Nonlinear Optimization based frameworks for Model Predictive Control, State-Estimation, Sensitivity Analysis, and Ill-posed Problems

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

Combining the regularization strategy and the SQP to solve MPCC — A MATLAB implementation

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.

In applications of (nonlinear) model predictive control a more and more common approach for the state estimation is to use moving horizon estimation, which employs (nonlinear) optimization directly on a model for a whole batch of data. This paper shows that horizon estimation may also be used for joint parameter estimation and state estimation, as long as a bias correction based on the Kalman filter is included. A procedure how to approximate the bias correction for nonlinear systems is outlined.

ABSTRACTThe proximal method of multipliers was proposed by Rockafellar [Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math Oper Res. 1976;1:97–116] for solving convex programming and it is a kind of proximal point method applied to convex programming. In this paper, we apply this method for solving nonlinear semidefinite programming problems, in which subproblems have better properties than those from the augmented Lagrange method. We prove that, under the linear independence constraint qualification and the strong second-order sufficiency optimality condition, the rate of convergence of the proximal method of multipliers, for a nonlinear semidefinite programming problem, is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold . Moreover, the rate of convergence of the proximal method of multipliers is superlinear when the parameter c increases to .

International audience

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.

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

NONLINEAR MODEL PREDICTIVE CONTROL BASED ON SEQUENTIAL MONTE CARLO STATE ESTIMATION

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.

In this paper we deal with strong stability of stationary solutions of nonlinear positive semidefinite programs. We prove two convergent properties of matrices sequences, and we give a sucient condition for strong stability under the Linear Independence Constraint Qualification (LICQ) and the transversality condition.

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 article, an L1-Norm model predictive controller with a dead band zone is proposed for the cut tobacco drum dryer system. The control objective is to make the drum dryer temperature, hot air temperature and cut tobacco outlet temperature meet the process constraints, and optimize the outlet moisture content of the cut tobacco. First, the cut tobacco drum dryer system is introduced, and the nonlinear open equation model is established. Then an L1-Norm moving horizon estimator (MHE) is designed to provide state and parameter estimation for the controller by using its ability to deal with nonlinearity and constraints. A model predictive control (L1-Norm zone MPC) for L1-Norm target tracking with a dead band zone is proposed for the cut tobacco drum dryer system. The simulation results show that the proposed L1-Norm zone MPC (L1-ZMPC) better-tracking performance and the controller's minimum action economic characteristics compared with the traditional setpoint tracking model predictive control.

Constrained state estimation in nonlinear/non-Gaussian processes has been the domain of optimization based methods such as moving horizon estimation (MHE). MHE has a Bayesian interpretation, but it is not practical to implement a recursive MHE without assumptions of Gaussianity and linearized dynamics at various stages. This paper presents the constrained cell filter (CCF) as an alternative to MHE, requiring no linearization, jacobians, or nonlinear program. The CCF computes a piecewise constant approximation of the state probability density function with support defined by constraints; thus, all point estimates are constrained. The CCF can be more accurate and orders of magnitude faster than MHE for problems of a size as investigated in this work.

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.