Non-fragile control for interval type-2 TSK fuzzy logic control systems with time-delay
Introduction
Time-delay problems have been well investigated in last decades both in theoretical and practical issues in the engineering field and information technology. Time-delay is the most common source of system instability. Thus, many researchers have considered the stability analysis and criteria, or controller design with time-delay as important topics. Some of them have proposed effective approaches for time-delay in different systems, for example, fuzzy systems [1], [2] and neural networks [3], [4].
Fuzzy systems have become useful tools in many applications since 1965, such as [5], [6], [7], [8], [9]. Fuzzy logic deals with approximate problems through IF-THEN rules, rather than fixed valued and exact reasoning. In the linear systems, many approaches are used to derive efficient criteria which can prove stability of the systems. However, when dealing with the nonlinear systems, it is a useful method to approximate and analyze through Takagi–Sugeno–Kang (TSK) fuzzy model, such as [10], [11], [12], [13]. A type-1 TSK was proposed by Tomohiro Takagi and Michio Sugeno in 1985 and Michio Sugeno and G.T. Kang in 1988, which is similar to the Mamdani method in many ways. Both methods have the same antecedent structures, fuzzifying the inputs, applying the fuzzy operator and fuzzy inference process. The difference is that the consequent of a TSK rule is either a linear function or a constant, whereas the consequent of a Mamdani rule is a fuzzy set.
Lotfi A. Zadeh introduced the concept of a type-2 (T2) fuzzy set as an extension of a type-1 (T1) fuzzy set in 1975. A T2 fuzzy set provide another dimension than a T1 fuzzy set, which has been known as a better tool to handle uncertainties. However, characterizing a T2 fuzzy set is not as easy as characterizing a T1 fuzzy set [14], [15], [16], [17]. The membership function (MF) of a T2 fuzzy set is three-dimensional, where the third dimension is called the secondary MF. If the amplitude of a secondary MF at each point on its two-dimensional domain is equal to the same everywhere (most of the cases are 1), which means no new information to be added in the third dimension, then that is an interval type-2 fuzzy set (IT2FS). Two-dimensional domain is named footprint of uncertainty (FOU), the union of all the primary MF. Since the third dimension is ignored, and only the FOU is used to describe an IT2FS. So FOU can be completely decided by its two bound functions, a lower membership function (LMF) and an upper membership function (UMF), both of which are T1 fuzzy sets.
A T2 TSK was introduced by Qilian Liang and Jerry M. Mendel in 1999 [18]. In a T1 fuzzy logic control system (T1FLCS) they are all type-1 fuzzy sets, whereas in an IT2FLCS at least one MF is an IT2 fuzzy set. The IT2FLCS can be classified into three classes of models [19]: A2-C0 means antecedent structures are IT2 fuzzy sets and consequent structures are crisp numbers; A2-C1 means antecedent structures are IT2 fuzzy sets and consequent structures are T1 Fuzzy sets; A2-C2 means antecedent and consequent structures are both IT2 fuzzy sets.
The typical fuzzy model approaches for designing feedback stabilization controller are worked out based on the parallel distributed compensation (PDC) method via Lyapunov–Krasovskii functional (LKF) approach, then transformed into an linear matrix inequality (LMI) problem. So sufficient conditions will be presented to guarantee FLCS stability [20], [21]. Normally, in the PDC framework the plant and the controller share the same MF, which simplifies the computation. In this paper, we use different type fuzzy sets to represent the plant and the controller in order to take the advantage of IT2 fuzzy sets. Precisely, we focus on an A2-C0 of IT2FLCS model with a greater degree of freedom and the ability to handle the uncertainties.
The fragile problem of controller has existed in a long time, which is related to the issue of accuracy of controller implementation. The fragile problem have attracted more attentions when an overview is given by Peter Dorato in 1998 [22]. The core of fragility problem is a trade-off between performance deterioration and implementation accuracy by tuning the controller parameters automatically. Some works have been done with respect to non-fragile controllers design of the T1FLCS TSK with time delay in [23], [24]. To the best of our knowledge, there is no previous work about non-fragile controllers of the IT2 FLCS with time delay.
Motivated by the above discussion, this paper pays attention to the controller design for the nonlinear time-delay system by using a T1FLCS TSK modeling the plant and an IT2FLCS TSK modeling the controller. The contributions of this paper are worth mentioning. First, its original to use IT2FLCS TSK to design the non-fragile controller showing more suitable than the traditional controller to uncertain environment. Second, Wirtinger-based integral inequality is employed to get a novel LKF in IT2FLCS TSK, which is much less conservative than the existing work using Jensen inequality [25].
The paper is organized as follows. Firstly, we will introduce the problem and describe the plant and the controller model in Section 2. Secondly, two controllers design are presented in Section 3 which based on a Lyapunov–Krasovskii stability theory. Finally, simulation examples in different cases will illustrate our statements.
Notations: Denote Rn as the n-dimensional Euclidean space, Rm×n as the set of all m × n real matrices. The matrix Q > 0 represents that Q is symmetric and positive definite. The matrix Q < 0 represents that Q is symmetric and negative definite. The matrix Q ≤ 0 represents that Q is symmetric and negative-semidefinite definite. The matrix I indicates the unit matrix of appropriate dimension. The symmetric matrix represents for . The superscripts and ``T'' denote the inverse matrix and transposed matrix, respectively.
Section snippets
Problem formulation
Consider a nonlinear plant described by the following T1 TSK fuzzy rules. Then, the ith fuzzy rule has the following form:
IF p1(t) is Mi1, p2(t) is Mi2, and ⋅⋅⋅ and pq(t) is Miq, then where Mij represent type-1 fuzzy sets, L is the rules number, ; x(t) ∈ Rn is the state vector; u(t) ∈ Rm is the control input vector; are known antecedent variables that may be functions of the state variables, external
Main results
Theorem 1 For a given positive scalar d, when and mean without uncertain parameters, the closed-loop IT2 TSK FLCS in expression (13) is asymptotically stable if there exist symmetric positive-definite matrices and any appropriately dimensioned matrices Yi such that the following LMIs holds
where
Simulation
Example 1 Consider the unstable nonlinear system with following differential equation[39]: where . Choose the state variable and the input variable as . It can be represented by two rules through the fuzzy modeling method in [40]: Rule 1: if x2(t)/0.5 is about 0, then ; Rule 2: if x2(t)/0.5 is about π or , then ; where
Conclusion
In this paper, the problem of stabilization criteria for IT2FLCS TSK with time-delay has been investigated. By using Theorem 1 and Theorem 2, LMI stability conditions for IT2 TSK FLCS have been derived and transformed into the formats that can be easily solved by using software tools such as MATLAB LMI toolbox. Finally, the proposed methods have been demonstrated by numerical examples in case 1 and case 2, and both are effective.
Acknowledgment
This work was supported in part by the National Natural Science Foundation of China under Grants 61433004, 61273027, in part by the State Key Laboratory of Integrated Automation for Process Industries Fundamental Research Funds under Grant 2013ZCX14, in part by Liaoning Province science and technology key project under Grant 2013219005, and in part by the Development Project of the Key Laboratory of Liaoning Province.
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