Fuzzy modeling
Multiple (local) linear model based process identification and control (Local and global issues)
Fuzzy identification is an effective tool for the approximation of uncertain nonlinear dynamic systems on the basis of measured input-output data. The Takagi-Sugeno (TS) fuzzy model is often used to represent nonlinear dynamic systems, by interpolating between local linear, time-invariant (LTI) ARX models. There are two approaches to extract a linear model from the fuzzy model around a given operating point. The first approach is based on the fact that the TS model interpolates between local linear models. Hence, the extracted linear model is obtained by interpolating the parameters of the local models in the TS model. The second approach extracts the parameters of the linear model by Taylor expansion. The locally interpreted interpolated model is not identical to the model obtained by this Taylor expansion based linearization of the fuzzy model. For the identification of a fuzzy model, also two approaches can be followed. With the global approach the parameters of all rule consequences are estimated within one identification problem, yielding an optimal predictor. The local parameter estimation does not estimate all parameters simultaneously. It rather divides this task into a set of weighted least-squares problems. As it has been shown in this project, only consequent parameters obtained by local estimation (weighted least squares) can be interpreted locally and used as a Linear Parameter Varying (LPV) model in the MPC where the extracted model parameters are interpolated. Parameters obtained by global identification do not lend themselves to the LPV interpretation. Such a model is only applicable where the extracted models are obtained by Taylor-series linearization.
Interpretable Semi-Mechanistic Fuzzy Models by Clustering, OLS and FIS Model Reduction
A semi-mechanistic fuzzy modeling technique is proposed to obtain compact and transparent process models based on small data-sets. Semi-mechanistic models are hybrid models that consist of a white box structure based on mechanistic relationships and black-box substructures to model less defined parts. First, it is shown that certain type of white-box models can be efficiently incorporated into a Takagi-Sugeno fuzzy rule structure. Next, the proposed models are identified from learning data and special attention is paid to transparency and accuracy aspects. The approach is based on a combination of (i) prior knowledge-based model structures, (ii) fuzzy clustering, (iii) orthogonal least-squares, and (iv) the modified Fisher’s interclass separability method. For the identification of the semimechanistic fuzzy model, a new fuzzy clustering method is proposed, i.e., clustering is achieved by the simultaneous identification of fuzzy sets defined on some of the scheduling variables and identification of the parameters of the local semimechanistic submodels. Subsequently, model reduction is applied to make the TS models as compact as possible, i.e., the most relevant consequent variables are selected by an orthogonal least squares method, and the modified Fisher’s interclass separability criteria is used for selection of relevant antecedent (scheduling) variables. The overall procedure is demonstrated by the development of a semimechanistic model for a biochemical process. Although the results do not carry over directly to other engineering fields, the main ideas and conclusions, will certainly hold for other application areas as well.
Fuzzy modeling with multivariate membership functions: gray-box identification and control design
A novel framework for fuzzy modeling and model-based control design is described. The fuzzy model is of the Takagi-Sugeno (TS) type with constant consequents. It uses multivariate antecedent membership functions obtained by Delaunay triangulation of their characteristic points. The number and position of these points are determined by an iterative insertion algorithm. Constrained optimization is used to estimate the consequent parameters, where the constraints are based on control-relevant a priori knowledge about the modeled process. Finally, methods for control design through linearization and inversion of this model are developed. The proposed techniques are demonstrated by means of two benchmark examples: identification of the well-known Box-Jenkins gas furnace and inverse model-based control of a pH process. The obtained results are compared with results from the literature.
Hybrid fuzzy convolution model and its application in predictive control
In this paper a new method for synthesising nonlinear, control-oriented process models is presented. The proposed hybrid fuzzy convolution model (HFCM) consists of a steady-state fuzzy model and a gain-independent impulse response model. The proposed HFCM is applied in model based predictive control of a laboratory-scale electrical water-heater. Simulation and real-time studies confirm that the method is capable of controlling this delayed and distributed parameter system with a strong nonlinear feature.
Incorporating Prior Knowledge in Fuzzy Model Identification
This paper presents an algorithm for incorporating a priori knowledge into data-driven identification of dynamic fuzzy models of the Takagi-Sugeno type. Knowledge about the modelled process such as its stability, minimal or maximal static gain, or the settling time of its step response can be translated into inequality constraints on the consequent parameters. By using input-output data, optimal parameter values are then found by means of quadratic programming. The proposed approach has been applied to the identification of a laboratory liquid level process. The obtained fuzzy model has been used in model-based predictive control. Real-time control results show that, when the proposed identification algorithm is applied, not only are physically justified models obtained but also the performance of the model-based controller improves with regard to the case where no prior knowledge is involved.
Combining First Principles Models and Neural Networks for Generic Model Control
Generic Model Control (GMC) is a control algorithm capable of using non-linear process model directly. In GMC, mostly, first-principles models derived from dynamic mass, energy and momentum balances are used. When the process is not perfectly known, the unknown parts of first principles models can be represented by black-box models, e.g. by neural networks. This paper is devoted to the application of such hybrid models in GMC. It is shown that the first principles part of the hybrid model determines the dominant structure of the controller, while the black-box elements are used as state and/or disturbance estimators. The sensitivity approach is used for the identification of the neural network elements of the control-relevant hybrid model. The underlying framework is illustrated by the temperature control of a continuous stirred tank reactor (CSTR) where a neural network is used to model the heat released by an exothermic chemical reaction.
Identification and Control of Nonlinear Systems Using Fuzzy Hammerstein Models
This paper addresses the identification and control of nonlinear systems by means of Fuzzy Hammerstein (FH) models, which consist of a static fuzzy model connected in series with a linear dynamic model. For the identification of nonlinear dynamic systems with the proposed FH models, two methods are proposed. The first one is an alternating optimization algorithm that iteratively refines the estimate of the linear dynamics and the parameters of the static fuzzy model. The second method estimates the parameters of the nonlinear static model and of the linear dynamic model simultaneously by using a constrained recursive least-squares algorithm. The obtained FH model is incorporated in a model-based predictive control scheme and a new constraint-handling method is presented. A simulated water-heater process is used as an illustrative example. A comparison with an affine neural network and a linear model is given. Simulation results show that the proposed FH modeling approach is useful for modular parsimonious modeling and model-based control of nonlinear systems.
Hybrid fuzzy convolution model and its application in model predictive control
In this paper a new method for synthesising nonlinear, control-oriented process models is presented. The proposed hybrid fuzzy convolution model (HFCM) consists of a steady-state fuzzy model and a gain-independent impulse response model. The proposed HFCM is applied in model based predictive control of a laboratory-scale electrical water-heater. Simulation and real-time studies confirm that the method is capable of controlling this delayed and distributed parameter system with a strong nonlinear feature.
Hybrid convolution model and its application in predictive pH control, Computers and Chemical Engineering
This paper presents a new method for synthesising chemical process models that combines prior knowledge and fuzzy models. The hybrid convolution model consists of a fuzzy model based steady-state, and an impulse response model based dynamic part. Prior knowledge enters to the dynamic part as a resident time distribution model of the process. The proposed approach is applied in the modelling and model based control of a highly nonlinear pH process.
Predictor corrector controller using Wiener fuzzy convolution model
This paper investigates the application of hybrid fuzzy models in modelling and model based predictive control of a delayed and distributed parameter system with a nonlinear feature. The presented hybrid fuzzy convolution model consists of a nonlinear fuzzy steady-state model and an impulse response model based dynamic part. The proposed non-linear block-oriented dynamic model is applied to form a predictor corrector controller. The control of a laboratory-sized heating system is chosen as a realistic nonlinear case study for the demonstration of the control algorithm. The proposed model based controller is shown to be capable of controlling the nonlinear process that operates over wide range.
Hybrid fuzzy convolution modelling and identification of chemical process systems
This paper looks at a new method of modelling nonlinear dynamic processes, using grid-type Takagi-Sugeno fuzzy models and a priori knowledge. The proposed hybrid fuzzy convolution dynamic model consists of a non-linear fuzzy steady-state static and a gainindependent impulse response model-based dynamic part. The modelling of nonlinear pH processes is chosen as a realistic case study for demonstration of the proposed modelling approach. The off-line identified hybrid fuzzy convolution model is shown to be capable of modelling the nonlinear process and providing better multiple-step prediction than the conventional grid-type Takagi-Sugeno fuzzy model.
Identification of nonlinear systems using gaussian mixture of local models
Identification of operating regime based models of nonlinear dynamic systems is addressed. The operating regimes and the parameters of the local linear models are identified directly and simultaneously based on the Expectation Maximization (EM) identification of Gaussian Mixture Model (GMM). The proposed technique is demonstrated by means of the identification of a neutralization reaction in a continuously stirred tank reactor
Local and global identification for fuzzy model based control
There are two approaches to extract a linear model from a Takagi-Sugeno fuzzy model for model based control. The first local approach obtains the linear model by interpolating the parameters of the local models in the TS model, while the second one is based on linearization by Taylor expansion. The locally interpreted interpolated model is not identical to the model obtained by the linearization of the fuzzy model. The paper analyzes the origin of this difference with regard to the applied identification method and the application of the resulted model in model predictive control. In order to keep the analysis simple and transparent, a fuzzy model of a Hammerstein system is studied.
Extraction of local linear models from Takagi-Sugeno fuzzy model with application to model-based predictive control
Abstract: MBPC is a nice technique to control multivariable systems while dealing with constraints and certain objective. Linear MBPC (LMBPC) is currently a settled theory and can be applied straightforward for linear processes. In this paper we deal with nonlinear systems, for which linear models that can be extracted. This way a time varying linear representaion is obtained which is used in LMBPC. Di erent schemes to obtain such local linear models are assessed in the light of the achieved performance of the predictive controller. Takagi-Sugeno (TS) fuzzy models are chosen, because the model structure as local linear models can be derived from the linear rule consequences in a direct way. Keywords: TS fuzzy models, Model-based predictive control, nonlinear systems, multivariable (MIMO) systems.
