MHE Based State and Parameter Estimation for Systems subjected to Non-Gaussian Disturbances

Nonlinear state estimation is a pre-requisite for advanced process control and fault diagnosis tasks. In literature, various recursive nonlinear filtering techniques have been proposed and used for state estimation of nonlinear systems. Over the last few years, Moving Horizon Estimation (MHE) is increasingly being used for state estimation of nonlinear systems. Moving horizon estimation works with a window of data and hence requires additional online computation compared to recursive nonlinear filters. However, MHE performs both smoothing and filtering and thus has the potential to obtain more accurate state estimates as compared to the recursive filters. Most of the available comparisons of MHE with recursive filters are based on simulation case studies where the true states and parameters, as well as noise processes are exactly known. In this work, we apply MHE to a three-tank experimental setup and compare its performance with various nonlinear filters available in literature such as Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) and Gaussian Sum EKF (GSEKF). We present some of the challenges in experimental implementation of state estimation approaches. These include presence of unknown disturbances, non-whiteness of noise signals as well as lack of accurate measurements. We then discuss the approach followed for obtaining model parameters and noise characterization to make the models amenable for filter implementation. It is found that EKF, GSEKF and UKF perform as well as MHE as far as accuracy is concerned, but require significantly lower computational efforts.

Nearly every model predictive control (MPC) algorithm is premised on knowledge of the system’s state. As a result, state estimation is vital to good MPC performance. Moving horizon estimation (MHE) is an optimization-based state estimation algorithm. Similar to MPC, it relies on the minimization of a sum of stage costs subject to a dynamic model. Unlike MPC, however, conditions under which MHE is robustly stable have been slow to emerge. Recently, several results have appeared about the robust stability of MHE. We generalize the result on the robust stability of MHE without a max-term presented in Muller (Automatica 79:306–314, 2017). using assumptions inspired by the result in Hu (Robust stability of optimization-based state estimation under bounded disturbances. ArXiv e-prints, 2017). Furthermore, we show that all systems that are covered by the assumptions used in those previous works satisfy a certain form of exponential incremental input/output-to-state stability.

NONLINEAR MODEL PREDICTIVE CONTROL BASED ON SEQUENTIAL MONTE CARLO STATE ESTIMATION

Model Predictive Control (MPC) and constrained Moving Horizon Estimation (MHE) are both optimization-based method where a constrained optimization problem is solved at each time instant. Recursive feasibility of the constrained optimization problem plays a key role in MPC as it ensures that the state is not driven into a "blind alley" where the constrained problem could become infeasible. For MHE the constrained optimization problem can become infeasible when erroneous measurements occur. In this thesis, we showed relationship between a trajectory advisor for MPC with tracking error bound and a measurement checker for constrained MHE. Akin to the concept of reference governor, the trajectory advisor can supply a new feasible reference signal to recover the feasibility of the MPC problem if the original reference signal makes the MPC problem infeasible. The measurement checker could be used to determine the admissibility of the measurements before the MHE is solved. MPC requires state feedback and if the full state is not measurable, a state estimator is required. Gaussian noise is often assumed in state estimation and the optimality of the estimators based on this assumption may be lost when the assumption is violated. Constrained MHE offers an approach to handle non Gaussian noise by using constraints. In this thesis, a novel moving horizon Maximum Likelihood Estimation (MLE) is proposed for Autoregressive-Moving-Average with eXogenous (ARMAX) process with t-distribution noise model. The thick tail property of t-distribution downs weigh outliers in the likelihood function so that the proposed estimator is robust to outliers. Instead of solving the maximization problem numerically, the Influence Function (IF), an analysis tool in robust statistics, is used to formulate a computationally efficient recursive solution to the maximization problem approximately. As expected, the recursive estimator is reduced to the traditional Moving Window Least-Squares Estimation when the noise is Gaussian. The moving horizon N can be used as tuning parameter for the proposed estimator. The formula for the variance of the estimate is derived. This formula shows explicitly how the variance of estimate is affected by the number of measurements and noise variance. In addition, a formula which predicts the effect of an outlier on estimate is derived. The simulations show the advantages of the proposed estimator over Moving Horizon Least-Squares Estimation in the sense of estimation variance and robustness to outlier, and the advantage over Kalman filter when model mismatch happens. The proposed estimator is applied to liquid level estimation of a Coupled-Tank Control Apparatus and indoor environment localization which show that the proposed estimator gives smaller estimation variance and is more robust to outlier than Moving Window Least-Squares Estimation.

The development of robust and efficient algorithms for nonlinear optimization is an important matter; because they make possible the solution of models with increasingly sizeand complexity. In this context, the Interior Point Optimizer (IPOPT) has been proven to be a competitive algorithm, because of its ability to handle large scale problems. However, there are some specific situations in which convergence can not be guaranteed. For instancethe violation of a regularity condition given by the Linear Independence Constraint Qualification (LICQ). This property is given in terms of the consistency of local linearizationsof the set of constraints, i.e. the gradients. If the set of gradients of equality and active bound inequality are linearly independent, then the LICQ holds. It turns out that if thisproperty is not satisfied, IPOPT would attempt to regularize the system by perturbing it, with the hope to overcome regions of local inconsistency. However, if the LICQ does nothold globally or if the system has severe ill-conditioned, the regularization strategies will fail. Thus, we present the `1-exact penalty-barrier strategy to deal with this lack of regularity. This works by penalizing the infeasibility with a parameter at the objective; and then by reformulating the problem to obtain a higher dimensional problem, the LICQ property holds at all points for which the new variables are nonzero. Thus achieving higher level of robustness as opposed to the regular IPOPT. Moreover, an unified framework for nonlinear model-based state-estimation and control with parametric sensitivity is presented. Typically, control strategy of a processes is tied to the ability to accommodate online computations; in this sense nonlinear optimization strategies like Moving Horizon Estimation (MHE) and Model Predictive Control (NMPC) are overlooked. Nevertheless, it has been proposed to use the parametric nature of these optimization problems to create a scheme in which the bulk of computations is partitioned into a background phase with all the most computationally expensive steps, and an online phase, in which the sensitivity information is used to make inexpensive calculations for the state estimate or controller input. These ideas have been put together into a Python framework, that streamlines the creation of a state-estimation and control strategy with the additional benefit of having parametric sensitivity embedded into it. This framework works by using the Pyomo libraries to create optimization objects and automatically discretize the models given by the user. We demonstrate the effectiveness of the frameworkin two problems of vastly different complexity. A benchmark distillation case study and a Bubbling Fluidized Bed Reactor (BFB) for the capture of CO2. The results show thatsensitivity is effective for online implementations of the controller and it was seamlessly enabled by our framework.

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Based on a simplified control-oriented hybrid linear delayed model, model predictive control (MPC) of a sewer network designed to reduce pollution during heavy rain events is presented. The lack of measurements at many parts of the system to update the initial conditions of the optimal control problems (OCPs) leads to the need for estimation techniques. A simple modification of the OCP used in the MPC iterations allows to formulate a state estimation problem (SEP) to reconstruct the full system state from a few measurements. Results comparing the system performance under the MPC controller using full-state measurements and a moving horizon estimation (MHE) strategy solving a finite horizon SEP at each time instant are presented. Closed-loop simulations are performed by using a detailed physically-based model of the network as virtual reality.

In this paper we consider a nonlinear constrained system observed by a sensor network and propose a distributed state estimation scheme based on moving horizon estimation (MHE). In order to embrace the case where the whole system state cannot be reconstructed from data available to individual sensors, we resort to the notion of MHE detectability for nonlinear systems, and add to the MHE problems solved by each sensor a consensus term for propagating information about estimates through the network. We characterize the error dynamics and provide conditions on the local exchanges of information in order to guarantee convergence to zero and stability of the state estimation error provided by any sensor. Copyright © 2010 John Wiley & Sons, Ltd.

There are varieties of time-varying processes in chemical engineering industrial applications. These processes are abundant among lumped and distributed parameter systems and in batch systems they involve time-dependent change of parameters and/or geometry within lumped and/or distributed parameter systems settings. The focus of this thesis is on optimal state estimation and tracking regulation of two fed-batch processes with time-varying parameters and geometry. The first process is nonlinear time-varying microalgae growth and lipid production. An optimal reference trajectory is identified for maximum lipid production and moving horizon estimator along with model predictive control is realized for reference trajectory tracking of lipid production model. The second process is Czochralski crystal growth process which has moving boundary parabolic partial deferential equation describing heat transfer as dynamic model coupled with a lumped parameter model of pulling dynamics. Galerkin’s method is used to reduce the distributed parameter model’s order and an observer is developed to reconstruct temperature distribution evolution over the entire crystal domain during growth process. The performance of the observer is examined by implementing the observer on finite element model of the heat transfer in crystal. Furthermore, finite element model of the heat transfer along with finite element model of anisotropic thermal stresses in growing crystal are utilized to identify

A Comparison of Moving Horizon and Bayesian State Estimators with an Application to a pH Process

On the use of shape constraints for state estimation in reaction systems

Combined nonlinear model predictive control and moving horizon estimation for a copolymerization process

This paper deals with load frequency control (LFC) problem for power system. The control is conducted to retain power demand supply balance by controlling output of generators in power network. In order to realize LFC with low power generation cost while maintaining control performance, this paper proposes frequency control method using Moving Horizon Estimation (MHE) and Model Predictive Control (MPC). First, MHE is used to estimate the state of the system. Next, optimization calculation is performed using MPC to determine the command values for each generator. Finally, simulation of load frequency control is carried out to verify the effectiveness of the proposed method.

In this paper, the navigational state estimation problem is investigated for a class of networked spatial-navigation systems with quantization effects, mixed time-delays, and network-based observations (i.e. complementary measurements and regional estimations). A decentralized moving horizon estimation approach, featuring complementary reorganization and recursive procedure, is proposed to tackle this problem. First, through the proposed reorganized scheme, a random delayed system with complementary observations is reconstructed into an equivalent delay-free one without dimensional augment. Second, with this equivalent system, a robust moving horizon estimation scheme is presented as a uniform estimator for the navigational states. Third, for the demand of real-time estimate, the recursive form of decentralized moving horizon estimation approach is developed. Furthermore, a collective estimation is obtained through the weighted fusion of two parts, i.e. complementary measurements based estimation, and regional estimations directly from the neighbors. The convergence properties of the proposed estimator are also studied. The obtained stability condition implicitly establishes a relation between the upper bound of the estimation error and two parameters, i.e. quantization density and delay occur probability. Finally, an application example to networked unmanned aerial vehicles is presented and comparative simulations demonstrate the main features of the proposed method.

Two approaches, extended Kalman filter (EKF) and moving horizon estimation (MHE), are discussed for state estimation for nonlinear dynamical systems over packet-dropping networks. For EKF, we provide sufficient conditions that guarantee a bounded EKF error covariance. For MHE, a natural scheme on organizing the finite horizon window is proposed to handle intermittent observations. A nonlinear programming software package, SNOPT, is employed in MHE and the formulation for constraints is discussed in detail. Examples and simulation results are presented.