Analysis and design of fuzzy control systems with random delays using invariant cones

This paper addresses the analysis and design of fuzzy control systems for a class of complex uncertain single input single output nonlinear systems. In the design procedure, we represent the nonlinear system using a Takagi-Sugeno fuzzy model and construct a global fuzzy logic controller by blending all local state feedback controllers with a sliding mode controller. Unlike the commonly used parallel distributed compensation technique, we can design a global stable fuzzy controller without finding a common Lyapunov function for all local control systems, and can obtain a good tracking performance. Furthermore, a stability analysis is carried out not for the fuzzy model but for a real nonlinear system with uncertainties. One simulation example is given to show the effectiveness and feasibility of the proposed fuzzy controller design method.

Over the past two decades, there has been a rapidly growing interest in approximating a nonlinear system by a Takagi-Sugeno (T-S) fuzzy model (Takagi & Sugeno, 1985). In general, this model is represented by using a set of fuzzy rules to describe a global nonlinear system in terms of a set of local linear models which are smoothly connected by fuzzy membership functions. Based on the T-S fuzzy model, recently, various fuzzy controllers have been developed under the so-called parallel-distributed compensation (PDC) scheme (in which each control rule is distributively designed for the corresponding rule of a T-S fuzzy model) and have been widely and successfully applied in fields ranging from aerospace to process control. The reason is because the fuzzy model-based control method provides a natural, simple and effective design approach to complement other nonlinear control techniques that require special and rather involved knowledge. Below are listed the main features of T-S model-based fuzzy control method: 1. It does not require severe structural assumptions on the plant model. 2. It preserves well-understood linear intuition. 3. It is naturally compatible with decompositions of the overall control problem. The decompositions are typically not hierarchical, and the interactions of sub-problems are captured by physical variables that are typically state variables in a more complete model of the overall system. 4. It enables control systems to respond rapidly to changing operating conditions. For this reason, it is important that the selected physical variables reflect changes in plant dynamics as operating conditions change. In fact, the T-S model-based fuzzy control method (of divide and conquer type) constructs a nonlinear controller, with certain required dynamic properties, by combining, in some sense, the members of appropriate family of linear time-invariant controllers. Here, nonlinear control design task is broken into a number of linear sub-problems, which enable linear design methods to be applied to nonlinear problems. Within the general framework of the T-S fuzzy model-based control method, a flurry of research activities have quickly yielded many important results on the design of fuzzy control systems by means of the following Lyapunov function approaches: 1. Common quadratic Lyapunov function approach (Tanaka & Sugeno, 1992; Tanaka et al, 1996; Wang et al, 1996; Cao & Frank, 2000; Assawinchaichote, 2004).

In this second part of the paper on the analysis and design of uncertain fuzzy control systems, we present a controller design method for a class of uncertain fuzzy control systems. This class of systems can be represented by the discrete-time uncertain fuzzy models discussed in Part I. Since this model representation considers system uncertainties and system disturbance, the controller design problem for this kind of model becomes a H/sub /spl infin// control problem for a nonlinear uncertain system. A new stability analysis and control system design approach are then developed for this kind of uncertain fuzzy model. Furthermore, a constructive algorithm is developed to obtain the H/sub /spl infin// feedback control law.

In the past several years, research results on type- 2 fuzzy control have started to emerge in the literature. None of them, however, are concerned with mathematical input-output relationship of a type-2 fuzzy controller. Establishing such relationship is important as it will deepen our understanding of type- 2 fuzzy controller structures, how they function, and their advantages and drawbacks as compared to type-1 fuzzy controllers and conventional controllers (e.g., the PID controller). The availability of such structure will also lay a foundation for more rigorous theoretical analysis and design of type-2 fuzzy control systems. In this paper, we derive the mathematical input-output relation of a type-2 fuzzy PD (or PI) controller that has one of the simplest possible configurations - only two interval type-2 fuzzy sets for each input variables, four fuzzy rules with Zadeh AND operator, and four interval type-2 output fuzzy sets. We relate the resulting structure to conventional control theory and show that it actually is a PD (or PI) controller with the proportional and derivative (or integral) gains changing as the input variable values vary.

The central point in designing a fuzzy system for modeling and control is the generation of a set of efficient and effective fuzzy rules. Generally, fuzzy rules can be generated either from heuristics or from experimental data. Heuristic fuzzy rule generation, as discussed in Chapter 1, consists of the following main steps: Determination of the structure of the fuzzy systems. This includes mainly the selection of the inputs and outputs. Usually, a multi-input multi-output fuzzy systems can be decomposed into a number of multi-input single-output fuzzy systems. Definition of linguistic terms and their fuzzy membership function for each fuzzy variables. If a Mamdani fuzzy system is designed, each input and output can be seen as a linguistic variable. Therefore, it is necessary to define the linguistic terms (e.g., Small, Medium, Large etc.) and their fuzzy membership function. Details about the definition of a linguistic variable is presented in Section 1.2.1. Determination of fuzzy inference method. For example, for fuzzy mapping rules, the fuzzy Cartesian product can be used to define the fuzzy relation. Refer to Section 1.2.2. Defuzzification. If the Mamdani type fuzzy rules are used, the fuzzy output derived from the fuzzy rule base should be defuzzified. Refer also to Section 1.2.2 for different defuzzification methods. If the Takagi-Sugeno fuzzy rules are used, the defuzzification is included in the fuzzy inference and a crisp output is obtained directly.

The paper discusses the controller design for a class of fuzzy systems. A new method is developed for the design of fuzzy control systems. The basic idea of this new method is to represent the fuzzy system in a family of local state space models, and then to construct a global controller by considering each local state feedback controller. It is shown that this controller guarantees global stability of the closed loop fuzzy system. The issue of a state observer for the fuzzy system is also addressed. A numerical example is given for demonstration of the performance of the method. >

This paper presents a method for fuzzy model reference adaptive control system design for a partially known multi-input multi-output (MIMO) linear dynamic plant. Differential or convolutional equations can describe the plant to be controlled, and the fuzzy controller is assumed to be from the MIMO sector-bounded nonstationary nonlinearities class. The task is to obtain robust adaptive stabilization. Some absolute stability theorems and integral evaluations of the state and control vectors are applied in the MIMO fuzzy adaptive controller design procedure for the first time. A method for automatically obtaining a near-optimal nonlinear dynamic reference model is described. A practical step-by-step procedure of the MIMO adaptive fuzzy controller design is proposed, resulting in highly interpretable MIMO fuzzy control rules based on triangular fuzzy sets and strong triangular fuzzy partition. An example of the fuzzy controller design according to the proposed procedure is given.

Gravitational search algorithm-based design of fuzzy control systems with a reduced parametric sensitivity

A systematic methodology for the design of a learning fuzzy logic control system is presented. The basic design idea is an indirect control approach where selection of control parameters relies on the estimates of process parameters. The control law consists of three components: an online fuzzy identifier, a desired transition model, and a fuzzy controller. The fuzzy version of the signal Hebbian learning law is introduced for adaptively identifying the process relation of the unknown plant. The desired transition model is constructed so that the control designer's goal can be achieved. A computationally efficient way to construct the transition model is provided via a forward-in-time method based on the concept of truncated policy space. Clear trade-offs between control performance and computational complexity are obtained. >

Based on the properties analysis of Takagi-Sugeno (T-S) fuzzy systems with two-overlapped fuzzy partition (TFP) inputs, an analytical design approach of T-S fuzzy control systems is proposed in this paper. This approach includes two parts: (1) a new method to check the stability of a T-S fuzzy control system which only requires to find a common positive-definite matrix in each maximal overlapped-rules group (MORG). This method can reduce the conservatism and difficulty of the former stability analysis methods. (2) The systematic design of T-S fuzzy controllers by means of the parallel distributed compensation (PDC) and linear matrix inequalities (LMI). The simulation results of a nonlinear mass-spring-damper system show that this approach is effective.

A new DNA-based immune genetic algorithm (DNA-IGA) is proposed to design a class of Takagi-Sugeno fuzzy control systems. The DNA-IGA is developed from the structures of biological immune system and DNA. The genetic operators of the method are based on the DNA genetic operations and evaluation of immune system. It uses DNA encoding method stemmed from the biological DNA to encode the design parameters of the fuzzy controllers. The DNA encoding method can significantly shorten the code length of antibodies and is suitable for complex knowledge representation. The proposed method has the local search ability and prevents the premature convergence. As a demonstration, we show how to implement the new method to optimize the design parameters of the TS fuzzy controller in control of a nonlinear system. Computer simulation indicates that the designed fuzzy controller is satisfactory.

Fuzzy systems based on fuzzy if-then rules have been applied to various problems. The main application area has been fuzzy control problems. In many cases, such fuzzy systems can handle only a few input variables. This is because the number of fuzzy if-then rules exponentially increases as the number of input variables increases. In this paper, we try to design fuzzy classification systems based on fuzzy if-then rules for multidimensional pattern classification problems with many attributes. For designing such fuzzy classification systems, we compare two frameworks in the area of genetics-based machine learning: the Michigan approach and the Pittsburgh approach. The performance of fuzzy rule-based classification systems is also compared with that of various pattern classification methods. In computer simulations, we use a wine classification problem with 13 attributes, a cancer diagnosis problem with 9 attributes, and a credit approval problem with 14 attributes. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(12): 10–19, 1997

Analysis and design of a stable fuzzy control system

This paper presents new stability conditions and LMI (linear matrix inequality) based designs for both continuous and discrete fuzzy control systems. First, Takagi and Sugeno's fuzzy models and some stability results are recalled. To design fuzzy control systems, nonlinear systems are represented by Takagi-Sugeno fuzzy models. The concept of parallel distributed compensation is employed to design fuzzy controllers from the Takagi-Sugeno fuzzy models. New stability conditions are obtained by relaxing the stability conditions derived in previous papers. The stability analysis of the feedback system is reduced to a problem of finding a common Lyapunov function for a set of linear matrix inequalities. Convex optimization techniques involving LMIs are utilized to find a common Lyapunov function. A procedure to design the fuzzy controller is constructed using the parallel distributed compensation and the relaxed stability conditions. Some stability issues are remarked. A simple example demonstrates the effects of the derived stability conditions.

"The curse of dimensionality" has become a significant bottleneck for fuzzy system identification and approximation. In this paper, we cast the Takagi–Sugeno (T–S) fuzzy system identification into a hierarchical sparse representation problem, where our goal is to establish a T–S fuzzy system with a minimal number of fuzzy rules, which simultaneously have a minimal number of nonzero consequent parameters. The proposed method, which is called hierarchical sparse fuzzy inference systems ( H-sparseFIS), explicitly takes into account the block-structured information that exists in the T–S fuzzy model and works in an intuitive way: First, initial fuzzy rule antecedent part is extracted automatically by an iterative vector quantization clustering method; then, with block-structured sparse representation, the main important fuzzy rules are selected, and the redundant ones are eliminated for better model accuracy and generalization performance; moreover, we simplify the selected fuzzy rules consequent with sparse regularization such that more consequent parameters can approximate to zero. This algorithm is very efficient and shows good performance in well-known benchmark datasets and real-world problems.