Subsystem decomposition and configuration for distributed state estimation

Distributed state estimation plays a very important role in process control. Improper subsystem decomposition for distributed state estimation may increase the computational burdens, degrade the estimation performance, or even deteriorate the observability of the entire system. The subsystem decomposition problem for distributed state estimation of nonlinear systems is investigated. A systematic procedure for subsystem decomposition for distributed state estimation is proposed. Key steps in the procedure include observability test of the entire system, observable states identification for each output measurement, relative degree analysis and sensitivity analysis between measured outputs and states. Considerations with respect to time‐scale multiplicity and direct graph are discussed. A few examples are used to illustrate the applicability of the methods used in different steps. The effectiveness of the entire distributed state estimation configuration procedure is also demonstrated via an application to a chemical process example used in coal handling and preparation plants. © 2016 American Institute of Chemical Engineers AIChE J, 62: 1995–2003, 2016     

Subsystem decomposition of process networks for simultaneous distributed state estimation and control

An appropriate subsystem configuration is a prerequisite for a successful distributed control/state estimation design. Existing subsystem decomposition methods are not designed to handle simultaneous distributed estimation and control. In this article, we address the problem of subsystem decomposition of general nonlinear process networks for simultaneous distributed state estimation and distributed control based on community structure detection. A systematic procedure based on modularity is proposed. A fast folding algorithm that approximately maximizes the modularity is used in the proposed procedure to find candidate subsystem configurations. Two chemical process examples of different complexities are used to illustrate the effectiveness and applicability of the proposed approach. © 2018 American Institute of Chemical Engineers AIChE J, 65: 904–914, 2019     

复杂非线性系统的子系统分解方法

利用社区发现算法研究了一种复杂非线性化工系统的子系统分解方法,并进行了分布式模型预测控制设计。引入信息图论的节点表示系统的状态、输入和输出变量,构建非线性过程系统的加权有向图,节点通过加权边连接,加权反映了节点间连接的强度,因而能够同时反映系统内部的连通性和连接强度。利用社区结构发现算法将所有变量分成子系统的群组,使得每个组内的关联比不同组间的相互作用强,从而得到复杂化工过程系统的子系统分解。针对连续搅拌反应釜过程,实施子系统分解,并设计分布式模型预测控制算法,结果表明,所提出的子系统分解方法更能考虑子系统之间的连接权重,得到更有利于分布式模型预测控制的子系统划分,提升系统控制的性能。     

Partition-based distributed extended Kalman filter for large-scale nonlinear processes with application to chemical and wastewater treatment processes

In this article, we address a partition-based distributed state estimation problem for large-scale general nonlinear processes by proposing a Kalman-based approach. First, we formulate a linear full-information estimation design within a distributed framework as the basis for developing our approach. Second, the analytical solution to the local optimization problems associated with the formulated distributed full-information design is established, in the form of a recursive distributed Kalman filter algorithm. Then, the linear distributed Kalman filter is extended to the nonlinear context by incorporating successive linearization of nonlinear subsystem models, and the proposed distributed extended Kalman filter approach is formulated. We conduct rigorous analysis and prove the stability of the estimation error dynamics provided by the proposed method for general nonlinear processes consisting of interconnected subsystems. A chemical process example is used to illustrate the effectiveness of the proposed method and to justify the validity of the theoretical findings. In addition, the proposed method is applied to a wastewater treatment process for estimating the full-state of the process with 145 state variables.     

Distributed state estimation from infrequent and delayed measurements in large-scale processes

Effective control and monitoring of a process usually require frequent and delay-free measurements of important process output variables. However, these measurements are often either not available or available infrequently with significant time delays. This article presents a method that allows for improving the performance of distributed state estimators implemented on large-scale manufacturing processes. The method uses a sample state augmentation approach that permits using delayed measurements in distributed state estimation. The method can be used with any state estimator, including unscented Kalman filters, extended Kalman filters, and moving horizon state estimators. The method optimally handles the tradeoff between computational time and estimation accuracy in distributed state estimation implemented using a computer with parallel processors. Its implementation and performance are shown using a few simulated examples.     

Consensus-based approach for parameter and state estimation of agro-hydrological systems

The development of advanced closed-loop irrigation systems requires accurate soil moisture information. In this work, we address the problem of soil moisture estimation for the agro-hydrological systems in a robust and reliable manner. A nonlinear state-space model is established based on the discretization of the Richards equation to describe the dynamics of the agro-hydrological systems. We consider that model parameters are unknown and need to be estimated together with the states simultaneously. We propose a consensus-based estimation mechanism, which comprises two main parts: (a) a distributed extended Kalman filtering algorithm used to estimate several model parameters; and (b) a distributed moving horizon estimation algorithm used to estimate the state variables and one remaining model parameter. Extensive simulations are conducted, and comparisons with existing methods are made to demonstrate the effectiveness and superiority of the proposed approach. In particular, the proposed approach can provide accurate soil moisture estimate even when poor initial guesses of the parameters and the states are used, which can be challenging to be handled using existing algorithms.     

Constrained particle filtering methods for state estimation of nonlinear process

Increasingly in practical applications, nonlinearity, non‐Gaussianity, and constraint must be considered to obtain good state estimation. A constrained particle filter (PF) approach for state estimation, which involves three alternative strategies to impose the constraints on the prior particles, posterior particles, and state estimation has been proposed. First, to impose constraints on prior particles, a constrained Gibbs sampling method with a constrained inverse transform sampling is proposed to restrict sampling within the constraint region under cases of both univariate and coupling constraints. Second, to ensure validity of posterior particles, resampling is confined to the valid prior particles and the violated ones are discarded, which results in a similar formulation as the existing acceptance/rejection constrained PF method in literature. Third, if the state estimation violates the constraint, different from the existing methods that either discard all violated particles or accept all of them by projecting them onto the constraint region, the proposed method makes a balance between the prior and the likelihood function by adjusting the weights of violated and valid particles, respectively. Compared with the existing methods, the proposed method provides better physical interpretation and involves no restrictive assumptions about the distributions. Simulation results demonstrate effectiveness of the proposed methods. © 2014 American Institute of Chemical Engineers AIChE J, 60: 2072–2082, 2014     

Machine learning-based distributed model predictive control of nonlinear processes

This work explores the design of distributed model predictive control (DMPC) systems for nonlinear processes using machine learning models to predict nonlinear dynamic behavior. Specifically, sequential and iterative DMPC systems are designed and analyzed with respect to closed-loop stability and performance properties. Extensive open-loop data within a desired operating region are used to develop long short-term memory (LSTM) recurrent neural network models with a sufficiently small modeling error from the actual nonlinear process model. Subsequently, these LSTM models are utilized in Lyapunov-based DMPC to achieve efficient real-time computation time while ensuring closed-loop state boundedness and convergence to the origin. Using a nonlinear chemical process network example, the simulation results demonstrate the improved computational efficiency when the process is operated under sequential and iterative DMPCs while the closed-loop performance is very close to the one of a centralized MPC system.     

Community detection based process decomposition and distributed monitoring for large-scale processes

Distributed architectures wherein multiple decision-making units are employed to coordinate their decision-making/actions based on real-time communication have become increasingly important for monitoring processes that have large scales and complex structures. Typically, the development of a distributed monitoring scheme involves two key steps, that is, the decomposition of the process into subsystems, and the design of local monitors based on the configured subsystem models. In this article, we propose a distributed process monitoring approach that tackles both steps for large-scale processes. A data-driven process decomposition approach is proposed by leveraging community structure detection to divide variables into subsystems optimally via finding a maximal value of the metric of modularity. A two-layer distributed monitoring scheme is developed where local monitors are designed based on the configured subsystems of variables using canonical correlation analysis. Inner-subsystem interactions and inter-subsystem interactions are tackled by the two layers separately, such that the sensitivity of this monitoring scheme to certain types of faults is improved. We utilize a numerical example to illustrate the effectiveness and superiority of the proposed method. It is then applied to a simulated wastewater treatment process.     

Parameter estimation in nonlinear chemical and biological processes with unmeasured variables from small data sets

We propose an algorithm for parameter estimation in nonlinear chemical and biological stochastic processes with unmeasured variables and small data sets. The algorithm relies on an iterative approach wherein random samples of parameters and unmeasured variables are generated, from their respective posterior density functions, through Markov chain Monte Carlo simulations. The random samples are then used in approximating the posterior density functions of the parameters. The effectiveness of the algorithm is demonstrated through two biological examples—a feed-forward loop genetic regulatory network and a JAK–STAT signal transduction pathway.     

Nonlinear stochastic modeling to improve state estimation in process monitoring and control

State estimation from plant measurements plays an important role in advanced monitoring and control technologies, especially for chemical processes with nonlinear dynamics and significant levels of process and sensor noise. Several types of state estimators have been shown to provide high‐quality estimates that are robust to significant process disturbances and model errors. These estimators require a dynamic model of the process, including the statistics of the stochastic disturbances affecting the states and measurements. The goal of this article is to introduce a design method for nonlinear state estimation including the following steps: (i) nonlinear process model selection, (ii) stochastic disturbance model selection, (iii) covariance identification from operating data, and (iv) estimator selection and implementation. Results on the implementation of this design method in nonlinear examples (CSTR and large dimensional polymerization process) show that the linear time‐varying autocovariance least‐squares technique accurately estimates the noise covariances for the examples analyzed, providing a good set of such covariances for the state estimators implemented. On the estimation implementation, a case study of a chemical reactor demonstrates the better capabilities of MHE when compared with the extended Kalman filter. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

Explicit distributed model predictive control design for chemical processes under constraints and uncertainty

In multivariable industrial processes, the common distributed model predictive control strategy is usually unable to deal with complex large-scale systems efficiently, especially under system constraints and high control performance requirements. Based on this situation, we use the distributed idea to divide the large-scale system into multiple subsystems and transform them into the state space form. Combined with the output tracking error term, we build an extended non-minimal state space model that includes output error and measured output and input. When dealing with system constraints, the new constraint matrix is divided into range and kernel space by using the explicit model predictive control algorithm, which reduces the difficulty of solving constraints in the extended system and further improves the overall control performance of the system. Finally, taking the coke furnace pressure control system as an example, the proposed algorithm is compared with the conventional distributed model predictive control algorithm using non-minimal state space, and the simulation results show the feasibility and superiority of this method.     

Maximum likelihood estimation of noise covariance matrices for state estimation of autonomous hybrid systems

A critical aspect of developing Bayesian state estimators for hybrid systems, that involve a combination of continuous and discrete state variables, is to have a reasonably accurate characterization of the stochastic disturbances affecting their dynamics. Recently, Bavdekar et al. (2011) have proposed a maximum likelihood (ML) based framework for estimation of the noise covariance matrices from operating input–output data when an EKF is used for state estimation. In this work, the ML framework is extended to estimation of the noise covariance matrices associated with autonomous hybrid systems, and, to a wider class of recursive Bayesian filters. Under the assumption that the innovations generated by an estimator form a white noise sequence, the proposed ML framework computes the noise covariance matrices such that they maximize the log-likelihood function of the estimator innovations. The efficacy of the proposed scheme is demonstrated through the simulation and experimental studies on the benchmark three-tank system.     

Guaranteed cost distributed fuzzy observer‐based control for a class of nonlinear spatially distributed processes

The guaranteed cost distributed fuzzy (GCDF) observer‐based control design is proposed for a class of nonlinear spatially distributed processes described by first‐order hyperbolic partial differential equations (PDEs). Initially, a T–S fuzzy hyperbolic PDE model is proposed to accurately represent the nonlinear PDE system. Then, based on the fuzzy PDE model, the GCDF observer‐based control design is developed in terms of a set of space‐dependent linear matrix inequalities. In the proposed control scheme, a distributed fuzzy observer is used to estimate the state of the PDE system. The designed fuzzy controller can not only ensure the exponential stability of the closed‐loop PDE system but also provide an upper bound of quadratic cost function. Moreover, a suboptimal fuzzy control design is addressed in the sense of minimizing an upper bound of the cost function. The finite difference method in space and the existing linear matrix inequality optimization techniques are used to approximately solve the suboptimal control design problem. Finally, the proposed design method is applied to the control of a nonisothermal plug‐flow reactor. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2366–2378, 2013     

State estimation for one-dimensional agro-hydrological processes with model mismatch

The importance of accurate soil moisture data for the development of modern closed-loop irrigation systems cannot be overstated. Due to the diversity of soil, it is difficult to obtain an accurate model for the agro-hydrological system. In this study, soil moisture estimation in one-dimensional (1D) agro-hydrological systems with model mismatch is the focus. To address the problem of model mismatch, a nonlinear state-space model derived from the Richards equation is utilized, along with additive unknown inputs. The determination of the number of sensors required is achieved through sensitivity analysis and the orthogonalization projection method. To estimate states and unknown inputs in real-time, a recursive expectation maximization (EM) algorithm derived from the conventional EM algorithm is employed. During the E-step, the extended Kalman filter (EKF) is used to compute states and covariance in the recursive Q-function, while in the M-step, unknown inputs are updated by locally maximizing the recursive Q-function. The estimation performance is evaluated using comprehensive simulations. Through this method, accurate soil moisture estimation can be obtained, even in the presence of model mismatch.     

Application of MHE to large-scale nonlinear processes with delayed lab measurements

The paper addresses nonlinear estimation problems on nonlinear processes containing several lab measurements sampled slowly and with long delay, which is the usual case in industrial polymerization applications. A moving horizon estimation algorithm is developed to compute the theoretical optimal solution given the multi-rate measurements. In this algorithm, the MHE window is recalculated as the new lab measurement becomes available. Simulation studies on a polymerization process with plant model mismatch are performed. Observability analysis and estimation results of MHE with and without lab measurements show that lab measurements help identify the disturbances and can improve the performance of both estimation and closed-loop control.