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 pseudoovervaluing 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 andGentina 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.


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][4][5][6][7][8][9][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][13][14][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: delayindependent criteria, which are applicable to delay of arbitrary size [17][18][19][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][23][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][26][27][28][29][30][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][33][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][37][38][39][40][41][42][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.

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 () ∈ R  is the state vector,  > 0 is the time delay, Φ : [−, 0] → R  is a differentiable vector valued initial function,   : R  → R  are sufficiently regular functions that are parametrized by the index set  = {1, 2, . . ., }, () : R  →  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 () = [ 1 (), . . .,   ()]  whose components   () are given by It is obvious that ∑  =1   () = 1 ∀ ≥ 0. 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 by where By using the product inference engine and the center of average defuzzification, the th TS fuzzy subsystem can be inferred as where () = [ Equation ( 8) can be written as In the sequel, ℎ  (()) will be simplified by ℎ  .

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.

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 𝜎(𝑡) = 𝑖, 𝑖 ∈ 𝐼, if matrix
is the opposite of an -matrix, where Proof.Let  ∈ R *  + with components (  > 0, ∀ = 1, . . ., ) and consider the radially unbounded candidate Lyapunov functional for each fuzzy submodel ∑   ,  given by where It is clear that ( 0 ) < ∞ and that a common Lyapunov functional for all the submodels can be given by The right Dini derivative of () along the trajectory of system (10) gives with sgn ( ()) = ( Then, Finally, we obtain where On the other hand, if we suppose that   is the opposite of an -matrix and according to the -matrices properties, we can find a vector  ∈ R *  + (  > 0 ∀ = 1, . . ., ) satisfying the relation: we can write This completes the proof of Theorem 5.

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 () ∈ R  is the state vector and  A change of variable under the form  +1 () =  () (),  = 0, . . .,  − 1, 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 gives where   =  −1   ,   =  −1   , and  is the corresponding passage matrix such that Arrow form matrices   ,  = 1, . . .,  and  ∈ , are given by with whereas matrices   are given by with Note that   ,  = 1, . . ., −1, are distinct constant parameters that can be chosen arbitrarily and    () is the instantaneous characteristic polynomial of matrix   given by Therefore, 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, . Consequently, we can construct for each individual submodel ∑   ,  the following overvaluing system: where ) Finally, the common overvaluing matrix is given by with 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   < 0, ∀ = 1, . . .,  − 1,   ̸ =   , ∀ ̸ = , 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 Δ  , are of alternating signs; that is, It is clear that, for an arbitrary choice of   < 0, relation ( 43) is satisfied for  = 1, . . .,  − 1.

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.
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, where we deduce that it suffices that  , is the opposite of an matrix to conclude to the asymptotic stability of system (52).

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 delaydependent stability criteria for systems described by the following multiple time-delayed differential equation: The same change of variable  +1 () =  () (),  = 0, . . ., −1, as in Section 3.2 yields to the new state space representation: where () = [ 1 (), . . .,   ()]  is the state vector and for each  ∈ ,  ∈ [1, . . ., ],   and  , are given by ) . (60) A change of base of the form () = () allows system (59) to be represented by the arrow form matrix: where   is given by (30) and  , becomes with Define the new polynomials: Finally, we can compute the common overvaluing matrix as follows: Hence, we can state Theorem 10.
Theorem 10.System (59) is globally asymptotically stable under an arbitrary switching rule () =  ∈ , if there exist Proof.The same proof is as in Theorem 6; just replace matrix   by  , .

Illustrative Examples
Example 1.Consider a switched system composed of three second-order subsystems ∑   ,  ,  ∈ {1, 2, 3}; each subsystem is represented by a TS fuzzy time-delay model as follows: Subsystem A transformation of state matrices under the arrow form gives For  ∈ {1, 2, 3} and  ∈ {1, 2}, parameters   and   are computed as follows: For an arbitrary choice of  = −1, we obtain, for example, for subsystem For ℎ 2 = 1−ℎ 1 and by using the center of average defuzzification, state matrix   = ℎ 1  1 + ℎ 2  2 and delayed-state matrix ) are given, respectively, by Denote by   ,  ∈ {1, 2} and  ∈ {1, 2, 3}, the minimal pseudoovervaluing matrix of the fuzzy submodel ∑   ,  such that where At 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 1.25 s to ensure the asymptotic stability of all the submodels individually.Taking into account this value, we can notice that ∑  23 can be chosen as the common comparison system for the six submodels ∑   ,  ( ∈ {1, 2} and  ∈ {1, 2, 3}).Then, the TS fuzzy switched time-delay system is asymptotically stable under an arbitrary switching law if The switching signal is plotted on Figure 1.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.
Denote by  , , ,  ∈ {1, 2}, the minimal overvaluing matrix of subsystem of index () and relative to the vector norm where Applying Theorem 10, we can draw Table 2 which gives a sufficient stability condition 0 <  2 <  () 2,max for each submodel.
Finally, we can construct the following common comparison system ∑  , such that Consequently, the TS fuzzy switched time-delay system is asymptotically stable under arbitrary switching if In this example, we can notice that maximum bounds found for the delays  1 and  2 are small.Indeed, this is justified by the multiplicity of time delay, by the choice of delay matrices  , , , ,  ∈ {1, 2}, 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  1 = 0.01 s and  2 = 0.033 s, particular values of weighting factors ℎ 1 = 0.2 and ℎ 2 = 0.8, and an initial state vector () = [1 −2]  ∀ ∈ [−0.033, 0], 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 by where is the local feedback gain vector.
The TS fuzzy switched system is then represented in the state space by where   = ( 0 1 0 −  ) and  = (0 1)  .
We can also write ℎ     ( − )} .(90) To tune parameters ( 0  , The pseudo-overvaluing matrix for each submodel is constructed as follows: with  (97) Finally, the common comparison matrix that verifies the properties of the opposite of an -matrix is given by This 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.

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 delaydependent 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.