H.B. Verbruggen

Rule-based modeling: precision and transparency

This article is a reaction to recent publications on rule-based modeling using fuzzy set theory and fuzzy logic. The interest in fuzzy systems has recently shifted from the seminal ideas about complexity reduction toward data-driven construction of fuzzy systems. Many algorithms have been introduced that aim at numerical approximation of functions by rules, but pay little attention to the interpretability of the resulting rule base. We show that fuzzy rule-based models acquired from measurements can be both accurate and transparent by using a low number of rules. The rules are generated by product-space clustering and describe the system in terms of the characteristic local behavior of the system in regions identified by the clustering algorithm. The fuzzy transition between rules makes it possible to achieve precision along with a good qualitative description in linguistic terms. The latter is useful for expert evaluation, rule-base maintenance, operator training, control systems design, user interfacing, etc. We demonstrate the approach on a modeling problem from a recently published article.

An overview of fuzzy modeling for control

Abstract In this article some aspects of fuzzy modeling are discussed in connection with nonlinear system identification and control design. Methods for constructing fuzzy models from process data are reviewed, and attention is paid to the choice of a suitable fuzzy model structure for the identification task. Some approaches to control design based on a fuzzy model are outlined.

Neuro-fuzzy methods for nonlinear system identification

Most processes in industry are characterized by nonlinear and time-varying behavior. Nonlinear system identification is becoming an important tool which can be used to improve control performance and achieve robust fault-tolerant behavior. Among the different nonlinear identification techniques, methods based on neuro-fuzzy models are gradually becoming established not only in the academia but also in industrial applications. Neuro-fuzzy modeling can be regarded as a gray-box technique on the boundary between neural networks and qualitative fuzzy models. The tools for building neuro-fuzzy models are based on combinations of algorithms from the fields of neural networks, pattern recognition and regression analysis. In this paper, an overview of neuro-fuzzy modeling methods for nonlinear system identification is given, with an emphasis on the tradeoff between accuracy and interpretability.

Fuzzy predictive control applied to an air-conditioning system

Abstract A method of designing a nonlinear predictive controller based on a fuzzy model of the process is presented. The Takagi-Sugeno fuzzy model is used as a powerful structure for representing nonlinear dynamic systems. An identification technique which enables the acquisition of the fuzzy model from process measurements is described. The fuzzy model is incorporated as a predictor in a nonlinear model-based predictive controller, using the internal model control scheme to compensate for disturbances and modeling errors. Since the model is nonlinear, a non-convex optimization problem must be solved at each sampling period. An optimization approach is proposed, that alleviates the computational burden of iterative optimization techniques, by using a combination of a branch-and-bound search technique, applied in a discretized space of the control variable, with an inverted fuzzy model of the process. The algorithm is applied to temperature control in air-conditioning system. Comparisons with a nonlinear predictive control scheme based on iterative numerical optimization show that the proposed method requires fewer computations and achieves better performance. Real-time control results are presented.

Fuzzy model-based predictive control using Takagi–Sugeno models

Abstract Nonlinear model-based predictive control (MBPC) in multi-input multi-output (MIMO) process control is attractive for industry. However, two main problems need to be considered: (i) obtaining a good nonlinear model of the process, and (ii) applying the model for control purposes. In this paper, recent work focusing on the use of Takagi–Sugeno fuzzy models in combination with MBPC is described. First, the fuzzy model-identification of MIMO processes is given. The process model is derived from input–output data by means of product-space fuzzy clustering. The MIMO model is represented as a set of coupled multi-input, single-output (MISO) models. Next, the Takagi–Sugeno fuzzy model is used in combination with MBPC. The critical element in nonlinear MBPC is the optimization routine which is nonconvex and thus difficult to solve. Two methods to deal with this problem are developed: (i) a branch-and-bound method with iterative grid-size reduction, and (ii) control based on a local linear model. Both methods have been tested and evaluated with a simulated laboratory setup for a MIMO liquid level process with two inputs and four outputs.

Effective optimization for fuzzy model predictive control

This paper addresses the optimization in fuzzy model predictive control. When the prediction model is a nonlinear fuzzy model, nonconvex, time-consuming optimization is necessary, with no guarantee of finding an optimal solution. A possible way around this problem is to linearize the fuzzy model at the current operating point and use linear predictive control (i.e., quadratic programming). For long-range predictive control, however, the influence of the linearization error may significantly deteriorate the performance. In our approach, this is remedied by linearizing the fuzzy model along the predicted input and output trajectories. One can further improve the model prediction by iteratively applying the optimized control sequence to the fuzzy model and linearizing along the so obtained simulated trajectories. Four different methods for the construction of the optimization problem are proposed, making difference between the cases when a single linear model or a set of linear models are used. By choosing an appropriate method, the user can achieve a desired tradeoff between the control performance and the computational load. The proposed techniques have been tested and evaluated using two simulated industrial benchmarks: pH control in a continuous stirred tank reactor and a high-purity distillation column.

Model-based predictive control for Hammerstein?Wiener systems

In this paper a model-based predictive control (MPC) algorithm is presented for Hammerstein?Wiener systems. This type of system consists of a linear dynamic block preceded and followed by a static non-linear block. These systems appear useful in modelling several non-linear processes encountered in industry. Directly using such a model in a MPC algorithm will in general lead to a non-linear optimization problem due to the static non-linearities. This can be avoided by exploiting the structure of these models. In this paper the non-linearities are transformed into polytopic descriptions. This procedure enables one to use robust linear MPC techniques for controlling these systems. In such a way a convex optimization problem is retained. For the presented MPC algorithm, which is stated as an optimization problem subject to linear matrix inequalities, nominal closed loop stability is proven. In two examples it is shown that by means of transforming the non-linearities into polytopic descriptions, as done in the presented MPC algorithm, a better tuning of the input?output behaviour of the plant is obtained, compared to removing the static non-linearities from the control problem by an inversion, a technique often used for these systems.

Genetic algorithms for optimization in predictive control

Abstract Genetic algorithms (GAs) are optimization methods inspired by natural biological evolution. GAs have been successfully applied to a variety of complex optimization problems where other techniques have often failed. The aim of this paper is to investigate the use of GAs for optimization in nonlinear model-based predictive control. Advanced genetic operators and other new features are introduced to increase the efficiency of the genetic search. In order to deal with real-time constraints, termination conditions are proposed to abort the evolution, once a defined level of optimality is reached. Simulated pressure dynamics of a batch fermenter are considered as an example of a highly nonlinear system. Simulation results with GAs are compared with the branch-and-bound method, in terms of the control accuracy and computational costs achieved.

Constructing fuzzy models by product space clustering

There are several different approaches to modeling of complex nonlinear systems. The main distinction can be made between global and local methods. Global methods describe the system under study using nonlinear functional relationships between the system’s variables. Examples are nonlinear state space models or input-output black-box models such as the popular NARX (Nonlinear AutoRegressive with eXogenous input) structure used often in connection with neural or wavelet networks. Local approaches, on the other hand, attempt to cope with complexity and nonlinearity of systems by decomposing the modeling problem into a number of simpler, in most cases, linear sub-problems (Johansen and Foss, 1993; Banerjee et al., 1995). These methods are conceptually simple and intuitively appealing, as they are close to the way human solve problems. Local models are usually more easily interpretable than complicated global models.

Fuzzy gain scheduling: controller and observer design based on Lyapunov method and convex optimization

Addresses model-based fuzzy control. A constructive and automated method for the design of a gain-scheduling controller is presented. Based on a given Takagi-Sugeno fuzzy model of the plant, the controller is designed such that stability and prescribed performance of the closed loop are guaranteed. These properties are valid in a wide working range around an equilibrium without restrictions to slowly varying trajectories. The synthesis is based on linear matrix inequalities and convex optimization techniques. If required, a fuzzy state estimator and an extended controller can be included, providing a zero steady-state error in the presence of disturbances and modeling errors. The proposed method has been applied to a control of a laboratory liquid-level process. Hence, the performance has been evaluated in simulations as well as in real-time control.