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Nawel Aoun, Marwen Kermani, Anis Sakly, "Delay-Dependent Stability Analysis of TS Fuzzy Switched Time-Delay Systems", Mathematical Problems in Engineering, vol. 2017, Article ID 8123830, 16 pages, 2017. https://doi.org/10.1155/2017/8123830
Delay-Dependent Stability Analysis of TS Fuzzy Switched Time-Delay Systems
This paper proposes a new approach to deal with the problem of stability under arbitrary switching of continuous-time switched time-delay systems represented by TS fuzzy models. The considered class of systems, initially described by delayed differential equations, is first put under a specific state space representation, called arrow form matrix. Then, by constructing a pseudo-overvaluing system, common to all fuzzy submodels and relative to a regular vector norm, we can obtain sufficient asymptotic stability conditions through the application of Borne and Gentina practical stability criterion. The stability criterion, hence obtained, is algebraic, is easy to use, and permits avoiding the problem of existence of a common Lyapunov-Krasovskii functional, considered as a difficult task even for some low-order linear switched systems. Finally, three numerical examples are given to show the effectiveness of the proposed method.
Switched systems, seen as an important class of hybrid dynamical systems, are defined as a family of subsystems described by differential or difference equations and a rule that governs the switching between them [1, 2]. They have a strong engineering background in various areas and are often used to suitably model a great number of real-world systems, such as mechanical systems, chemical processes, communication networks, robotic systems, and aircraft and air traffic.
On the other hand, since the significant development that has witnessed the field of fuzzy modeling and control and more particularly model-based control, a new theory about fuzzy switched systems has emerged as an answer to more complicated real systems analysis and synthesis requirements such as multiple nonlinear systems, switched nonlinear systems, and second-order nonholonomic systems [3–10].
Originally inspired from the concept of sector nonlinearity, the main idea of Takagi-Sugeno fuzzy modeling is to partition the nonlinear system dynamics into several locally linearized models so that the overall nonlinear system could be represented by a sufficiently accurate approximation . Hence, model-based control is considered as a powerful universal approximation tool and a reliable approach to deal with complex and ill-determined systems [12–15].
A fuzzy switched system is defined as a switched system which involves fuzzy models among its subsystems.
Despite the considerable efforts made for the analysis of nonlinear systems, stability study of switched systems and in particular of fuzzy switched systems is still complex. Indeed, the example of asymptotically stable subsystems which, due to a specific switching sequence, yield to an unstable behavior of the overall system is well known. Besides, the case of unstable subsystems that, via a particular stabilizing switching law, lead to a stable global system also exists.
The stability problem of such systems becomes more challenging when time delay is involved . In fact, time delay often occurs in many dynamical systems, namely, in biological systems, chemical systems, metallurgical processing systems, and network systems. Its existence, whether in the state variables, control inputs, or the measurement outputs, is frequently a source of instability and poor performance.
The existing results on stability of TS fuzzy switched time-delay systems can be classified into two types: delay-independent criteria, which are applicable to delay of arbitrary size [17–20] and delay-dependent criteria, which include information on the size of the delay . It is generally recognized that delay-dependent results are usually less conservative than delay-independent ones, especially when the size of the delay is small.
The first is related to the asymptotic stability of switched systems under arbitrary switching. An important result in this area states that a sufficient condition for the asymptotic stability of this class of systems is that all the subsystems share a common Lyapunov function [25–31]. Sufficient stability conditions are then derived through the resolution of a set of Linear Matrix Inequalities (LMIs).
In the case of switched time-delay systems, a common Lyapunov-Krasovskii functional is searched [32–34]. However, checking the existence of such a functional is a hard task even for some simple cases. Moreover, the method becomes less reliable when the number of subsystems switching between each other is important or when the number of fuzzy rules required to model each subsystem with a good accuracy is high.
A second problem concerns the stabilization of TS fuzzy switched systems by restricting the class of admissible switching signals to those in which the interval between any two consecutive switching instants is no smaller than a number called dwell time .
To overcome limitations due to the existence of Lyapunov-Krasovskii functionals, we propose, in this paper, to study the stability of TS fuzzy switched systems through the convergence of a regular vector norm, associated with a specific characteristic matrix, called arrow form matrix [36–43]. The proposed method is based on the construction of a common overvaluing/comparison system for all the fuzzy submodels and whose stability permits concluding to that of the original system. The obtained results, valuable for the case of arbitrary switching, are expressed in terms of simple algebraic conditions and explicitly involve the time delay.
The remainder of the paper is organized as follows: Section 2 presents the problem formulation and some preliminaries. In Section 3, new delay-dependent stability conditions are derived for a class of TS fuzzy switched time-delay systems described by differential equations and having a single constant delay. Section 4 generalizes the main result to the case of TS fuzzy switched systems with multiple delays. Three numerical examples are provided in Section 5 to demonstrate the effectiveness of the proposed method. Finally, some concluding remarks are given in Section 6.
Notations. The notations used throughout this paper are fairly standard. denotes the -dimensional Euclidean space, is the identity matrix with appropriate dimensions, and denotes Euclidean vector norm. For any , , we define the scalar product of the vectors and as . , and are its transpose and its inverse, respectively. We denote with if and if and , .
2. Problem Formulation and Preliminaries
2.1. Problem Formulation
Let us consider the unforced switched nonlinear time-delay systems that are described by a differential equation of the form:where is the state vector, is the time delay, is a differentiable vector valued initial function, are sufficiently regular functions that are parametrized by the index set , is a piecewise constant function depending on time, called switching signal and assumed to be available in real time, and is the number of subsystems.
The switching sequence is defined through a switching vector whose components are given byIt is obvious that .
Therefore, the switched system is composed of subsystems expressed as
In addition, each subsystem is described by a set of IF-THEN rules; each rule is related to a region of the state space where the subsystem could be approximated by a local linear model.
Thus, the th fuzzy rule associated with the th subsystem is given bywhere are the premise variables, and ( and ) are matrices of appropriate dimensions, () are the fuzzy sets, and and are the number of fuzzy rules and premise variables, respectively.
By using the product inference engine and the center of average defuzzification, the th TS fuzzy subsystem can be inferred as where and are the normalized weighting functions expressed by is the firing strength of the membership function . It is assumed that and .
Hence, and .
Finally, the fuzzy switched system (3) can be represented by Equation (7) can also be written asBy using the Newton-Leibniz formula, Equation (8) can be written as In the sequel, will be simplified by .
This subsection recalls some of the definitions and remarks that will be useful throughout this paper.
Kotelyanski Lemma (see ). The real parts of the eigenvalues of matrix , with non-negative off-diagonal elements, are less than a real number if and only if all those of matrix ; are positive, with the identity matrix.
Definition 1 (see ). The matrix is called an -matrix, if the following conditions are met:(i) , (; );(ii)the principal minors of are all positive:(iii)for any positive real vector , the algebraic equations have a positive solution .
Remark 2. is the opposite of an -matrix if is an -matrix.
Remark 3. A continuous-time system characterized by is stable if is the opposite of an -matrix. In this case, the main minors of are of alternating signs (the first is negative) and the Kotelyanski lemma permits concluding to the stability of the system characterized by .
Definition 4 (see [43, 55]). If is a pseudo-overvaluing matrix of the system with respect to the vector norm , the following inequality is satisfied: where denotes the right hand derivative operator.
Consequently, the stability of the comparison system, with the initial conditions such as , implies the same property for the initial system.
3. Delay-Dependent Stability Conditions for TS Fuzzy Switched Systems with Single Delay
3.1. Main Results
In this section, new delay-dependent conditions for global asymptotic stability of system (10) under arbitrary switching are stated.
Theorem 5. System (10) is globally asymptotically stable under an arbitrary switching rule , , if matrix is the opposite of an -matrix, where
Proof. Let with components (, ) and consider the radially unbounded candidate Lyapunov functional for each fuzzy submodel given by where It is clear that and that a common Lyapunov functional for all the submodels can be given byThe right Dini derivative of along the trajectory of system (10) giveswithThen,Finally, we obtainwhereOn the other hand, if we suppose that is the opposite of an -matrix and according to the -matrices properties, we can find a vector ( ) satisfying the relation: , .
Knowing thatwe can writeThis completes the proof of Theorem 5.
3.2. Extension of the Results to the Case of TS Fuzzy Switched Systems Described by Delayed Differential Equations with Single Delay
In this section, we consider the class of TS fuzzy switched time-delay systems that are governed by the following differential equation:where is the state vector and and (; ; and ) are constant coefficients.
A change of variable under the form , , allows system (25) to be represented in the state space as follows:where matrices and are given by A change of base of the form of (26) under the arrow form matrix giveswhere , , and is the corresponding passage matrix such thatArrow form matrices , and , are given bywith whereas matrices are given bywithNote that , , are distinct constant parameters that can be chosen arbitrarily and is the instantaneous characteristic polynomial of matrix given byTherefore, the expression of the common comparison matrix in the new base is given through the sum of the following terms:The new polynomial is defined by Notice that can be simplified. In fact, . However, . Consequently, we can construct for each individual submodel the following overvaluing system: whereFinally, the common overvaluing matrix is given bywith At this level, we can state the following theorem.
Theorem 6. System (26) is globally asymptotically stable under an arbitrary switching rule , , if there exist , , , , satisfying the following condition:
Proof. It is sufficient to check that the matrix is the opposite of an -matrix by verifying that all the successive principal minors of , denoted by , are of alternating signs; that is,It is clear that, for an arbitrary choice of , relation (43) is satisfied for .
Condition (43) becomes for By dividing (44) by , we obtain This achieves the proof of Theorem 6.
Remark 7. The maximum value of ensuring the asymptotic stability of each individual model () is computed by
Remark 8. Expression (46) can be widely simplified. We can give, for instance, two of the possible combinations.
Case 1. One hasThen, stability conditions will depend on the sign of as follows:Case 2. One has
Proof. See the Appendix.
4. Delay-Dependent Stability Conditions for TS Fuzzy Switched Systems with Multiple Delays
Stability criteria in Theorem 5 can be generalized to systems with multiple delays.
4.1. Main Result
Consider a switched system composed of subsystems; each subsystem is a TS fuzzy time-delay system as shown in the following differential equation:where , (; ; and ) are constant matrices of appropriate dimensions, are fuzzy weighting factors, previously defined in (6), and .
System (52) can also be put in the following form:where
Theorem 9. System (52) is globally asymptotically stable under arbitrary switching rule , if matrixis the opposite of an -matrix.
Proof. Following the same steps as those given in the proof of Theorem 5 and by choosing the common radially unbounded Lyapunov functional as follows,wherewe deduce that it suffices that is the opposite of an -matrix to conclude to the asymptotic stability of system (52).
4.2. Extension of the Results to the Case of TS Fuzzy Switched Systems Described by Differential Equations with Multiple Delays
The same method is applied to determine delay-dependent stability criteria for systems described by the following multiple time-delayed differential equation:The same change of variable , , as in Section 3.2 yields to the new state space representation:where is the state vector and for each , , and are given by A change of base of the form allows system (59) to be represented by the arrow form matrix:where is given by (30) and becomeswithDefine the new polynomials:Finally, we can compute the common overvaluing matrix as follows:withHence, we can state Theorem 10.
Theorem 10. System (59) is globally asymptotically stable under an arbitrary switching rule , if there exist , , , such that
Proof. The same proof is as in Theorem 6; just replace matrix by .
5. Illustrative Examples
Example 1. Consider a switched system composed of three second-order subsystems , ; each subsystem is represented by a TS fuzzy time-delay model as follows: Subsystem Subsystem Subsystem A transformation of state matrices under the arrow form and ( and ) with givesFor and , parameters and are computed as follows:For an arbitrary choice of , we obtain, for example, for subsystem For and by using the center of average defuzzification, state matrix and delayed-state matrix () are given, respectively, by