Subsystem decomposition and distributed state estimation of nonlinear processes with implicit time-scale multiplicity

In this work, we propose a subsystem decomposition approach and a distributed estimation scheme for a class of implicit two-time-scale nonlinear systems. Taking the advantage of the time scale separation, these processes are decomposed into fast subsystem and slow subsystem according to the dynamics. In the proposed method, an approach that combines the approximate solutions obtained from both the fast and slow subsystems to form a composite solution of the original system is proposed. Also, based on the fast and slow subsystems, a distributed state estimation scheme is proposed to handle the implicit time-scale multiplicity. In the proposed design, an extended Kalman filter (EKF) is designed for the fast subsystem and a moving horizon estimator (MHE) is designed for the slow subsystem. In the design, the slow subsystem is only required to send information to the fast subsystem one-directionally. The fast subsystem estimator does not send out any information. The estimators use different sampling times, that is, fast sampling of the fast state variables is considered in the fast EKF and slow sampling of the slow state variables is considered in the slow MHE. Extensive simulations based on a chemical process are performed to illustrate the effectiveness and applicability of the proposed subsystem decomposition and composite estimation architecture.     

State estimation for high-dimensional chemical processes

The application of conventional observer designs for high-dimensional systems may not always be practical due to high computational requirements or the resulting observers being too sensitive to measurement noise. In order to address these issues, this paper presents two observer design techniques for state estimation of high-dimensional chemical processes. One technique is used for systems with inputs, whereas the other one is specifically geared towards systems that are not excited from the outside. Both of these observers are applicable to linear and with a modification to non-linear systems.The main idea behind the presented observer designs is that a reduced-order observer is implemented instead of a conventional state estimator. The motivation is that subspaces, which are close to being unobservable, cannot be correctly reconstructed in a realistic setting due to measurement noise and inaccuracies in the model. The presented approaches make use of this observation and only reconstruct the parts of the system where accurate state estimation is possible. The observer designs are illustrated on a 30-tray distillation column model. Additionally, it has been shown that the location of process measurements has a major effect on the performance of the presented reduced-order observers.     

Data-driven state observation for nonlinear systems based on online learning

For controlling nonlinear processes represented by state-space models, a state observer is needed to estimate the states from the trajectories of measured variables. While model-based observer synthesis is traditionally challenging due to the difficulty of solving pertinent partial differential equations, this article proposes an efficient model-free, data-driven approach for state observation, which is suitable for data-driven nonlinear control without accurate nonlinear models. Specifically, by using a Chen–Fliess series representation of the observer dynamics, state observation is endowed with an online least squares regression formulation that can be solved by gradient flow with performance guarantees. When the target state trajectories for regression are unavailable, by exploiting the Kazantzis–Kravaris/Luenberger observer structure, state observation is reduced to a dimensionality reduction problem amenable to an online implementation of kernel principal component analysis. The proposed approach is demonstrated by a limit cycle dynamics and a chaotic system.     

Composite fast‐slow MPC design for nonlinear singularly perturbed systems

The design of a composite control system for nonlinear singularly perturbed systems using model predictive control (MPC) is described. Specifically, a composite control system comprised of a “fast” MPC acting to regulate the fast dynamics and a “slow” MPC acting to regulate the slow dynamics is designed. The composite MPC system uses multirate sampling of the plant state measurements, i.e., fast sampling of the fast state variables is used in the fast MPC and slow‐sampling of the slow state variables is used in the slow MPC. Using singular perturbation theory, the stability and optimality of the closed‐loop nonlinear singularly perturbed system are analyzed. A chemical process example which exhibits two‐time‐scale behavior is used to demonstrate the structure and implementation of the proposed fast–slow MPC architecture in a practical setting. © 2012 American Institute of Chemical Engineers AIChE J, 58: 1802–1811, 2012     

Detection and management of actuator faults in controlled particulate processes using population balance models

This paper presents a methodology for the design of an integrated fault detection and fault-tolerant control (FD-FTC) architecture for particulate processes described by population balance models (PBMs) with control constraints, actuator faults and a limited number of process measurements. The architecture integrates model-based fault detection, state estimation, nonlinear feedback and supervisory control on the basis of an appropriate reduced-order model that captures the dominant dynamics of the process and is obtained through application of the method of weighted residuals. The architecture comprises a family of control configurations together with a fault detection filter and a supervisor. For each configuration, a stabilizing output feedback controller with well-characterized stability properties is designed through the combination of a state feedback controller and a state observer that uses the available measurements of principal moments of the particle size distribution (PSD) and the continuous-phase variables to provide appropriate state estimates. A fault detection filter that simulates the behavior of the fault-free, reduced-order model is designed, and its discrepancy from the behavior of the actual process state estimates is used as a residual for fault detection. Finally, a switching law based on the stability regions of the constituent control configurations is derived to reconfigure the control system in a way that preserves closed-loop stability in the event of fault detection. Appropriate fault detection thresholds and control reconfiguration criteria that account for model reduction and state estimation errors are derived for the implementation of the FD-FTC architecture on the particulate process. Finally, the methodology is applied to the problem of constrained, actuator fault-tolerant stabilization of an unstable steady-state of a continuous crystallizer.     

一种新型神经网络观测器的设计与应用

针对传统线性观测器只在操作点附近具有工作满意度区间和传统非线性观测器对模型准确度依赖较大的问题,提出一种非传统的神经网络观测器设计方法。该神经网络是一个三层前馈网络,采用带修正项的误差反传算法进行训练以保证控制的精度和权值的有界。为降低对系统模型精度的依赖度,采用神经网络去识别系统的非线性部分,再结合传统的龙伯格观测器去重构系统的状态。利用Lyapunov直接法保证基于权值误差的非传统观测器的稳定性。最后将该观测器应用于机器人轨迹跟踪控制中,仿真结果表明:该方法适用于模型精度较低的非线性系统,可以满足控制的要求。     

应用非线性降阶观测器在线估计连续搅拌反应釜系统状态

本文以实际非线性连续搅拌反应釜(CSTR)为背景,利用 Lyapunov稳定性和指数可观系统概念,提出并设计了一种适合于工程实际应用的非线性降阶观测器,并实际用作实时在线观测估计CSTR系统的不可测状态。结果表明,这种观测器计算量小、结构简单、适应性强,且便于实时在线估算,即使系统处于随机干扰的环境,亦能获得满意的结果。     

Two observer-based nonlinear control approaches for temperature control of a class of continuous stirred tank reactors

In this paper, two nonlinear observer based controllers for temperature control of a continuous stirred tank reactor in which a special class of parallel exothermic reactions take place are proposed. A reduced order nonlinear observer is constructed to estimate the concentration in the reactor. The observer is coupled with two nonlinear controllers, designed based on two well-known techniques, namely input-output linearization and backstepping for controlling the reactor temperature. For dampening the effect of observer error dynamics, a compensating term is used in each control law. The asymptotical stability of the closed-loop system is shown by the Lyapunov's stability theorem. The effectiveness of the proposed controllers has been demonstrated through computer simulations.     

Nonlinear model reduction for dynamic analysis of cell population models

Transient cell population balance models consist of nonlinear partial differential-integro equations. An accurate discretized approximation typically requires a large number of nonlinear ordinary differential equations that are not well suited for dynamic analysis and model based controller design. In this paper, proper orthogonal decomposition (also known as the method of empirical orthogonal eigenfunctions and Karhunen Loéve expansion) is used to construct nonlinear reduced-order models from spatiotemporal data sets obtained via simulations of an accurate discretized yeast cell population model. The short-term and long-term behavior of the reduced-order models are evaluated by comparison to the full-order model. Dynamic simulation and bifurcation analysis results demonstrate that reduced-order models with a comparatively small number of differential equations yield accurate predictions over a wide range of operating conditions.     

Studies in the control of tubular reactors—III stabilization by observer design

The stabilization of an unstable nonlinear distributed chemical reactor system is examined when concentration measurements are not possible. The linearized form of the finite dimensional approximate model developed in Parts I[1] and II[2] is used to show that the observability index is equal to two. Furthermore it is shown that the dynamical characteristics of the reactor are such that, by a proper design of the Luenberger observer, the concentration estimation at a given collocation point can be made independently of the estimation at other collocation points.For purposes of control a one-dimensional observer is designed to directly estimate the control variable. Simulations of the nonlinear model of the reactor show that the observer design is quite successful in the stabilization of an unstable steady state when only temperature measurements along the reactor are available.     

Long range pipeline leak detection and localization using discrete observer and support vector machine

A realistic pipeline modeled by a nonlinear coupled first-order hyperbolic partial differential equations (PDEs) system is studied for the long transportation pipeline leak detection and localization. Based on the so-called water hammer equation, a linear distributed parameter system is obtained by linearization. The structure and energy preserving time discretization scheme (Cayley–Tustin) is used to realize a discrete infinite-dimensional hyperbolic PDEs system without spatial approximation or model order reduction. In order to reconstruct pressure and mass flow velocity evolution with limited measurements, a discrete-time Luenberger observer is designed by solving the operator Riccati equation. Based on this distributed observer system, data on different normal and leakage conditions (various leak amounts and positions) are generated and fed to train a support vector machine model for leak detection, amount, and position estimation. Finally, the leak detection, amount estimation, and localization effectiveness of the developed method are proved by a set of simulations. © 2019 American Institute of Chemical Engineers AIChE J, 65: e16532 2019     

一类不匹配不确定动态系统的鲁棒执行器故障检测与重构

针对一类不匹配不确定性动态系统,将不匹配不确定性的滑模控制方法与线性矩阵不等式（LMI）方法结合,设计一种新的鲁棒滑模观测器,提出了不匹配不确定动态系统滑模观测器稳定的充分必要条件以及LMI的存在定理,并证明了对系统不确定性以及外界干扰具有鲁棒性。无须对动态系统进行规范化处理,直接利用LMI方法求解鲁棒观测器增益矩阵,简化了滑模观测器设计过程。根据上述设计的鲁棒滑模观测器,应用等价输出误差介入原理和LMI方法,设计重构执行器故障的优化策略,提出在线获取故障信息的鲁棒执行器故障检测与重构方法,实现执行器故障的检测与重构。数字仿真验证了执行器故障重构方法的可靠性。     

Nonlinear observers for distillation columns

A full-order nonlinear observer for distillation columns is presented. Temperatures are measured at different points of the column and compared to the observer's output temperatures. The error is weighed by a spatially distributed function. The resulting term is used to correct the mathematical model of the observer. The estimated temperature and concentration profiles are then given by the state of the observer. The mathematical model used includes material balances on each tray as well as thermodynamic state-functions. It consists of dynamic and algebraic equations, exhibits severe nonlinearities and is of high order. Physical insight is used to design the observer. The movement of the regions of high mass transfer rate determines the essential dynamic quality of a column. Knowing the location of these mass transfer regions allows observer design and positioning of measurement points. The obtained observer was successfully tested in several simulation studies. It exhibits dynamic behaviour and robustness properties.     

基于扩张状态观测器的换热器动态面控制

针对带有延时、大惯性换热器温度控制问题,设计基于扩张状态观测器（ESO）动态面控制器。换热器温度控制是非线性的、时变的和不确定的动态过程,采用扩张状态观测器不仅可以获取系统不可测状态还能实时估计系统动态;采用动态面控制律,实现对温度的精确快速跟踪。利用Lyapunov函数证明控制系统的稳定性。通过对经典PID控制方法、动态矩阵控制方法和基于扩张状态观测器的动态面控制方法进行比较,仿真结果表明基于扩张状态观测器的动态面控制方法具有更优越的控制效果和鲁棒性能,且在被控系统设定值和扰动发生变化时具有更好的动态响应性能。     

Design and assessment of delay timer alarm systems for nonlinear chemical processes

In process and manufacturing industries, alarm systems play a critical role in ensuring safe and efficient operations. The objective of a standard industrial alarm system is to detect undesirable deviations in process variables as soon as they occur. Fault detection and diagnosis systems often need to be alerted by an industrial alarm system; however, poorly designed alarms often lead to alarm flooding and other undesirable events. In this article, we consider the problem of industrial alarm design for processes represented by stochastic nonlinear time‐series models. The alarm design for such complex processes faces three important challenges: (1) industrial processes exhibit highly nonlinear behavior; (2) state variables are not precisely known (modeling error); and (3) process signals are not necessarily Gaussian, stationary or uncorrelated. In this article, a procedure for designing a delay timer alarm configuration is proposed for the process states. The proposed design is based on minimization of the rate of false and missed alarm rates—two common performance measures for alarm systems. To ensure the alarm design is robust to any non‐stationary process behavior, an expected‐case and a worst‐case alarm designs are proposed. Finally, the efficacy of the proposed alarm design is illustrated on a non‐stationary chemical reactor problem. © 2017 American Institute of Chemical Engineers AIChE J, 63: 77–90, 2018     

Using surrogate models for efficient optimization of simulated moving bed chromatography

A new approach of using computationally cheap surrogate models for efficient optimization of simulated moving bed (SMB) chromatography is presented. Two different types of surrogate models are developed to replace the detailed but expensive full-order SMB model for optimization purposes. The first type of surrogate is built through a coarse spatial discretization of the first-principles process model. The second one falls into the category of reduced-order modeling. The proper orthogonal decomposition (POD) method is employed to derive cost-efficient reduced-order models (ROMs) for the SMB process. The trust-region optimization framework is proposed to implement an efficient and reliable management of both types of surrogates. The framework restricts the amount of optimization performed with one surrogate and provides an adaptive model update mechanism during the course of optimization. The convergence to an optimum of the original optimization problem can be guaranteed with the help of this model management method. The potential of the new surrogate-based solution algorithm is evaluated by examining a separation problem characterized by nonlinear bi-Langmuir adsorption isotherms. By addressing the feed throughput maximization problem, the performance of each surrogate is compared to that of the standard full-order model based approach in terms of solution accuracy, CPU time and number of iterations. The quantitative results prove that the proposed scheme not only converges to the optimum obtained with the full-order system, but also provides significant computational advantages.     

On-line estimation of VFA concentration in anaerobic digestion via methane outflow rate measurements

This paper deals with the design of a robust nonlinear observer as a software sensor to achieve the on-line estimation of the concentration of Volatile Fatty Acids (VFA) in a class of continuous anaerobic digesters (AD). Taking into account the limited availability of on-line sensors for AD process, in this contribution is assumed that only the methane outflow rate is available for on-line measurement. The estimation method is based on a modified version for a two-dimensional mathematical model of AD process. From the differential algebraic observability approach it is shown that the VFA concentration is detectable from the methane outflow rate measurements. The observer convergence is analyzed by using Lyapunov stability techniques. Numerical simulations illustrate the effectiveness of the proposed estimation method for a four-dimensional AD model with uncertainties associated with unmodeled dynamics and disturbances in the inflow composition.     

Cascade closed-loop optimization and control of batch reactors

In this paper, a cascade closed-loop optimization and control strategy for batch reactors is proposed. Based on the reduction of a physical conservation model a cascade system is developed, which can effectively combine optimization and control to achieve good on-line optimization and tracking performance under the common condition where incomplete knowledge of the reaction system exists. A two-tier estimation scheme using a nonlinear observer for heat production rate and reaction rates is also developed. In the reaction rate estimation, calorimetric information is used. The on-line closed-loop optimization strategy uses a descending horizon dynamic optimization algorithm based on nonlinear programming and an additive unknown disturbance for feedback. A simple adaptive nonlinear tracking system is designed based on the generic model control concept. The efficiency of this strategy is demonstrated through simulations on a batch reactor under various operation conditions, such as noisy measurements, varying initial states and model mismatch.     

Parameter identification for state estimation—application to bioprocess software sensors

Together with some on-line measurements, a reliable process model is the key ingredient of a successful state observer design. In common practice, the model parameters are inferred from experimental data so as to minimize a model prediction error, e.g. so as to minimize an output least-squares criterion. In this procedure, no care is actually exercised to ensure that the unmeasured model states are sensitive to the measured states. In turn, if sensitivity is too low, the resulting state observer will probably generate poor estimates of the unmeasured states. To alleviate these problems, a new parameter identification procedure is proposed in this study, which is based on a cost function combining a conventional prediction error criterion with a state estimation sensitivity measure. Minimization of this combined cost function produces a model dedicated to state estimation purposes. A thorough analysis of the procedure is presented in the context of bioreactor modeling, including parameter identification, model validation and design of extended Kalman filters and full horizon observers.