in existing MHE schemes. Second, it is demonstrated through the application of the proposed MHE to a gas-phase reactor and a continuous stirred tank reactor that the proposed MHE scheme is less sensitive to the accuracy of the approximated arrival cost compared with the classical MHE schemes because of the additional information provided by the deterministic nonlinear observer. This proposed approach gives us another option to compensate for the effects of errors in the arrival cost approximation, and it can be used together with different arrival cost approximation techniques to further improve state estimates.

[1] F. Thau. Observing the state of nonlinear dynamic systems. International Journal of Control, 17:471-479, 1973.
[2] A. J. Krener and A. Isidori. Linearization by output injection and nonlinear observers. Systems & Control Letters, 3:47-52, 1983.