An algorithm for the estimation of parameters in models with stochastic differential equations

An algorithm for the estimation of parameters of stochastic differential equations (SDEs) is presented. It is based on a nonlinear weighted least-squares formulation, in which the objective function is evaluated based on mean values of the measured variables predicted through an Euler discretisation of the SDEs and their integration by Monte-Carlo simulation. The problem is solved using a Levenberg-Marquardt algorithm. The presence of simulation noise is handled by choosing a convergence criterion based on the noise level and by ensuring that the optimality criterion is met for a large simulation size and hence a low noise level. In order to increase the reliability of the algorithm and to decrease its computational cost, stochastic sensitivity equations are derived. Furthermore, the number of trajectories used in the Monte-Carlo simulations is changed adaptively throughout the execution of the algorithm. This leads to a significant decrease in computational requirements. These concepts are illustrated on a simple example and a more complex model of polymer rheology. In all cases, parameter estimates close to the true parameter values are identified.     

Feedback control of dissipative PDE systems using adaptive model reduction

The problem of feedback control of spatially distributed processes described by highly dissipative partial differential equations (PDEs) is considered. Typically, this problem is addressed through model reduction, where finite dimensional approximations to the original infinite dimensional PDE system are derived and used for controller design. The key step in this approach is the computation of basis functions that are subsequently utilized to obtain finite dimensional ordinary differential equation (ODE) models using the method of weighted residuals. A common approach to this task is the Karhunen‐Loève expansion combined with the method of snapshots. To circumvent the issue of a priori availability of a sufficiently large ensemble of PDE solution data, the focus is on the recursive computation of eigenfunctions as additional data from the process becomes available. Initially, an ensemble of eigenfunctions is constructed based on a relatively small number of snapshots, and the covariance matrix is computed. The dominant eigenspace of this matrix is then utilized to compute the empirical eigenfunctions required for model reduction. This dominant eigenspace is recomputed with the addition of each snapshot with possible increase or decrease in its dimensionality; due to its small dimensionality the computational burden is relatively small. The proposed approach is applied to representative examples of dissipative PDEs, with both linear and nonlinear spatial differential operators, to demonstrate its effectiveness of the proposed methodology. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

基于状态观测器的电镀液温度状态反馈控制系统

针对电镀液温度控制系统,引入状态反馈控制.相对于传统的反馈控制,本方法可以获得更优异的性能.另外,考虑到镀液温度控制具有纯迟延的特点,提出了一种消去纯迟延的方法.仿真验证结果显示,控制效果良好.     

Output feedback control of dissipative PDE systems with partial sensor information based on adaptive model reduction

We address the problem of control of spatially distributed processes in the presence of measurement constraints. Specifically, we assume the availability of sensors that measure part of the state spatial profile. The measurements are utilized for the derivation and on‐demand update of reduced order models (ROM) based on an extension of the adaptive proper orthogonal decomposition (APOD) method using a snapshot reconstruction technique. The proposed Gappy‐APOD methodology constructs locally accurate low‐dimensional ROM thus resulting in a computationally efficient alternative to using a large‐dimensional ROM with global validity. Based on the low‐dimensional ROM and continuous measurements available from point sensors a Lyapunov‐based static output feedback controller is subsequently designed. The proposed controller design method is illustrated on an unstable process modeled by the Kuramoto‐Sivashinsky equation, when the designed controller successfully stabilizes the process even in the presence of model uncertainty. © 2012 American Institute of Chemical Engineers AIChE J, 59: 747–760, 2013     

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     

Optimal mechano-electric stabilization of cardiac alternans

Alternation of normal action-potential morphology in the myocardium is a condition with a beat-to-beat oscillation in the length of the electric wave which is linked through electromechanical coupling to the cardiac muscle contraction, and is believed to be the first manifestation of the onset of life threatening ventricular arrhythmias and sudden cardiac death. In this work, the effects of electrical and mechanical stimuli are utilized in alternans annihilation problem. Electrical stimuli that alter the action-potential morphology are represented by a pacer located at the domain's boundary, while mechanical stimuli are distributed within the spatial domain and affect the action potential by altering intracellular calcium kinetics. Alternation of action potential is described by the small amplitude of alternans parabolic partial differential equation (PDE). Spatially uniform unstable steady state of the alternans amplitude PDE is stabilized by optimal control methods through boundary and spatially distributed actuation. Mixed boundary and spatially distributed actuation is manipulated by a linear quadratic regulator (LQR) in the full-state-feedback control structure and in a compensator design with a finite-dimensional Luenberger-type observer, and it achieves exponential stabilization in a finite size tissue cable length. The proposed control problem formulation and the performance and robustness of the closed-loop system under the proposed linear controller are evaluated through simulations.     

Predictive control of transport-reaction processes

This work focuses on the development of computationally efficient predictive control algorithms for nonlinear parabolic and hyperbolic PDEs with state and control constraints arising in the context of transport-reaction processes. We first consider a diffusion-reaction process described by a nonlinear parabolic PDE and address the problem of stabilization of an unstable steady-state subject to input and state constraints. Galerkin’s method is used to derive finite-dimensional systems that capture the dominant dynamics of the parabolic PDE, which are subsequently used for controller design. Various model predictive control (MPC) formulations are constructed on the basis of the finite dimensional approximations and are demonstrated, through simulation, to achieve the control objectives. We then consider a convection-reaction process example described by a set of hyperbolic PDEs and address the problem of stabilization of the desired steady-state subject to input and state constraints, in the presence of disturbances. An easily implementable predictive controller based on a finite dimensional approximation of the PDE obtained by the finite difference method is derived and demonstrated, via simulation, to achieve the control objective.     

连续化聚酯装置的产量波动与控制

阐述了连续化聚酯装置产量波动对生产的影响 ,分析了影响产量波动的因素。结果表明保持产量控制泵转速、出口压力、酯化物量比及酯化率的稳定有利于装置产量的稳定。     

The exact discrete model of a system of linear stochastic differential equations driven by fractional noise

Abstract. This paper derives the exact discrete model (EDM) of a kth‐order system of stochastic differential equations driven by a vector fractional noise under fixed initial conditions. The EDM can be used for the Gaussian estimation and forecasting with long‐memory discrete‐time equispaced data. Detailed formulae which are necessary for the construction and numerical evaluation of the Gaussian likelihood under two observation schemes are established. State variables can be observed either at equispaced points in time or as integrals over the observational interval.     

Predictive control of parabolic PDEs with boundary control actuation

This work focuses on predictive control of linear parabolic partial differential equations (PDEs) with boundary control actuation subject to input and state constraints. Under the assumption that measurements of the PDE state are available, various finite-dimensional and infinite-dimensional predictive control formulations are presented and their ability to enforce stability and constraint satisfaction in the infinite-dimensional closed-loop system is analyzed. A numerical example of a linear parabolic PDE with unstable steady state and flux boundary control subject to state and control constraints is used to demonstrate the implementation and effectiveness of the predictive controllers.     

不确定离散线性切换系统的鲁棒控制

利用共同李亚普诺夫函数方法和线性矩阵不等式处理方法,研究了一类含有结构参数不确定性的离散时间切换系统的鲁棒渐近稳定问题。基于凸组合条件,通过各个子系统的状态反馈控制器和相应切换规律的设计,在有限控制器之间进行切换使得该系统是全局渐近稳定的。文中用数学方法证明了该设计方法的有效性,所得结果为线性切换系统的分析提供了更深的视角,有助于控制系统其它性能实现。     

Low-order ODE approximations and receding horizon control of surface roughness during thin-film growth

The problem of feedback controller synthesis with objective to control the microstructure during thin-film growth is considered. The problem of the non-availability of closed form dynamic models for the evolution of the microstructure is circumvented by deriving low-order state-space models that approximate the underlying kinetic Monte Carlo simulations. Initially, a finite set of “coarse” observables is identified from spatial correlation functions to represent the coarse microscopic state and capture the dominant characteristics of the microstructure during the deposition process. Subsequently, a state-space model is identified, employing proper orthogonal decomposition and Carleman linearization, that describes the evolution of the coarse observables. The state-space model is subsequently employed to design receding horizon controllers that regulate the surface roughness of the thin-film at a specified set-point during the growth process by manipulating the substrate temperature. The above approach is applied to: (i) a deposition process modeled using solid-on-solid model on a one-dimensional lattice; and (ii) an anisotropic deposition process on a two-dimensional lattice. Closed-loop simulations at various growth rates and in the presence of disturbances are performed to demonstrate the effectiveness of the proposed controller design scheme.     

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.     

Real‐time control of a semi‐industrial fed‐batch evaporative crystallizer using different direct optimization strategies

This article presents a model‐based control approach for optimal operation of a seeded fed‐batch evaporative crystallizer. Various direct optimization strategies, namely, single shooting, multiple shooting, and simultaneous strategies, are used to examine real‐time implementation of the control approach on a semi‐industrial crystallizer. The dynamic optimizer utilizes a nonlinear moment model for on‐line computation of the optimal operating policy. An extended Luenberger‐type observer is designed to enable closed‐loop implementation of the dynamic optimizer. In addition, the observer estimates the unmeasured process variable, namely, the solute concentration, which is essential for the intended control application. The model‐based control approach aims to maximize the batch productivity, as satisfying the product quality requirements. Optimal control of crystal growth rate is the key to fulfill this objective. This is due to the close relation of the crystal growth rate to product attributes and batch productivity. The experimental results suggest that real‐time application of the control approach leads to a substantial increase, i.e., up to 30%, in the batch productivity. The reproducibility of batch runs with respect to the product crystal size distribution is achieved by thorough seeding. The simulation and experimental results indicate that the direct optimization strategies perform similarly in terms of optimal process operation. However, the single shooting strategy is computationally more expensive. © 2010 American Institute of Chemical Engineers AIChE J, 57: 1557–1569, 2011