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Abstract—In this paper, the problem of switching stabilization for a class of switched nonlinear systems is studied by using average dwell time (ADT) switching ...
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Stabilization for a Class of Switched Nonlinear Systems With Novel Average Dwell Time Switching by T–S Fuzzy Modeling Xudong Zhao, Yunfei Yin, Ben Niu, and Xiaolong Zheng

Abstract—In this paper, the problem of switching stabilization for a class of switched nonlinear systems is studied by using average dwell time (ADT) switching, where the subsystems are possibly all unstable. First, a new concept of ADT is given, which is different from the traditional definition of ADT. Based on the new proposed switching signals, a sufficient condition of stabilization for switched nonlinear systems with unstable subsystems is derived. Then, the T–S fuzzy modeling approach is applied to represent the underlying nonlinear system to make the obtained condition easily verified. A novel multiple quadratic Lyapunov function approach is also proposed, by which some conditions are provided in terms of a set of linear matrix inequalities to guarantee the derived T–S fuzzy system to be asymptotically stable. Finally, a numerical example is given to demonstrate the effectiveness of our developed results. Index Terms—Average dwell time (ADT), multiple quadratic Lyapunov function, switched nonlinear systems, Takagi–Sugeno (T–S) fuzzy modeling.

I. I NTRODUCTION Many physical systems encountered in practice involve a coupling between continuous dynamics and discrete events. For example, a thermostat turning the heat on and off, a valve or a power switch open and close, and a server switching among buffers in a queueing network always coexist and interact two kinds of dynamics. Such systems are generally termed as hybrid systems which constitute a relatively new and very active study area. Meanwhile, switched systems that can be viewed as higher-level abstractions of hybrid systems, have also attracted considerable attention by a large number of researchers in recent years [13], [15], [21]. Switched systems are composed of continuous-time or discrete-time subsystems and discrete switching events [8]. The discrete switching events that may be either autonomous or controlled can yield some interesting and special properties to switched systems. For instance, the switching signals can make a switched system composed of all unstable subsystems stable, and may also make a switched system composed of all stable subsystems unstable. In the past a few years, great effort has been paid to the study of switched systems due to their numerous applications in areas such as power electronics, flight control systems, network control Manuscript received December 4, 2014; revised April 12, 2015; accepted July 12, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61573069, Grant 61203123 and Grant 61304054, in part by the Shandong Provincial Natural Science Foundation, China, under Grant ZR2012FQ019, and in part by the Program for Liaoning Provincial Excellent Talents in University, China, under Grant LJQ2014122. This paper was recommended by Associate Editor W.-Y. Wang. X. Zhao is with the College of Engineering, Bohai University, Jinzhou 121013, China, and also with Chongqing SANY High-Intelligent Robots Company, Ltd., Chongqing 401120, China (e-mail: [email protected]). Y. Yin and X. Zheng are with the College of Engineering, Bohai University, Jinzhou 121013, China (e-mail: [email protected]; [email protected]). B. Niu is with the College of Mathematics and Physics, Bohai University, Jinzhou 121013, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCYB.2015.2458896

systems, etc. Among these studies, major efforts are devoted to the robust control [22], H∞ filtering [18], and particularly the stability [9], [23], [29]. As far as the stability of switched systems is concerned, it is well known that traditional common Lyapunov functions for switched systems may give conservative stability conditions, whereas Zhao et al. [26] presented a new and efficient method for stability analysis of switched systems. Then, Zhang and Gao [24] applied a class of Lyapunov-like function to study the problem of stabilization for switched systems comprising unstable subsystems. It is worth noting that the aforementioned works all require the existence of (at least one) stable modes to guarantee the stability of a switched system. The basic idea of these works is to activate the stable subsystem for sufficiently large time that we could call slow switching, to compensate the state divergence caused by unstable subsystems. However, when a switched system is composed of subsystems that may be all unstable, this promising idea clearly is not feasible. Therefore, it will be very meaningful and challenging to carry out the studies on stabilization of switched systems with possibly all unstable subsystems. In [27], such a problem was solved by introducing the invariant subspace theory. On the other hand, it is worth mentioning that the aforementioned works are all focused on switched linear systems, and up to now, few results have been reported for switched nonlinear systems [12], [20]. Switched nonlinear systems can be found in various domains [17], [25], [30], such as mobile robots, network control systems, automotive, dc converters, and so on. Note that linear control methods rely on the key assumption of small range operation for the linear model to be valid. During the past decade, in order to handle nonlinearities, the Takagi–Sugeno (T–S) fuzzy model [16] has been introduced, which can approximate smooth nonlinear functions to any arbitrary accuracy. It has been shown that T–S fuzzy models can be used to analyze and synthesize nonlinear systems effectively [5], [6], [11], [28]. Following the remarkable developments in both theory and application of fuzzy control systems, many control issues for switched nonlinear systems have also been investigated based on T–S fuzzy modeling approaches. To list a few, the problems of stability analysis and stabilization were considered in [14]; state-feedback controller design for discrete-time switching fuzzy systems has been studied in [2]; H∞ controller design for fuzzy dynamic systems based on a piecewise Lyapunov function was investigated in [4]. However, these works did not consider the case that all the subsystems are unstable. As can be seen from the above illustrations, it is of both theoretical and practical importance to investigate the problem of stabilization for switched nonlinear systems with possibly all unstable subsystems. In this case, an efficient switching signal needs to be designed to stabilize the system [3], [19]. Note that time constraint switching signal which can also be called slow switching is viewed as an important class of switching signals, and has been extensively investigated during the past several decades. The slow switching means that a switching signal dwells on each subsystem longer than a specified constant time [10]. This constant time is generally termed as dwell time. However, in some circumstances, specifying a fixed dwell time

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may be restrictive. Thus, it is of interest to extend the concept of dwell time to the average dwell time (ADT) that allows the possibility of switching signals with dwell time being occasionally less than the specified constant [7]. It is noted that ADT turns out to be very useful for analysis and synthesis of switched nonlinear systems. However, to the best of our knowledge, up to now, there is not any literature studying the problem of stabilization of switched nonlinear systems composed of unstable subsystems via ADT switching design. All the above observations motivate us to carry out this paper: based on the multiple Lyapunov function (MLF) theory and T–S fuzzy modeling approach, propose a novel ADT switching that is different from the traditional ADT, and can be used to solve the switching stabilization for a given switched nonlinear system composed of possibly all unstable modes to improve some existing results on this issue. In this paper, the problems of stability and stabilization for switched nonlinear systems with a new class of switching signals will be studied in continuous-time context. A novel ADT is first proposed for switching stabilization design for switched nonlinear systems composed of unstable subsystems. In addition, the stability and stabilization conditions of the systems with proposed ADT switching are derived, and numerically easily verified stabilization conditions of the corresponding switched nonlinear systems are also formulated in terms of a set of linear matrix inequalities by using T–S fuzzy modeling and a new class of multiple quadratic Lyapunov functions. The remainder of this paper is organized as follows. Section II reviews a necessary definition on stability analysis of switched systems, and defines the concept of a novel ADT which characterizes a different set of switching signals from traditional ADT. In Section III, stabilization criteria for switched nonlinear systems with the proposed ADT are derived, upon which some improved conditions for stability and stabilization of considered systems are also developed by using T–S fuzzy modeling and establishing a new type of Lyapunov functions. Section IV provides a numerical example to demonstrate the feasibility and effectiveness of the proposed techniques, and Section V concludes this paper.

The switched nonlinear system (1) can be described by fuzzy rules, and the pth fuzzy subsystem is represented as follows. i and · · · and θ (t) is M i , THEN Model Rule Rip : IF θ1 (t) is Mp1 l pl x˙ (t) = Api x(t), t ≥ t0 , i ∈ R = {1, 2, . . . , r}, p ∈ S

In this paper, the notations used are standard. R and Rn represent the field of real numbers and n-dimensional Euclidean space, respectively; the notation  ·  refers to the Euclidean norm. C1 denotes the space of continuously differentiable functions, and a function α : [0, ∞) → [0, ∞) is said to be of class K if it is continuous, strictly increasing, and α(0) = 0. Class K∞ denotes the subset of K consisting of all those functions that are unbounded. A function β : [0, ∞) × [0, ∞) → [0, ∞) is said to be of class KL if β(·, t) is of class K for each fixed t > 0 and β(r, t) is decreasing to zero as t → ∞ for each fixed r ≥ 0. In addition, the notation P > 0(≥ 0) means that P is a real symmetric and positive definite (semi-positive definite) matrix. II. P ROBLEM F ORMULATION AND P RELIMINARIES This section presents some definitions and preliminary results which will be used throughout this paper. Consider the following switched nonlinear system: m 

δp (σ (t))fp (x(t), t), x(t0 ) = x0 , t ≥ t0

(1)

p=1

where x(t) ∈ Rn is the state vector, and x0 and t0 ≥ 0 denote the initial state and initial time, respectively; σ (t) is a switching signal

(3)

i ( j = 1, 2, . . . , l) is the fuzzy where x(t) ∈ Rn is the state vector; Mpj set, and r is the number of IF-THEN rules; θ1 (t), θ2 (t) . . . θp (t) are the premise variables. Furthermore, Api , i ∈ R, p ∈ S is a real matrix with appropriate dimensions. Thus, through the use of fuzzy blending, the global model of the pth fuzzy subsystem can be given by

x˙ (t) = A(h(t))x(t) r  hpi (θ(t))Api x(t), p ∈ S =

(4)

i=1

hpi (θ(t)) are the normalized membership functions satisfying l i j=1 Mpj (θj (t)) ≥0 hpi (θ(t)) = r l i i=1 j=1 Mpj (θj (t)) r 

(5)

hpi (θ(t)) = 1

i=1 i (θ (t)) represent the grade of the membership function where Mpj j i . Finally, we can describe switched of premise variable θj (t) in Mpj nonlinear system (1) in the following form:

x˙ (t) =

A. Notations

x˙ (t) =

which is a piecewise constant function from the right of time and takes its values in the finite set S = {1, . . . , m}, where m > 1 is the number of subsystems. fp : Rn × R −→ Rn are smooth functions for any σ (t) = p ∈ S. Moreover, all the subsystems in system (1) may be unstable. Also, for a switching sequence, 0 < t1 < . . . < tk < tk+1 < . . . , σ (t) may be either autonomous or controlled. When t ∈ [tk , tk+1 ), we say σ (tk )th mode is active, i.e., the indication functions δp (σ (t)) satisfy  1, if σ (t) = p (2) δp (σ (t)) = 0, otherwise.

m  r 

δp (σ (t))hpi (θ(t))Api x(t).

(6)

p=1 i=1

Remark 1: Here, it should be pointed out that T–S fuzzy models can approximate any smooth switched nonlinear system to any accuracy on a compact set, i.e., a switched system composed of subsystems described by smooth nonlinear functions can be represented by (3) via choosing appropriate fuzzy rules. Definition 1 [8]: The equilibrium x = 0 of switched system (1) is global asymptotically stable (GAS) under a certain switching signal σ (t) if there exists a KL function β such that the solution of the system satisfies the inequality x(t) ≤ β(x(t0 ), t), ∀t ≥ t0 , with any initial conditions x(t0 ). In this paper, our goal is to find a set of admissible switching signals with ADT property, such that the switched system (1) is GAS. For this purpose, let us first define a new class of ADT switching. Definition 2: For a switching signal σ (t) and each T ≥ t ≥ 0, let Nσ (T, t) denote the number of discontinuities of σ (t) in the interval (t, T). We say that σ (t) has an ADT τa if there exist two positive numbers N0 (we call N0 the chatter bound here) and τa such that T −t , ∀T ≥ t ≥ 0. (7) τa Remark 2: In Definition 2, it should be noted that unlike the traditional ADT requiring Nσ (T, t) ≤ N0 + (T − t/τa ), ∀T ≥ t ≥ 0, our proposed ADT requires Nσ (T, t) ≥ N0 + (T − t/τa ), ∀T ≥ t ≥ 0. Nσ (T, t) ≥ N0 +

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Since all the subsystems may be unstable, it is impossible to use the decrement energy of Lyapunov function produced by staying in stable subsystems to compensate the increment energy of Lyapunov function yielding in unstable subsystems. So, in this paper, we turn to the idea of compensating the increment energy of the Lyapunov function by designing switching behavior with our proposed ADT property. Also, it is worth noting in Definition 2 that Nσ (T, t) ≥ N0 + (T − t/τa ) ⇐⇒ (T − t/Nσ (T,t) − N0 ) ≤ τa , ∀T ≥ t ≥ 0, which means that on average the dwell time between any two consecutive switchings is no more than a common constant τa for all system modes. Lemma 1 [1]: For given positive scalars ηpk > 1, cpk and μ satisfying 0 < μ < (cpk ηpk /ηpk + 1), define  εpk  

(ηpk − 1)

1

arctan h cp k

c2p μ2 k 4 − ηpk

2 (ηpk + 1) − 2μ

c2p μ2 k 4 − ηpk

.

(8)

Let ϕ(t) be the solution of the following initial value problem: ⎧ ⎨ ϕ˙ (t) = − εpk (ϕ 2 (t) − c ϕ (t) + μ2 ), t ∈ [t , t p pk p k k+1 ] tk+1 −tk p ηpk (9) ⎩ ϕp (tk ) = μ . ηpk

It is clear that this function is piecewise differentiable along solutions of (1). When t ∈ [tk , tk+1 ), we get from (13) that ˙ W(t) = −λe−λt Vp (x(t), t) + e−λt V˙p (x(t), t) ≤ −λe−λt Vp (x(t), t) + e−λt λVp (x(t), t) = 0.

(18)

Thus W(t) is nonincreasing when t ∈ [tk , tk+1 ). This together with (14) gives that



+ + + + = e−λtk+1 Vp x tk+1 , tk+1 W tk+1

− − − , tk+1 ≤ μe−λtk+1 Vp x tk+1

− = μW tk+1 ≤ μW(tk ).

(19)

By integrating this for t ∈ [tk , tk+1 ), it yields     W T − ≤ W tNδ ≤ μW tN−  δ  ≤ μW tNδ −1 ...

Then, for t ∈ [tk , tk+1 ] 

ϕ˙ p (t) ≥ 0, μ = ϕ (t ) ≤ ϕp (t) ≤ ϕp (tk+1 ) = μ. p k ηp

≤ μNδ W(t0 ). (10)

k

Moreover, if tk+1 −tk ≥ τ ∗ , it can be obtained that for t ∈ [tk , tk+1 ]



εp ϕ˙p (t) ≤ − ∗k ϕp2 (t) − cpk ϕp (t) + . τ ηpk μ2

(11)

In this section, we consider the problem of stabilization for switched nonlinear systems described in the previous section. Next, we are in a position to provide the first version of stabilization conditions for switched nonlinear systems (1) in the following theorem by designing ADT switching signals with our proposed property. Theorem 1: Consider switched nonlinear system (1). Suppose that there exist a switching sequence ξ = {t0 , t1 , . . . , tk , . . . , tNσ (t)} satisfying (7), a set of C1 non-negative functions Vp : Rn ×R → R, p ∈ S, two class K∞ functions α1 and α2 , and two positive numbers λ > 0 and 0 < μ < 1 such that ∀(p × q) ∈ S × S α1 (x(t)) ≤ Vp (x(t), t) ≤ α2 (x(t)) (12) ˙ Vp (x(t), t)) ≤ λVp (x(t), t) (13)



(14) Vq x tk+ , tk+ ≤ μVp x tk− , tk− −lnμ τa ≤ . (15) λ Then switched system (1) is globally asymptotically stable under the switching sequence ξ generated by σ (t). Proof: Without loss of generality, we denote ξ = {t0 , t1 , . . . , tk , . . . , tNσ (t)} as the switching sequence on time interval [0, T] for any T > 0, t0 = 0. Then, we establish an MLF for switched nonlinear system (1) as follows: m 

δp (σ (t))Vp (x(t), t).

(16)

δp (σ (t))Vp (x(t), t).

(17)

p=1

Then, we consider the function W(t) = e−λt

m  p=1

One can easily obtain from the definition of W(t) that e−λT Vδ(T − ) (x(T), T) ≤ μNδ Vδ(t0 ) (x(t0 ), t0 ).

(21)

Moreover, it can be derived from (7) and (21) that Vδ (T − ) (x(T), T) ≤ eλT eNδ ln μ Vδ(t0 ) (x(t0 ), t0 )

III. M AIN R ESULTS

V(x(t), t) =

(20)



N0 + τTa ln μ λT Vδ(t0 ) (x(t0 ), t0 ) ≤e e

ln μ λ+ T τa = eN0 ln μ e

× Vδ(t0 ) (x(t0 ), t0 ).

(22)

Finally, we can conclude from (22) that, if τa satisfies the condition in (15), then Vδ(T − ) (x(T), T) exponentially converges to zero as −1

T → ∞. By (12), we can get x(T)α1 (μN0 eλT α2 (x0 ), which verifies the global asymptotic stability by Definition 1. We conclude that switched nonlinear system (1) is asymptotically stabilized by our proposed ADT switching signals (7) satisfying (15) if the conditions (12)–(14) hold. This completes the proof. Remark 3: For a given switched nonlinear system composed of unstable subsystems, a basic problem raised by Theorem 1 is how to find a switching sequence to guarantee the system stable under the corresponding set of proposed ADT switching signals. However, for a general switched nonlinear system model composed of unstable subsystems, the switching sequence ξ is not fixed in advance. Therefore, it is hard to check condition (14) for all switching instants tk as k → ∞. As a result, we utilize T–S fuzzy modeling approach to represent nonlinear system (1), to develop more applicable results. Next, the following theorem for switched T–S fuzzy system (6) can be given on the basis of the Theorem 1. Theorem 2: Consider switched T–S fuzzy system (6), and let λ > 0, ηp > 1, 0 < μ < 1 satisfying 0 < μ < {(cp ηp /ηp + 1)}, and τ ∗ > 0 be given constants. If there exists a set of matrices Pp > 0, Qp > 0, and Gp , p ∈ S such that ∀i ∈ R, p = q, ∀(p × q) ∈ S × S ATpi Pp + Pp Api + GTp + Gp − Qp − λPp +

εp cp ≤0 τ∗

(23)

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⎡ ⎣

ε

− τ p∗ Pp GTp

⎤ Gp   ⎦≥0 ε μ2 − τp∗ η Pp + μQp

(24)

p

Pp − ηp μPq ≤ 0

(25)

 where εp = (1/ (c2p /4) − (μ2 /ηp )) arctan h((ηp − 1)  (c2p /4) − (μ2 /ηp ))(cpk /2)(ηp + 1) − 2μ, then, the system is GAS for any switching signal with ADT satisfying

−lnμ τ ∗ ≤ τa ≤ . (26) λ Proof: Here, we construct a multiple quadratic Lyapunov function for switched T–S fuzzy system (6) as follows: Vp (x(t), t) = ϕp (t)xT (t)Pp x(t), ∀p ∈ S

(27)

where ϕp (t) is defined in (9) satisfying ϕp (tk+ ) = ϕp (tk ) for t ∈ (tk , tk+1 ). From the Lemma 1, it can be seen that ⎧ μ   ⎨ η = ϕp (tk ) ≤ ϕp (t) ≤ ϕp tk+1 = μ p 2 (28) ⎩ 0 ≤ ϕ˙p (t) ≤ − εp∗ ϕp2 (t) − cp ϕp (t) + μ ηp . τ Moreover, we obtain from (25), (27) and (28) that

Vp x tk+ , tk+ = ϕp (tk )xT (tk )Pp x(tk ) μ T = x (tk )Pp x(tk ) ηp ≤ μ2 xT (tk )Pp x(tk ) = μϕq tk− xT (tk )Pp x(tk )

= μVq x tk− , tk− .

It follows from (23), (30), and (31) that: V˙ p (x(t), t) − λVp (x(t), t) r  ≤ hpi (θ(t))xT (t)ϕp (t) ATpi Pp + Pp Api + GTp + Gp i=1

− Qp + ≤ 0.

IV. N UMERICAL E XAMPLE

(29)

V˙ p (x(t), t) − λVp (x(t), t)

We provide the following numerical example in this section to verify our main results developed in this paper. Using a T–S fuzzy model to represent a given switched nonlinear system composed of all unstable subsystems, we will design a switching signal with our proposed ADT property to asymptotically stabilize the system. Consider the switched nonlinear system composed of the following two subsystems:

1 =

= ϕ˙p (t)xT (t)Pp x(t) + ϕp (t)˙xT (t)Pp x(t) + ϕp (t)xT (t)Pp x˙ (t) − λϕp (t)xT (t)Pp x(t)

εp μ2 T 2 = − ∗ ϕp (t) − cp ϕp (t) + x (t)Pp x(t) + ϕp (t) τ ηp r

 × hpi (θ (t))xT (t) ATpi Pp + Pp Api − λPp x(t) i=1

=

i=1

2 =



εp hpi (θ (t))xT (t) − ∗ ϕp2 (t)Pp + ϕp (t) τ

cp εp × ATpi Pp + Pp Api + ∗ Pp − λPp τ  εp μ2 − ∗ Pp x(t). τ ηp

⎧ 1 x˙ 1 (t) = −6.48x1 (t) − 7.32 x (t) ⎪ ⎪ 1+e−(x1 +4) 1 ⎪ ⎪ 1 ⎨ x2 (t) + 4.98x2 (t) + 5.52 1+e−(x1 +4)

1 ⎪ x (t) x˙ 2 (t) = −5.48x1 (t) − 6.12 ⎪ ⎪ 1+e−(x1 +4) 1 ⎪ ⎩ 1 x (t) + 4.23x2 (t) + 4.62 1+e−(x1 +4) 2 ⎧ 1 x˙ 1 (t) = 5.77x1 (t) + 3.36 x (t) ⎪ ⎪ 1+e−(x1 +4) 1 ⎪ ⎪ 1 ⎪ ⎨ − 6.82x2 (t) − 3.96 −(x1 +4) x2 (t) 1+e

1 ⎪ x˙ 1 (t) = 7.52x1 (t) + 4.36 x (t) ⎪ ⎪ 1+e−(x1 +4) 1 ⎪ ⎪ 1 ⎩ − 8.92x2 (t) − 5.16 −(x1 +4) x2 (t). 1+e

(30) From (24), one can obtain that   T  εp P Gp ϕp (t)x(t) τ∗ p 2 ε μ x(t) GTp τp∗ η Pp − μQp p   ϕp (t)x(t) × + (μ − ϕp (t))xT (t)Qp x(t) x(t)  εp μ2 = xT (t) ϕp2 (t) ∗ Pp + ϕp (t) τ ηp 

εp T 2 × Gp + Gp − Qp + ϕp (t) ∗ Pp x(t) τ ≥ 0.

(32)

Finally, one can readily conclude by Theorem 1 that switched T–S fuzzy system (6) is GAS for any switching signal with our proposed ADT (7). Remark 4: In Theorem 2, based on a novel type of Lyapunov functions, a set of switching signals with our proposed ADT property is designed to ensure the asymptotic stability of a given switched T–S fuzzy system composed of possibly all unstable subsystems, whose stability cannot be guaranteed under arbitrary switching. More important, the obtained stability condition is formulated in terms of linear matrix inequlities, which compared with Theorem 1, has the advantage of being efficiently solved by LMI toolbox.

On the other hand, it can be derived from (27) and (28) that

r 

cp εp Pp − λPp x(t) ∗ τ

The state trajectories as shown in Figs. 1 and 2 demonstrate that both the subsystems 1 and 2 are unstable. Next, we are interested in designing a kind of switching signal σ (t) with property (7) to asymptotically stabilize the system. First, we can formulate the T–S fuzzy model of switched nonlinear system in the following. When p = 1, the system 1 can be written as  1 −6.48 − 7.32 1+e−(x1 +4) x˙ (t) = 1 −5.48 − 6.12 1+e−(x1 +4)    1 4.98 + 5.52 x1 (t) 1−e−(x1 +4) . 1 x2 (t) 4.23 + 4.62 −(x +4) 1+e

(31)

1

For the nonlinear term (1/1 + e−(x1 +4) ), define θ (t) (1/1 + e−(x1 +4) ). Then, we have   −6.48 − 7.32θ(t) 4.98 + 5.52θ(t) x˙ (t) = −5.48 − 6.12θ(t) 4.23 + 4.62θ(t)   x × 1 . x2

=

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Fig. 1.

State response of the subsystem

5



Fig. 3. State responses of switched nonlinear system under switching signal σ (t) with τa = 0.6.

1.

where 1 M11 (θ(t)) =1− M11 (θ(t)) + M12 (θ(t)) 1 + e−(x1 +4) 1 M12 (θ(t)) = h12 (θ(t)) = . M11 (θ(t)) + M12 (θ(t)) 1 + e−(x1 +4) h11 (θ(t)) =

Similarly, the second nonlinear subsystem 2 is represented by the following fuzzy model. Model Rule R12 : IF θ(t) is about 0, THEN x˙ (t) = A21 x(t). Model Rule R22 : IF θ(t) is about 1, THEN x˙ (t) = A22 x(t) Fig. 2.

State response of the subsystem

where



2.

Next, calculate the minimum and maximum values of θ(t). The minimum and maximum values of θ (t) are 0 and 1, respectively. From the minimum and maximum values, θ (t) can be represented by θ(t) = (1/1 + e−(x1 +4) ) = M11 (θ(t)) × 0 + M12 (θ(t)) × 1, where M11 (θ (t)) + M12 (θ (t)) = 1. Therefore, the membership functions can be calculated by M11 (θ (t)) = 1 −

1 1 + e−(x1 +4)

, M12 (θ(t)) =

1 1 + e−(x1 +4)

.

Then, the first nonlinear subsystem 1 is represented by the following fuzzy model. Model Rule R11 : IF θ (t) is about 0, THEN x˙ (t) = A11 x(t). Model Rule R21 : IF θ (t) is about 1, THEN x˙ (t) = A12 x(t). Here A11 =



−6.48 −5.48

 4.98 , 4.23

 A12 =

−13.8 −11.6

 10.5 . 8.85

Thus, through the use of fuzzy blending, the global mode of the 1th fuzzy subsystem can be given by x˙ (t) = A(h(t))x(t) 2  h1i (θ(t))A1i x(t) = i=1

A21 =



5.77 7.52

 −6.82 , −8.92

 A22 =

9.13 11.88

 −10.78 . −14.08

Then, by using Theorem 2, if we choose c1 = 1.1, c2 = 1.2, η1 = 2.2, η2 = 2.3, τ ∗ = 0.3, μ = 0.6, λ = 0.7, the feasible solutions are obtained as follows:     0.2363 −0.1804 0.2649 −0.1885 , P2 = P1 = −0.1804 0.2430 −0.1885 0.2206     0.1177 0.2984 −0.3165 −0.1537 , G2 = G1 = −0.0566 −0.3519 0.2290 0.2134     0.6383 0.0755 7.3372 0.0825 , Q2 = . Q1 = 0.0755 1.2117 0.0825 0.6420 Furthermore, we generate one possible switching sequence by our proposed ADT switching (τa = 0.6 < −(lnλ/λ) = 0.73). The corresponding state responses of the system under initial state condition x(0) = [10, 15]T , are shown in Fig. 3, from which we can see that the switched nonlinear system is stable under the designed ADT switching. V. C ONCLUSION The problems of stabilization for switched nonlinear systems composed of unstable subsystems are studied in this paper by using a new defined class of ADT switching. The proposed ADT switching is different from the traditional ADT switching in the literature. The stabilization result for switched nonlinear systems composed of all unstable subsystems is first derived on the basis of our proposed switching signals. Moreover, the T–S fuzzy modeling approach

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