Delay-Dependent Stability Analysis of TS Fuzzy Switched Time-Delay Systems

Aoun, Nawel; Kermani, Marwen; Sakly, Anis

doi:https://doi.org/10.1155/2017/8123830

Mathematical Problems in Engineering

On this page

Abstract Introduction Preliminaries Conclusion Appendix Conflicts of Interest References Copyright Related Articles

Special Issue

Advances in Modelling, Analysis, and Design of Delayed Systems

View this Special Issue

Research Article | Open Access

Volume 2017 | Article ID 8123830 | https://doi.org/10.1155/2017/8123830

Delay-Dependent Stability Analysis of TS Fuzzy Switched Time-Delay Systems

Nawel Aoun,^1,2Marwen Kermani ,¹and Anis Sakly¹

Academic Editor: Radek Matušů

Received14 Feb 2017

Revised17 Apr 2017

Accepted07 May 2017

Published07 Jun 2017

Abstract

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.

1. Introduction

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 [11]. 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 [16]. 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 [21]. 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.

Stability analysis of switched systems in general has been conducted mainly on the basis of Lyapunov stability theory and concerns two major problems [22–24].

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 [35].

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 application of vector norms to switched systems has already been introduced in [44–47]. It has been extended later to switched time-delay systems in [48–53].

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 .

2.2. Preliminaries

This subsection recalls some of the definitions and remarks that will be useful throughout this paper.

Kotelyanski Lemma (see [54]). 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 [43]). 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, byDenote by , and , the minimal pseudo-overvaluing matrix of the fuzzy submodel such thatwhereAt this level, we can derive sufficient conditions for the asymptotic stability of each submodel individually as shown in Table 1.
From Table 1, we can already see that it is sufficient to have a time delay inferior to to ensure the asymptotic stability of all the submodels individually. Taking into account this value, we can notice that can be chosen as the common comparison system for the six submodels ( and ). Then, the TS fuzzy switched time-delay system is asymptotically stable under an arbitrary switching law ifThe switching signal is plotted on Figure 1. For a time delay equal to , a final simulation time , weighting factors , and an initial state vector , the evolution of state vectors, the system’s trajectory, and the state’s norm are illustrated on Figures 2, 3, and 4, respectively.
This example shows that the obtained stability conditions are sufficient and very close to be necessary. Moreover, the proposed method makes it possible to avoid searching a common Lyapunov-Krasovskii functional, which is very difficult in this case.

Example 2. Consider the TS fuzzy switched system with two delays and as follows:Subsystem Subsystem For , a transformation under the arrow form matrices givesDenote by , , the minimal overvaluing matrix of subsystem of index () and relative to the vector norm such thatwhereApplying Theorem 10, we can draw Table 2 which gives a sufficient stability condition for each submodel.
Finally, we can construct the following common comparison system such thatConsequently, the TS fuzzy switched time-delay system is asymptotically stable under arbitrary switching ifIn this example, we can notice that maximum bounds found for the delays and are small. Indeed, this is justified by the multiplicity of time delay, by the choice of delay matrices , , and by the fact that arbitrary switching strategy brings conservativeness because of the “strict” condition that requires finding a comparison system which is common to all fuzzy local models.
Nevertheless, the simplicity of vector norms based approach makes the study almost feasible in most cases in the opposite of Lyapunov-based method which proves to be difficult when dealing with the problem of stability under arbitrary switching.
For a switching signal as shown on Figure 5, a choice of time delays and , particular values of weighting factors and , and an initial state vector , the evolution of state responses is illustrated on Figure 6 whereas Figures 7 and 8 show the system’s trajectory and the norm’s state, respectively.

Example 3 (see [38, 45]). Consider a DC motor with separate excitation (Figure 9). The system acts under variable mechanical load and is subject to a retarded state feedback control input. Our aim is to synthesize a state feedback control for each local model so that the overall system is asymptotically stable under arbitrary switching despite the time delay.
A PDC controller scheme is employed such that the lth controller fuzzy rule of the th subsystem is given bywhere , , is the local feedback gain vector.
The TS fuzzy switched system is then represented in the state space bywhere and .
We can also writeHence, Denote , and the problem is then reduced to determining admissible values of parameters , , for a given value of time delay .
A transformation under the arrow form matrix with yieldsTo tune parameters , suppose that for , we have Thus, we can verify that Since all the assumptions of Remark 8 (Case 2) are met, this allows us, according to inequality (50), to write Therefore, each fuzzy model is asymptotically stable individually if The pseudo-overvaluing matrix for each submodel is constructed as follows:withFor instance, if the time delay is equal to , , , , and , and local feedback gains are chosen as , , , and , we obtain Finally, the common comparison matrix that verifies the properties of the opposite of an -matrix is given byThis example has shown that the proposed approach is appropriate for the synthesis of a stabilizing state feedback retarded control for an initially unstable nonlinear system described by a set of linear TS fuzzy local linear models.

6. Conclusion

In this paper, new delay-dependent stability conditions by using the vector norms approach have been presented. In fact, most of the research works that have been carried out on stability analysis of TS fuzzy switched time-delay systems under arbitrary switching are based on the research of a common Lyapunov-Krasovskii functional for all the fuzzy models, which is considered as a hard task. The idea of the proposed method consists in putting the switched system under a special form of state space representation using arrow form matrices and then finding a common pseudo-overvaluing system for all the constituent submodels. The stability analysis of this comparison system, based on the aggregation techniques and the -matrices properties, permits concluding to that of the original TS fuzzy switched system.

Vector norms approach, applied to TS fuzzy switched time-delay systems, whether with single or multiple delay, is suitable for the case of arbitrary switching and provides delay-dependent algebraic criteria. The applicability of the obtained conditions is shown through three numerical examples. It would be interesting to generalize the study to the case of systems with control input.

Appendix

Demonstrations of (48), (50), and (51) are based on the following expressions:Considering assumptions of Case 1 and the condition , these relations permit us to writeConsidering assumptions of Case 2 and the same condition , it becomesThe same steps will be followed for the other cases.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

D. Liberzon, Switching in Cystems and Control, Birkhäuser, Boston, Mass, USA, 2003.
View at: Publisher Site | MathSciNet
Z. Sun and S. S. Ge, Switched Linear Systems-Control and Design, Springer, 2005.
K. Tanaka, M. Iwasaki, and H. O. Wang, “Switching control of an R/C hovercraft: Stabilization and smooth switching,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 31, no. 6, pp. 853–863, 2001.
View at: Publisher Site | Google Scholar
R. Palm and D. Driankov, “Fuzzy switched hybrid systems-modeling and identification,” in Proceedings of the IEEE International Symposium on Intelligent Control, pp. 130–135, September 1998.
View at: Google Scholar
V. Ojleska, T. K. Gugulovska, and G. M. Dimirovski, “Superior performance of switched fuzzy control systems: an overview and simulation experiments,” International Journal of Simulation Systems, Science and Technology, vol. 12, no. 2, pp. 19–29, 2011.
View at: Google Scholar
J. Dong and G.-H. Yang, “State feedback control of continuous-time T-S fuzzy systems via switched fuzzy controllers,” Information Sciences. An International Journal, vol. 178, no. 6, pp. 1680–1695, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
V. Ojleska and G. Stojanovski, “Switched fuzzy systems: overview and perspectives,” in Proceedings of the 9th International PhD Workshop on Systems and Control: Young Generation Viewpoint, Izola, Slovenia, October 2008.
View at: Google Scholar
K. Tanaka, M. Iwasaki, and H. O. Wang, “Stable switching fuzzy control and its application to a hovercraft type vehicle,” in Proceedings of the FUZZ-IEEE 2000: 9th IEEE International Conference on Fuzzy Systems, vol. 2, pp. 804–809, May 2000.
View at: Google Scholar
H. Yang, G. M. Dimirovski, and J. Zhao, “Switched fuzzy systems: Representation modelling, stability analysis, and control design,” IEEE Intelligent Systems, pp. 306–311, 2006.
View at: Publisher Site | Google Scholar
J. P. Hespanha, D. Liberzon, and A. S. Morse, “Logic-based switching control of a nonholonomic system with parametric modeling uncertainty,” Systems & Control Letters, vol. 38, no. 3, pp. 167–177, 1999.
View at: Publisher Site | Google Scholar | MathSciNet
T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Transactions on Systems, Man and Cybernetics, vol. 15, no. 1, pp. 116–132, 1985.
View at: Google Scholar
K. Tanaka and M. Sano, “A Robust Stabilization Problem of Fuzzy Control Systems and Its Application to Backing up Control of a Truck-Trailer,” IEEE Transactions on Fuzzy Systems, vol. 2, no. 2, pp. 119–134, 1994.
View at: Publisher Site | Google Scholar
K. Tanaka, T. Ikeda, and H. O. Wang, “Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, H^∞ control theory, and linear matrix inequalities,” IEEE Transactions on Fuzzy Systems, vol. 4, no. 1, pp. 1–13, 1996.
View at: Publisher Site | Google Scholar
G. Feng, “A survey on analysis and design of model-based fuzzy control systems,” IEEE Transactions on Fuzzy Systems, vol. 14, no. 5, pp. 676–697, 2006.
View at: Publisher Site | Google Scholar
K. Tanaka and H. O. Wang, Fuzzy Control Systems Design and Analysis: A LMI Approach, John Wiley & Sons, Inc., New York, NY, USA, 2001.
View at: Publisher Site
J.-P. Richard, “Time-delay systems: an overview of some recent advances and open problems,” Automatica. A Journal of IFAC, the International Federation of Automatic Control, vol. 39, no. 10, pp. 1667–1694, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Cao and P. M. Frank, “Analysis and synthesis of nonlinear time-delay systems via fuzzy control approach,” IEEE Transactions on Fuzzy Systems, vol. 8, no. 2, pp. 200–211, 2000.
View at: Publisher Site | Google Scholar
Y.-Y. Cao and P. M. Frank, “Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi-Sugeno fuzzy models,” Fuzzy Sets and Systems. An International Journal in Information Science and Engineering, vol. 124, no. 2, pp. 213–229, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
B. Chen and X. P. Liu, “Fuzzy guaranteed cost control for nonlinear systems with time-varying delay,” IEEE Transactions on Fuzzy Systems, vol. 13, no. 2, pp. 238–249, 2005.
View at: Publisher Site | Google Scholar
M. Kermani and A. Sakly, “Delay-independent stability criteria under arbitrary switching of a class of switched nonlinear time-delay systems,” Advances in Difference Equations, 2015:225, 20 pages, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
B. Chen and X. Liu, “Delay-dependent robust H_∞ control for T-S fuzzy systems with time delay,” IEEE Transactions on Fuzzy Systems, vol. 13, no. 4, pp. 544–556, 2005.
View at: Publisher Site | Google Scholar
L. Liu, Y. Yin, L. Wang, and R. Bai, “Stability analysis for switched positive T-S fuzzy systems,” Neurocomputing, vol. 173, pp. 2009–2013, 2016.
View at: Publisher Site | Google Scholar
J.-S. Chiou, C.-J. Wang, C.-M. Cheng, and C.-C. Wang, “Analysis and synthesis of switched nonlinear systems using the T-S fuzzy model,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 34, no. 6, pp. 1467–1481, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
K. Tanaka, M. Iwasaki, and H. O. Wang, “Stability and smoothness conditions for switching fuzzy systems,” in Proceedings of the American Control Conference (ACC '00), pp. 2474–2478, Chicago, Ill, USA, June 2000.
View at: Publisher Site | Google Scholar
L. Vu and D. Liberzon, “Common Lyapunov functions for families of commuting nonlinear systems,” Systems & Control Letters, vol. 54, no. 5, pp. 405–416, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
D. Liberzon, J. P. Hespanha, and A. S. Morse, “Stability of switched systems: a Lie-algebraic condition,” Systems & Control Letters, vol. 37, no. 3, pp. 117–122, 1999.
View at: Publisher Site | Google Scholar | MathSciNet
H. Lin and P. J. Antsaklis, “Stability and stabilizability of switched linear systems: A short survey of recent results,” in Proceedings of the 20th IEEE International Symposium on Intelligent Control, ISIC '05 and the 13th Mediterranean Conference on Control and Automation, MED '05, pp. 24–29, cyp, June 2005.
View at: Publisher Site | Google Scholar
D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Systems Magazine, vol. 19, no. 5, pp. 59–70, 1999.
View at: Publisher Site | Google Scholar
J. L. Mancilla-Aguilar and R. A. García, “A converse Lyapunov theorem for nonlinear switched systems,” Systems & Control Letters, vol. 41, no. 1, pp. 67–71, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
J. Zhao and G. M. Dimirovski, “Quadratic stability of a class of switched nonlinear systems,” IEEE Transactions on Automatic Control, vol. 49, no. 4, pp. 574–578, 2004.
View at: Publisher Site | Google Scholar
R. N. Shorten and K. S. Narendra, “On the stability and existence of common Lyapunov functions for stable linear switching systems,” in Proceedings of the 37th IEEE Conference on Decision and Control, pp. 3723-3724, December 1998.
View at: Google Scholar
F. Ahmida and E. H. Tissir, “Exponential stability of uncertain T-S fuzzy switched systems with time delay,” International Journal of Automation and Computing, vol. 10, no. 1, pp. 32–38, 2013.
View at: Publisher Site | Google Scholar
H. Gassara, A. El Hajjaji, and M. Chaabane, “Robust control of T-S fuzzy systems with time-varying delay using new approach,” International Journal of Robust and Nonlinear Control, vol. 20, no. 14, pp. 1566–1578, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
G. Yu, Y. Mao, H. Zhang, and Y. Qin, “Robust stability analysis of uncertain switching fuzzy systems with time-varying delays,” in Proceedings of the 31st Chinese Control Conference (CCC '12), pp. 3419–3422, Hefei, China, July 2012.
View at: Google Scholar
J. P. Hespanha and A. S. Morse, “Stability of switched systems with average dwell-time,” in Proceedings of the 38th IEEE Conference on Decision and Control (CDC '99), pp. 2655–2660, December 1999.
View at: Google Scholar
M. Benrejeb, M. Gasmi, and P. Borne, “New stability conditions for TS fuzzy continuous nonlinear models,” Nonlinear Dynamics and Systems Theory—An International Journal of Research and Surveys, vol. 5, no. 4, pp. 369–379, 2005.
View at: Google Scholar | MathSciNet
M. Benrejeb, D. Soudani, A. Sakly, and P. Borne, “New discrete Tanaka-Sugeno-Kang fuzzy systems characterization and stability domain,” International Journal of Computers, Communications & Control, vol. 1, no. 4, pp. 9–19, 2006.
View at: Publisher Site | Google Scholar
M. Benrejeb, A. Sakly, K. B. Othman, and P. Borne, “Choice of conjunctive operator of {TSK} fuzzy systems and stability domain study,” Mathematics and Computers in Simulation, vol. 76, no. 5-6, pp. 410–421, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
M. Benrejeb, P. Borne, and F. Laurent, “Sur une application de la représentation en flèche à l'analyse des processus,” Rairo Automatique/Systems Analysis and Control, vol. 16, pp. 133–146, 1982.
View at: Google Scholar
S. Elmadssia, K. Saadaoui, and M. Benrejeb, “New delay-dependent stability conditions for linear systems with delay,” Systems Science and Control Engineering: An Open Access Journal, vol. 1, no. 1, pp. 2–11, 2013.
View at: Publisher Site | Google Scholar
L. T. Grujić, J. C. Gentina, and P. Borne, “General aggregation of large-scale systems by vector Lyapunov functions and vector norms,” International Journal of Control, vol. 24, no. 4, pp. 529–550, 1976.
View at: Publisher Site | Google Scholar | MathSciNet
F. Robert, “Recherche d’une M-matrice parmi les minorantes d’un opérateur linéaire,” Numerische Mathematik, vol. 9, pp. 189–199, 1966.
View at: Publisher Site | Google Scholar | MathSciNet
P. Borne, J. P. Richard, and N. E. Radhy, Systèmes non linéaires, tome 2 : stabilité-stabilisation, chapitre 2, Editions Masson, Paris, France, 1993.
A. Sakly and M. Kermani, “Stability and stabilization studies for a class of switched nonlinear systems via vector norms approach,” ISA Transactions, 2014.
View at: Publisher Site | Google Scholar
M. Kermani and A. Sakly, “Robust stability and stabilization studies for uncertain switched systems based on vector norms approach,” International Journal of Dynamics and Control, vol. 4, no. 1, pp. 76–91, 2016.
View at: Publisher Site | Google Scholar | MathSciNet
M. Kermani, A. Sakly, and F. M'Sahli, “Stability analysis and stabilization of switched linear systems based on vector norms approach,” in Proceedings of 2nd International Conference on Communications Computing and Control Applications, CCCA 2012, fra, December 2012.
View at: Publisher Site | Google Scholar
M. Kermani, A. Sakly, and F. M'Sahli, “A new stability analysis and stabilization of uncertain switched linear systems based on vector norms approach,” in Proceedings of 2013 10th International Multi-Conference on Systems, Signals and Devices, SSD 2013, tun, March 2013.
View at: Publisher Site | Google Scholar
M. Kermani and A. Sakly, “A new robust pole placement stabilization for a class of time-varying polytopic uncertain switched nonlinear systems under arbitrary switching,” Control Engineering and Applied Informatics, vol. 17, no. 2, pp. 20–31, 2015.
View at: Google Scholar
A. Jaballi, A. Sakly, and A. El Hajjaji, “Delay-independent stability criterion for switched T-S fuzzy systems with multiple delays,” in Proceedings of the 16th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering (STA '15), pp. 870–875, Monastir, Tunisia, December 2015.
View at: Publisher Site | Google Scholar
A. Jaballi, A. Sakly, and A. E. Hajjaji, “M-matrix based robust stability and stabilization for uncertain discrete-time switched TS fuzzy systems with time-varying delays,” ISA Transactions, vol. 63, pp. 60–68, 2016.
View at: Publisher Site | Google Scholar
M. Kermani and A. Sakly, “On stability analysis of discrete-time uncertain switched nonlinear time-delay systems,” Advances in Difference Equations, 2014:233, 22 pages, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
M. Kermani and A. Sakly, “Stability analysis of switched nonlinear time-delay systems,” Systems Science & Control Engineering: An Open Access Journal, vol. 2, no. 1, pp. 80–89, 2014.
View at: Google Scholar
M. Kermani, A. Jaballi, and A. Sakly, “A new stability analysis for discrete-time switched time-delay systems,” International Journal of Control, Energy and Electrical Engineering, vol. 1, pp. 68–74, 2014.
View at: Google Scholar
D. M. Kotelyanski, “Some properties of matrices with positive elements,” Matematicheskii Sbornik, vol. 31, pp. 497–505, 1952.
View at: Google Scholar | MathSciNet
P. Borne, “Nonlinear system stability: vector norm approach system and control,” in Encyclopedia, vol. 5, pp. 3402–3406, 1987.
View at: Google Scholar

Copyright

Copyright © 2017 Nawel Aoun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

781

Downloads

695

Citations