Order and delay selection for neural network modelling by identification of linearized models

An approach to selecting the order and delay for neural network modelling of nonlinear dynamic systems is proposed by identifying local linear models at points spanning the system operating range. The method is based on relationships between linear and nonlinear models and is developed for three popular nonlinear model structures; nonlinear autoregressive with exogenous inputs (NARX), NARX with a linear noise model and nonlinear autoregressive moving-average with exogenous inputs (NARMAX). Simulation results illustrate the application of the method, and the suitability of the orders and delays selected are demonstrated by nonlinear system identification using radial basis function neural networks. The method is also shown to indicate the suitability of a particular nonlinear model structure for representing a system.     

Multiple model approach to nonlinear system identification with an uncertain scheduling variable using EM algorithm

This paper deals with system identification of general nonlinear dynamical systems with an uncertain scheduling variable. A multi model approach is developed; wherein, a set of local auto regressive exogenous (ARX) models are first identified at different process operating points, and are then combined to describe the complete dynamics of a nonlinear system. An expectation-maximization (EM) algorithm is used for simultaneous identification of local ARX models, and for computing the probability associated with each of the local ARX models taking effect. A smoothing algorithm is used to estimate the distribution of the hidden scheduling variables in the EM algorithm. If the dynamics of the scheduling variables are linear, Kalman smoother is used; whereas, if the dynamics are nonlinear, sequential Monte-Carlo (SMC) method is used. Several simulation examples, including a continuous stirred tank reactor (CSTR) and a distillation column, are considered to illustrate the efficacy of the proposed method. Furthermore, to highlight the practical utility of the developed identification method, an experimental study on a pilot-scale hybrid tank system is also provided.     

Hinging hyperplane based regression tree identified by fuzzy clustering and its application

Hierarchical fuzzy modeling techniques have great advantage since model accuracy and complexity can be easily controlled thanks to the transparent model structures. A novel tool for regression tree identification is proposed based on the synergistic combination of fuzzy c-regression clustering and the concept of hierarchical modeling. In a special case (c = 2), fuzzy c-regression clustering can be used for identification of hinging hyperplane models. The proposed method recursively identifies a hinging hyperplane model that contains two linear submodels by partitioning operating region of one local linear model resulting a binary regression tree. Novel measures of model performance and complexity are developed to support the analysis and building of the proposed special model structure. Effectiveness of proposed model is demonstrated by benchmark regression datasets. Examples also demonstrate that the proposed model can effectively represent nonlinear dynamical systems. Thanks to the piecewise linear model structure the resulted regression tree can be easily utilized in model predictive control. A detailed application example related to the model predictive control of a water heater demonstrate that the proposed framework can be effectively used in modeling and control of dynamical systems.     

Model predictive control of plant transitions using a new identification technique for interpolating nonlinear models

This paper presents a model based controller design approach for plants that operate in several distinct operating regimes and make transitions between them. Often it is difficult to identify a single global model that describes plant behavior in all the regimes. In the present work we propose an identification method that builds linear models for the individual regimes, and then interpolates nonlinear models in between these local models to match plant dynamics during transitions. The identification technique is shown to work well with transition data which lack excitation. A model predictive controller based on the local and the transition models is then presented and applied to a reactor.     

液位系统的多模型广义预测控制研究

用多模型的广义预测控制器对复杂的非线性液位系统进行仿真控制。通过在覆盖工况的若干个平衡点采用最小二乘法离线辨识建立多个线性模型,形成非线性系统的多模型表示,然后对各个子模型分别设计子控制器,采用基于相对残差的方法来实现控制增量的加权以获取控制增量。通过对单容液位系统的仿真,表明该方法的有效性。     

Application of Model Predictive Control suitable for closed-loop re-identification to a polymerization reactor

Recently, a linear Model Predictive Control (MPC) suitable for closed-loop re-identification was proposed, which solves the potential conflict between the persistent excitation of the system (necessary to perform a suitable identification) and the control, and guarantees recursive feasibility and attractivity of an invariant region of the closed-loop. This approach, however, needs to be extended to account for a proper robustness to moderate-to-severe model mismatches, given that re-identifications are necessary when the system is not close to the operating point where the current linear model was identified. In this work, new results on robustness are presented, and an exhaustive application of the new MPC suitable for closed-loop re-identification to a nonlinear polymerization reactor simulator is made to explore the difficulties arising from a real life identification. Furthermore, several closed-loop re-identification are performed in order to clearly show that the proposed controller provides uncorrelated input–output data sets, which together with the guaranteed stability, constitute the main controller benefit.     

Temperature control of industrial gas phase polyethylene reactors

The performance of linear and nonlinear temperature control schemes is assessed for an open-loop unstable gas-phase polyethylene reactor (GPPER), based on speed, damping, robustness and the ability to maintain closed-loop stability in different operating regimes. An existing industrial GPPER model is improved by modelling the temperature states in the external heat exchanger using linear and nonlinear driving force models with varying numbers of heat transfer stages. Differences in heat exchanger models do not produce gain mismatch but do result in phase mismatch. It is shown that the nonlinear error trajectory controller (ETC) exhibits significantly superior responses in terms of speed, damping and robustness compared with an optimally-tuned PID controller. Therefore, substantial benefits could be realized using nonlinear controllers because they can provide good disturbance rejection capabilities and ensure closed-loop stability over a wide range of operating conditions. An approach is presented for tuning ETCs for minimum-phase processes of arbitrary relative degree.     

Dynamics identification and control of nonlinear MIMO coupled plant using supervised neural gas and comparison with recurrent neural controller

The dynamics identification and subsequent control of a nonlinear system is not a trivial issue. The application of a neural gas network that is trained with a supervised batch version of the algorithm can produce identification models in a robust way. In this paper, the neural model identifies each local transfer function, demonstrating that the local linear approximation can be done. Moreover, other parameters are analyzed in order to obtain a correct modeling. Furthermore, the algorithm is applied to control a nonlinear multi-input multi-output system composed of tanks. In addition, this plant is a coupled system where the manipulated input variables are influencing all the output variables. The aim of the work is to demonstrate that the supervised neural gas algorithm is able to obtain linear models to be used in a state space design scenario to control nonlinear coupled systems and guarantee a robust control method. The results are compared with the common approach of using a recurrent neural controller trained with a dynamic backpropagation algorithm. Regarding the steady-state errors in disturbance rejection, reference tracking and sensitivity to simple process changes, the proposed approach shows an interesting application to control nonlinear plants.

     

Evolving principal component clustering with a low run-time complexity for LRF data mapping

In this paper a new approach called evolving principal component clustering is applied to a data stream. Regions of the data described by linear models are identified. The method recursively estimates the data variance and the linear model parameters for each cluster of data. It enables good performance, robust operation, low computational complexity and simple implementation on embedded computers. The proposed approach is demonstrated on real and simulated examples from laser-range-finder data measurements. The performance, complexity and robustness are validated through a comparison with the popular split-and-merge algorithm.     

Identification of complex nonlinear processes based on fuzzy decomposition of the steady state space

A methodology for identification and control of complex nonlinear plants using multi-model approach is presented in this paper. The proposed methodology is based on fuzzy decomposition of the steady state map. It is shown that such a decomposition strategy facilitates the design of input perturbation signals and helps in identifying linear or simple nonlinear models for each local region. A composition strategy to aggregate the local model predictions is proposed and shown to give excellent cross validation as well as to facilitate smooth switching between the local models. A novel control scheme that is based on the multi model strategy is proposed. The practicality of the identification and control scheme presented here is demonstrated by application to the continuous fermenter of Henson and Seborg (M.A. Henson, D.E. Seborg, Nonlinear control strategies for continuous fermenter, in: Proceedings of 1990 American Control Conference, San Diego, 1990), which exhibits severe nonlinearities and gain directionality changes.     

Fuzzy model based adaptive control for a class of nonlinear systems

A fuzzy model based adaptive control algorithm for a class of continuous-time nonlinear dynamic systems is presented. The fuzzy model consisting of a set of linear fuzzy local models that are combined using a fuzzy inference mechanism is used to model a class of nonlinear systems. Each fuzzy local model represents a linearized model corresponding to the operating point of the controlled nonlinear system. The proposed control algorithm employs the fuzzy controller that is designed by considering the linear state feedback controller corresponding to the fuzzy local model with the maximum weight and the switching-σ modification adaptive controller to adaptively compensate for the plant nonlinearities. Stability robustness of the closed-loop system is analyzed in Lyapunov sense. It is shown, that the proposed control algorithm guarantees global stability of the system with the output of the system approaching the origin if there are no disturbances and uncertainties, converging to the neighborhood of the origin for all realizations of uncertainties and disturbances. The simulation examples for controlling inverted pendulum system are given to illustrate the effectiveness of the proposed method     

Local model network identification with Gaussian processes.

A Bayesian Gaussian process (GP) modeling approach has recently been introduced to model-based control strategies. The estimate of the variance of the predicted output is the most useful advantage of GPs in comparison to neural networks (NNs) and fuzzy models. However, the GP model is computationally demanding and nontransparent. To reduce the computation load and increase transparency, a local linear GP model network is proposed in this paper. The proposed methodology combines the local model network principle with the GP prior approach. A novel algorithm for structure determination and optimization is introduced, which is widely applicable to the training of local model networks. The modeling procedure of the local linear GP (LGP) model network is demonstrated on an example of a nonlinear laboratory scale process rig.     

An intelligent multiple models based predictive control scheme with its application to industrial tubular heat exchanger system

The purpose of this paper is to deal with a novel intelligent predictive control scheme using the multiple models strategy with its application to an industrial tubular heat exchanger system. The main idea of the strategy proposed here is to represent the operating environments of the system, which have a wide range of variation in the span of time by several local explicit linear models. In line with this strategy, the well-known linear generalized predictive control (LGPC) schemes are initially designed corresponding to each one of the linear models of the system. After that, the best model of the system and the LGPC control action are precisely identified, at each instant of time, by an intelligent decision maker scheme (IDMS), which is playing the so important role in realizing the finalized control action for the system. In such a case, as soon as each model could be identified as the best model, the adaptive algorithm is implemented on the both chosen model and the corresponding predictive control schemes. In conclusion, for having a good tracking performance, the predictive control action is instantly updated and is also applied to the system, at each instant of time. In order to demonstrate the effectiveness of the proposed approach, simulations are carried out and the results are compared with those obtained using a nonlinear GPC (NLGPC) scheme as a benchmark approach realized based on the Wiener model of the system. In agreement with these results, the validity of the proposed control scheme can tangibly be verified.     

A gap metric based multiple model approach for nonlinear switched systems

Many applications in chemical engineering often exhibit a switching character due to the presence of discrete modes in the course of their operation. First principles models of such systems constructed using process simulators are far too complex for use in online applications, especially in model-based control. For such systems, numerous control-relevant modeling approaches have been reported in the literature such as mixed logic dynamical (MLD) models [1] and piece wise affine (PWA) [2] models among others. These models describe the evolution of states in each discrete mode using linear equations. Fewer control-relevant models have been reported that address the nonlinear behavior of switched systems. To model nonlinear hybrid systems, Nandola and Bhartiya [3] proposed a multiple linear model approach wherein multiple linear models are used to describe the dynamic behavior in each mode of the hybrid system. However, no guidelines were provided to select the number of models necessary in each mode and their region of validity. In this work, we address these lacunae by presenting a systematic multiple model approach to describe nonlinear switched systems. The method involves a trajectory based linearization and employs a model bank with a set of local linear models for each discrete operational mode. The model bank is generated by linearizing the first principles model across a carefully designed trajectory based on accuracy of multi-step ahead predictions. The numerous models thus obtained are clustered using the gap metric as the distance measure and representative models are selected. The selected linear models are aggregated using Bayesian or Fuzzy approaches to obtain the global model for the nonlinear switched system. A simulation case study of spherical two-tank system and an experimental case study of a benchmark problem consisting of three tanks are used to validate the proposed modeling strategy.     

A method for high order linear system reduction and nonlinear system simplification

Efficient least-squares based algorithms are presented for reducing the complexity of a system. One algorithm is capable of reducing a high order linear system in the frequency domain while another is developed for simplifying a nonlinear system in the time domain. In both cases an optimal simplified model is obtained from the set of all admissible models of the system. In the case of linear systems, the admissible set of models are those which are formed from linear combinations of any and all of the partial fractions (coefficients in the expansion are not needed) associated with the system transfer function. In the nonlinear case, linear combinations of the nonlinear terms form the admissible set of models. In either situation, each possible model structure is then represented as a node in a tree. To avoid designing and testing all possible models a cost is assigned, based on a least square error criterion, as a measure of the reduction in error that results when a term is added to a particular model. Basically, this assigns a weight to the relative importance of each component of the model. These costs appear on the branches of the tree and are derived such that the optimal n term model is the maximum cost path from the root to a node at depth n. The above approach insures that no cumbersome matrix inversion is done and so it allows the consideration of a large number of model candidates in a short period of time. Finally, the reduction of an eighth order linear system and the simplification of a three-state nonlinear system with eight nonlinear terms are presented.     

Variable selection algorithm for the construction of MIMO operatingpoint dependent neurofuzzy networks

An input variable selection procedure is introduced for the identification and construction of multi-input multi-output (MIMO) neurofuzzy operating point dependent models. The algorithm is an extension of a forward modified Gram-Schmidt orthogonal least squares procedure for a linear model structure which is modified to accommodate nonlinear system modeling by incorporating piecewise locally linear model fitting. The proposed input nodes selection procedure effectively tackles the problem of the curse of dimensionality associated with lattice-based modeling algorithms such as radial basis function neurofuzzy networks, enabling the resulting neurofuzzy operating point dependent model to be widely applied in control and estimation. Some numerical examples are given to demonstrate the effectiveness of the proposed construction algorithm     

基于支持向量机的非线性离散动力学系统控制

提出了一种用支持向量机辨识系统状态空间模型的非线性离散动力学系统控制新方法. 在本方法中, 采用最小二乘支持向量机在每一个工作点辨识非线性系统的局部最优线性化模型. 针对该模型, 采用常规的线性控制方法在每个工作点设计局部线性控制器, 并在整个控制任务的每个工作点重复此设计过程.用该方法对两个典型的非线性离散系统采用极点配置技术进行了仿真验证, 结果显示系统对参考输入具有满意的跟踪性能, 证明该方法是有效和可行的.     

Design and analysis of gain-scheduled control using local controller networks

This paper describes a nonlinear control structure known as a local controller network. The structure consists of a weighted combination of a number of individual controllers, each of which is valid locally in the state space of the plant. Local controller designs are based upon local models valid in operating regimes which do not necessarily contain any physical equilibria. Consequently, the transient performance can be improved. Some 'scheduling' variables determine the current operating regime, and a validity function is assigned to each local controller. A 'feedforward' component may be used in each local controller in order to compensate directly for the operating-point-dependent model offsets. The application of the local controller network approach to a nonlinear control problem, that of longitudinal vehicle dynamics control, is described. A stability analysis for the discrete-time local controller network is given in this paper and the results are compared with known theoretical guidelines for related control approaches such as gain scheduling and feedback linearization.     

非线性时变系统的一种即时局部模型及参数估计算法

针对包含未知和不可测量的确定性扰动的非线性时变系统的辨识和预测,提出了一种简便实用的线性化即时局部模型,给出并证明了这种即时模型的存在性定理。为了跟踪快速变化的模型参数,利用最新的多个线性局部模型进行外推,提出了一种滚动多模型加权平均参数估计算法。仿真结果表明了这种即时局部模型和参数估计算法的可行性。     

A Sliding Mode Multiobserver Based on an Uncoupled Multimodel: An Application on a Transesterification Reaction

This paper concerns the design of robust sliding mode multiobserver for nonlinear systems. A discrete uncoupled multimodel structure is retained for the modeling of nonlinear systems. Unlike the classically used multimodel structures, the retained uncoupled multimodel is known by its flexibility of modeling, thus, the structures of the partial models are adapted to the complexity of the local models in each operating zone. Sufficient conditions are provided, in terms of linear matrix inequalities (LMIs), to ensure the asymptotic stability of the proposed sliding mode multiobserver. A convergence analysis is achieved to obtain the convergence radius. A numerical example and a real time application on a transesterification reactor are carried out to illustrate, once again, the performance of the proposed sliding mode multiobserver in terms of precision and rapidity of convergence.