Exponential stability analysis and L2 -gain control synthesis for positive switched T–S fuzzy systems

Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91

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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs

Exponential stability analysis and L2 -gain control synthesis for positive switched T–S fuzzy systems Shuo Li, Zhengrong Xiang * School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

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Article history: Received 11 October 2016 Accepted 7 August 2017

Keywords: Positive switched systems T–S fuzzy modeling Weighted L2 -gain performance Stabilization Piecewise quadratic copositive Lyapunov function

a b s t r a c t In this paper, the problems of exponential stability analysis and weighted L2 -gain control synthesis for positive switched T–S fuzzy systems are addressed. Firstly, by proposing a piecewise quadratic copositive Lyapunov function, a less conservative stability condition for positive switched T–S fuzzy systems is derived for the first time. Then, sufficient conditions for the positive switched T–S fuzzy system to be exponentially stable with a weighted L2 -gain performance are obtained. Based on the obtained results, a state-feedback controller by the parallel distributed compensation scheme is designed to stabilize the positive switched T–S fuzzy system, while guaranteeing the prescribed weighted L2 -gain performance and positivity in closed-loop. Finally, two numerical examples are presented to demonstrate the effectiveness of the obtained theoretical results. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction In practice, physical systems involve variables that have nonnegative property, such as population levels, absolute temperature, concentration of substances in chemical processes and so on. Such systems are referred to as positive systems [1,2], whose states and outputs are nonnegative whenever the initial conditions and inputs are nonnegative. Positive systems are frequently encountered in various fields, for instance, pharmacokinetics [3], chemical reaction [4] and internet congestion control [5]. The positivity will bring some interesting properties to positive systems, while it will also lead to some difficulties in analysis and synthesis of such systems. For positive systems, the controller design is more difficult than that for general systems since the positivity constraints need to be considered. Otherwise, the mathematical model of such systems might move into infeasible regions, causing loss of stability or performance. In recent years, much efforts have been devoted to positive systems such as realizability [6], reachability/controllability [7], particularly stability [8,9], and control synthesis [10–12]. Many practical systems are always subjected to abrupt variations in their structures and parameters, such as failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment changes. Switched system, which consists of a finite number of subsystems and a logical rule orchestrating the switching between these subsystems, can be used to characterize such kind of systems [13–15]. When positive systems experience the switching-type changes in their parameters, they become positive switched systems, which have been highlighted by many researchers for their broad applications in communication systems [16], formation flying [17], and systems theories [18]. The stability and stabilization problems are the main concerns in the field of positive switched systems [19–21]. In the Lyapunov stability framework, the diagonal quadratic Lyapunov function, the quadratic Lyapunov function and the copositive Lyapunov function have been investigated and widely used for positive switched systems.

*

Corresponding author. E-mail address: [email protected] (Z. Xiang).

http://dx.doi.org/10.1016/j.nahs.2017.08.006 1751-570X/© 2017 Elsevier Ltd. All rights reserved.

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S. Li, Z. Xiang / Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91

It is noted that these results are only applicable to linear positive switched systems, and few perfect theoretical results have been obtained on nonlinear positive switched systems. An important reason for this is that the existing approaches for analyzing linear positive switched systems, such as common linear copositive Lyapunov function or multiple linear copositive Lyapunov function, cannot be directly applied to nonlinear positive switched systems. Also, the nonlinearity problem has been a hard topic in the science and engineering fields, and how to define the positivity of a nonlinear system is not an easy work. As nonlinear features widely exist in many practical cases, the stability and control problems need to be investigated for nonlinear positive switched systems. On the other hand, Takagi and Sugeno have proposed the Takagi– Sugeno (T–S) fuzzy modeling approach in 1985 [22], it can approximate a complex nonlinear system to any accuracy on a compact set, and provides an efficient method to tackle some problems of nonlinear systems [23]. The main idea of the T–S fuzzy modeling is to obtain linear time-invariant models close to the nonlinear system in some regions of the state space and then combine these linear models using nonlinear fuzzy membership functions. Through such a modeling approach, a nonlinear system can be transformed into a framework composed of several linear subsystems [24]. As a result, some existing techniques developed for linear systems can be used for nonlinear systems [25]. In recent years, numerous results have been obtained for nonlinear systems by using T–S fuzzy modeling approach [26–32], and some results are on nonlinear positive systems [29–32]. It should be recognized that the switching features, which widely exist in practice, were not considered in the aforementioned results of positive T–S fuzzy systems [29–32]. For efficient control design, the switching phenomenon is essential to be taken into consideration, and the results need to be extended to positive switched T–S fuzzy systems. Due to the coupling between the fuzzy characteristics, the switching features and the positivity constraints of state valuables, positive switched T–S fuzzy systems become difficult and complex to study. Moreover, the aforementioned results on positive T–S fuzzy systems [29–32] are based on nominal systems without taking exogenous disturbances into account. As exogenous disturbances are frequently encountered in practice, it is necessary and significant to further consider positive switched T–S fuzzy systems with this phenomenon. In consideration of exogenous disturbances, the system’s performance analysis is also an important issue for positive switched T–S fuzzy systems, since one might consider whether the system can tolerate a certain amount of disturbances. When taking exogenous disturbances into account, the problems of stability analysis and control synthesis become more complicated and challenging. However, until now, the problems of exponential stability analysis and weighted L2 -gain control synthesis for positive switched T–S fuzzy systems have not been studied. It should be pointed out that those developed methods employed in the above-mentioned works, such as the popular quadratic Lyapunov function and the excellent linear copositive Lyapunov function methods will lead to conservative stability results to a certain extent, if we directly use them to positive switched T–S fuzzy systems. A piecewise quadratic copositive Lyapunov function (PQCLF) is proposed in this paper to avoid these weaknesses. The PQCLF is a new type of quadratic Lyapunov function that generalizes the linear copositive Lyapunov function and the quadratic Lyapunov function. All the above observations lead to that the topic addressed in this paper is interesting but full of challenge. In this paper, we investigate the problems of exponential stability analysis and weighted L2 -gain control synthesis for positive switched T–S fuzzy systems. The main contributions of this paper lie in the following aspects: (1) Different from the general quadratic Lyapunov function, a PQCLF is proposed for the first time in this paper; (2) By applying the PQCLF approach, a less conservative stability condition is firstly derived for a class of positive switched T–S fuzzy systems; (3) In consideration of the exogenous disturbances, the system’s disturbance attenuation performance (the weighted L2 -gain performance) is analyzed for the first time for the exponentially stable positive switched T–S fuzzy systems; (4) A fuzzy state-feedback controller via the parallel distributed compensation (PDC) scheme is designed such that the corresponding closed-loop system is exponentially stable with a prescribed weighted L2 -gain performance while preserving positivity in closed-loop. The remainder of the paper is organized as follows. The problem formulation and some necessary definitions are provided in Section 2. The problems of exponential stability analysis, weighted L2 -gain performance analysis and controller design are addressed in Section 3. Section 4 provides two numerical examples to demonstrate the effectiveness of the obtained theoretical results. Concluding remarks are given in Section 5. Notations Z and Z+ are the sets of nonnegative and positive integers, respectively. A≻0(≺, ≻, ≺) means that all entries of matrix A are nonnegative (nonpositive, positive, negative); A ≻ B(A≻B) means that A − B ≻ 0(A − B≻0) . AT means the transpose of matrix A. R(R+ ) is the set of all real (positive real) numbers; Rn (Rn+ ) is the set of all n-dimensional real (positive m T n {m} real) vector space; Rm×n is the set of all m × n-dimensional real matrices. x[m] = [xm xm is a base vector 1 2 · · · xn ] , ∀x ∈ R ; x (n−m+1)×n containing all homogeneous monomials of degree m in x. Im = [0(n−m+1)×(m−1) |In−m+1 ] ∈ R is a matrix composed of the rows m, m + 1, . . . , n of In ; ϑ (n, m) ≡ (n − m + 1)!/((n − 1)!m!). P > 0(≥ 0) means that P is a real symmetric and positive definite (semi-positive definite) matrix. Aij means the (i, j)th entry of a matrix A, and xi means the ith entry of a vector x. The matrix A is said to be Metzler if it satisfies Aij ≥ 0 for i ̸ = j. All matrices are assumed to have compatible dimensions, if their dimensions are not explicitly stated. 2. Preliminaries and problem formulation Takagi and Sugeno have proposed a fuzzy model to approximate a nonlinear systems [22]. This fuzzy dynamic model is described by fuzzy IF-THEN rules which represent local linear input–output relations of a nonlinear system. It is proved that the T–S fuzzy model is a universal approximator.

S. Li, Z. Xiang / Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91

79

Now consider the following switched T–S fuzzy system described by its ith rule as follows: Model rule i for subsystem σ (t) = n: [n ] [n]i [n ]i [n] [n]i [n ] IF θ1 (t) is M1 and θ2 (t) is M2 and · · · and θp (t) is Mp , THEN [n]

[n ]

[n]

x˙ (t) = Ai x(t) + Bi u(t) + Ei w (t), [n]

z(t) = Ci x(t),

i ∈ R = {1, 2, . . . , r },

[n]

[n ]

[n]

(1) [n]i

where θ (t) = [θ1 (t), θ2 (t), . . . , θp (t)] is the premise variable vector assumed independent of input u(t), Mj , i = 1, 2, ..r, j = 1, 2, . . . , p are fuzzy sets for subsystem n ∈ N, r is the number of IF-THEN rules, p is the number of premise variables. x(t) ∈ Rnx is the system state, z(t) ∈ Rnz is the controlled output, u(t) ∈ Rnu is the control input and w(t) ∈ Rnw is the exogenous disturbance signal. σ (t) : [0, ∞) → N = {1, 2, . . . , N } denotes the switching signal, which takes its value in finite set N; σ (t) = n ∈ N implies that ∑ the nth subsystem is activated, N is the amount of all subsystems. Correspondingly, the switching sequence is denoted by = {(t0 , σ (t0 )), (t1 , σ (t1 )), . . . , (tk , σ (tk )), . . .}, where t0 = 0 is [n] [n] [n] [n] the initial instant, tk is the kth switching instant. Ai , Bi , Ei and Ci , i ∈ R, n ∈ N, are system matrices with appropriate dimensions. By using the fuzzy inference method with a singleton fuzzier, product inference and a center-average defuzzifier, together with the switching signal σ (t), subsystem n can be inferred as: [n ]

T

x˙ (t) = A[n] (h(t))x(t) + B[n] (h(t))u(t) + E [n] (h(t))w (t), z(t) = C [n] (h(t))x(t),

(2)

where A[n] (h(t)) =

E [n] (h(t)) =

r ∑

[n ]

[n]

hi (θ [n] (t))Ai ,

B[n] (h(t)) =

i=1

i=1 r ∑

[n]

[n]

hi (θ [n] (t))Ei ,

C [n] (h(t)) =

and hi (θ

[n ]

[n ]

[n]

[n ]

hi (θ [n] (t))Ci ,

i=1

(t)) are the normalized membership functions: p

Πj=1 Mj[n]i (θj[n] (t))

[n ]

hi (θ [n] (t)) = ∑r

p

[n ]i [n ] (θj (t)) i=1 Πj=1 Mj

[n ]i

[n]

hi (θ [n] (t))Bi ,

r ∑ i=1

[n]

r ∑

≥ 0,

r ∑

[n]

hi (θ [n] (t)) = 1,

(3)

i=1

[n ]

[n]

[n]i

with Mj (θj (t)) representing the grade of membership of premise variable θj (t) in Mj Thus, the final switched T–S fuzzy system is inferred as follows:

.

x˙ (t) = A[σ (t)] (h(t))x(t) + B[σ (t)] (h(t))u(t) + E [σ (t)] (h(t))w (t), z(t) = C [σ (t)] (h(t))x(t),

(4)

where A[σ (t)] (h(t)) =

r ∑

[σ (t)]

hi

[σ (t)]

(θ [σ (t)] (t))Ai

,

B[σ (t)] (h(t)) =

r ∑

i=1 r

E [σ (t)] (h(t)) =

∑

[σ (t)]

hi

[σ (t)]

(θ [σ (t)] (t))Bi

,

i=1 r

[σ (t)]

hi

[σ (t)]

(θ [σ (t)] (t))Ei

,

C [σ (t)] (h(t)) =

i=1

∑

[σ (t)]

hi

[σ (t)]

(θ [σ (t)] (t))Ci

.

i=1

For convenience, system (4) with u(t) = 0 is introduced. x˙ (t) = A[σ (t)] (h(t))x(t) + E [σ (t)] (h(t))w (t), z(t) = C [σ (t)] (h(t))x(t).

(5)

Next, we recall some definitions and lemmas that will be used throughout the paper. Definition 1 ([2]). System (4) is said to be positive if, for any initial conditions x(t0 ) ⪰ 0, any inputs u(t) ⪰ 0, w (t) ⪰ 0, and any switching signals σ (t), we have x(t) ⪰ 0 and z(t) ⪰ 0, ∀t ≥ 0. Lemma 1 ([2]). A linear system x˙ (t) = Ax(t) + Bu(t) + E w (t) is positive, if and only if A are Metzler matrices, B ⪰ 0 and E ⪰ 0. [n ]

Lemma 2. System (4) is positive, if Ai

[n ]

are Metzler matrices, Bi

⪰ 0, Ei[n] ⪰ 0 and Ci[n] ⪰ 0, for all σ (t) = n ∈ N, i ∈ R.

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S. Li, Z. Xiang / Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91 [n ]

Remark 1. Due to the fact that 0 ≤ hi (θ [n] (t)) ≤ 1, Lemma 2 can be directly obtained by Lemma 1. Remark 2. Lemmas 1 and 2 mean that a fuzzy system is positive if and only if each of its local linear subsystem is positive. Definition 2. System (4) is said to be controlled positive relative to an initial condition x(t0 ) ⪰ 0 and w (t) ⪰ 0 for any switching signals σ (t), if there exists a control input u(t) such that we have x(t) ⪰ 0, z(t) ⪰ 0, ∀t ≥ 0. Remark 3. It should be pointed out that it is important to introduce the concept of ‘‘controlled positive’’. Consider the [n ] switched T–S fuzzy system (4) with u(t) ̸ = 0, when Ai is not a Metzler matrix, we can still design a controller to guarantee the positivity of the closed-loop system for correct approximation. Definition 3. System (5) with w (t) = 0 is said to be exponentially stable under switching signal σ (t) if, for a given constant α > 0, the solution x(t) satisfies

∥x(t)∥2 ≤ e−α(t −t0 ) h(x(t0 )),

∀t ≥ t0 ,

(6)

where h(x(t0 )) ≥ 0 is a function. Definition 4 ([33]). For any switching signal σ (t) and any T2 > T1 ≥ 0, let Nσ (T1 , T2 ) denotes the number of switches of σ (t) over the interval [T1 , T2 ). For given Ta > 0 and N0 ≥ 0, if the inequality Nσ (T1 , T2 ) ≤ N0 +

T2 − T1

(7)

Ta

holds, then the positive constant Ta is called the average dwell time and N0 is called the chattering bound. Remark 4. As stated in [34], if N0 = 1, then (7) implies that σ (t) cannot switch twice on any interval of length smaller than Ta . Switching signals with this property are exactly switching signals with dwell time Ta . Note also that N0 = 0 corresponds to the case that σ (t) cannot switch at all on any interval of length smaller than Ta . In general, if we discard the first N0 switches (more precisely, the smallest integer greater than N0 ), the average time between consecutive switches is at least Ta . If N0 increases, the allowed number of switches will increase on some certain intervals. As commonly used in the literature, we choose N0 = 0 in this paper. Definition 5. For given scalars α > 0 and γ > 0, system (5) is said to have a prescribed weighted L2 -gain performance level γ if, there exists a switching signal σ (t) such that the following conditions are satisfied,

• (i) System (5) is exponentially stable when w(t) = 0; • (ii) Under zero initial condition, system (5) satisfies ∫ ∞ ∫ ∞ wT (t)w(t)dt , e−α (t −t0 ) z T (t)z(t)dt ≤ γ 2 t0

w(t) ̸= 0.

(8)

t0

Remark 5. In Definition 5, the weighted L2 -gain performance characterizes systems suppression to exogenous disturbance. The smaller the value of γ is, the better the performance of the system is, i.e., the lesser the effect of the disturbance input to the control output is. In this paper, we consider the exponential stability analysis and weighted L2 -gain stabilization problem for positive switched T–S fuzzy system (4). We establish less conservative stability conditions and address the weighted L2 -gain performance analysis for positive switched T–S fuzzy system (5), upon which a control input u(t) is designed such that the resulting closed-loop system (4) is controlled positive, exponentially stable with a prescribed weighted L2 -gain performance level. 3. Main results 3.1. Exponential stability analysis In this section, by exploring a PQCLF, we will focus on exponential stability analysis for positive switched T–S fuzzy system (5). An improved stability condition is firstly provided in the following theorem by the PQCLF approach. Theorem 1. For a given positive constant α , if there exist real matrices P [n] ∈ Rnx ×nx , Θ [n] ∈ Rnx ×nx , Θ [n ] Φi[n] ∈ Rnx ×nx , Φ i ∈ R(ϑ (nx ,2)−nx )×(ϑ (nx ,2)−nx ) , such that, ∀n ∈ N, ∀i ∈ R,

− P [n ] − Θ [n ] ≤ 0 ,

[n ]

∈ R(ϑ (nx ,2)−nx )×(ϑ (nx ,2)−nx ) ,

(9)

S. Li, Z. Xiang / Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91

Θ

[n ]

≤ 0,

(10)

[n ]

[n]

(Ai )T P [n] + P [n] Ai

+ α P [n] − Φi[n] ≤ 0,

(11)

[n ]

Φ i ≤ 0,

(12) [n]

[n ]

where Θ [n] = [Θpq ](p,q)∈nx ×nx with Θpp = 0, Θ [n ]

Φi =

81

[n]i

[n]i

[n]i diag {Φ 1 , Φ 2 , . . . , Φ nx −1 },

[n ]

[n ]

[n ]

[n ]

[n ]i [n]i = 0, = diag {Θ 1 , Θ 2 , . . . , Θ nx −1 }, Φi[n] = [Φpq ](p,q)∈nx ×nx with Φpp

with

[n ]

[n ] [n] [n] [n ] Θ k = diag {Θk(k +1) + Θ(k+1)k , . . . , Θk(nx ) + Θ(nx )k }, k ∈ {1, 2, . . . , nx − 1}, [n ]i

Φk

[n ]i [n ]i [n]i [n]i = diag {Φk(k +1) + Φ(k+1)k , . . . , Φk(nx ) + Φ(nx )k }, k ∈ {1, 2, . . . , nx − 1},

then the positive switched T–S fuzzy system (5) is exponentially stable for any switching signal σ (t) with average dwell time Ta > Ta∗ =

ln µ

α

,

(13)

where µ ≥ 1 satisfies P [n] + Θ [n] ≤ µ(P [s] + Θ [s] ),

µΘ

[s]

[n]

≤Θ ,

∀ n, s ∈ N .

(14)

Proof. We choose the following PQCLF candidate: {2} T

√

V (x(t)) = Vσ (t) (x(t)) = ( x(t)

P [σ (t)] + Θ [σ (t)]

[

)

0

−Θ

0

[σ (t)]

]

√

{2}

( x(t)

),

(15)

where ∀x ∈ Rnx , x{2} = [x[2] ; x1 (I2 x); x2 (I3 x); · · · ; xnx −1 (Inx x)]. Therefore, it is clear that if (9) and (10) hold, ∀x(t) ⪰ 0, V (x(t)) ≥ 0.

(16)

When t ∈ [tk , tk+1 ), we have σ (t) = n ∈ N. Define ˜ x(t) = [x1 (I2 x); x2 (I3 x); · · · ; xnx −1 (Inx x)], one can get that [n ] (x[2] (t))T Θ [n] x[2] (t) = (x[2] (t))T [Θpq ](p,q)∈nx ×nx x[2] (t)

=

nx nx ∑ ∑

[n ] 2 Θpq xp (t)x2q (t)

p=1 q=1 nx −1

=

nx ∑ ∑

[n] [n] 2 (Θpq + Θqp )xp (t)x2q (t)

p=1 q=p+1

[n ]

[n]

[n ]

x(t) =˜ xT (t)diag {Θ 1 , Θ 2 , . . . , Θ nx −1 }˜ [n]

=˜ xT (t)Θ ˜ x(t),

(17)

and similarly, xT (t)Θ [n] x(t) =

√ T [n] √ ˜ ˜ x(t) Θ x(t).

(18)

With (18), one has from (15) that, ∀x(t) ⪰ 0, V˙ (x(t)) + α V (x(t)) d

√ T [n ] √ ˜ [xT (t)(P [n] + Θ [n] )x(t) − ˜ x(t) Θ x(t)] dt √ √ T [n] ˜ + α[xT (t)(P [n] + Θ [n] )x(t) − ˜ x(t) Θ x(t)] ) ( r r ∑ [n] ∑ [n ] T [n ] [n] [n] [n ] T [n ] [n] [n ] = x (t) hi (θ (t))(Ai ) P + P hi (θ (t))Ai + α P x(t) =

i=1

=

r ∑

i=1

(

[n ]

[n]

[n ]

hi (θ [n] (t))xT (t) (Ai )T P [n] + P [n] Ai

) + α P [n] x(t)

i=1

=

r ∑ i=1

[n ]

(

(

[n ]

[n ]

hi (θ [n] (t)) xT (t) (Ai )T P [n] + P [n] Ai

) √ T [n ] √ ) ˜ + α P [n] − Φi[n] x(t) + ˜ x(t) Φ i x(t)

82

S. Li, Z. Xiang / Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91

=

r ∑

[n ]

hi ( θ

[n]

√

(t))( x(t)

{2} T

[

)

[n ]

[n ]

(Ai )T P [n] + P [n] Ai

+ α P [n] − Φi[n]

]

[n]

Φi

0

i=1

0

√

{2}

( x(t)

).

(19)

From (11)–(12), it is obvious that V˙ (x(t)) + α V (x(t)) ≤ 0, and when t ∈ [tk , tk+1 ), we have Vσ (t) ≤ e−α (t −tk ) Vσ (tk ) (tk ).

(20)

From (14) and (15), we can easily obtain Vσ (tk ) (tk ) ≤ µVσ (t − ) (tk− ), k

∀k = 1, 2, . . . .

(21)

Therefore, it follows from (13), (20), (21) and the relation k = Nσ (t0 , t) ≤

t −t0 Ta

that

Vσ (t) (t) ≤ e−α (t −tk ) µVσ (t − ) (tk− ) ≤ e−α (t −tk−1 ) µVσ (tk−1 ) (tk−1 ) k

−(α− lnT µ )(t −t0 )

a ≤ ··· ≤ e [ P [n] + Θ [n] Define ε1 = λmin

0

Vσ (t0 ) (t0 ).

]

0

−Θ

[n]

and ε2 = λmax

(22)

[

P [n] + Θ [n] 0

]

0

−Θ

[n]

, ∀n ∈ N, it yields that

Vσ (t) (t) ≥ ε1 ∥x{2} (t)∥2 ≥ ε1 ∥x(t)∥2 ,

(23)

Vσ (t0 ) (t0 ) ≤ ε2 ∥x{2} (t0 )∥2 ,

(24)

where ∥x(t)∥2 is the Euclidean norm of x(t). Combine (22)–(24), we obtain

∥x(t)∥2 ≤

ε2 −(α− lnT µ )(t −t0 ) {2} a ∥x (t0 )∥2 , e ε1

(25)

which means that the positive switched T–S fuzzy system (5) is exponentially stable. This completes the proof. □ Remark 6. In Theorem 1, PQCLF is essentially a quadratic Lyapunov function that satisfies the following two inequalities: V (x(t)) = xT (t)P [n] x(t) ≥ 0, V˙ (x(t)) =

r ∑

∀x(t) ∈ Rn+x ,

(

[n]

[n ]

[n ]

hi (θ [n] (t))xT (t) (Ai )T P [n] + P [n] Ai

)

x(t) ≤ 0,

∀x(t) ∈ Rn+x .

i=1

[n ]

[n]

Note that we do not require P [n] ≥ 0 and (Ai )T P [n] + P [n] Ai

≤ 0, ∀n ∈ N, ∀i ∈ R.

Remark 7. It should be pointed out that, compared with general quadratic Lyapunov function and copositive Lyapunov function methods that used in the stability analysis of systems, a PQCLF is proposed in (15) to reduce the conservativeness and results in less conservative stability conditions for system (5), and we will give the following illustrations to clarify this point. Moreover, the obtained stability condition in Theorem 1 is in the terms of linear matrix inequality (LMI), which can be conveniently solved by LMI toolbox. [n ]

Remark 8. When Θ [n] = 0 and Θ = 0 in (15), we can get the general quadratic Lyapunov function, which is a special case of the one in (15), and if we can find a general quadratic Lyapunov function to guarantee the stability of system (5), then we can also find a Lyapunov function in (15), but not vice versa. [n ] On the other side, when P [n] = ν [n] (ν [n] )T , Θ [n] = 0 and Θ = 0 in (15), the Lyapunov function in (15) will reduce to T [n] [n] T V (x(t)) = x (t)ν (ν ) x(t) as a special case, and if we can find a linear copositive Lyapunov function V (x(t)) = (ν [n] )T x(t) to guarantee the stability of system (5), then we can always find a Lyapunov function V (x(t)) = xT (t)ν [n] (ν [n] )T x(t), and the [n ] feasible solutions are P [n] = ν [n] (ν [n] )T , Θ [n] = 0, Φi = 0. In the absence of switching feature, i.e., N = 1, we can get the following corollary. Corollary 1. For a given positive constant α , if there exist real matrices P ∈ Rnx ×nx , Θ ∈ Rnx ×nx , Θ ∈ R(ϑ (nx ,2)−nx )×(ϑ (nx ,2)−nx ) , Φi ∈ Rnx ×nx , Φ i ∈ R(ϑ (nx ,2)−nx )×(ϑ (nx ,2)−nx ) , such that, ∀i ∈ R,

− P − Θ ≤ 0, Θ ≤ 0,

S. Li, Z. Xiang / Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91

83

ATi P + PAi − Φi + α P ≤ 0,

Φ i ≤ 0, i i = 0, Φ i = where Θ = [Θpq ](p,q)∈nx ×nx with Θpp = 0, Θ = diag {Θ 1 , Θ 2 , . . . , Θ nx −1 }, Φi = [Φpq ](p,q)∈nx ×nx with Φpp i

i

i

diag {Φ 1 , Φ 2 , . . . , Φ nx −1 }, with

Θ k = diag {Θk(k+1) + Θ(k+1)k , . . . , Θk(nx ) + Θ(nx )k }, k ∈ {1, 2, . . . , nx − 1}, i

i i i i Φ k = diag {Φk(k +1) + Φ(k+1)k , . . . , Φk(nx ) + Φ(nx )k }, k ∈ {1, 2, . . . , nx − 1},

then the positive T–S fuzzy system (5) is exponentially stable. 3.2. Weighted L2 -gain performance analysis In consideration of exogenous disturbances, to analyze the disturbance attenuation performance, we consider the weighted L2 -gain performance for positive switched T–S fuzzy system (5) in this section. A sufficient condition of exponential stability with a weighted L2 -gain performance is proposed in the following theorem. [n ]

Theorem 2. For given positive constants α and γ , if there exist real matrices P [n] ∈ Rnx ×nx , Θ [n] ∈ Rnx ×nx , Θ ∈ [n] [n ] R(ϑ (nx ,2)−nx )×(ϑ (nx ,2)−nx ) , Ψ i ∈ R(ϑ ((nx +nw ),2)−(nx +nw ))×(ϑ ((nx +nw ),2)−(nx +nw )) , Ψi ∈ R(nx +nw )×(nx +nw ) , such that, ∀n ∈ N, ∀i ∈ R,

− P [n] − Θ [n] ≤ 0, Θ

(26)

[n ]

≤ 0,

(27)

[n ]

− Ψi[n] ≤ 0,

(28)

Wi

[n ]

Ψ i ≤ 0,

(29)

where

⎡ [n] T [n]

[n ]

Wi

⎢(A ) P =⎣ i

[n ] [n ]

+ P Ai + α P

[n ]

[n] T

+ (Ci )

r ∑

⎤ [n ]

[n]

hi (θ (t))Ci

P

[n] [n]

Ei ⎥

i=1

[n]

(Ei )T P [n]

−γ 2 I

⎦,

[n ] [n] [n ]i [n ]i Θ [n] = [Θpq ](p,q)∈nx ×nx with Θpp = 0, Ψi[n] = [Ψpq ](p,q)∈(nx +nw )×(nx +nw ) with Ψpp = 0, Θ [n] [n]i [n ]i [n]i Ψ i = diag {Ψ 1 , Ψ 2 , . . . , Ψ (nx +nw )−1 }, with

[n ]

[n ]

[n]

[n ]

= diag {Θ 1 , Θ 2 , . . . , Θ nx −1 },

[n ]

[n ] [n] [n] [n ] Θ k = diag {Θk(k +1) + Θ(k+1)k , . . . , Θk(nx ) + Θ(nx )k }, k ∈ {1, 2, . . . , nx − 1}, [n]i

Ψk

[n ]i [n]i [n ]i [n]i = diag {Φk(k +1) + Φ(k+1)k , . . . , Φk(nx +nw ) + Φ(nx +nw )k }, k ∈ {1, 2, . . . , (nx + nw ) − 1},

then the positive switched T–S fuzzy system (5) is exponentially stable with a prescribed weighted L2 -gain performance level γ for any switching signal σ (t) with average dwell time (13). Proof. First, note that (28)–(29) implies (11)–(12). When w (t) = 0, by Theorem 1, if (26)–(29) have feasible solutions, then the positive switched T–S fuzzy system (5) is exponentially stable. Next, when w (t) ̸ = 0, it is required to prove that positive switched T–S fuzzy system (5) satisfies the weighted L2 -gain performance. Construct the PQCLF candidate (15), similar to the proof in Theorem 1, when t ∈ [tk , tk+1 ) and σ (t) = n, one has V˙ (x(t)) + α V (x(t)) + z T (t)z(t) − γ 2 w T (t)w (t)

= x˙ T (t)P [n] x(t) + xT (t)P [n] x˙ (t) + α xT (t)P [n] x(t) ( r )T r ∑ [n] ∑ [n ] [n ] [n] [n ] + hi (θ (t))Ci x(t) hi (θ [n] (t))Ci x(t) − γ 2 w T (t)w (t) i=1

( =

r ∑ i=1

[n]

hi ( θ

i=1

[n ]

[n ]

(t))Ai x(t) +

r ∑ i=1

)T [n]

hi ( θ

[n ]

[n]

(t))Ei w (t)

P [n] x(t)

84

S. Li, Z. Xiang / Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91

( T

+ x (t)P

[n]

r ∑

[n]

hi (θ

[n ]

[n ]

(t))Ai x(t) +

i=1

) [n]

[n ]

)T

r ∑

hi (θ

+ α x (t)P x(t) + [n]

r ∑

[n ]

hi (θ

[n]

[n ]

(t))Ci x(t)

i=1 r ∑

=

[n ]

hi (θ [n] (t))

[n]

(t))Ei w (t)

i=1

( T

r ∑

[ (

[n ]

[n]

hi (θ [n] (t))Ci x(t) − γ 2 w T (t)w (t)

i=1

[n ]

[n]

Ai x(t) + Ei w (t)

)T

(

[n ]

[n]

P [n] x(t) + xT (t)P [n] Ai x(t) + Ei w (t)

)

i=1

(

[n ]

+ α x (t)P x(t) + Ci x(t) T

[n]

]

r )T ∑

[n]

hi ( θ

[n ]

[n ]

(t))Ci x(t) − γ w (t)w (t) 2

T

i=1 r ∑

=

( [n ]

hi ( θ

[n]

[n] T [n]

T

(t))[x (t) (Ai ) P

[n] [n]

+ P Ai + α P

[n]

[n ] T

+ (Ci )

i=1

r ∑

) [n]

hi ( θ

[n ]

[n ]

(t))Ci

x(t)

i=1

( ) ( ) + xT (t) P [n] Ei[n] w(t) + wT (t) (Ei[n] )T P [n] x(t) − γ 2 wT (t)w(t)] [ ]T r ∑ x(t) [n] [n] = hi (θ (t)) w(t) i=1 ⎡ ⎤ r ∑ ] [n ] T [n ] [n] T [n] [n] [n] [n ] [n ] [n ] [n] [n] [ hi (θ (t))Ci P Ei ⎥ x(t) ⎢(Ai ) P + P Ai + α P + (Ci ) ⎣ ⎦ i=1 w(t) [n] (Ei )T P [n] −γ 2 I r ∑ [n ] [n] = hi (θ [n] (t))X T (t)Wi X (t) i=1 r ∑

=

( [n ]

hi ( θ

[n ]

[n ]

[n]

=

hi ( θ

[n]

− Ψi

{2} T

[

[n ]

(t)) X (t) Wi

i=1 r

∑

(

T

√

(t))( X (t)

)

[n ]

Wi

x(t)

) √ √ T [n] ˜ ˜ X (t) X (t) + X (t) Ψ i

− Ψi[n]

0 [n]

Ψi

0

i=1

[

)

]

√

( X (t)

{2}

),

(30)

]

where X (t) = w(t) . [n ] Since 0 ≤ hi (θ [n] (t)) ≤ 1 and X (t) ⪰ 0, from (28)–(29), it is obvious that V˙ (x(t)) +α V (x(t)) + z T (t)z(t) −γ 2 w T (t)w (t) ≤ 0 for all t ≥ 0, and when t ∈ [tk , tk+1 ), we have V (t) ≤ e−α (t −tk ) V (tk ) −

∫

t

e−α (t −s) Λ(s)ds,

(31)

tk

where Λ(s) = z T (s)z(s) − γ 2 w T (s)w (s). Similar to inequality (22) in Theorem 1, we have from (31) that V (t) ≤ e−α (t −t0 )+Nσ (t0 ,t) ln µ V (t0 ) −

∫

t

e−α (t −s)+Nσ (s,t) ln µ Λ(s)ds.

(32)

t0

Under zero initial condition, from (32), we have

∫

t

e−α (t −s)+Nσ (s,t) ln µ z T (s)z(s)ds ≤ γ 2

t0

∫

t

e−α (t −s)+Nσ (s,t) ln µ w T (s)w (s)ds.

(33)

t0

Following the proof in Theorem 2 in [35] leads to inequality (8), which implies that the effect of disturbance input w (t) on the controlled output z(t) is attenuated with a prescribed performance γ . This completes the proof. □ 3.3. Weighted L2 -gain control synthesis In this section, the problem of weighted L2 -gain stabilization for positive switched T–S fuzzy system (4) is addressed. Our objective is to design a state-feedback controller for system (4) to ensure the exponential stability with a prescribed weighted L2 -gain performance while imposing the positivity in closed-loop.

S. Li, Z. Xiang / Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91

85

Here, the well-known PDC technique is employed to design the fuzzy controller [36], and the control law is defined as follows: Model rule i for subsystem σ (t) = n: [n ] [n]i [n] [n ]i [n ] [n]i IF θ1 (t) is M1 and θ2 (t) is M2 and · · · and θp (t) is Mp , THEN [n ]

u(t) = Ki x(t),

i ∈ R,

(34)

Then, the nth subsystem of closed-loop system (4) can be rewritten as follows: x˙ (t) = A[n] (h(t))x(t) + B[n] (h(t))K [n] (h(t))x(t) + E [n] (h(t))w (t) [n]

= A (h(t))x(t) + E [n] (h(t))w(t), z(t) = C [n] (h(t))x(t),

(35)

where [n ]

A (h(t)) =

r ∑

[n ]

hi (θ [n] (t))

r ∑

i=1

[n ]

[n ]

hj (θ [n] (t))(Ai

+ B[i n] Kj[n] ).

j=1

Theorem 3. For given positive constants α , γ and ϵij , if there exist real matrices P [n] ∈ Rnx ×nx , Θ [n] ∈ Rnx ×nx , Θ

∈ ∈ R(nx +nw )×(nx +nw ) , Π ij ∈ R(ϑ ((nx +nw ),2)−(nx +nw ))×(ϑ ((nx +nw ),2)−(nx +nw )) , such that, ∀n ∈ N, ∀(i, j) ∈

R(ϑ (nx ,2)−nx )×(ϑ (nx ,2)−nx ) , Πij R × R,

[n ]

− P [n] − Θ [n] ≤ 0, Θ

[n ]

(36)

≤ 0,

⎡

[n ]

[n]

(37) [n]

− Πij[n]12

P [n ] E i

T1

⎣(E [n] )T P [n] − Π [n]21 i

T3 0

ij

T4

⎤

T2

⎦ ≤ 0, 0 [n] −1 −ϵij (P )

(38)

[n ]

Π ij ≤ 0,

(39)

− ep (A[i n] + B[i n] Kj[n] )eTq ≤ 0,

∀(p, q) ∈ nx × nx ,

p ̸ = q,

(40)

where [n]11

T1 = −2P [n] − Πij

− ϵij P [n] + α P [n] + (Ci[n] )T

r ∑

[n ]

[n]

hi (θ [n] (t))Ci ,

i=1

[n]

T2 = ϵij I + (Ai

+ B[i n] Kj[n] + I)T , [n]22

T3 = −γ 2 I − Πij [n]

T4 = ϵij I + (Ai

,

+ B[i n] Kj[n] + I), [n]ij

[n]ij

[n] [n] Θ [n] = [Θpq ](p,q)∈nx ×nx with Θpp = 0, Πij[n] = [Πpq ](p,q)∈(nx +nw )×(nx +nw ) with Ψpp [n]

[n]ij

[n]ij

Π ij = diag {Π 1 , Π 2 [n ]

[n]11

Πij = [Πij

[n]ij , . . . , Π (nx +nw )−1 },

[n]12

, Πij

[n]21

; Πij

[n]22

, Πij

= 0, Θ

[n ]

[n ]

[n]

[n ]

= diag {Θ 1 , Θ 2 , . . . , Θ nx −1 },

with

],

[n ]

[n ] [n] [n] [n ] Θ k = diag {Θk(k +1) + Θ(k+1)k , . . . , Θk(nx ) + Θ(nx )k }, k ∈ {1, 2, . . . , nx − 1}, [n]ij

Πk

[n]ij [n]ij [n]ij [n]ij = diag {Πk(k +1) + Π(k+1)k , . . . , Πk(nx +nw ) + Π(nx +nw )k }, k ∈ {1, 2, . . . , (nx + nw ) − 1},

then there exists a fuzzy state-feedback controller (34) such that the closed-loop switched T–S fuzzy system (4) is controlled positive and exponentially stable with a prescribed weighted L2 -gain performance level γ for any switching signal σ (t) with average dwell time (13). [n ]

[n]

[n]

[n]

Proof. It can be derived from (40) that Ai + Bi Kj , ∀(i, j) ∈ R × R, are Metzler matrices. Also note that 0 ≤ hi (θ [n] (t)) ≤ 1, [n]

∀i ∈ R. Therefore, it is seen thatA (h(t)), are Metzler matrices, ∀t > 0. One can conclude by Lemma 2 and Definition 2 that the closed-loop system (4) is controlled positive.

86

S. Li, Z. Xiang / Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91

On the other hand, construct the PQCLF candidate (15), similar to the proof in Theorem 1, when t ∈ [tk , tk+1 ) and σ (t) = n, one has V˙ (x(t)) + α V (x(t)) + z T (t)z(t) − γ 2 w T (t)w (t)

=

r ∑

[n ]

hi ( θ

[n]

(t))

i=1

r ∑

[n ]

hj ( θ

[n]

j=1

]T [

[

x(t) (t)) w(t)

F1 [n ] (Ei )T P [n]

[n]

P [n ] E i

][

−γ 2 I

]

x(t) , w(t)

(41)

where [n ]

F1 = (Ai

+ B[i n] Kj[n] )T P [n] + P [n] (A[i n] + B[i n] Kj[n] ) + α P [n] + (Ci[n] )T

r ∑

[n]

[n ]

hi (θ [n] (t))Ci .

i=1

[n]

Since 0 ≤ hi (θ (t)) ≤ 1 and [x (t) w (t)] ⪰ 0, V˙ (x(t)) + α V (x(t)) + z T (t)z(t) − γ 2 w T (t)w (t) ≤ 0, if ∀(i, j) ∈ R × R, [n ]

T

T

T

(39) holds and

[

] − Πij[n]12 ≤ 0, −γ 2 I − Πij[n]22 [n ]

P [n] Ei

F2 [n ]

[n]21

(Ei )T P [n] − Πij

(42)

where [n]

F2 = (Ai

+ B[i n] Kj[n] )T P [n] + P [n] (A[i n] + B[i n] Kj[n] ) + α P [n] + (Ci[n] )T

r ∑

[n]

[n ]

hi (θ [n] (t))Ci

− Πij[n]11 .

i=1

It is true that [n]

(Ai

+ B[i n] Kj[n] )T P [n] + P [n] (A[i n] + B[i n] Kj[n] ) + α P [n] + (Ci[n] )T

r ∑

[n]

[n ]

hi (θ [n] (t))Ci

− Πij[n]11

i=1

[n]

[n ]

[n ]

= (Ai + Bi Kj

T

+ I) P

+ α P [n] + (Ci[n] )T

r ∑

[n ]

[n ]

[n ]

[n ]

[n]

+ P (Ai + Bi Kj

[n ]

[n]

+ I) − 2P

[n]

− Πij[n]11 .

hi (θ [n] (t))Ci

(43)

i=1

Therefore (42) holds if and only if there exist sufficiently small scalars εij > 0, ∀(i, j) ∈ R × R, such that

[ [n ]

] − Πij[n]12 ≤ 0, −γ 2 I − Πij[n]22 [n ]

P [n] Ei

F3 [n]21

(Ei )T P [n] − Πij

(44)

where [n]

F3 = (Ai

+ B[i n] Kj[n] + I)T P [n] + P [n] (A[i n] + B[i n] Kj[n] + I) − 2P [n]

+ εij (A[i n] + B[i n] Kj[n] + I)T P [n] (A[i n] + B[i n] Kj[n] + I) + α P [n] + (Ci[n] )T

r ∑

[n ]

[n]

hi (θ [n] (t))Ci

− Πij[n]11 ,

i=1

which is equivalent to ∀(i, j) ∈ R × R,

[ [n ]

] − Πij[n]12 ≤ 0, −γ 2 I − Πij[n]22 [n ]

P [n] Ei

F4 [n]21

(Ei )T P [n] − Πij

(45)

where [n ]

F4 = (Ai

+ B[i n] Kj[n] + I)T P [n] + P [n] (A[i n] + B[i n] Kj[n] + I)

+ εij (A[i n] + B[i n] Kj[n] + I)T P [n] (A[i n] + B[i n] Kj[n] + I) + εij−1 P [n] − εij−1 P [n] − 2P [n] + α P [n] + (Ci[n] )T

r ∑

[n]

[n ]

hi (θ [n] (t))Ci

− Πij[n]11

i=1

= [I + εij (A[i n] + B[i n] Kj[n] + I)]T εij−1 P [n] [I + εij (A[i n] + B[i n] Kj[n] + I)] − εij−1 P [n] − 2P [n] + α P [n] + (Ci[n] )T

r ∑ i=1

[n]

[n ]

hi (θ [n] (t))Ci

− Πij[n]11 .

S. Li, Z. Xiang / Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91

87

It then follows by Schur complement that (45) holds, if and only if ∀(i, j) ∈ R × R,

⎡

[n]

− Πij[n]12 2 −γ I − Πij[n]22

F6

⎤

0

⎦ ≤ 0,

0

−εij (P [n] )−1

P [n ] E i

F5

⎣(E [n] )T P [n] − Π [n]21 i

ij

F7

(46)

where [n]11

F5 = −2P [n] − Πij

− εij−1 P [n] + α P [n] + (Ci[n] )T

r ∑

[n]

[n ]

hi (θ [n] (t))Ci ,

i=1

[n ]

+ B[i n] Kj[n] + I)T ,

[n ]

+ B[i n] Kj[n] + I).

F6 = I + εij (Ai F7 = I + εij (Ai

Next, by setting εij = ϵij−1 and performing a congruence transformation to (46) via diag(I , I , ϵij I), we can obtain that (46) is equivalent to (38). Finally, from Theorem 2 and the above discussions, one can conclude that if (36)–(40) are feasible, then there exist a [n] state-feedback controller in the form of (34) with gains Ki such that the closed-loop system (4) is controlled positive and exponentially stable with a prescribed weighted L2 performance level γ . This completes the proof. □ The conditions of Theorem 3 are nonconvex. However, such a problem can be solved by using the cone complementarity linearization (CCL). Next, the following algorithm solves the nonconvex problem formulated in Theorem 3. [n](0)

Algorithm 1. Step 1. For given positive constants α , γ and ϵij , find a feasible solution denoted as (Ki P [n](0) , Q [n](0) ) for the linear matrix inequalities (36)–(37), (39)–(40), and ∀n ∈ N, ∀(i, j) ∈ R × R,

⎡

[n]

P [n ] E i

M1

⎣(E [n] )T P [n] − Π [n]21 i

ij

[

P [n ] I

I

M] 4

Q [n ]

− Πij[n]12

M2

⎤

⎦ ≤ 0, 0 [n ] −ϵij Q

M3 0

, Πij[n](0) , Θ [n](0) ,

(47)

≥ 0,

(48)

∑ −ϵij P [n] +α P [n] +(Ci[n] )T ri=1 h[i n] (θ [n] (t))Ci[n] , M2 = ϵij I +(A[i n] +B[i n] Kj[n] +I)T , M3 = −γ 2 I −Πij[n]22 , [n ] [n] [n] M4 = ϵij I + (Ai + Bi Kj + I). Then, set k = 0, P [n](0) , and Q [n](0) as the iterative initial values. [n]11

where M1 = −2P [n] −Πij

Step 2. Find a feasible solution with respect to the convex minimization problem: min

[n](k+1) [n](k+1) [n](k+1) [n](k+1) [n](k+1) (Ki ,Πij ,Θ ,P ,Q )

s.t.

trace{P [n](k+1) Q [n](k) + Q [n](k+1) P [n](k) }

(36)–(37), (39)–(40), and (47)–(48) [n]

[n]

[n](k+1)

Step 3. If (36)–(40) are satisfied for (Ki , Πij , Θ [n] , P [n] , Q [n] ) = (Ki

[n ]

Ki

, Πij[n](k+1) , Θ [n](k+1) , P [n](k+1) , Q [n](k+1) ), then

is a set of stabilizing controllers, EXIT. Step 4. If k > N0 (N0 is the prescribed iteration number), EXIT. Otherwise, k = k + 1, and go to Step 2.

Remark 9. From Algorithm 1, Step 1 and every Step 2 and Step 3 are all simple LMI problems, which can be solved by standard LMI solution packages. However, it should be pointed out that there are five parameters in Step 3 to be verified, and many iterations may be needed before obtaining the result. Therefore, to find a better algorithm to solve this problem will be considered in the future work. 4. Examples In this section, two numerical examples are provided to demonstrate the effectiveness and applicability of the developed theoretical results. Example 1. First of all, we introduce the following continuous-time nonlinear positive switched system with two subsystems:

⎧ ⎪ x˙ (t) = −x1 (t) + (2 + sin2 x1 (t))x2 (t) + (1.1 − 0.1sin2 x1 (t))u1 (t) + 0.1w (t), ⎪ ⎨ 1 x˙ 2 (t) = (1 + 2sin2 x1 (t))x1 (t) − (1 + 1sin2 x1 (t))x2 (t) Subsystem1 : ⎪ + (1.2 − 0.1sin2 x1 (t))u2 (t) + 0.1w (t), ⎪ ⎩ z(t) = x1 (t) + 0.1x2 (t),

88

S. Li, Z. Xiang / Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91

⎧ ⎨x˙ 1 (t) = −2x1 (t) + (2 + sin2 x1 (t))x2 (t) + u1 (t) + 0.1w(t), Subsystem2 : x˙ 2 (t) = 3x1 (t) − (2 + sin2 x1 (t))x2 (t) + u2 (t) + 0.1w (t), ⎩ z(t) = x1 (t), Next, we consider its T–S fuzzy model as follows. Denote xT (t) = [x1 (t) sin2 (x1 (t)), n = 1, 2, with two fuzzy rules: Modelrule 1: IF θ [n] (t) is 0, THEN [1]

[1]

[1 ]

[2]

[2 ]

[2 ]

[1]

[1 ]

[1]

[2]

[2 ]

[2]

{

x˙ (t) = A1 x(t) + B1 u(t) + E1 w (t), [1 ] z(t) = C1 x(t),

{

x˙ (t) = A1 x(t) + B1 u(t) + E1 w (t), [2 ] z(t) = C1 x(t),

Subsystem1 : Subsystem2 :

x2 (t)], uT (t) = [u1 (t)

u2 (t)]. Let θ [n] (t) =

Modelrule 2: IF θ [n] (t) is 1, THEN

{

x˙ (t) = A2 x(t) + B2 u(t) + E2 w (t), [1 ] z(t) = C2 x(t),

{

x˙ (t) = A2 x(t) + B2 u(t) + E2 w (t), [2] z(t) = C2 x(t),

Subsystem1 :

Subsystem2 :

[n ]

[n ]

where the normalized membership functions are h1 (θ [n] (t)) = 1 − sin2 (x1 (t)), h2 (θ [n] (t)) = sin2 (x1 (t)), and parameter matrices of the T–S fuzzy system are given as follows: [1]

A1 = [1]

A2 = [2]

A1 = [2]

A2 =

[ −1 1

[ −1 3

[ −2 3

[ −2

]

2 , −1

]

3 , −2

]

[1 ]

[1 ]

1 0

[

[2 ]

1 0

B1 =

]

3

[

[1 ]

B2 =

2 , −2 3 , −3

1.1 0

[

[1]

B1 =

[2 ]

[

B2 =

[2 ]

1 0

]

0 , 1.2

]

0 , 1.1

]

0 , 1

[2]

]

[

[1 ]

E2 = E1 =

0 , 1

0.1 , 0.1

[

[1]

E1 =

[2]

E2 =

]

0.1 , 0.1

[1]

C1

]

[1 ]

C2

0.1 , 0.1

C1

0.1 , 0.1

C2

[

[

[2]

]

]

[1 ]

0.1 ,

[ = 1

]

0.1 ,

[ = 1

]

[2]

[ = 1

0 ,

[2]

[ = 1

0 .

[1]

[2]

]

]

[2 ]

[1 ]

[1 ]

[2]

[2]

[1 ]

[1]

[2]

[2 ]

Note that A1 , A2 , A1 and A2 are Metzler matrices, B1 , B2 , B1 , B2 , C1 , C2 , C1 , C2 , E1 , E2 , E1 , E2 are all nonnegative, so the considered switched T–S fuzzy system is a positive switched T–S fuzzy system. However, the state trajectory shown in Fig. 1 demonstrates that the open-loop system is unstable. Next, we are interested in designing a controller in the form of (34) such that the resulting closed-loop system is positive and exponentially stable. Let α = 0.7, γ = 0.5, w (t) = 0.5e−0.5t , then applying Algorithm 1 to the obtained T–S fuzzy [1] [1] [2 ] [2 ] system, the feedback gains K1 , K2 , K1 and K2 are obtained as follows: [1 ]

K1

[2 ]

K1

[ −0.6925 −0.7260 [ −0.1647 = −1.7733

=

] −1.6831 , 0.1073 ] −1.4799 , 0.8677

[1 ]

K2

[2 ]

K2

[

−0.5582 −0.6804

[

0.1853 −0.9881

= =

] −1.7446 , −0.1068 ] −1.9313 , −0.0364

and Ta∗ = 0.9506 with µ = 1.9454. According to Theorem 3, we can conclude that the closed-loop system is controlled positive and exponentially stable with a prescribed weighted L2 -gain performance for any switching signal satisfying Ta > 0.9506. The simulation results are shown in Figs. 2 and 3, where the initial condition is x(0) = [8 10]T . Fig. 2 shows the switching signal with average dwell time Ta = 1, and the corresponding state responses of the closed-loop system are shown in Fig. 3, from which one can see that the obtained controller exponentially stabilize the system, while imposing the positivity in closed-loop. This demonstrates the effectiveness of the proposed results. Example 2. Introduce the following nonlinear positive system without switching feature, i.e., N = 1,

⎧ ⎨x˙ 1 (t) = −(1 + 9sin2 x1 (t))x1 (t) + (2 + 8sin2 x1 (t))x3 (t), x˙ (t) = (2 − 2sin2 x1 (t))x1 (t) − (1 + 9sin2 x1 (t))x2 (t), ⎩ 2 x˙ 3 (t) = (10sin2 x1 (t))x2 (t) − (10 − 9sin2 x1 (t))x3 (t). Consider its T–S fuzzy model in the following. Denote xT (t) = [x1 (t) x2 (t) rules:

x3 (t)]. Let θ (t) = sin2 (x1 (t)) with two fuzzy

S. Li, Z. Xiang / Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91

89

Fig. 1. State responses of the corresponding open-loop system.

Fig. 2. Switching signal.

Modelrule 1: IF θ (t) is 0, THEN x˙ (t) = A1 x(t), Modelrule 2: IF θ (t) is 1, THEN x˙ (t) = A2 x(t), where the normalized membership functions are h1 (θ (t)) = 1 − sin2 (x1 (t)), h2 (θ (t)) = sin2 (x1 (t)), and parameter matrices of the T–S fuzzy system are given as follows:

[ −1

0 −1 0

2 0

A1 =

2 0 , −10

[ −10

]

A2 =

0 0

0 −10 10

10 0 . −1

]

Note that A1 and A2 are Metzler matrices, so the considered T–S fuzzy system is a positive T–S fuzzy system. It should be pointed out that the stability result in Corollary 1 can be applied to this positive T–S fuzzy system, and the stability problem for this system has also been studied in the literature [32]. According to Theorem 3.1 in the paper [32], no feasible solutions can be found, however, by Corollary 1 given in this paper, a feasible solution is achieved with 0.7889 P = 10 × −0.5529 −1.9530

[

6

−0.5529 5.0523 −1.7468

1.6527 −1.3525 , 0.0009

]

and noted that the matrix P is not constrained to be symmetric positive definite. Thus, the stability condition presented in this paper is less conservative than the one given in the literature [32].

90

S. Li, Z. Xiang / Nonlinear Analysis: Hybrid Systems 27 (2018) 77–91

Fig. 3. State responses of the corresponding closed-loop system.

5. Conclusions The problems of exponential stability analysis and weighted L2 -gain control synthesis for a class of positive switched T–S fuzzy systems have been addressed in this paper. Different from the general quadratic Lyapunov function, a PQCLF is proposed for the first time. A less conservative stability condition is firstly established for the positive switched T–S system by applying the PQCLF. Then, the weighted L2 -gain performance is analyzed for the given positive switched T–S fuzzy system. Furthermore, a fuzzy state-feedback controller ensuring the exponential stability with a prescribed weighted L2 -gain performance and positivity in closed-loop is designed for the underlying system. Finally, two numerical examples are provided to verify the applicability of the proposed results. It is known that time-delay and uncertainties are frequently encountered in practice and have attracted much attentions in recent years. To achieve more practical results, it would be interesting to generalize the obtained results to robust stabilization problem of positive switched T–S fuzzy systems with time delays in our future work. Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 61273120, 61673219), Jiangsu Six Talents Peaks Project of Province (No. XNYQC-CXTD-001), and Tianjin Major Projects of Science and Technology (No. 15ZXZNGX00250). References [1] L. Farina, S. Rinaldi, Positive Linear Systems, Wiley, New York, 2000. [2] T. Kaczorek, Positive 1D and 2D Systems, Springer-Verlag, Berlin, Germany, 2002. [3] I. Sandberg, On the mathematical foundations of compartmental analysis in biology, medicine, and ecology, IEEE Trans. Circuits Syst. 25 (5) (1978) 273–279. [4] G. Silva-Navarro, J. Alvarez-Gallegos, Sign and stability of equilibria in quasi-monotone positive nonlinear systems, IEEE Trans. Automat. Control 42 (3) (1997) 403–407. [5] R. Shorten, F. Wirth, D. Leith, A positive systems model of TCP-like congestion control: asymptotic results, IEEE/ACM Trans. Netw. 14 (3) (2006) 616–629. [6] T. Kaczorek, Realization problem for positive continuous-time systems with delays, Int. J. Comput. Intell. Appl. 6 (2) (2006) 289–298. [7] C. Commault, A simple graph theoretic characterization of reachability for positive linear systems, Systems Control Lett. 52 (3–4) (2004) 275–282. [8] O. Mason, R. Shorten, On linear copositive Lyapunov functions and the stability of switched positive linear systems, IEEE Trans. Automat. Control 52 (7) (2007) 1346–1349. [9] X. Zhao, L. Zhang, P. Shi, M. Liu, Stability of switched positive linear systems with average dwell time switching, and design, Automatica 48 (6) (2012) 1132–1137. [10] X. Zhao, S. Yin, H. Li, B. Niu, Switching stabilization for a class of slowly switched systems, IEEE Trans. Automat. Control 60 (1) (2015) 221–226. [11] X. Zhao, L. Zhang, P. Shi, M. Liu, Stability and stabilization of switched linear systems with mode-dependent average dwell time, IEEE Trans. Automat. Control 57 (7) (2012) 1809–1815. [12] X. Zhao, H. Liu, J. Zhang, Multiple-mode observer design for a class of switched linear systems linear systems, IEEE Trans. Autom. Sci. Eng. 12 (1) (2015) 272–280. [13] D. Liberzon, Switching in Systems and Control, Birkhauser, Boston, 2003. [14] Z. Sun, S. Ge, Switched Linear Systems: Control and Design, Springer, New York, 2005. [15] D. Wang, W. Wang, P. Shi, X. Sun, Controller failure analysis for systems with interval time-varying delay: a switched method, Circuits Systems Signal Process. 28 (3) (2009) 389–407. [16] R.N. Shorten, D.J. Leith, J. Foy, R. Kilduff, Analysis and design of congestion control in synchronised communication networks, Automatica 41 (1) (2005) 725–730.

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