Modeling of chromatographic separation process with Wiener-MLP representation

Dynamic input–output-models have been identified for columns of an industrial sequential ion-exclusive chromatographic separation unit. Models are aimed at describing motion and form transformation of the fronts of different substances in the columns so that changes in “limit cycles” dynamics and drifts to undesired disturbed states could be observed on-line with model based simulations. The model structure has been innovated on the basis of classical Wiener representation, in which nonlinear dynamic system is described with a combination of linear Laguerre dynamics and static nonlinear mapping. The static mapping is realized here with MLP-type neural network. A separate delay model is needed for describing the movement of the front. The delay time adapts on variations of the process flow rate. Form transformation of the front is described with a dispersion model, which is smoother type Wiener-MLP model. Forward and backward Laguerre presentations are calculated with Laguerre filters. These Laguerre presentations are mapped to the output with a neural network. Dynamics of “salt” and two important compounds have been modeled on the basis of analyzed samples, which were taken in a factory experiment during normal production. A priori information about the process dynamics can be included in the dispersion model by choosing a suitable Laguerre parameter, but otherwise representativeness of the identification data determines validity of the model.     

A spatio-temporal Volterra modeling approach for a class of distributed industrial processes

It is difficult to model a distributed parameter system (DPS) due to the infinite-dimensional time/space nature and unknown nonlinear uncertainties. A low-dimensional and simple nonlinear model is often required for practical applications. In this paper, a spatio-temporal Volterra model is proposed with a series of spatio-temporal kernels for modeling unknown nonlinear DPS. To estimate these kernels, they are expanded onto spatial and temporal bases with unknown coefficients. To reduce the model dimension and parametric complexity in the spatial domain, the Karhunen–Loève (KL) method is used to find the dominant spatial bases. To reduce the parametric complexity in the temporal domain, the Laguerre polynomials are selected as temporal bases. Next, using the Galerkin method, this spatio-temporal modeling becomes a linear regression problem. Then unknown parameters can be easily estimated using the least-squares method in the temporal domain. After the time/space synthesis, the spatio-temporal Volterra model can be constructed. The convergence of parameter estimation can be guaranteed under certain conditions. This model has a low-dimensional and simple nonlinear structure, which is useful for the prediction and control of the DPS. The simulation and experiment demonstrate the effectiveness of the proposed modeling method.     

Pipelined robust M-estimate adaptive second-order Volterra filter against impulsive noise

The main limits on adaptive Volterra filters are their computational complexity in practical implementation and significant performance degradation under the impulsive noise environment. In this paper, a low-complexity pipelined robust M-estimate second-order Volterra (PRMSOV) filter is proposed to reduce the computational burdens of the Volterra filter and enhance the robustness against impulsive noise. The PRMSOV filter consists of a number of extended second-order Volterra (SOV) modules without feedback input cascaded in a chained form. To apply to the modular architecture, the modified normalized least mean M-estimate (NLMM) algorithms are derived to suppress the effect of impulsive noise on the nonlinear and linear combiner subsections, respectively. Since the SOV-NLMM modules in the PRMSOV can operate simultaneously in a pipelined parallelism fashion, they can give a significant improvement of computational efficiency and robustness against impulsive noise. The stability and convergence on nonlinear and linear combiner subsections are also analyzed with the contaminated Gaussian (CG) noise model. Simulations on nonlinear system identification and speech prediction show the proposed PRMSOV filter has better performance than the conventional SOV filter and joint process pipelined SOV (JPPSOV) filter under impulsive noise environment. The initial convergence, steady-state error, robustness and computational complexity are also better than the SOV and JPPSOV filters.     

Comparative nonlinear modeling of renal autoregulation in rats:Volterra approach versus artificial neural networks

In this paper, feedforward neural networks with two types of activation functions (sigmoidal and polynomial) are utilized for modeling the nonlinear dynamic relation between renal blood pressure and flow data, and their performance is compared to Volterra models obtained by use of the leading kernel estimation method based on Laguerre expansions. The results for the two types of artificial neural networks and the Volterra models are comparable in terms of normalized mean square error (NMSE) of the respective output prediction for independent testing data. However, the Volterra models obtained via the Laguerre expansion technique achieve this prediction NMSE with approximately half the number of free parameters relative to either neural-network model. However, both approaches are deemed effective in modeling nonlinear dynamic systems and their cooperative use is recommended in general     

Nonlinear input/output modelling

This paper gives a highly abbreviated overview of some of the key issues in empirical nonlinear modelling for chemical process applications. This task is complicated by the inherent nature of nonlinearity: the term describes a class of systems by the one feature they lack. In fact, this division — linear vs. nonlinear — suggests a ‘unity’ or ‘homogeneity’ of the class of nonlinear systems that does not exist. Consequently, this review will focus on specific sub-classes of nonlinear models that have analytically useful structural characteristics, and comparisons will be made both between these classes and with the more familiar linear models. Length limitations restrict these discussions somewhat, but it is hoped that the range of examples will be great enough to demonstrate how nonlinear model identification is both similar to and different from linear model indentification. The general conclusion of this paper is that nonlinear input/output modelling is a vitally important practical art with many unresolved issues; the principal objective of this paper is to elucidate some of these issues.     

A note on the optimal expansion of Volterra models using Laguerre functions

This work tackles the problem of expanding Volterra models using Laguerre functions. A strict global optimal solution is derived when each multidimensional kernel of the model is decomposed into a set of independent orthonormal bases, each of which parameterized by an individual Laguerre pole intended for representing the dominant dynamic of the kernel along a particular dimension. It is proved that the solution derived minimizes the upper bound of the squared norm of the error resulting from the practical truncation of the Laguerre series expansion into a finite number of functions. This is an extension of the results in Campello, Favier and Amaral [(2004). Optimal expansions of discrete-time Volterra models using Laguerre functions. Automatica, 40, 815-822.], where an optimal solution was obtained for the usual yet particular case in which a single Laguerre pole is used for expanding a given kernel along all its dimensions. It is also proved that the particular and extended solutions are equivalent to each other when the Volterra kernels are symmetric.     

Volterra models and three-layer perceptrons

This paper proposes the use of a class of feedforward artificial neural networks with polynomial activation functions (distinct for each hidden unit) for practical modeling of high-order Volterra systems. Discrete-time Volterra models (DVMs) are often used in the study of nonlinear physical and physiological systems using stimulus-response data. However, their practical use has been hindered by computational limitations that confine them to low-order nonlinearities (i.e., only estimation of low-order kernels is practically feasible). Since three-layer perceptrons (TLPs) can be used to represent input-output nonlinear mappings of arbitrary order, this paper explores the basic relations between DVMs and TLPs with tapped-delay inputs in the context of nonlinear system modeling. A variant of TLP with polynomial activation functions-termed "separable Volterra networks" (SVNs)-is found particularly useful in deriving explicit relations with DVM and in obtaining practicable models of highly nonlinear systems from stimulus-response data. The conditions under which the two approaches yield equivalent representations of the input-output relation are explored, and the feasibility of DVM estimation via equivalent SVN training using backpropagation is demonstrated by computer-simulated examples and compared with results from the Laguerre expansion technique (LET). The use of SVN models allows practicable modeling of high-order nonlinear systems, thus removing the main practical limitation of the DVM approach.     

Comments on: “Efficient robust constrained model predictive control with a time varying terminal constraint set” by Wan and Kothare

We present an algorithm that modifies the original formulation proposed in Wan and Kothare [Efficient robust constrained model predictive control with a time-varying terminal constraint set, Systems Control Lett. 48 (2003) 375–383]. The modified algorithm can be proved to be robustly stabilizing and preserves all the advantages of the original algorithm, thereby overcoming the limitation pointed out recently by Pluymers et al. [Min–max feedback MPC using a time-varying terminal constraint set and comments on “Efficient robust constrained model predictive control with a time-varying terminal constraint set”, Systems Control Lett. 54 (2005) 1143–1148].     

基于谱分析的一类截尾二阶非线性模型盲辨识的方法(英)

提出一种基于自相关谱分析的对一类截尾Volterra非线性模型-二阶非线性模型进行盲辨识的算法．理论分析和计算机仿真均显示，该算法准确通用，适合于实际工程应用．     

On Convergence of Volterra Series Expansion of a Class of Nonlinear Systems

A fundamental issue in conducting the analysis and design of a nonlinear system via Volterra series theory is how to ensure the excitation magnitude and/or model parameters will be in the appropriate range such that the nonlinear system has a convergent Volterra series expansion. To this aim, parametric convergence bounds of Volterra series expansion of nonlinear systems described by a NARX model, which can reveal under what excitation magnitude or within what parameter range a given NARX system is able to have a convergent Volterra series expansion subject to any given input signal, are investigated systematically in this paper. The existing bound results often are given as a function of the maximum input magnitude, which could be suitable for single‐tone harmonic inputs but very conservative for complicated inputs (e.g. multi‐tone or arbitrary inputs). In this study, the output response of nonlinear systems is expressed in a closed form, which is not only determined by the input magnitude but also related to the input energy or waveform. These new techniques result in more accurate bound criteria, which are not only functions of model parameters and the maximum input magnitude but also consider a factor reflecting the overall input energy or wave form. This is significant to practical applications, since the same nonlinear system could exhibit chaotic behavior subject to a simple single‐tone input but might not with respect to other different input signals (e.g. multi‐tone inputs) of the same input magnitude. The results provide useful guidance for the application of Volterra series‐based theory and methods from an engineering point of view. The Duffing equation is used as a benchmark example to show the effectiveness of the results.     

Robust stabilizing control laws for a class of second-order switched systems

For a class of second-order switched systems consisting of two linear time-invariant (LTI) subsystems, we show that the so-called conic switching law proposed previously by the present authors is robust, not only in the sense that the control law is flexible (to be explained further), but also in the sense that the Lyapunov stability (resp., Lagrange stability) properties of the switched system are preserved in the presence of certain kinds of vanishing perturbations (resp., nonvanishing perturbations). The analysis is possible since the conic switching laws always possess certain kinds of “quasi-periodic switching operations”. We also propose for a class of nonlinear second-order switched systems with time-invariant subsystems a switching control law which locally exponentially stabilizes the entire nonlinear switched system, provided that the conic switching law exponentially stabilizes the linearized switched systems (consisting of the linearization of each nonlinear subsystem). This switched control law is robust in the sense mentioned above.     

The formation of learners’ semiosphere by authentic inquiry with an integrated learning object “Young Scientist”

An integrated learning object, a web-based inquiry environment “Young Scientist” for basic school level is introduced by applying the semiosphere conception for explaining learning processes. The study focused on the development of students’ (n = 30) awareness of the affordances of learning objects (LO) during the 3 inquiry tasks, and their ability of dynamically reconstructing meanings in the inquiry subtasks through exploiting these LO affordances in “Young Scientist”. The problem-solving data recorded by the inquiry system and the awareness questionnaire served as the data-collection methods.It was demonstrated that learners obtain complete awareness of the LO affordances in an integrated learning environment only after several problem-solving tasks. It was assumed that the perceived task-related properties and functions of LOs depend on students’ interrelations with LOs in specific learning contexts. Learners’ overall awareness of certain LO affordances, available in the inquiry system “Young Scientist”, developed with three kinds of patterns, describing the hierarchical development of the semiosphere model for learners. The better understanding of the LO affordances, characteristic to the formation of the functioning semiosphere, was significantly related to the advanced knowledge construction during these inquiry subtasks that presumed translation of information from one semiotic system to another. The implications of the research are discussed in the frames of the development of new contextual gateways for learning with virtual objects. It is assumed that effective LO-based learning has to be organized through pedagogically constrained gateways by manifesting certain LO affordances in the context in order to build up the dynamic semiosphere model for learners.     

Visualizing “typical” and “exotic” Internet traffic data

The threat of cyber attacks motivates the need to monitor Internet traffic data for potentially abnormal behavior. Due to the enormous volumes of such data, statistical process monitoring tools, such as those traditionally used on data in the product manufacturing arena, are inadequate. “Exotic” data may indicate a potential attack; detecting such data requires a characterization of “typical” data. We devise some new graphical displays, including a “skyline plot,” that permit ready visual identification of unusual Internet traffic patterns in “streaming” data, and use appropriate statistical measures to help identify potential cyberattacks. These methods are illustrated on a moderate-sized data set (135,605 records) collected at George Mason University.     

二阶Volterra变记忆长度LMP算法

针对实际应用中非线性系统记忆长度未知致使Volterra自适应滤波器可能无法达到最优性能的问题,提出一种二阶Volterra变记忆长度LMP算法。利用Volterra滤波器二阶权系数矩阵的对称性和对称矩阵可对角化分解性质,推导得到了一阶权系数与二阶权系数个数相同的信号矢量与权系数矢量内积的二阶Volterra滤波器输出信号表达式;提出了基于DCT的二阶Volterra自适应滤波器（CSVF）及其LMP算法（CSVLMP）;采用FIR抽头长度的自适应调整思想,提出了基于DCT的二阶Volterra变记忆长度LMP算法（CSVMLMP）。记忆长度未知的非线性系统辨识的仿真结果表明,在[α]稳定分布噪声背景下,该算法在收敛速度、稳态性能和计算复杂度之间达到了较好的折中。     

Nonlinear Laguerre–Volterra observer‐controller and its application to process control

By expanding each kernel using the orthonormal Laguerre series, a Volterra functional series is used to represent the input/output relation of a nonlinear dynamic system. With the feedback of the modeling error, we design a novel nonlinear state observer, based on which an output feedback controller is derived for both the stabilization and tracking problems. The stability of the closed‐loop system is analyzed theoretically. The algorithm is effectively applied on the continuous stirring tank reactor and chemical reactor temperature control system. Copyright © 2009 John Wiley & Sons, Ltd.     

A “mixed” small gain and passivity theorem in the frequency domain

We show that the negative feedback interconnection of two causal, stable, linear time-invariant systems, with a “mixed” small gain and passivity property, is guaranteed to be finite-gain stable. This “mixed” small gain and passivity property refers to the characteristic that, at a particular frequency, systems in the feedback interconnection are either both “input and output strictly passive”; or both have “gain less than one”; or are both “input and output strictly passive” and simultaneously both have “gain less than one”. The “mixed” small gain and passivity property is described mathematically using the notion of dissipativity of systems, and finite-gain stability of the interconnection is proven via a stability result for dissipative interconnected systems.     

The characteristics of a “stop–flow” mode of sensor array operation using data with the best classification performance

The paper presents an approach to characterizing a “stop–flow” mode of sensor array operation. The considered operation mode involves three successive phases of sensors exposure: flow (in a stream of measured gas), stop (in zero flow conditions) and recovery (in a stream of pure air). The mode was characterized by describing the distribution of information, which is relevant for classification of measured gases in the response of sensor array. The input data for classifier were the sets of sensors output values, acquired in discrete time moments of the measurement. Discriminant Function Analysis was used for data analysis. Organic vapours of ethanol, acetic acid and ethyl acetate in air were measured and classified. Our attention was focused on data sets which allowed for 100% efficient recognition of analytes. The number, size and composition of those data sets were examined versus time of sensor array response. This methodology allowed to observe the distribution of classification-relevant information in the response of sensor array obtained in “stop–flow” mode. Hence, it provided for the characterization of this mode.     

Control relevant model reduction of Volterra series models

This paper presents a two-step method for control-relevant model reduction of Volterra series models. First, using the nonlinear IMC design as a basis, an explicit expression relating the closed-loop performance to the open-loop modeling error is obtained. Secondly, an optimization problem that seeks to minimize the closed-loop error subject to the restriction of a reduced-order model is posed. By showing that model reduction of kernels with different degrees can be decoupled in the problem formulation, the optimization problem is simplified into a mathematically more convenient form which can be solved with significantly less computational effort. The effectiveness of the proposed method is illustrated on a polymerization reactor example where a second-order Volterra model with 85 parameters is reduced to a Hammerstein model with 3 parameters. Despite the lower ‘open-loop’ predictive ability of the control-relevant model, the closed-loop performance of the reduced-order control system closely mimics that of the full order model.     

Computationally efficient nonlinear Min–Max Model Predictive Control based on Volterra series models—Application to a pilot plant

The mathematical model used in Min–Max MPC (MMMPC) to predict the future trajectory of the system explicitly considers disturbances and uncertainties. Based on the future trajectory, the control sequence is computed minimizing the worst case cost with respect to all possible trajectories of the disturbances and uncertainties. This approach leads to a more robust control performance but also complicates the practical implementation of MMMPC due to the high computational burden required to solve the optimization problem. This computational burden is even worse if a nonlinear prediction model is used. In fact, to the best of the authors’ knowledge, there have not yet been reported any applications of nonlinear MMMPC to real processes. In this paper a nonlinear MMMPC strategy based on a second order Volterra series model is presented. The particular structure of the used prediction model allows to obtain an explicit formulation of the worst case cost and its computation in polynomial time. Real time applications with typical prediction and control horizons are possible because of the reduced complexity of the proposed control strategy. Furthermore, input-to-state practical stability for the proposed control strategy is guaranteed under certain conditions. The MMMPC strategy is implemented and validated in experiments with a continuous stirred tank reactor whose temperature dynamics are approximated by a second order Volterra series model. The control performance of the proposed MMMPC strategy is illustrated by the obtained experimental results.     

Feedforward based transient control in solid oxide fuel cells

In SOFCs, transient control of fuel utilization is achievable via input-shaping. In this paper, the approach is generalized to a feedforward control problem for second-order LTI systems with two inputs and one output. One is a measurable, time-varying, exogenous input and the other is a control input. The problem studied is exact tracking of a constant reference using the plant's DC gain vector. The problem considers plant models that can be divided into known and unknown parts, and where feedback is unavailable. Although SOFCs have nonlinear dynamics, the linear abstraction nevertheless helps predict the observed effectiveness of input-shaping.