Finite time annular domain robust stability analysis and controller design for T-S fuzzy interval positive systems

This paper is concerned with the finite time annular domain robust stability (FTADRS) analysis and controller design for T-S fuzzy positive systems with interval uncertainties. The concept of finite time annular domain stability is first introduced for positive systems. Based on this and using the copositive Lyapunov function approach, some sufficient conditions for FTADRS are derived. Subsequently, the finite time annular domain robust controller is designed via the linear programming technique. Finally, two numerical examples and an application example are employed to show the effectiveness of our results.

stability (FTS) is proposed, and many important results have 23 been reported [13]- [17]. However, in some special cases, it 24 requires that system states do not only exceed a specific upper 25 bound, but also do not fall below a specific lower bound in 26 finite time. For example, it is always needed to take multiple 27 medical measures for diabetes patients to keep their blood 28 sugar levels within a safe range(i.e., 70-180 mg/dL) [18], 29 [19]. If the blood glucose concentration is not in this range, 30 it will lead to a series of serious complications and even where x(t) ∈ R n is the state of the system; u(t) ∈ R s is the 118 input of the system; ϑ i (t) are the known premise variables 119 and M ij (i = 1, 2, · · · , r, r is the number of model rules, j = 120 1, 2, · · · , l, l is the number of premise variables) is fuzzy set, 121 respectively. A i ∈ R n×n , B i ∈ R n×s are uncertain system 122 matrices, but with known bounds, i.e., where A i , B i and A i , B i are the upper and lower bound 125 matrices of A i , B i . By fuzzy blending, we can obtain the 126 following overall T-S fuzzy model: Definition 2. For given positive scalars ξ 1 , ξ 2 , ξ 3 , ξ 4 , T with ξ 2 > ξ 4 > ξ 3 > ξ 1 ≥ 0, and a vector R 0, if T ], then positive system (4) is FTADS w.r.t. (ξ 1 , ξ 2 , ξ 3 , ξ 4 , T, R).
[20] If there exist a nonnegative function g(t) and nonnegative constants a, b such that holds, then g(t) ≥ ae bt , ∀t ∈ [0, T ].

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In this section, a sufficient condition is given to test the 163 FTADRS of positive system (4) when u(t) = 0.

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Proof. We use two steps to prove Theorem 1.
Consider the following copositive Lyapunov function: By (6), we can obtain Multiplying both sides of (13) by e −αt and integrating from 0 to t, one has By Lemma 4, we have: Considering (8) and (9), one has By (12) and (15), the following inequality holds: Considering (14)-(16), we have Then, by the fact of Step 2.
Multiplying both sides of (18) by e −βt and integrating from 187 0 to t, one has VOLUME 4, 2016 According to Lemma 5, we can obtain: By (12) and (15), one has Considering (11) (15) (20) and (21), we have Then, by the fact of Hence, combining the above two steps, we obtain that (i.e., Theorem 1). Therefore, the values of α and β must be 198 appropriately selected. Considering (2) and (6)-(7), we have Then, by (10) and (11), 0 ≤ 212 there exist a vector P 0 and positive constants ω 1 , ω 2 such 215 that (8)-(11) and the following inequalities hold: Proof. Letting A i = A i , inequalities (6) and (7) immediately 219 reduce to (23) and (24).  Proof. Letting A i = A i and referring to the proof process of Step 1 in Theorem 1, we get that In this section, based on the above stability result, a state 230 feedback controller is designed for positive system (4).

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Consider the following state feedback controller: where K j ∈ R s×n . Then, the closed-loop system is obtained The controller gain matrices K j can be obtained by the 235 following theorem.

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In this case, the controller gain matrices can be obtained 248 as: Proof. Considering (30), we have Then, by Lemma 3, A i + B i K j are M matrices. With the fact that Q j 0, we obtain Hence, by Lemma 2, it is easy to prove that system (25) is 251 positive.

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In this case, controller gain matrices can be obtained as:  In this case, K j can also be obtained by (43).

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In this section, we use two numerical examples and an 300 application example to illustrate the above results.
By Lemma 2, it is easy to prove that system (44) is positive. x T (t)R of system (A i ) is not higher than ξ 2 . Hence, positive By Lemma 2, it is easy to prove that system (45) is positive.

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By the algorithm in Figure 1, we obtain a feasible region of α and β, as shown in Figure 5. Choosing α = 0.6, β = 0.1 and solving (8)- (11) and (26) Then, the controller gain matrices are obtained as: By Lemma 2, it can be easily verified that the closed-

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loop system is positive. Figure 6 shows the trajectory of the Lotka-Volterra model [1]: parameters, and one may refer to [1] for more details.