A new LMI-based approach to relaxed quadratic stabilization of T-S fuzzy control systems
TL;DR: The condition is represented in the form of linear matrix inequalities (LMIs) and is shown to be less conservative than some relaxed quadratic stabilization conditions published recently in the literature and to include previous results as special cases.
Abstract: This paper proposes a new quadratic stabilization condition for Takagi-Sugeno (T-S) fuzzy control systems. The condition is represented in the form of linear matrix inequalities (LMIs) and is shown to be less conservative than some relaxed quadratic stabilization conditions published recently in the literature. A rigorous theoretic proof is given to show that the proposed condition can include previous results as special cases. In comparison with conventional conditions, the proposed condition is not only suitable for designing fuzzy state feedback controllers but also convenient for fuzzy static output feedback controller design. The latter design work is quite hard for T-S fuzzy control systems. Based on the LMI-based conditions derived, one can easily synthesize controllers for stabilizing T-S fuzzy control systems. Since only a set of LMIs is involved, the controller design is quite simple and numerically tractable. Finally, the validity and applicability of the proposed approach are successfully demonstrated in the control of a continuous-time nonlinear system.
Citations
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TL;DR: The result provides a set of progressively less conservative sufficient conditions for proving positivity of fuzzy summations of Polya's theorems on positive forms on the standard simplex.
582 citations
01 Jun 2008
TL;DR: To investigate the system stability, an interval type-2 Takagi-Sugeno (T-S) fuzzy model is proposed to represent the nonlinear plant subject to parameter uncertainties, which allows the introduction of slack matrices to handle the parameter uncertainties in the stability analysis.
Abstract: This paper presents the stability analysis of interval type-2 fuzzy-model-based (FMB) control systems. To investigate the system stability, an interval type-2 Takagi-Sugeno (T-S) fuzzy model, which can be regarded as a collection of a number of type-1 T-S fuzzy models, is proposed to represent the nonlinear plant subject to parameter uncertainties. With the lower and upper membership functions, the parameter uncertainties can be effectively captured. Based on the interval type-2 T-S fuzzy model, an interval type-2 fuzzy controller is proposed to close the feedback loop. To facilitate the stability analysis, the information of the footprint of uncertainty is used to develop some membership function conditions, which allow the introduction of slack matrices to handle the parameter uncertainties in the stability analysis. Stability conditions in terms of linear matrix inequalities are derived using a Lyapunov-based approach. Simulation examples are given to illustrate the effectiveness of the proposed interval type-2 FMB control approach.
382 citations
TL;DR: An IT2 Takagi-Sugeno (T-S) fuzzy model is employed to represent the dynamics of nonlinear systems of which the parameter uncertainties are captured by IT2 membership functions characterized by the lower and upper membership functions.
Abstract: This paper focuses on designing interval type-2 (IT2) control for nonlinear systems subject to parameter uncertainties. To facilitate the stability analysis and control synthesis, an IT2 Takagi-Sugeno (T-S) fuzzy model is employed to represent the dynamics of nonlinear systems of which the parameter uncertainties are captured by IT2 membership functions characterized by the lower and upper membership functions. A novel IT2 fuzzy controller is proposed to perform the control process, where the membership functions and number of rules can be freely chosen and different from those of the IT2 T-S fuzzy model. Consequently, the IT2 fuzzy-model-based (FMB) control system is with imperfectly matched membership functions, which hinders the stability analysis. To relax the stability analysis for this class of IT2 FMB control systems, the information of footprint of uncertainties and the lower and upper membership functions are taken into account for the stability analysis. Based on the Lyapunov stability theory, some stability conditions in terms of linear matrix inequalities are obtained to determine the system stability and achieve the control design. Finally, simulation and experimental examples are provided to demonstrate the effectiveness and the merit of the proposed approach.
311 citations
TL;DR: This paper presents a systematic approach for decreasing conservativeness in stability analysis and control design for Takagi-Sugeno (TS) systems based on the idea of multiple Lyapunov functions together with simple techniques for introducing slack matrices.
294 citations
TL;DR: Two procedures for designing state-feedback control laws are given: one casts the controller design into a convex optimization by introducing some over design and the other utilizes the cone complementarity linearization idea to cast the controllerDesign into a sequential minimization problem subject to linear matrix inequality constraints, which can be readily solved using standard numerical software.
Abstract: This paper investigates the problem of stabilization for a Takagi-Sugeno (T-S) fuzzy system with nonuniform uncertain sampling. The sampling is not required to be periodic, and the only assumption is that the distance between any two consecutive sampling instants is less than a given bound. By using the input delay approach, the T-S fuzzy system with variable uncertain sampling is transformed into a continuous-time T-S fuzzy system with a delay in the state. Though the resulting closed-loop state-delayed T-S fuzzy system takes a standard form, the existing results on delay T-S fuzzy systems cannot be used for our purpose due to their restrictive assumptions on the derivative of state delay. A new condition guaranteeing asymptotic stability of the closed-loop sampled-data system is derived by a Lyapunov approach plus the free weighting matrix technique. Based on this stability condition, two procedures for designing state-feedback control laws are given: one casts the controller design into a convex optimization by introducing some over design and the other utilizes the cone complementarity linearization idea to cast the controller design into a sequential minimization problem subject to linear matrix inequality constraints, which can be readily solved using standard numerical software. An illustrative example is provided to show the applicability and effectiveness of the proposed controller design methodology.
253 citations
References
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01 Jan 1985
TL;DR: A mathematical tool to build a fuzzy model of a system where fuzzy implications and reasoning are used is presented and two applications of the method to industrial processes are discussed: a water cleaning process and a converter in a steel-making process.
Abstract: A mathematical tool to build a fuzzy model of a system where fuzzy implications and reasoning are used is presented. The premise of an implication is the description of fuzzy subspace of inputs and its consequence is a linear input-output relation. The method of identification of a system using its input-output data is then shown. Two applications of the method to industrial processes are also discussed: a water cleaning process and a converter in a steel-making process.
18,803 citations
"A new LMI-based approach to relaxed..." refers background in this paper
...Recently, based on Takagi–Sugeno (T–S) fuzzy model [ 13 ], [14], there have appeared in the literature a great number of results concerning stability analysis and design [1]–[3], [10], [14]–[17]....
[...]
...T–S fuzzy control system [ 13 ], [14] is one of the most popular and promising research platforms in the model-based fuzzy control....
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Book•
20 Aug 19962,938 citations
"A new LMI-based approach to relaxed..." refers methods or result in this paper
...It has been applied to various industrial fields [12], [ 17 ]....
[...]
...Recently, based on Takagi–Sugeno (T–S) fuzzy model [13], [14], there have appeared in the literature a great number of results concerning stability analysis and design [1]–[3], [10], [14]–[ 17 ]....
[...]
TL;DR: The authors represent a nonlinear plant with a Takagi-Sugeno fuzzy model with a model-based fuzzy controller design utilizing the concept of the so-called "parallel distributed compensation" and presents a design methodology for stabilization of a class of nonlinear systems.
Abstract: Presents a design methodology for stabilization of a class of nonlinear systems. First, the authors represent a nonlinear plant with a Takagi-Sugeno fuzzy model. Then a model-based fuzzy controller design utilizing the concept of the so-called "parallel distributed compensation" is employed. The main idea of the controller design is to derive each control rule so as to compensate each rule of a fuzzy system. The design procedure is conceptually simple and natural. Moreover, the stability analysis and control design problems can be reduced to linear matrix inequality (LMI) problems. Therefore, they can be solved efficiently in practice by convex programming techniques for LMIs. The design methodology is illustrated by application to the problem of balancing and swing-up of an inverted pendulum on a cart.
2,534 citations
TL;DR: The fuzzy block diagrams and the stability analysis are applied to the design problems of a model-based fuzzy controller and a new design technique of a fuzzy controller is proposed.
2,266 citations
"A new LMI-based approach to relaxed..." refers background or result in this paper
...T–S fuzzy control system [13], [ 14 ] is one of the most popular and promising research platforms in the model-based fuzzy control....
[...]
...Recently, based on Takagi–Sugeno (T–S) fuzzy model [13], [14], there have appeared in the literature a great number of results concerning stability analysis and design [1]–[3], [10], [ 14 ]–[17]....
[...]
...Recently, based on Takagi–Sugeno (T–S) fuzzy model [13], [ 14 ], there have appeared in the literature a great number of results concerning stability analysis and design [1]–[3], [10], [14]–[17]....
[...]
TL;DR: New relaxed stability conditions and LMI- (linear matrix inequality) based designs for both continuous and discrete fuzzy control systems are applied to design problems of fuzzy regulators and fuzzy observers.
Abstract: This paper presents new relaxed stability conditions and LMI- (linear matrix inequality) based designs for both continuous and discrete fuzzy control systems. They are applied to design problems of fuzzy regulators and fuzzy observers. First, Takagi and Sugeno's fuzzy models and some stability results are recalled. To design fuzzy regulators and fuzzy observers, nonlinear systems are represented by Takagi-Sugeno's (TS) fuzzy models. The concept of parallel distributed compensation is employed to design fuzzy regulators and fuzzy observers from the TS fuzzy models. New stability conditions are obtained by relaxing the stability conditions derived in previous papers, LMI-based design procedures for fuzzy regulators and fuzzy observers are constructed using the parallel distributed compensation and the relaxed stability conditions. Other LMI's with respect to decay rate and constraints on control input and output are also derived and utilized in the design procedures. Design examples for nonlinear systems demonstrate the utility of the relaxed stability conditions and the LMI-based design procedures.
1,625 citations
"A new LMI-based approach to relaxed..." refers background or methods or result in this paper
...It has been shown in [4] that Theorems 1 and 2 are more relaxed than the famous relaxed stabilization conditions proposed in [ 10 ] and [11]....
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...Reference [ 10 ] proposed a so-called relaxed stabilization condition and then applied the condition to design fuzzy controllers for T–S fuzzy systems....
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...Digital Object Identifier 10.1109/TFUZZ.2006.876331 [ 10 ] as special cases....
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...In this paper, a new LMI-based stabilization condition is obtained by relaxing the results in [4], [8], and [ 10 ]....
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...Recently, based on Takagi–Sugeno (T–S) fuzzy model [13], [14], there have appeared in the literature a great number of results concerning stability analysis and design [1]–[3], [ 10 ], [14]–[17]....
[...]