(615b) Moving Horizon Estimation: Efficient Methods for Fast Implementations

Model based control schemes, such as nonlinear model predictive control or globally linearizing control, assume at the design stage that the true states are available for feedback control. In practice, however, the only available information of the system is obtained through a set of noisy measurements that seldom include all of the state variables [1]. Thus, the unmeasured states need to be inferred from these measurements in combination with a dynamic model of the process. Chemical processes are usually highly nonlinear and large scale. Therefore, they require the use of state estimation methods that can handle them. Moving Horizon Estimation (MHE) represents a powerful framework for nonlinear state estimation that can handle large scale nonlinear systems [2]. Many efforts have been used to both reduce the size of the problems and the computational time needed to solve them. In this work we will explore two different approaches that deal with these issues in different ways.

A first approach reduces the size of the estimation problems by better representing the prior information through the use of particle filters in combination with MHE. From the stochastic point of view the prior information is represented by a unknown probability density function which very hard to determine analytically. The effects of errors introduced by improperly approximating this distribution are reduced by increasing the number of past measurements included in the MHE horizon, and thus increasing the problem size [3]. By better approximating the prior information using particles it is possible to reduce the size of said problem. This is shown with a few small examples. Unfortunately, these types of methods do not scale well for very large problems. This forces the use of a simple approximation (e.g., Gaussian) of the distribution, and therefore, compensating by adding more measurements, becomes the best solution.

A second approach is examined for these cases. Here Nonlinear Programming (NLP) sensitivity theory is used to generate MHE methods that allow us to perform fast approximations of the first two moments of the prior distribution. The most common way in which MHE is implemented requires the use of the Extended Kalman Filter Equations to propagate the covariance along the horizon [4]. This requires multiple inversions of possibly very large matrices. On the other hand, the sensitivity based method takes advantage of newly developed tools in Ipopt (the NLP solver used) that allow us to extract the Reduced Hessian at the solution directly from the optimality conditions of the NLP [5]. This method requires a single back-solve for each column of the Reduced Hessian that is extracted. Also, the Reduced Hessian can be shown to correspond to the posterior covariance of the state estimate, or the smoothed covariance of the smoothed estimates in the horizon. In this way we can obtain fast approximations of the covariance of the prior distribution that can be used for the MHE calculations.

References

[1] Jazwinski, A. H. Stochastic Processess and Filtering Theory Dover Publications, Inc., 2007

[2] Rao, C. V. & Rawlings, J. B. Constrained Process Monitoring: Moving-Horizon Approach. AIChE Journal, 2002, 48, 97-109

[3] Lopez-Negrete, R, Patwardhan, SC, and Biegler, LT, Constrained Particle Filter Approach for Approximation of the Arrival Cost in Moving Horizon Estimation, Submitted for publication to JPC, May 201

[4] Tenny, M. J. & Rawlings, J. B. Efficient Moving Horizon Estimation and Nonlinear Model Predictive Control Proceedings of the American Control Conference, 2002, 4475-4480

[5] Zavala, V. M.; Laird, C. D. & Biegler, L. T. A Fast Moving Horizon Estimation Algorithm Based on Nonlinear Programming Sensitivity. Journal of Process Control, 2008, 18, 876-884