Nonlinear Programming Sensitivity Based Methods for Constrained State Estimation

Model based control schemes, such as nonlinear model predictive control, assume that the full state vector of the plant is known for feedback control. However, in reality this is not always true. Most times only a set of noisy measurements are available, and thus, the unmeasured states need to be inferred from these measurements. This is done in combination with a detailed model of the system. The most common nonlinear state estimation methods do not have a means to deal with bounds or constraints on the states in an efficient or systematical way. These bounds and constraints are important in chemical engineering processes since states usually have physical meaning, for example, concentrations, molecular weights, and conversions are always positive. Therefore, state estimates must be physically feasible. Since Moving Horizon Estimation (MHE) is optimization based it has become a superior strategy for constrained state estimation because bounds are handled optimally by theNonlinear Programming (NLP) solver. In the presentworkwe develop strategies for MHE based on NLP sensitivity to reduce the on-line computational expense of solving these problems. These formulations are intended to make the on-line application of MHE feasible, by reducing the potential of delays due to the computational expense of solving the associated NLP.

Here we discuss two approaches to update certain tuning parameters in MHE: one of them allows us to reduce the size of the NLP that is being solved, while the other provides a fast approximation of the covariance of the initial condition. The former method is only suitable for small and medium sized problems, while the latter one is better suited for large-scale systems. Additionally, we also discuss NLP sensitivity theory and extensions that apply to the Interior Point solver we use (i.e., IPOPT). With these extensions we are able to develop fast on-line strategies for NMPC and MHE. However, in this work we focus only in the application of these developments to the latter.

To reduce the horizon window we propose methods to approximate the initial condition parameters based on particle filters and sample based statistics to approximate the conditional probability density function (or its parameters) of the initial condition of the states in the MHE horizon window (i.e., the so-called arrival cost). As mentioned above, this approach is suitable mostly for only certain classes of systems that have few states or almost Gaussian behaviors. Therefore,we also develop othermethods for on-lineMHE suitable for large-scale systems that usesmoremeasurements to reduce any effects of the simplifications done to approximate the initial condition terms. Thus, using NLP sensitivity we develop strategies that leverage the parametric properties of the MHE problem to formulate fast on-line methods applicable to large-scale systems. Moreover, using NLP sensitivity also allows us to relate the optimality conditions of the associated NLP problem to the stochastic origin of MHE. For example, we show the relationship of the covariance of the state estimates with the reduced Hessian matrix of the NLP. This information can also be used to update the parameters if the initial condition penalty term. Moreover, we also discuss the use of Robust M-Estimators to reduce the effects of outliers or gross errors in the measurements. Finally, we illustrate the use and benefits of these strategies through several small and large-scale examples taken from the literature.