DELAY-DEPENDENT STABILIZATION ... - Semantic Scholar

International Journal of Innovative Computing, Information and Control Volume 7, Number 4, April 2011

c ICIC International ⃝2011 ISSN 1349-4198 pp. 1533–1547

DELAY-DEPENDENT STABILIZATION CONDITIONS OF CONTROLLED POSITIVE T-S FUZZY SYSTEMS WITH TIME VARYING DELAY Abdellah Benzaouia1 and Ahmed El Hajjaji2 1

LAEPT-EACPI, Dep. of Physics, Faculty of Sciences Semlalia University Cadi Ayyad B.P. 2390, Marrakech 40000, Morocco [email protected]

2

LAEPT-EACPI, Dep. of Physics, Faculty of Sciences Semlalia University of Picardie Jules Vernes MIS Laboratory, 7 Rue de Moulin Neuf 8000 Amiens, France [email protected]

Received November 2009; revised March 2010 Abstract. This paper deals with the problem of delay-dependent stability and the stabilization of Takagi-Sugeno (T-S) fuzzy systems with a time-varying delay while imposing positivity in closed-loop. The stabilization conditions are derived using the single Lyapunov-Krasovskii Functional (LKF) combining the introduction of free-single matrices. A memory feedback control is also used in case the delay matrix is not nonnegative. An example of a real plant is studied to show the advantages of the design procedures. Keywords: LMI, Parallel distributed compensation, Lyapunov-Krasovskii functional, T-S fuzzy systems, Stabilization, Time-delay, Positive systems

1. Introduction. Since the introduction of T-S fuzzy models by Takagi and Sugeno [34] in 1985, fuzzy model control has been extensively studied [3, 7, 8, 10, 15, 17, 30, 31, 32, 33] because T-S fuzzy models can provide an effective representation of complex non linear systems. However, all the aforementioned results are proposed for time-delay free T-S fuzzy systems. In practice, time-delay often occurs in the transmission of information or material between different parts of a system. Transportation systems, communication systems, chemical processing systems, environmental systems and power systems are examples of time-delay systems. Also, it has been shown that the existence of time-delay usually becomes the source of instability and deteriorates the performance of systems. Therefore, the T-S fuzzy model has been extended to deal with nonlinear systems with time-delay. The existing results of stability and stabilization criteria for this class of T-S fuzzy systems can be classified into two types: delay-independent, which is applicable to delays of arbitrary size [12, 24, 36], and delay-dependent, which includes information on the size of delays [9, 13, 22, 23, 25, 28, 29]. It is generally recognized that the delaydependent results are usually less conservative than delay-independent ones, especially when the size of the delay is small. We notice that all the results concerning the analysis and synthesis of delay-dependent methods cited previously are based on single LKF that bring conservativeness in establishing the stability and stabilization tests. Moreover, ∫t the model transformation x(t − τ (t)) = x(t) − t−τ (t) x(s)ds, ˙ the conservative inequalities T T T −1 −2c d ≤ c Xc + d X d and the so-called Moon’s inequality [30] for bounding cross terms are all used in the derivation processes, which introduce the conservatism of the results. More recently, [38] have used a fuzzy LKF combining the introduction of free 1533

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A. BENZAOUIA AND A. EL HAJJAJI

weighting matrices, which improves existing ones in [13, 28] without imposing any bounding techniques on some cross product terms. Other conditions involving only one tuning parameter were also developed [19, 20, 21]. This paper considers an additional problem usually found in dynamical systems: the nonnegativity of the states. The study of systems with nonnegative states is important in practice because many chemical, physical and biological processes involve quantities that have intrinsically constant and nonnegative signs: the concentration of substances, the level of liquids, etc, are always nonnegative. In the literature, systems whose states are nonnegative whenever the initial conditions are nonnegative are referred to as positive [18]. The design of controllers for these positive systems has been studied by [1, 2], where the authors provide a new treatment for the stabilization of positive linear systems. All the proposed conditions are necessary and sufficient, and expressed in terms of Linear Programming (LP). These results were then extended to systems with delay by [26, 27]. One could think that LMI techniques can easily handle this new constraint of nonnegativity of the states. Nevertheless, this is not usually possible without taking care of the use of the adequate Lyapunov function. The model of a real plant is used to show the need for such controllers in practice, especially for fuzzy systems where the model is global involving the whole state and not a state of variation around a set point. This idea, which was earlier used for positive switching systems in [5, 6], has a different impact on positive fuzzy systems due to the form of the obtained global matrix in closed-loop. Sufficient conditions of asymptotic stability, with multiple Lyapunov functions, for positive discrete-time fuzzy systems, represented by Takagi-Sugeno models, were obtained for the first time in [4]. In this paper, we are studying the asymptotic stabilization of continuous-time T-S fuzzy systems with state delay by imposing positivity in closed-loop. We focus on the delay-dependent stabilization synthesis based on the PDC scheme [35, 37]. The delaydependent stabilization conditions obtained in this paper are presented in terms of LMIs without involving any tuning parameter. A memory feedback control is also used in case the delay matrix Aτ is not nonnegative. Thus, in this work, stabilization conditions for continuous-time T-S fuzzy systems with delays, are obtained for the first time. The rest of this paper is organized as follows. In Section 2, we give the description of T-S fuzzy models with time varying state delay and the fuzzy control law based on the PDC structure. New delay dependent stabilization conditions are established for positive systems in Section 3. In Section 4, an example of a real plant is given to show the need of such controllers. Some conclusions are given in Section 5. 1.1. • • •

Notation and definitions. Rn+ denotes the non-negative orthant of n-dimensional real space Rn . M T denotes the transpose of a real matrix M . A matrix M ∈ Rn×n is called a Metzler matrix if its off-diagonal elements are nonnegative. That is, if M = {mij }ni,j=1 , M is Metzler if mij ≥ 0 when i ̸= j. • A matrix M (or a vector) is said to be nonnegative if all its components are nonnegative (by notation M ≥ 0). It is said to be positive if all its components are positive (M > 0).

2. Problem Formulation. Consider a nonlinear system with state-delay which could be represented by a T-S fuzzy time-delay model described by: Plant Rule i (i = 1, 2, · · · , r): If θ1 is αi1 and · · · and θp is αip THEN x(t) ˙ = Ai x(t) + Aτ i x(t − τ (t)) + Bi u(t) x(t) = ψ(t), t ∈ [−µ, 0],

(1)

POSITIVE DELAYED T-S FUZZY SYSTEMS

1535

where θj (x(t)) and αij (i = 1, · · · , r, j = 1, · · · , p) are respectively the premise variable and the fuzzy sets; ψ(t) is the initial conditions; x(t) ∈ Rn is the state and u(t) ∈ Rm is the control input. r is the number of IF-THEN rules; the time delay, τ (t), is a time-varying continuous function that satisfies 0 ≤ τ (t) ≤ µ, τ˙ (t) ≤ β

(2)

By using the commonly used center-average defuzzifier, product interference and singleton fuzzifier, the T-S fuzzy systems can be inferred as r ∑ x(t) ˙ = hi (θ(x(t)))[Ai x(t) + Aτ i x(t − τ (t)) + Bi u(t)] i=1

= A(θ)x(t) + Aτ i (θ)x(t − τ (t)) + B(θ)u(t)

(3)

where θ(x(t)) = [θ1 (x(t)), · · · , θp (x(t))]T and νi (θ(x(t))) : Rp → [0, 1], i ∈ L = {1, · · · , r} is the membership function of the system with respect to the ith plant rule. Denote νi (θ(x(t))) hi (θ(x(t))) = ∑r i=1 νi (θ(x(t))) It is obvious that hi (θ(x(t))) ≥ 0 and

r ∑

hi (θ(x(t))) = 1

i=1

The design of state feedback stabilizing fuzzy controllers for fuzzy system (1) is based on the parallel distributed compensation Controller Rule i (i = 1, 2, · · · , r): If θ1 is αi1 and · · · and θp is αip THEN u(t) = Ki x(t) The overall state feedback control law is represented by r ∑ u(t) = K(θ)x(t) = hi (θ(x(t)))Ki x(t)

(4)

(5)

i=1

In what follows, for brevity we use hi to denote hi (θ(x(t))). The closed-loop system is then written as: r ∑ r [ ] ∑ b x(t) ˙ = hi hj Aij x(t) + Aτ i x(t − τ (t)) i=1 j=1

b = A(θ)x(t) + Aτ i (θ)x(t − τ (t)) x(t) = ψ(t), t ∈ [−µ, 0],

(6)

with bij := Ai + Bi Kj , A

(7)

Hence, the problem we are dealing with consists of designing a gain K(θ) that stabilizes the closed-loop system (6) while ensuring positivity of the state at each time. Our goal is to propose an algorithm that uses the LMI framework which will facilitate the computation of the feedback control gain. The following definitions given in [14] for switched systems can be extended to fuzzy systems: Definition 2.1. System (3) with Ai Metzler and Aτ i nonnegative, is said to be positive if, given any nonnegative initial state x(t) = ψ(t) > 0, t ∈ [−µ, 0] and any input function u(t) ≽ 0, the corresponding trajectory remains in the positive orthant for all t: x(t) ∈ Rn+ .

1536


Definition 2.2. System (3) is said to be controlled positive relative to an initial state x(t) = ψ(t) > 0, t ∈ [−µ, 0] if there exists a control strategy such that the corresponding trajectory remains in the positive orthant for all t: x(t) ∈ Rn+ . The aim of this work is to present new sufficient conditions of existence of state feedback controllers allowing the state to be always nonnegative for continuous-time fuzzy systems with time varying delay. 2.1. Preliminary results. In this section, we characterize the stability of the class of positive systems. In order to adequately characterize positive systems, we propose the following examples. Consider a linear system given by: x(t) ˙ = Ax(t).

(8)

Consider system (8) with, [ A=

−1 −1 1 −2

] .

Even this system is not positive, a diagonal matrix P can be found satisfying AT P +P A < 0 with the following matrix [ ] 27.0459 0 P = (9) 0 16.5505 On the other hand, for a positive system with matrix A given by: [ ] −1 0.5 A= 0.2 −1 a non diagonal matrix P can also be found such that AT P + P A < 0: [ ] 56.9067 16.8728 P = 16.8728 52.3971

(10)

(11)

Further, the positive system (8), with A given by (10), admits the following positive diagonal matrix P : [ ] 33.0230 0 P = . 0 30.4061 However, the non positive system (8) with A given by: [ ] −4 2 A= , −5 0 does not admit a diagonal positive matrix P satisfying AT P + P A < 0, even if it is asymptotically stable. In order to distinguish between these two cases, one has to note the following: • For an asymptotically stable positive linear system, it is always possible to find a diagonal positive matrix P satisfying AT P + P A < 0. • It is not always possible to find a diagonal positive matrix P satisfying AT P +P A < 0 for a non positive asymptotically stable linear system.


1537

Consider the following continuous-time fuzzy autonomous system: x(t) ˙ = A(θ)x(t) + Aτ i (θ)x(t − τ (t)) r ∑ = hi (t) [Ai x(k) + Aτ i x(t − τ (t))]

(12)

i=1

x(t) = ψ(t), t ∈ [−µ, 0]. According to Definition 2.1, we need to find the condition under which the delayed system (12) is positive (see for example [11]). Lemma 2.1. System (12) is positive (i.e., x(t) ∈ Rn+ ) if and only if Ai is a Metzler matrix and Aτ i is a nonnegative matrix ∀i ∈ L. 3. Main Results. In this section, four main results of this work are presented: the first and the second deal with the delay dependent conditions of asymptotic stability and stabilizability of fuzzy systems respectively, the third deals with the corresponding LMI while the fourth extends the previous ones to the case of memory controllers to take care of systems with arbitrary matrices Aτ i . 3.1. Time-delay dependent stability conditions. First, we derive the stability condition for the unforced system (12). In order to do so, consider the following assumption: Assumption 1: Matrices Aτ i are nonnegative. Theorem 3.1. System (12) is asymptotically stable, if there exist a diagonal matrix P = P T > 0, matrices R = RT > 0, Q = QT > 0, G and S satisfying the following LMIs   P Ai + ATi P + R + G + GT P Aτ i − G + S T ATi Q −G  ∗ −(1 − β)R − S − S T ATτi Q −S    (13)  0  0, matrices R = RT > 0, Q = QT > 0, G, S and Ki satisfying the following conditions for i, j = 1, 2, · · · , r and i ≤ j, Φij + Φji < 0 Ai + Bi Kj is Metzler where

 bTij Q −G A Φ12 Φ11 i ij  ∗ Φ22 ATτi Q −S  , Φij =   ∗ ∗ − µ1 Q 0  ∗ ∗ ∗ − µ1 Q

(20) (21)



in which T b bT Φ11 ij = P Aij + Aij P + R + G + G T Φ12 i = P Aτ i − G + S Φ22 = −(1 − β)R − S − S T

(22)


1539

Then, System (6) is asymptotically stable. Proof: The same reasoning from (14) to (16) is used. Inequality (17) is now bounded differently as pointed out in [13]:   bij +A bji ) bij +A bji )T bij +A bji )T r r A A A ( ( ( (A +A ) ∑∑ Q Q τi 2 τj  2 2 2 x(t) ˙ T Qx(t) ˙ ≤ hi hj  (23) b b T T (Aij +Aji ) (Aτ i +Aτ j ) (Aτ i +Aτ j ) (Aτ i +Aτ j ) Q Q i=1 j=1 2 2 2 2 [ ] [ ] T Note ζ(t)T = x(t)T , x(t − τ (t))T and let W T = GT , S T , we obtain: r ∑ r [ ] ∑ ˜ ij + τ W Q−1 W T ζ(t) V˙ (x(t)) ≤ hi hj ζ(t)T Φ i=1 j=1

∫

[ ] [ ]T ζ(t)T W + x(υ) ˙ T Q Q−1 ζ(t)T W + x(υ) ˙ T Q dυ

t

−

(24)

t−τ (t)

[

˜ 11 Φ ij ∗

˜ ij = Φ in which

(

]

˜ 12 Φ ij ˜ 22 Φ ij

bij + A bji A

(25) )T

(

bij + A bji A

)

˜ 11 = P A bij + A bT P + R + µ Φ Q + G + GT ij ij 2 2 ( )T b b Aij + Aji (Aτ i + Aτ j ) ˜ 12 = P Aτ i + µ Φ Q − G + ST ij 2 2 T ˜ 22 = −(1 − β)R + µ (Aτ i + Aτ j ) Q (Aτ i + Aτ j ) − S − S T Φ ij 2 2 By applying the Schur complement ( ) ˜ ij + τ W Q−1 W T < 0 Φ is equivalent to b ij = 1 (Φij + Φji ) < 0 Φ 2 where

      b ij =  Φ     

( b 12 b 11 Φ Φ ij ij ∗

b 22 Φ

∗

∗

∗

∗

bij + A bji A



)T

Q 2 (Aτ i + Aτ j )T Q 2 1 − Q µ ∗

(26)

−G     −S     0   1  − Q µ

in which bij + A bT P + R + G + GT b 11 = P A Φ ij ij b 12 = P Aτ i − G + S T Φ ij 22 b Φ = −(1 − β)R − S − S T

(27)

1540


Finally, closed-loop system (6) is positive by virtue of condition (21). Since matrix P is diagonal, therefore, from (20) we get V˙ (x(t)) ≤ 0. Our objective is to transform the conditions in Theorem 3.2 into LMI terms which can easily be solved by using existing solvers such as the LMI TOOLBOX in Matlab software. ¯ = R ¯ T > 0, Theorem 3.3. If there exist a diagonal matrix X = X T > 0, matrices R T ¯ ¯ ¯ ¯ Q = Q > 0, G, S and Yi satisfying the following LMIs for i ≤ j = 1, 2, · · · , r,

where

Ξij + Ξji < 0 Ai X + Bi Yj is Metzler

(28) (29)

 ¯ ξij11 ξi12 [Ai X + Bi Yj ]T −G  ∗ ξ 22 XATτi −S¯   Ξij =  1 ¯  ∗ ∗ − µ (2X − Q) 0  ¯ ∗ ∗ ∗ − µ1 Q

(30)



in which ¯+G ¯+G ¯T ξij11 = Ai X + YjT BiT + XATi + Bi Yj + R ¯ + S¯T ξi12 = Aτ i X − G ¯ − S¯ − S¯T . ξ 22 = −(1 − β)R Then System (6) is asymptotically stable. In this case, ¯ −1 , Ki = Yi X −1 , R = X −1 RX ¯ −1 , P = X −1 , Q = X −1 QX i = 1, · · · , r Proof: Pre-and post multiplying Φij in (20) by diag[X, X, Q−1 , X],  bT XA ij T T T bij X + X A b +R ¯+G ¯+G ¯ ¯ + S¯ T  A Aτ i X − G XA ij τi  ¯ − S¯ − S¯T  1 ¯ −1 ∗ −(1 − β)R  − XQ X  ∗ ∗ µ  ∗ ∗ ∗

(31) we get  ¯ −G −S¯    0  < 0 (32)  1¯  − Q µ

¯ = XM X. Using the fact that (X − Q) ¯ Q ¯ −1 (X − Q) ¯ > 0 implies −X Q ¯ −1 X < where, M ¯ one can obtain: −2X + Q,   ¯ bT −G XA ij ¯+G ¯+G ¯T ¯ + S¯T b X + XA bT + R  A Aτ i X − G −S¯  XATτi ij   ij T ¯ ¯ ¯   1 ∗ −(1 − β)R − S − S ¯  − (2X − Q) 0  < 0 (33)   ∗ ∗ µ   1 ∗ ∗ ¯ ∗ − Q µ Using (31), we get LMIs (28) together with (30). In addition, condition (29) implies, by post-multiplying by X −1 which is positive and using (31), that matrices Ai + Bi Kj are Metzler. Note that if matrix X is only positive definite and not diagonal, matrix X −1 will not always be positive. This completes the proof.


1541

3.3. Synthesis of controllers with memory. In order to avoid the assumption of nonnegativity of matrices Aτ i , the previous results can be extended to controllers with memory as is now shown. The feedback law is now given by: u(t) = K(θ)x(t) + F (θ)x(t − r)

(34)

where Ki , Fi ∈ Rm×n . This control law has to be designed, in this paper, in such a way that the closed-loop system defined by x(t) ˙ = [A(θ) + B(θ)K(θ)] x(t) + [Aτ (θ) + B(θ)F (θ)] x(t − τ (t)),

(35)

satisfies the following problem: find sufficient conditions on matrices Ai , Aτ i ∈ Rn×n and Bi ∈ Rn×m such that there exist gains Ki and Fi ∈ Rm×n satisfying closed-loop stability and positivity. That is: • Ai + Bi Kj are Metzler matrices. • Aτ i + Bi Fj are nonnegative matrices. • System (35) is asymptotically stable. Using the delayed state adds a degree of freedom to the controller that makes the positive stabilization of a wider set of plants possible: For example, to impose non negativeness in systems where Aτ i has negative elements. ¯ = R ¯ T > 0, Corollary 3.1. If there exist diagonal matrix X = X T > 0, matrices R ¯=Q ¯ T > 0, G, ¯ S, ¯ Yi and Zi satisfying the following LMIs for i ≤ j = 1, 2, · · · , r, Q Ξij + Ξji < 0 Ai X + Bi Yj is Metzler Aτ i X + Bi Zj ≥ 0 where



¯ ξij11 ξi12 [Ai X + Bi Yj ]T −G T 22  ∗ ξ [Aτ i X + Bi Zj ] −S¯   1 ¯ Ξij =  ∗ 0 ∗ − (2X − Q)  µ  1¯ ∗ ∗ ∗ − Q µ

(36) (37) (38)       

(39)

in which ¯+G ¯+G ¯T ξij11 = Ai X + YjT BiT + XATi + Bi Yj + R ¯ + S¯T ξi12 = Aτ i X + Bi Zj − G ¯ − S¯ − S¯T . ξ 22 = −(1 − β)R Then System (6) is asymptotically stable. In this case, Ki = Yi X −1 , Fi = Zi X −1 , ¯ −1 , P = X −1 , ¯ −1 , Q = X −1 QX R = X −1 RX i = 1, · · · , r

(40)

Proof: The proof is similar to that of Theorem 3.3, while replacing Aτ i by Aτ i + Bi Fj and noting Fj X = Zj . Comments: • The theoretic conditions in Theorem 3.2 are transformed into LMI terms in Theorem 3.3 and Corollary 3.1, which can be easily solved using existing solvers such as the LMI TOOLBOX in Matlab software. Hence, the obtained results in this work are easily applicable in practice as will be indicated in the following section.

1542


• Since the presented results on the stabilization of continuous-time T-S fuzzy systems with state delay by imposing positivity in closed-loop, are obtained for the first time in this paper, no comparison study can be addressed. 4. Application to a Real Plant Model. Example 4.1. In order to show the need for this study on positive systems, consider the process composed of two linked tanks of capacity 22 liters each. This system can be described by the following balance equations [16]: x˙ 1 (t) = u1 (t) − q12 (t) − q1 (t) x˙ 2 (t) = u2 (t) + q12 (t) − q2 (t) where xi holds for the level in liters of tank i, uj represents the flow in liter/mn of pump j, q12 is the variation of the flow between the two tanks and qi the loss flow of each tank. Applying the Torricelli law, one obtains: √ q1 = γ1 σ1 2gx1 √ q2 = γ1 σ2 2gx2 √ q12 = γ12 σ1 2g|x1 − x2 |sign(x1 − x2 ) where γi and γij are physical constants, σi is the tank section and g the gravity acceleration. The process model is then as follows: √ √ x˙ 1 (t) = u1 − R1 x1 − R12 |x1 − x2 |sign(x1 − x2 ) √ √ x˙ 2 (t) = u2 − R2 x2 + R12 |x1 − x2 |sign(x1 − x2 ) The obtained model is then nonlinear. Note that levels xi must always be positive. To obtain a T-S fuzzy representation for this nonlinear system, the classical transfor√ mation xi = √xxi i = xi zi with zi = √1xi is used. In this case, √ 1 = √ z12z2 2 . The |x1 −x2 |

|z2 −z1 |

corresponding model is then given by: x(t) ˙ = A(z1 , z2 )x(t) + Bu(t), y(t) = Cx(t)

(41)

where matrix A(z1 , z2 ) has the general following form:   R12 z1 z2 R12 z1 z2 √ −R1 z1 − √ |z12 −z22 | |z12 −z22 | , A(z1 , z2 ) =  R z z R12 z1 z2 12 1 2 √ 2 2 −R2 z2 − √ 2 2 |z1 −z2 |

|z1 −z2 |

which is Metzler by construction, B = I2 and C = I2 . This system is known to have transport delay. As a first approximation, System (41) can be rewritten as a time delay system given by: x(t) ˙ = (1 − ε)A(z1 , z2 )x(t) + ε|A(z1 , z2 )|x(t − τ (t)) + Bu(t), y(t) = Cx(t)

(42)

with ε ∈ [0, 1] and the delay assumed to satisfy: τ (t) = µ + β |sin(t)|. The objective is that the output y tracks a given reference, yr . Thus, keeping in mind the notation of (3), the following control is used: u(t) = K(θ)x(t) + L(θ)yr , where controller gain K(θ) ensures the asymptotic stability together with the positivity in closed-loop while controller gain L(θ) achieves the tracking objective. In this case, by assuming τ constant,


1543

ˆ ˆ one obtains X(s) = (sI − A(θ) − Aτ (θ)e−sτ )−1 BL(θ)Yr (s), where matrix A(θ) = (1 − ε)A(θ) + BK(θ). That is, [ ]−1 ˆ − Aτ (θ)e−sτ C sI − A(θ) BL(θ)yr Y (s) = s [ ]−1 ˆ + Aτ (θ) Using the final value theorem, one can deduce y(∞) = −C A(θ) BL(θ)yr . ˆ − Aτ (θ), matrices B and C being equal to the identity in our If one chooses L(θ) = −A(θ) ˆ case, the tracking objective will be reached with y(∞) = yr . Further, L(θ) = −A(θ)−A τ (θ) ˆ holds if Li = −Ai − Aτ i , i = 1, · · · , r. This means that for a time varying delay, the tracking error will not be equal to zero. By considering that zi ∈ [ai ; bi ], the four following rules are taken into account: IF z1 (k) is about a1 and z2 (k) is about a2 , THEN, A(z1 , z2 ) = A(a1 , a2 ) = A1 . IF z1 (k) is about a1 and z2 (k) is about b2 , THEN, A(z1 , z2 ) = A(a1 , b2 ) = A2 . IF z1 (k) is about b1 and z2 (k) is about a2 , THEN, A(z1 , z2 ) = A(b1 , a2 ) = A3 . IF z1 (k) is about b1 and z2 (k) is about b2 , THEN, A(z1 , z2 ) = A(b1 , b2 ) = A4 . The membership functions are given by: α1 (k) = f11 (k)f21 (k), α2 (k) = f11 (k)f22 (k), i α3 (k) = f12 (k)f21 (k), α4 (k) = f12 (k)f22 (k); where fi1 (k) = zia(k)−b and fi2 (k) = 1 − i −bi i (k) fi1 (k) = aia−z , i = 1, 2. i −bi For this real system, matrix B is common for all the subsystems. The consequence is that the number of LMIs (28) and (29) to be used is considerably reduced by letting i = j. Parameters R1 , R2 , R12 are experimentally estimated as R1 = R2 = 0.95, R12 = 0.52. While a1 = 0.2236, b1 = 0.4472 (volume of tank 1 between 5 and 20 liters), a2 = 0.2582, b2 = 0.4082 (volume of tank 2 between 6 and 15 liters). The desired reference is yr = [18; 15]T . In order to stabilize the T-S fuzzy system while imposing positivity in closed-loop, we solve the LMIs of Theorem 3.3. If these LMIs are feasible, one can compute the required controllers Ki without memory and the corresponding Lyapunov function given by P . Using LMI TOOLBOX in Matlab software, LMIs (28) and (29) are feasible for ϵ = 0.2; µ = 2mn; β = 2. The obtained solutions are as follows:

[

] [ ] −0.4654 0.0001 −0.3647 0.0545 P = , K1 = , K2 = , 0.0003 −0.4422 0.0595 −0.4785 [ ] [ ] −0.5148 0.0806 −0.2475 −0.2569 K3 = , K4 = 0.0781 −0.3986 −0.2542 −0.2666 0.0048 0 0 0.0047

]

[

Matrices in closed-loop are obtained as: [ ] −0.8214 0.1861 Aˆ1 = , 0.1863 −0.8244 [ ] −0.7959 0.1917 ˆ A3 = , 0.1893 −0.8200

[ Aˆ2 = [ Aˆ4 =

−0.8361 0.1861 0.1910 −0.8063 −1.0031 0.1589 0.1616 −0.9926

] , ] .

One can notice that matrix P is diagonal while matrices in closed-loop are all Metzler as required by Theorem 3.3. Figure 1 plots the evolution of states x1 and x2 in liters starting at x(t) = [5; 6]T , t ∈ [−2, 0], The desired reference is reached in 10mn while the state remains always positive. Figure 2 plots the evolution of the two pump flows in liter/mn. As expected, the time varying delay, which was not taken into account in the tracking computation, prevents the output from reaching reference yr perfectly.

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16

14

x1,x2 in liter

12

10

8

6

4

0

2

4

6

8

10

t in mn

Figure 1. This figure plots the evolution of the states x1 and x2 in liter 8

7

u1,u2 in liter/mn

6

5

4

3

2

1

0

2

4

6

8

10

t in mn

Figure 2. This figure plots the evolution of the two pump flows in liter/mn Assume now that System (41) can be rewritten as a time delay system by: x(t) ˙ = (1 − ε)A(z1 , z2 )x(t) + εA(z1 , z2 )x(t − τ (t)) + Bu(t), y(t) = Cx(t)

(43)

with ε ∈ [0, 1] and the delay assumed to satisfy: τ (t) = µ. The objective is to stabilize the T-S fuzzy system while imposing positivity in closed-loop by choosing controllers with memory given by u(t) = Kx(t) + F x(t − τ (t)). For this, we solve the LMIs of Corollary 3.1. If these LMIs are feasible, one can compute the required controllers Ki , Fi with memory and the corresponding Lyapunov function given by P . The ˆ − Aˆτ (θ) or same technique as before to ensure the tracking objective with, L(θ) = −A(θ) Li = −Aî − Aˆτ i , i = 1, · · · , r, is followed. Using LMI TOOLBOX in Matlab software, LMIs (36) – (38) are feasible for µ = 2mn, ε = 0.2 while β is chosen to be equal to zero to overcome static error. The desired reference here is yr = [15; 10]T . The obtained solutions are as follows: [ ] [ ] [ ] 0.2555 0 −0.0297 −0.0369 0.0858 0.0176 P = , K1 = , K2 = , 0 0.3310 −0.0793 −0.0448 −0.0249 −0.0992 [ ] [ ] −0.1045 0.0380 0.3700 −0.2666 K3 = , K4 = −0.0045 −0.0056 −0.3091 0.2990


[ F1 = [ F3 =

0.2052 0.0953 0.0540 0.2210 0.1865 0.1140 0.0727 0.2308

]

[ ,

F2 =

]

[ ,

Matrices in closed-loop are obtained as:

F4 =

1545

0.2341 0.1089 0.0676 0.2074

]

0.3051 0.0378 −0.0034 0.3070

[

−0.3856 Aˆ1 = Aˆ2 = Aˆ3 = Aˆ4 = 0.1067 [ 0.1162 Aˆτ 1 = Aˆτ 2 = Aˆτ 3 = Aˆτ 4 = 0.1005

0.1492 −0.4270 0.1418 0.1255

, ] .

] , ] .

One can notice that matrix P is diagonal while matrices in closed-loop Aî are all Metzler and matrices Aˆτ i are nonnegative as required by Corollary 3.1. 15 14 13

x1,x2 in liter

12 11 10 9 8 7 6

0

5

10

15 t in mn

20

25

30

Figure 3. This figure plots the evolution of the states x1 and x2 in liter 7

6

4

1 2

u ,u in liter/mn

5

3

2

1

0

5

10

15 t in mn

20

25

30

Figure 4. This figure plots the evolution of the two pump flows in liter/mn Figure 3 plots the evolution of the states x1 and x2 in liters starting at x(t) = [5; 6]T , t ∈ [−2, 0]. The desired reference is reached in 30mn while the state always remains positive. Figure 4 plots the evolution of the two pump flows in liter/mn.

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5. Conclusion. In this paper, the delay-dependent design of state feedback stabilizing fuzzy controllers for T-S fuzzy systems with time varying delay, while imposing positivity in closed-loop, is investigated. The proposed method which was used to reduce the conservatism and the computational burden at the same time in [20], is used to govern the closed-loop system in the positive orthant only. However, the delay-dependent stabilization conditions obtained in this paper are presented in terms of LMIs without involving any tuning parameter. A memory feedback control is also used in case delay matrices Aτ i are not nonnegative. Finally, the model of a real plant is used to show the need for such controllers in practice, especially for fuzzy systems, where the model is global involving the whole state and not a state of variation around a set point. Acknowledgment. This research has been supported by the FEDER and ”Region Picardie” in the CHAMP project (low-Carbon Hybrid Advanced Motive Power) through the European INTERREG IVA . The authors gratefully acknowledge the support of these institutions. REFERENCES [1] M. A. Rami and F. Tadeo, Linear programming approach to impose positiveness in closed-loop and estimated states, Proc. of the 17th Int. Symp. on Mathematical Theory of Networks and Systems, Kyoto, Japan, 2006. [2] M. A. Rami and F. Tadeo, Controller synthesis for positive linear systems with bounded controls, IEEE Trans. on Circuits and Systems II, vol.54, no.2, pp.151-155, 2007. [3] A. Benzaouia, A. El Hajjaji and M. Naib, Stabilization of a class of constrained fuzzy systems: A positive invariance approach, International Journal of Innovative Computing, Information and Control, vol.2, no.4, pp.749-760, 2006. [4] A. Benzaouia, A. Hmamed and A. El Hajjaji, Stabilization of positive discrete-time fuzzy systems by state feedback control, Int. J. Adap. Cont. Signal Processing, 2010. [5] A. Benzaouia and F. Tadeo, Stabilization of positive switching linear discrete-time systems, International Journal of Innovative Computing, Information and Control, vol.6, no.6, pp.2427-2437, 2010. [6] A. Benzaouia and F. Tadeo, Stabilization of positive discrete-time switching systems by output feedback, The 16th Med. Conf. on Control and Automation, Ajaccio, France, 2008. [7] A. Benzaouia, D. Mehdi, A. El Hajjaji and M. Nachidi, Piecewise quadratic lyapunov function for nonlinear systems with fuzzy static output feedback control, European Control Conference, Kos, Greece, 2007. [8] E. Boukas and A. El Hajjaji, On stabilizability of stochastic fuzzy systems, Proc. of the 2006 American Control Conference, Minneapolis, MN, pp.4362-4366, 2006. [9] E.-K. Boukas, Free-weighting matrices delay-dependent stabilization for systems with time-varying delays, ICIC Express Letters, vol.2, no.2, pp.167-173, 2008. [10] M. Chadli and A. El Hajjaji, A observer-based robust fuzzy control of nonlinear systems with parametric uncertaintie, Fuzzy Sets and Systems, vol.157, no.9, pp.1276-1281, 2006. [11] V. Chellaboina, W. M. Haddad, J. Ramakrishnan and J. M. Bailey, On monotonocity of solutions of nonnegative and compartmental dynamical systems with time delays, Proc. of Conference on Decision and Control, Hawaii, HI, pp.4008-4013, 2003. [12] B. Chen and X. Liu, Fuzzy guaranteed cost control for nonlinear systems with time-varying delay, IEEE Trans. on Fuzzy Systems, vol.13, no.2, pp.238-249, 2005. [13] B. Chen and X. Liu, Delay-dependent robust H∞ control for T-S fuzzy systems with time delay, IEEE Trans. on Fuzzy Systems, vol.13, no.2, pp.238-249, 2005. [14] E. de Santis and P. Giordano, Positive switching systems, Proc. of the Multidisciplinary International Symposium on Positive Systems, Grenoble, France, 2006. [15] A. El Hajjaji, A. Benzaouia and M. Naib, Stabilization of fuzzy systems with constrained controls by using positively invariant sets, Mathematical Problems for Engineering, pp.1-17, 2006. [16] A. El Hajjaji and M. Chadli, Commande basée sur la modélisation floue de type Takagi-Sugeno d’un procédé expérimental á quatres cuves, CIFA Conference, Bucarest, 2008.


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