Computing Arrival Cost Parameters in Moving Horizon Estimation Using Sampling Based Filters

Moving horizon estimation (MHE) is a numerical optimization based approach to state estimation, where the joint probability density function (pdf) of a finite state trajectory is sought, which is conditioned on a moving horizon of measurements. The joint conditional pdf depends on the a priori state pdf at the start of the horizon, which is a prediction pdf based on historical data outside the horizon. When the joint pdf is maximized, the arrival cost is a penalty term based on the a priori pdf in the MHE objective function. Traditionally, the a priori pdf is assumed as a multivariate Gaussian pdf and the extended Kalman filter (EKF) and smoother are used to recursively update the mean and covariance. However, transformation of moments through nonlinearity is poorly approximated by linearization, which can result in poor initialization of MHE. Sampling based nonlinear filters completely avoid Taylor series approximations of nonlinearities and attempt to approximate the non-Gaussian state pdf using samples and associated weights or probability mass points. The performance gains of sampling based filters over EKF motivate their use to formulate the arrival cost in MHE. The a priori mean and covariance are more effectively propagated through nonlinearities and the resulting arrival cost term can help to keep the horizon small. It is also possible to find closed-form approximations to the non-Gaussian a priori pdf from the sampling based filters. Thus, more realistic nonparametric arrival cost terms can be included by avoiding the Gaussian assumption. In this paper the use of the deterministic sampling based unscented Kalman filter, the class of random sampling based particle filter and the aggregate Markov chain based cell filter are discussed for initializing MHE. Two simulation examples are included to demonstrate the benefits of these methods over the traditional EKF approach

A Direct Sampling Particle Filter from Approximate Conditional Density Function Supported on Constrained State Space

Constraints on the state vector must be taken into account in the state estimation problem. Recently, acceptance/rejection and projection methods are proposed in the particle filter framework for constraining the particles. A weighted least squares formulation is used for constraining samples in unscented and ensemble Kalman filters. In this paper, direct sampling from an approximate conditional probability density function (pdf) is proposed. It is obtained by approximating the a priori pdf as a Gaussian. The support of the conditional density is a subset of the intersection of two supports, the 3-sigma bounds of the priori Gaussian and the constrained state space. A direct sampling algorithm is proposed for handling linear and nonlinear equality and inequality constraints. The algorithm uses the constrained mode for nonlinear constraints

Computing Arrival Cost Parameters in Moving Horizon Estimation Using Sampling Based Filters

Moving horizon estimation (MHE) is a numerical optimization based approach to state estimation, where the joint probability density function (pdf) of a finite state trajectory is sought, which is conditioned on a moving horizon of measurements. The joint conditional pdf depends on the a priori state pdf at the start of the horizon, which is a prediction pdf based on historical data outside the horizon. When the joint pdf is maximized, the arrival cost is a penalty term based on the a priori pdf in the MHE objective function. Traditionally, the a priori pdf is assumed as a multivariate Gaussian pdf and the extended Kalman filter (EKF) and smoother are used to recursively update the mean and covariance. However, transformation of moments through nonlinearity is poorly approximated by linearization, which can result in poor initialization of MHE. Sampling based nonlinear filters completely avoid Taylor series approximations of nonlinearities and attempt to approximate the non-Gaussian state pdf using samples and associated weights or probability mass points. The performance gains of sampling based filters over EKF motivate their use to formulate the arrival cost in MHE. The a priori mean and covariance are more effectively propagated through nonlinearities and the resulting arrival cost term can help to keep the horizon small. It is also possible to find closed-form approximations to the non-Gaussian a priori pdf from the sampling based filters. Thus, more realistic nonparametric arrival cost terms can be included by avoiding the Gaussian assumption. In this paper the use of the deterministic sampling based unscented Kalman filter, the class of random sampling based particle filter and the aggregate Markov chain based cell filter are discussed for initializing MHE. Two simulation examples are included to demonstrate the benefits of these methods over the traditional EKF approach

The Use of a Cell Filter for State Estimation in Closed-Loop NMPC of Low Dimensional Systems

Combining variants of the Kalman filter and moving horizon estimation (MHE) with nonlinear MPC has been studied before. The MHE is appealing due to its ability to impose constraints and demonstrated superiority over extended Kalman filter. However, nonlinear MPC based on MHE requires solutions to two back to back nonlinear programs. In this paper we propose to use the cell filter (CF) to provide state feedback to the MPC regulator. The cell filter is a piecewise constant approximation of the conditional probability density of the states, whose temporal evolution is modeled by an aggregate Markov chain. Since the CF is based on discretized state, input and output spaces, the curse of dimensionality limits its application to low dimensional and constrained systems. In this paper we present simulation examples of closed-loop MPC for a nonlinear reactor and agricultural pest control based on state feedback from both CF and MHE

Time-Varying System Identification Using Modulating Functions and Spline Models With Application to Bio-Processes

Time dependent parameters are frequently encountered in many real processes which need to be monitored for process modeling, control and supervision purposes. Modulating functions methods are especially suitable for this task because they use the original continuous-time differential equations and avoid differentiation of noisy signals. Among the many versions of the method available, Pearson–Lee method offers a computationally efficient alternative. In this paper, Pearson–Lee method is generalized for non-stationary continuous-time systems and the on-line version is developed. The time dependent parameters are modeled as polynomial splines inside a moving data window and recursion formulae using shifting properties of sinusoids are formulated. The simple matrix update relations considerably reduce the number of computations required when compared with repeatedly using FFT. The method is illustrated for estimating the kinetic rates and yield factors as time-varying parameters in a fermentation process. The Monod law along with temperature dependency models were used to simulate the data. The simulation study shows that it is not necessary to assume a growth model in order to estimate the kinetic rate parameters

Bayesian Estimation Via Sequential Monte Carlo Sampling-Constrained Dynamic Systems

Nonlinear and non-Gaussian processes with constraints are commonly encountered in dynamic estimation problems. Methods for solving such problems either ignore the constraints or rely on crude approximations of the model or probability distributions. Such approximations may reduce the accuracy of the estimates since they often fail to capture the variety of probability distributions encountered in constrained linear and nonlinear dynamic systems. This article describes a practical approach that overcomes these shortcomings via a novel extension of sequential Monte Carlo (SMC) sampling or particle filtering. Inequality constraints are imposed by accept/reject steps in the algorithm. The proposed approach provides samples representing the posterior distribution at each time point, and is shown to satisfy the same theoretical properties as unconstrained SMC. Illustrative examples show that results of the proposed approach are at least as accurate as moving horizon estimation, but computationally more efficient and in addition, the approach indicates the uncertainty associated with these estimates

The Use of a Cell Filter for State Estimation in Closed-Loop NMPC of Low Dimensional Systems

Combining variants of the Kalman filter and moving horizon estimation (MHE) with nonlinear MPC has been studied before. The MHE is appealing due to its ability to impose constraints and demonstrated superiority over extended Kalman filter. However, nonlinear MPC based on MHE requires solutions to two back to back nonlinear programs. In this paper we propose to use the cell filter (CF) to provide state feedback to the MPC regulator. The cell filter is a piecewise constant approximation of the conditional probability density of the states, whose temporal evolution is modeled by an aggregate Markov chain. Since the CF is based on discretized state, input and output spaces, the curse of dimensionality limits its application to low dimensional and constrained systems. In this paper we present simulation examples of closed-loop MPC for a nonlinear reactor and agricultural pest control based on state feedback from both CF and MHE