Batch process modelling with mixtures of Gaussian processes

In recent years Gaussian processes have attracted a significant amount of interest with the particular focus being that of process modelling. This has primarily been a consequence of their good predictive performance and inherent analytical properties. Gaussian processes are a member of the family of non-parametric Bayesian regression models and can be derived from the perspective of neural networks. Their behaviour is controlled through the structure of the covariance function. However, when applied to batch processes, whose data exhibits different variance structures throughout the duration of the batch, a single Gaussian process may not be appropriate for the accurate modelling of its behaviour. Furthermore there are issues with respect to the computational costs of Gaussian processes. The implementation of a Gaussian process model requires the repeated computation of a matrix inverse whose order is the cubic of the number of training data points. This renders the algorithm impractical when dealing with large data sets. To address these two issues, a mixture model of Gaussian processes is proposed. The resulting prediction is attained as a weighted sum of the outputs from each Gaussian process component, with the weights determined by a Gaussian kernel gating network. The model is implemented through a Bayesian approach utilising Markov chain Monte Carlo algorithms. The proposed methodology is applied to data from a bench-mark batch simulation polymerization process, methyl methacrylate (MMA), and the results are compared with those from a single Gaussian process to illustrate the advantages of the proposed mixture model approach.     

Multi-model multivariate Gaussian process modelling with correlated noises

A composite multiple-model approach based on multivariate Gaussian process regression (MGPR) with correlated noises is proposed in this paper. In complex industrial processes, observation noises of multiple response variables can be correlated with each other and process is nonlinear. In order to model the multivariate nonlinear processes with correlated noises, a dependent multivariate Gaussian process regression (DMGPR) model is developed in this paper. The covariance functions of this DMGPR model are formulated by considering the “between-data” correlation, the “between-output” correlation, and the correlation between noise variables. Further, owing to the complexity of nonlinear systems as well as possible multiple-mode operation of the industrial processes, to improve the performance of the proposed DMGPR model, this paper proposes a composite multiple-model DMGPR approach based on the Gaussian Mixture Model algorithm (GMM-DMGPR). The proposed modelling approach utilizes the weights of all the samples belonging to each sub-DMGPR model which are evaluated by utilizing the GMM algorithm when estimating model parameters through expectation and maximization (EM) algorithm. The effectiveness of the proposed GMM-DMGPR approach is demonstrated by two numerical examples and a three-level drawing process of Carbon fiber production.     

Adaptive linear finite-element method for modelling nonlinear dynamic systems

This paper proposes a method of identifying nonlinear dynamic models with observation data (or a training data set) which exhibits a simple structure, adaptive input-space partition and fast convergence. The method employs the multiscale approximation concepts which have been introduced in numerical analysis motivated by wavelet analysis concepts. The partition or equivalently scaled basis functions are determined and selected adaptively in a sequenced and ordered manner. This method may be also considered as a single-layer neural network but with adaptive neural neurons. The number of multiscale basis functions required depends on the degree of nonlinearity of the system being modelled. The method is compared with the cerebellar model with interpolation (CEINT) and the cerebellar model articulation control (CMAC) methods and has been shown to achieve comparative modelling accuracies but with a reduced memory space and a concomitantly reduced training set.     

An ontological process modelling framework for stochastic systems

Most process modelling techniques exist without any firm theoretical foundation. This results in a lack of model validation, which can be in-terms of model consistency, feasibility and goal compliance. Moreover, these techniques are mostly deterministic in nature and not applicable to stochastic systems. In this paper, we propose an ontology-based stochastic process modelling framework that further provides a specialization to failure and reliability issues. The framework is notation independent, and is primarily rooted in Bunge’s ontology. The well-established theory of reliability constructs are also mapped to facilitate the modelling of failure prone systems.     

非线性离散系统的伴随辨识法

对非线性离散系统辨识的研究,通常是将其转化成一个非线性优化问题.为此需要计算目标函数对参数向量的梯度,以往的方法需要求解一个矩阵差分方程,计算量颇大.依据系统输出量测值来确定含在系统中的未知参数向量,首先引进伴随状态向量,代替矩阵差分方程求解的是计算一个向量差分方程,从而大大简化计算,然后将这种梯度计算方法结合进拟牛顿信赖域法中.最后给出了应用此方法的一个实际例子,数值仿真的结果说明方法是有效的.     

A Gaussian approximation recursive filter for nonlinear systems with correlated noises

This paper proposes a Gaussian approximation recursive filter (GASF) for a class of nonlinear stochastic systems in the case that the process and measurement noises are correlated with each other. Through presenting the Gaussian approximations about the two-step state posterior predictive probability density function (PDF) and the one-step measurement posterior predictive PDF, a general GASF framework in the minimum mean square error (MMSE) sense is derived. Based on the framework, the GASF implementation is transformed into computing the multi-dimensional integrals, which is solved by developing a new divided difference filter (DDF) with correlated noises. Simulation results demonstrate the superior performance of the proposed DDF as compared to the standard DDF, the existing UKF and EKF with correlated noises.     

Convex aggregative modelling of infinite memory nonlinear systems

The convex aggregation technique is applied for modelling general class of nonlinear systems with unknown structure and infinite memory. The finite sample size properties of the algorithm are formally established and compared to the standard least-squares counterpart of the method. The proposed algorithm demonstrates its advantages when the a-priori knowledge and the measurement data are both scarce, that is, when the information about the actual system structure is unknown or uncertain and the measurement set is small and disturbed by a noise. Numerical experiments illustrate application and practical benefits of the method for various nonlinear systems.     

A flexible,object-centric approach for business process modelling

Mainstream business process modelling techniques often promote a design paradigm wherein the activities that may be performed within a case, together with their usual execution order, form the backbone on top of which other aspects are anchored. This Fordist paradigm, while effective in standardised and production-oriented domains, breaks when confronted with processes in which case-by-case variations and exceptions are the norm. We contend that the effective design of flexible processes calls for a substantially different modelling paradigm. Motivated by requirements from the human services domain, we explore the hypothesis that a framework consisting of a small set of coordination concepts, combined with established object-oriented modelling principles, provides a suitable foundation for designing highly flexible processes. Several human service delivery processes have been designed using this framework, and the resulting models have been used to realise a system to support these processes in a pilot environment.     

Parametric modelling approach using bacterial foraging algorithms for modelling of flexible manipulator systems

This paper presents current work on biologically-inspired optimisation techniques based on bacterial foraging algorithms (BFAs) and their application to modelling of a single-link flexible manipulator. The objective of this work is to develop a single-link flexible manipulator model based on modified BFAs. First, three adaptation mechanisms of the chemotactic step size mechanism of BFA are proposed. New approaches of adaptable chemotactic step size are based on linear, quadratic and exponential functions of cost function value. Then, these three adaptive BFAs are used to develop three single-input single-output models to characterise a flexible manipulator from torque input to hub-angle, hub velocity and end-point acceleration responses. The performances of the adaptive BFAs are compared to that of standard BFA based on convergence to optimum value, the optimum value achieved and time-domain and frequency domain responses of the developed models.     

Multi-objective nonlinear predictive control of process systems: A dual-mode tracking control approach

There typically exist different and often conflicting control objectives, e.g., reference tracking, robustness and economic performance, in many chemical processes. The current work considers the multi-objective control problems of continuous-time nonlinear systems subject to state and input constraints and multiple conflicting objectives. We propose a new multi-objective nonlinear model predictive control (NMPC) design within the dual-mode paradigm, which guarantees stability and constraint satisfaction. The notions of utopia point and compromise solution are used to reconcile the confliction of the multiple objectives. The designed controller minimizes the distance of its cost vector to a vector of independently minimized objectives, i.e., the steady-state utopia point. Recursive feasibility is established via a particular terminal region formulation while stabilizing the closed-loop system to the compromise solution via the dual-mode control principle. In order to derive the terminal region as large as possible, a terminal control law with free-parameters is constructed by using the control Lyapunov functions (CLFs) technique. Two examples of multi-objective control of a CSTR and a free-radical polymerization process are used to illustrate the effectiveness of the new multi-objective NMPC and to compare their performance.     

An efficient chemical systems modelling approach

Systems of stiff chemical reactions are often associated with atmospheric chemistry modelling, which plays a very important role in the studies of stratospheric ozone depletion, tropospheric air pollution problems, and future chemistry-climate feedbacks and interactions. This paper revisits an open-source stiff system solver SVODE and presents its efficient use in modelling different levels of complexity of a range of chemical systems. The chemical systems discussed here are the Lotka–Volterra (predator–prey) model, the Brusselator model, the Oregonator model, and the Lorenz model. The first two models consist of two variables, while the remaining two models consist of three variables. Finally, an application of this modelling approach to a generalised organic/NOx mechanism for characterising air pollution development is presented. Since the SVODE is an open-source code, and the simulations were run on a Linux PC (with g77 compiler), all results discussed in this paper can be easily reproduced. Most importantly, the approach shown here can be readily extended to other larger scale applications such as the three-dimensional air pollution modelling.     

An effective trust-region-based approach for symmetric nonlinear systems

This paper presents a new trust-region procedure for solving symmetric nonlinear systems of equations having several variables. The proposed approach takes advantage of the combination of both an effective adaptive trust-region radius and a non-monotone strategy. It is believed that the selection of an appropriate adaptive radius and the application of a suitable non-monotone strategy can improve the efficiency and robustness of the trust-region framework as well as decrease the computational costs of the algorithm by decreasing the required number of subproblems to be solved. The global convergence and the quadratic convergence of the proposed approach are proved without the non-degeneracy assumption of the exact Jacobian. The preliminary numerical results of the proposed algorithm indicating the promising behaviour of the new procedure for solving nonlinear systems are also reported.     

Geometric estimation of nonlinear process systems

An estimation approach to jointly design the estimation model, data assimilation structure, and algorithm in the light of particular estimation objectives is developed within a constructive framework. On the basis of the detectability property which underlies the Lie derivative-based geometric estimator (GE) in conjunction with singular perturbation and robust stability tools, the GE is redesigned to remove its Lie derivation applicability obstacle. Then, the equivalence between the GE and the extended Kalman filter (EKF) is established, and the GE is endowed with uncertainty assessment capability. The resulting GE has: (i) a simple construction in terms of model Jacobians, (ii) a nonlocal convergence criterion coupled with easy to apply tuning guidelines, and (iii) the model and its detectability structure as key design degrees of freedom. The proposed methodology is illustrated and tested with an experimental binary distillation column.     

A robust fault detection approach for nonlinear systems

In this paper, we study the robust fault detection problem of nonlinear systems. Based on the Lyapunov method, a robust fault detection approach for a general class of nonlinear systems is proposed. A nonlinear observer is first provided, and a sufficient condition is given to make the observer locally stable. Then, a practical algorithm is presented to facilitate the realization of the proposed observer for robust fault detection. Finally, a numerical example is provided to show the effectiveness of the proposed approach.     

A neural differential evolution identification approach to nonlinear systems and modelling of shape memory alloy actuator

This paper proposes a hybrid modified differential evolution plus back‐propagation (MDE‐BP) algorithm to optimize the weights of the neural network model. In implementing the proposed training algorithm, the mutation phase of the differential evolution (DE) is modified by combining two mutation strategies rand/1 and best/1 to create trial vectors instead of only using one mutation operator or rand/1 or best/1 as the standard DE. The modification aims to balance the global exploration and local exploitation capacities of the algorithm in order to find potential global optimum solutions. Then the local searching ability of the back‐propagation (BP) algorithm is applied in that region so as to swiftly converge to the optimum solution. The performance and efficiency of the proposed method is tested by identifying some benchmark nonlinear systems and modeling the shape memory alloy actuator. The proposed training algorithm is compared with the other algorithms, such as the traditional DE and BP algorithm. As a result, the proposed method can improve the accuracy of the identification process.     

A practical approach for nonlinear analysis of tensegrity systems

Tensegrity systems are lightweight structures composed of cables and struts. The nonlinear behavior of tensegrity systems is critical; therefore, the design of these types of structures is relatively complex. In the present study, a practical and efficient approach for geometrical nonlinear analysis of tensegrity systems is proposed. The approach is based on the point iterative method. Static equilibrium equations are given in nodes for subsystems, thus the maximum unknown displacement number in each step is three. Pre-stress forces in the system are taken into account in a tangent stiffness matrix, while similar calculations are carried out for each node in the system which has a minimum of one degree of freedom. In each iteration step, the values found in previous steps are used. When it reaches permissible tolerance of calculation, final displacements and internal forces are obtained. The structural behavior of the tensegrity systems were evaluated by the proposed method. The results show that the method can be used effectively for tensegrity systems.     

Estimation-based approach for real-time optimisation of uncertain nonlinear systems

In this paper, the minimisation of an unknown but measurable cost function with uncertain dynamics is considered. The drift term of the uncertain dynamical system and the gradient of the objective function are treated as unknown time-varying parameters. A novel estimation scheme based on almost invariant manifolds is proposed to estimate the time-varying parameters. A direct gradient-based adaptive extremum-seeking controller is designed to solve the uncertain optimisation problem. This approach is shown to improve the transient performance of real-time optimisation control systems.     

Gaussian process approach to spiking neurons for inhomogeneous Poisson inputs.

This article presents a new theoretical framework to consider the dynamics of a stochastic spiking neuron model with general membrane response to input spike. We assume that the input spikes obey an inhomogeneous Poisson process. The stochastic process of the membrane potential then becomes a gaussian process. When a general type of the membrane response is assumed, the stochastic process becomes a Markov-gaussian process. We present a calculation method for the membrane potential density and the firing probability density. Our new formulation is the extension of the existing formulation based on diffusion approximation. Although the single Markov assumption of the diffusion approximation simplifies the stochastic process analysis, the calculation is inaccurate when the stochastic process involves a multiple Markov property. We find that the variation of the shape of the membrane response, which has often been ignored in existing stochastic process studies, significantly affects the firing probability. Our approach can consider the reset effect, which has been difficult to deal with by analysis based on the first passage time density.