A method to construct reduced‐order parameter‐varying models

This paper describes a method to construct reduced‐order models for high‐dimensional nonlinear systems. It is assumed that the nonlinear system has a collection of equilibrium operating points parameterized by a scheduling variable. First, a reduced‐order linear system is constructed at each equilibrium point using state, input, and output data. This step combines techniques from proper orthogonal decomposition, dynamic mode decomposition, and direct subspace identification. This yields discrete‐time models that are linear from input to output but whose state matrices are functions of the scheduling parameter. Second, a parameter‐varying linearization is used to connect these linear models across the various operating points. The key technical issue in this second step is to ensure the reduced‐order linear parameter‐varying system approximates the nonlinear system even when the operating point changes in time. Copyright © 2016 John Wiley & Sons, Ltd.     

Output regulator problem of a nuclear reactor system

A control problem of distributed reactor systems is investigated. After kinetic equations are expanded into Kaplan modes, the output regulator with structural stability is applied to the spatial mode system. In particular, the fundamental mode is regarded as an exogeneous input signal to the spatial control system due to a variation of control rods from the power control system. The output to be regulated is the neutron flux deviation from the equilibrium value at some specified points where the fuels in burn-up are located. Some numerical simulation results are presented to show the effectiveness of the proposed method.     

Operability analysis of nonlinear processes based on incremental dissipativity

Process operability can be defined as the ability of a process to reject disturbances at a specified operating point and/or to move quickly and smoothly from one operating point to another operating point using a feedback control system. Unlike linear processes, the properties of nonlinear processes (e.g., stability, minimum phase condition, etc.) are different around different equilibria. Most existing operability analysis for nonlinear systems focuses on one particular operating point of interest. This paper addresses the issues of dynamic process operability at various operating points, including the reachability of all equilibrium points or output trajectories in an operating region, regardless of initial conditions. In this work, a nonlinear analysis approach is developed based on the concept of incremental stability. Conditions for incremental stability are derived based on incremental dissipativity. The links between input and output multiplicities and incremental dissipativity are explored. The dynamic control performance achievable in terms of the speed of the response of the closed-loop system and offset minimization is studied. A method for determination of incremental dissipativity using Linear Differential Inclusion (LDI) is also presented, to facilitate the dissipativity based operability analysis.     

Invariant based modeling and control of multi-phase reactor systems

We propose a modeling framework for stable simulation of multi-phase reactor systems operating at thermodynamic equilibrium. The model framework can be used to determine system characteristics, explore parameter sensitivity and test control system strategies. The thermodynamic equilibrium assumption and the use of reaction invariants make it computationally inexpensive. We show that the feedback control approach based on the overall inventories of the system can be effectively used for improved performance of such reactor systems. Two multi-phase reactor systems – the vapor recovery reactor used in carbothermic aluminum reduction and the gasification reactor used in IGCC are considered to demonstrate the efficacy of the proposed modeling and control approach.     

Stability analysis of non-interacting control of continuous stirred tank reactor†

A continuous stirred tank reactor operating around a stable equilibrium point and having interaction may become unstable when made noii-intcractivo by tho addition of compensators. Procedures for selecting the settings of tho feedback controllers to obtain any desired degree of stability in the non-interacting system are worked out in this paper. A numerical example of a continuous stirred tank reactor with an irreversible, exothermic first-order reaction is taken and controller settings for its stable non-interacting operation worked out.     

A new feedback neural network with supervised learning

A model is introduced for continuous-time dynamic feedback neural networks with supervised learning ability. Modifications are introduced to conventional models to guarantee precisely that a given desired vector, and its negative, are indeed stored in the network as asymptotically stable equilibrium points. The modifications entail that the output signal of a neuron is multiplied by the square of its associated weight to supply the signal to an input of another neuron. A simulation of the complete dynamics is then presented for a prototype one neuron with self-feedback and supervised learning; the simulation illustrates the (supervised) learning capability of the network.     

Systematic design of an optimal control system for the SHARON-Anammox process

A systematic design of an optimal control structure for the SHARON-Anammox nitrogen removal process is studied. The methodology incorporates two novel features to assess the controllability of the design variables candidate for the regulatory control layer: (i) H_∞ control method, which formulates the control problem as a mathematical optimization problem, and (ii) close-loop disturbance gain (CLDG) plots. It is shown that the methodology is especially appropriate for bioreactors. The solution of the mixed sensitivity stacked H_∞ control problem ranked the combinations of controlled variables (CVs). The best candidates to CVs were paired with the manipulated variables using the relative gain array. The proposed control structure was further analyzed and verified for disturbance rejection using the CLDG plots. The optimal pairing of CVs with the actuators (k_La and acid/base addition) is found to be dissolved oxygen (DO) and pH in the SHARON reactor. Furthermore, to relate the controller actions to process operation objective, nitrogen removal efficiency, two cascade control systems are designed. The first cascade loop controls TNN/TAN ratio in the influent to the Anammox reactor by adjusting the set point for DO in the regulatory layer, while the second cascade loop controls the nitrogen removal efficiency (i.e. effluent TNN and TAN) by adjusting the TNN/TAN ratio at the effluent of the SHARON reactor. The control system is evaluated and benchmarked using a set of realistic dynamic scenario simulations, demonstrating that the different control strategies successfully maintain stable and high nitrogen removal efficiency. The nested cascade control structure shows the best performance, removing up to 95% of the influent ammonia. Both the control design methodology and the resulting optimal control structures are expected to contribute to stable operation and control of these emerging nitrogen removal technologies.     

Estimation of a system pulse transfer function in the presence of noise

Statistical estimation theory is applied to derive effective techniques for measurement of the pulse transfer function of a linear system from normal operating records obscured by additive noise. It is shown that the problem is equivalent to that of fitting a hyperplane to a set of observed points with random errors in certain coordinates. A method of Koopmans is applied to obtain generalized least squares estimates which are also maximum likelihood estimates when the noise is white and Gaussian. The estimates of the coefficients are obtained as the components of the eigenvector corresponding to the smallest eigenvalue of a matrix equation involving the sample auto- and cross-correlation functions of the input and output records and the covariance matrix of the corresponding noise components. Expressions for the sampling variances and biases are given. The properties of the simpler standard least squares estimates are also considered. The appropriate modifications for nonwhite noise are described.     

Nonlinear feedback regulator design for the Cuk converter

The method of extended linearization is used for designing stabilizing nonlinear proportional-integral (PI) feedback controllers which regulate to a constant set-point value either the average output inductor current, the average input inductor current, or the average transfer capacitor voltage of a pulse width modulation (PWM) controlled Cuk converter. The design is carried out on the basis of the frequency-domain Ziegler-Nichols method, applied to a family of transfer function models of the linearized average PWM controlled circuit, parametrized by constant operating equilibrium points     

Wheel torque proportioning, rear steering, and normal forcecontrol: a structural investigation

This paper is concerned with various analysis and design issues for a possible set of control channels which may be used to improve the maneuverability and handling of automobiles. Of the possible set of controls, the current paper is limited to variable wheel torque proportioning, rear steering, and normal force control. The analysis and design issues are studied using an extensive linear model representing the lateral, longitudinal, and vertical dynamics, including their interactions. For the investigation of the finite-time behavior of system dynamics, we first introduce the notion of a reachable set where the attainability of a particular state vector at a particular time instant is considered. Next, we examine the attainability of a particular output trajectory within a finite-time interval using the notion of output function controllability. Several structural properties of these concepts have been studied. Various degenerate operating points are identified, where the effectiveness of the control channels deteriorates. In the steady-state analysis, the reference tracking and regulation problems are considered. It is shown that for some input/output subsystems those problems are generically solvable due to the minimum phase structure of the subsystems. For some others, those problems are solvable provided that the equilibrium point is not a degenerate equilibrium point. Finally, we show that for some input/output subsystems those problems are generically unsolvable     

Economic model predictive control of parabolic PDE systems: Addressing state estimation and computational efficiency

In a previous work [20], an economic model predictive control (EMPC) system for parabolic partial differential equation (PDE) systems was proposed. Through operating the PDE system in a time-varying fashion, the EMPC system demonstrated improved economic performance over steady-state operation. The EMPC system assumed the knowledge of the complete state spatial profile at each sampling period. From a practical point of view, measurements of the state variables are typically only available at a finite number of spatial positions. Additionally, the basis functions used to construct a reduced-order model (ROM) for the EMPC system were derived using analytical sinusoidal/cosinusoidal eigenfunctions. However, constructing a ROM on the basis of historical data-based empirical eigenfunctions by applying Karhunen-Loève expansion may be more computationally efficient. To address these issues, several EMPC systems are formulated for both output feedback implementation and with ROMs based on analytical sinusoidal/cosinusoidal eigenfunctions and empirical eigenfunctions. The EMPC systems are evaluated using a non-isothermal tubular reactor example, described by two nonlinear parabolic PDEs, where a second-order reaction takes place. The model accuracy, computational time, input and state constraint satisfaction, and closed-loop economic performance of the closed-loop tubular reactor under the different EMPC systems are compared.     

Structured parametric epidemic models

A stage-structured model for a theoretical epidemic process that incorporates immature, susceptible and infectious individuals in independent stages is formulated. In this analysis, an input interpreted as a birth function is considered. The structural identifiability is studied using the Markov parameters. Then, the unknown parameters are uniquely determined by the output structure corresponding to an observation of infection. Two different birth functions are considered: the linear case and the Beverton–Holt type to analyse the structured epidemic model. Some conditions on the parameters to obtain non-zero disease-free equilibrium points are given. The identifiability of the parameters allows us to determine uniquely the basic reproduction number ?₀ and the stability of the model in the equilibrium is studied using ?₀ in terms of the model parameters.     

Small Helicopter Control Design Based on Model Reduction and Decoupling

In this paper a complete nonlinear mathematical model of a small scale helicopter is derived. A coupling between input and output variables, revealed by the model, is investigated. The influences that particular inputs have on particular outputs are examined, and their dependence on flying conditions is shown. In order to demonstrate this dependence, the model is linearized in various operating points, and linear, direct and decoupling, controllers are determined. Simulation results, presented at the end of the paper, confirm that the proposed control structure could be successfully used for gain scheduling or switching control of a small scale helicopter in order to provide acrobatic flight by using simple linear controllers.     

约束非线性系统稳定经济模型预测控制

考虑约束非线性系统,提出一种新的具有稳定性保证的经济模型预测控制（Economic model predictive control,EMPC）策略.由于经济性能指标的非凸性和非正定性,引入关于经济最优平衡点的正定辅助函数.利用辅助函数的最优值函数定义原始EMPC优化问题的稳定性约束.应用终端约束集、终端代价函数和局部控制器三要素,建立闭环系统关于经济最优平衡点的渐近稳定性和渐近平均性能.进一步,结合多目标理想点概念,将提出的控制策略用于多个经济性能指标的优化控制,得到稳定多目标EMPC策略.最后,以连续搅拌反应器为例,比较仿真结果验证本文策略的有效性.     

Modelling and multi-objective optimal control of batch processes using recurrent neuro-fuzzy networks

In this paper, the modelling and multi-objective optimal control of batch processes, using a recurrent neuro-fuzzy network, are presented. The recurrent neuro-fuzzy network, forms a "global" nonlinear long-range prediction model through the fuzzy conjunction of a number of "local" linear dynamic models. Network output is fed back to network input through one or more time delay units, which ensure that predictions from the recurrent neuro-fuzzy network are long-range. In building a recurrent neural network model, process knowledge is used initially to partition the processes non-linear characteristics into several local operating regions, and to aid in the initialisation of corresponding network weights. Process operational data is then used to train the network. Membership functions of the local regimes are identified, and local models are discovered via network training. Based on a recurrent neuro-fuzzy network model, a multi-objective optimal control policy can be obtained. The proposed technique is applied to a fed-batch reactor.     

Some extensions of a new method to analyze complete stability of neural networks

In a recent work, a new method has been introduced to analyze complete stability of the standard symmetric cellular neural networks (CNNs), which are characterized by local interconnections and neuron activations modeled by a three-segment piecewise-linear (PWL) function. By complete stability it is meant that each trajectory of the neural network converges toward an equilibrium point. The goal of this paper is to extend that method in order to address complete stability of the much wider class of symmetric neural networks with an additive interconnecting structure where the neuron activations are general PWL functions with an arbitrary number of straight segments. The main result obtained is that complete stability holds for any choice of the parameters within the class of symmetric additive neural networks with PWL neuron activations, i.e., such a class of neural networks enjoys the important property of absolute stability of global pattern formation. It is worth pointing out that complete stability is proved for generic situations where the neural network has finitely many (isolated) equilibrium points, as well as for degenerate situations where there are infinite (nonisolated) equilibrium points. The extension in this paper is of practical importance since it includes neural networks useful to solve significant signal processing tasks (e.g., neural networks with multilevel neuron activations). It is of theoretical interest too, due to the possibility of approximating any continuous function (e.g., a sigmoidal function), using PWL functions. The results in this paper confirm the advantages of the method of Forti and Tesi, with respect to LaSalle approach, to address complete stability of PWL neural networks.     

Stability Analysis and Control of Nonholonomic Systems with Potential Fields

This paper presents a theoretical analysis on the stability of nonholonomic systems when a potential field method is applied to them, and shows a numerical example for a nonholonomic underwater vehicle among obstacles. Nonholonomic systems with a potential function have an infinite number of equilibrium points, because the motion of the systems cannot always be generated exactly along the gradient vector of the potential function. By utilizing the component of the input that does not increase or decrease the potential function, the equilibrium points other than the critical points of the function can be destabilized, if the controllability of the systems is satisfied with the first-order Lie brackets of input vector fields. A time-invariant controller is proposed based on the theoretical analysis on the stability of equilibrium points, and applied to an underwater vehicle among obstacles. When the potential function has saddles as its critical points, the potential function is modified to be time-varying near the saddles in order to prevent the system from being trapped in the saddles. Numerical simulation results demonstrate that the underwater vehicle with the proposed controller converges to the desired point without collision with obstacles.     

乙酸丁酯反应精馏过程模拟设计

反应精馏偶合了反应和精馏两种单元操作,通过精馏促进反应,可以提高反应转化率和收率,为可逆反应的化工过程生产提供了新的设计途径。基于严格热力学分析计算,利用计算机模拟和优化手段。提出了乙酸丁酯反应精馏、分离纯化的生产流程。采用UNIQUAC方程表征乙酸-正丁醇-乙酸丁酯-水四元非理想体系的汽液平衡,首先,根据实验数据回归了热力学模型中的交互作用参数,并预测了体系中5个共沸物组成,模型的计算结果与实际数据吻合。基于平衡级模型,提出了由平衡反应器、反应精馏塔、倾析器和纯化塔构成的可行流程,对提出的设计流程进行了模拟、优化,得到了操作工艺参数。模拟结果对工业过程的设计和改造具有一定的指导意义。     

关于神经网络的能量函数

能量函数在神经网络的研究中有着非常重要的作用,人们普遍认为：只要能量函数沿着网络的解是下降的,能量函数的导数为零的点是网络的平衡态,能量函数有下界,则网络是稳定的且网络的平衡态是能量函数的极小点,文中取反例说明上述条件下不能保证网络的稳定性,并取例说明即使网络稳定也不能保证网络的平衡态与能量函数的极小点,证明了在网络具有上述条件的能量函数的情况下网络稳定的充分必要条件是网络的解有界,讨论了网络的平