2017 IEEE International Conference on Systems, Man, and Cybernetics (SMC) Banff Center, Banff, Canada, October 5-8, 2017
Improved Fuzzy H∞ Filter Design Method for Nonlinear Systems with Time-Varing Delay Qianqian Ma∗ , Li Li† , Junhui Shen∗ , Haowei Guan‡ , Guangcheng Ma∗ , and Hongwei Xia∗ ∗ School of Astronautics Harbin Institute of Technology, Harbin 150001, P. R. China Email: maqq222008@hit. edu.cn † School of Information Science and Engineering Harbin Institute of Technology at Weihai, Weihai 264200, P. R. China Email:
[email protected] ‡ Shanghai Institute of Satellite Engineering, Shanghai 201109 Email:
[email protected]
Abstract—This paper investigates the fuzzy H∞ filter design issue for nonlinear systems with time-varying delay. In order to obtain less conservative fuzzy H∞ filter design method, a novel integral inequality is employed to replace the conventional Lebniz-Newton formula to analyze the stability conditions of the filtering error system. Besides, the information of the membership functions is introduced in the criterion to further relax the derived results. The proposed delay dependent filter design method is presented as LMI-based conditions, and corresponding definite expressions of fuzzy H∞ filter are given as well. Finally, a simulation example is provided to prove the effectiveness and superiority of the designed fuzzy H∞ filter.
I. I NTRODUCTION Filtering is playing a critical role in signal processing, and during last decades, a variety of filters have been developed, like Kalman filter [1], H2 filter [2], and H∞ filter [3, 4]. Among them, H∞ filter has attracted considerable attention from researchers, since it can deal with systems with uncertainty, and it has no specific requirement for external noises. Moreover, time-delay often appears in practical systems, which can destroy the system performance and even cause instability of the control system. Therefore, the study of timedelay systems is of great importance [3–5]. When dealing with analysis and synthesis problems of nonlinear systems with time-delay, the T-S fuzzy model[6, 7] is often employed, which can represent the nonlinear system as the weighted sum of some local linear models. Motivated by the parallel distribution compensation (PDC) methodology [8], we assume that the fuzzy H∞ filter to be designed and the T-S fuzzy model have the same premise membership functions and the same number of rules in this paper. Thus, the stability analysis and synthesis can be facilitated. As the fuzzy H∞ filter has to guarantee the whole filtering system is asymptotically stable, the Lyapunov stability theory [9] is applied, which will make the derived results conservative. To reduce conservatism, researchers have developed various methods [3, 5, 6, 9]. To mention a few, in [5], a delaydependent fuzzy H∞ filter design method was proposed for T-S fuzzy-model-based system with time-varying delay. However, in this paper, the Lyapunov-Krasovskii function candidate 978-1-5386-1645-1/17/$31.00 ©2017 IEEE
was chosen as a single Lyapunov function, to obtain more relaxed results, the literature [4] adopted a fuzzy Lyapunov function to analyze the stability condition. In [3], the fuzzy H∞ filter design approach was improved by estimating the upper bound of the derivative of Lyapunov function without ignoring any useful terms. On the basis of [3], literature [6] proposed a technique to obtain more accurate upper bound of the derivative of Lyapunov function. However, all the aforementioned literature used the inequalities derived from the Leibniz-Newton formula [3] to derive stability conditions, just like in some other literature [10–12]. Though such methods can solve the fuzzy H∞ filter design problem, the derived results are conservative and there is little room left to further reduce the conservatism. Therefore, in this paper, we aim to propose a new less conservative fuzzy H∞ filter design method which is not based on the conventional LeibnizNewton formula. Besides, to the best of our knowledge, no existing fuzzy H∞ filter design method has considered the information of membership functions, which motivate us to investigate a membership function dependent design method. To achieve our goal, the fuzzy H∞ filter to be designed and the T-S fuzzy model will be assumed to have the same premise membership functions and the same number of fuzzy rules. To reduce the conservatism of the design method, a novel integral inequality [13] will be employed to replace the traditional Leibniz-Newton formula to deal with integral term Rβ T x ˙ (s)R x(s)ds ˙ in stability analysis. Besides, the informaα tion of the membership functions will be taken into account in the derived criteria to further relax the derived results. II. P RELIMINARIES Consider a nonlinear system involving time-varying delay, which is described by the following p-rule T-S fuzzy model: Plant Rule i: IF ψ1 (t) is Mi1 and ψ2 (t) is Mi2 and . . . and ψm (t) is Mim , THEN
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x(t) ˙
= Ai x(t) + Aτ i x(t − τ (t)) + Bi w(t),
y(t)
= Ci x(t) + Cτ i x(t − τ (t) + Di w(t),
z(t)
= Ei x(t) + Eτ i x(t − τ (t)),
x(t)
= φ(t),
∀t ∈ [−τ0 , 0],
(1)
where i = 1, 2, . . . , p. ψα (t)(α = 1, 2, . . . , m) is the premise variable. Miα is the fuzzy term of rule i which corresponds to the function ψα . m is a positive integer. And x(t) ∈ Rn is the system state, z(t) ∈ Rq is the unknown signal to be estimated, y(t) ∈ Rm is the system output, w(t) ∈ Rp is the noise signal which is assumed to be arbitrary and satisfies w(t) ∈ L2 ∈ [0, ∞). Ai , Aτ i , Bi , Ci , Cτ i , Di , Ei , Eτ i are given system matrices. Time delay τ (t) is a continuously differentiable function, satisfying the conditions: 0 ≤ τ (t) < h,
τ˙ (t) ≤ ρ.
(2)
x(t) ˙ =
i=1 j=1
p X p X
Aτ i 0 υi (ψ(t))υj (ψ(t)) ˆ Bj Cτ i 0 i=1 j=1 p X p X Bi ¯ = B(t) , υi (ψ(t))υj (ψ(t)) ˆ Bj Di A¯τ (t) =
i=1 j=1
¯ = E(t)
p p X X
−Cˆj ],
υi (ψ(t))υj (ψ(t))[Ei
i=1 j=1 p X p X
By fuzzy blending, the system dynamics can be presented as p X
where ζ(0) = [φ(t), x ˆ0 ] for ∀t ∈ [−τ0 , 0], and p p XX Ai 0 ¯ = A(t) υi (ψ(t))υj (ψ(t)) ˆ Bj Ci Aˆj
¯τ (t) = E
υi (ψ(t))υj (ψ(t))[Eτ i
0].
i=1 j=1
υi (ψ(t))[Ai x(t) + Aτ i x(t − τ (t)) + Bi w(t)],
Ergo, the fuzzy H∞ filter design problem that will be resolved in this paper can be summarized as follows: Fuzzy H∞ filter issue: Design a fuzzy filter in the form of y(t) = υi (ψ(t))(Ci x(t) + Cτ i x(t − τ (t) + Di w(t)), (5) satisfying the following two conditions: i=1 p X (1) If w(t) = 0, the filtering system (6) is asymptotically υi (ψ(t))(Ei x(t) + Eτ i x(t − τ (t))). z(t) = stable; i=1 (2) For a given scalar γ > 0, if ζ(t) ≡ 0 for t ∈ [−h, 0], the (3) P following H∞ performance can be satisfied for all the T > 0 p m (ψα (t)) where υi (ψ(t)) = Πm α=1 µMiα (ψα (t))/ k=1 Πα=1 µMk α and w(t) ∈ L2 [0, ∞). is Pp the normalized membership function satisfying: Z T Z T i=1 υi (ψ(t)) = 1, υi (ψ(t)) > 0; and µMiα (ψα (ψ(t))) is ke(t)k2 dt ≤ γ 2 kw(t)k2 dt. (7) the grade of membership function which corresponds to the 0 0 fuzzy term Miα . Besides, the following lemma is useful for the later deducMotivated by the parallel distribution compensation (PDC) tion of the main results. methodology [8], the fuzzy H∞ filter is assumed to have the Lemma 1 [13]: It is assumed that x is a differentiable same premise membership functions and the same number of function: [α, β] → Rn . For N1 , N2 , N3 ∈ R4n×n , and fuzzy rules as the fuzzy model, which can be presented as: R ∈ Rn×n > 0, the following inequality holds: j j Z β Filter Rule j: IF ψ1 (t) is M1 and ψ2 (t) is M2 and . . . and ψm (t) is Mjm , THEN x˙ T (s)Rx(s)ds ˙ ≤ ξ T Ωξ, (8) − i=1 p X
α
ˆj y(t), x ˆ˙ (t) = Aˆj x ˆ(t) + B zˆ(t) = Cˆj x ˆ(t),
where (4)
where j = 1, 2, . . . , p. x ˆ(t) ∈ Rn and zˆ(t) ∈ Rq are the state ˆj , Cˆj and output of the fuzzy H∞ filter respectively. And Aˆj , B are the filter matrices of that will be designed. Similarly, through fuzzy blending, the fuzzy H∞ filter to be designed can be presented as x ˆ˙ (t) =
zˆ(t) =
p X j=1 p X
ˆj y(t)), υj (ψ(t))(Aˆj x ˆ(t) + B (5) υj (ψ(t))Cˆj x ˆ(t).
j=1
According to (3) and (5), and define the augmented state vector as ζ(t) = [xT (t), x ˆ(t)]T and e(t) = z(t) − zˆ(t), we can obtain the H∞ filtering system as follows ˙ = A(t)ζ(t) ¯ ¯ ζ(t) + A¯τ (t)ζ(t − τ (t)) + B(t)w(t)), ¯ ¯τ (t)ζ(t − τ (t))), e(t) = E(t)ζ(t) +E
(6)
1 Ω = τ (N1 R−1 N1T + N2 R−1 N2T + 3 + Sym{N1 ∆1 + N2 ∆2 + N3 ∆3 }, ei = 0n×(i−1)n In 0n×(4−i)n , Π2 = e1 + e2 − 2e3 , T
T
ξ = [x (β) x (α)
1 N3 R−1 N3T ) 5 τ = β − α, Π1 = e 1 − e 2 ,
Π3 = e1 − e2 − 6e3 + 6e4 , RβRs Rβ T T x (s)ds d22 α α xT (u)duds] . α
1 d
III. M AIN R ESULTS First, a sufficient stability condition for the filtering system (6) will be derived. Lemma 2: Given constants h, ρ and γ > 0, the system (6) is asymptotically stable with w(t) ≡ 0, and satisfies the prescribed H∞ performance requirement (7), if there exist matrices P = P T ∈ R2n×2n , Y = Y T ∈ R2n×2n , Z = Z T ∈ R2n×2n , such that the following inequality is feasible. √ T Ξ(t) hΓ1 P ΓT2 (t) Φ(t) = ∗ (9) −P Z −1 P 0 < 0, ∗ ∗ −1
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where Ξ(t) = Λ + Ξ3 (t), −3Z Z 12Z −10Z 0 ∗ −3Z −8Z 10Z 0 3 ∗ ∗ −64Z 60Z 0 Λ= , h ∗ ∗ ∗ −60Z 0 ∗ ∗ ∗ ∗ 0 T ¯ + A¯ (t)P + Y P A¯τ (t) 0 P A(t) ∗ −Y 0 ∗ ∗ 0 Ξ3 (t) = ∗ ∗ ∗ ∗ ∗ ∗ ¯ ¯ ¯ Γ1 (t) = A(t) Aτ (t) 0 0 B(t) , ¯ ¯τ (t) 0 0 . E Γ2 (t) = E(t)
So according to (2) and (6), we can acquire ¯ ¯ V˙ (t) < 2ζ T (t)P [A(t)ζ(t) + A¯τ (t)ζ(t − τ (t)) + B(t)w(t)] − (1 − d)ζ T (t − τ (t))Y ζ(t − τ (t)) ˙ T Z ζ(t) ˙ + µT (t)(Ξ1 + Ξ2 )µ(t). + hζ(t) (13) After some algebra, (13) can be expressed as the following compact form: ¯ 0 P B(t) V˙ (t) + eT (t)e(t) − γ 2 wT (t)w(t) < µT (t)(Ξ1 + Ξ2 (14) 0 0 + Ξ3 (t) + hΓT1 (t)ZΓ1 (t) + ΓT2 (t)Γ2 (t))µ(t), 0 0 , 0 0 where Ξ3 (t), Γ1 (t), Γ2 (t) are defined in (9). ∗ −γ 2 So if Ξ1 + Ξ2 + Ξ3 (t) + hΓT1 ZΓ1 (t) + ΓT2 Γ2 (t) < 0, there will have
Proof: Constructing the Lyapunov-Krasovskii function as follows: Z t T ζ T (s)Y ζ(s)ds V (t) =ζ (t)P ζ(t) + t−τ (t) (10) Z 0 Z t T ˙ ˙ ζ(s) Z ζ(s)dsdθ. + −h
(15)
t+θ
Then the derivative of V (t) can be obtained as ˙ − (1 − τ˙ (t))ζ T (t − τ (t))Y ζ(t − τ (t)) V˙ (t) =2ζ T (t)P ζ(t) Z t T ˙ T Z ζ(s)ds. ˙ ˙ ˙ ζ(s) + hζ(t) Z ζ(t) − (11) Applying Lemma 1 to the last term in the right hand side of (11), we can obtain Z t Z t T ˙ ˙ ˙ ζ˙ T (s)Z ζ(s)ds ζ (s)Z ζ(s)ds < − − t−τ (t) t−h τ (t) T M2 Z −1 M2T ≤ µ (t) τ (t)M1 Z −1 M1T + 3 τ (t) −1 T M3 Z M3 + Sym{M1 Π1 + M2 Π2 + M3 Π3 } µ(t) + 5 h h < µT (t) hM1 Z −1 M1T + M2 Z −1 M2T + M3 Z −1 M3T 3 5 + Sym{M1 Π1 + M2 Π2 + M3 Π3 } µ(t) = µT (t) (Ξ1 + Ξ2 ) µ(t), (12) µ(t) = Rt 1 ζ T (s)ds θ1 w(t)]T , [ζ T (t) ζ T (t − τ (t)) τ (t) t−τ (t) Z s Z t 2 ζ T (u)duds, θ1 = 2 τ (t) t−τ (t) t−τ (t) h h Ξ1 = hM1 Z −1 M1T + M2 Z −1 M2T + M3 Z −1 M3T , 3 5 Ξ2 = Sym{M1 Π1 + M2 Π2 + M3 Π3 }, ei = 02n×(i−1)2n I2n 02n×(5−i)2n .
(16)
which can be converted to Z L (ke(t)k2 −γ 2 kw(t)k2 )dt+V (t)|t=L −V (t)|t=0 ≤ 0, (17) 0
as V (t)|t=0 = 0 and V (t)|t=L ≥ 0. So we have Z L Z L 2 γ 2 kw(t)k2 )dt, ke(t)k dt ≤ 0
t−h
where
V˙ (t) + eT (t)e(t) − γ 2 wT (t)w(t) < 0,
(18)
0
for all L > 0, and any nonzero w(t) ∈ L2 [0, ∞), which means the H∞ performance requirement is satisfied. Besides, to reduce computational complexity, we will eliminate free matrices by assuming 1 T M1 = [−Z Z 0 0 0] , h 3 T (19) M2 = [−Z − Z 2Z 0 0] , h 5 T M3 = [−Z Z 6Z − 6Z 0] . h Then Ξ1 + Ξ2 can be written as Λ, where Λ is defined in (9). Applying the Schur Complement lemma, we can transfer (15) to (9). Moreover, from inequalities (14) and (15), we can get V˙ (t) < 0 when w(t) ≡ 0, which means the filtering system (6) is aymptotically stable. Thus, the proof of Lemma 2 is completed. Remark 1: It can be seen from the proof process that a novel integral inequality is applied to deal with the integral term Rt ˙ which is tighter than other existing − t−h ζ˙ T (s)Z ζ(s)ds, ones. Therefore, the derived results can be less conservative. Besides, the stability condition obtained has a simpler form, which means the proposed method can be more practical. Usually, the Leibniz-Newton formula [10] is used to do this work, the introduction of the novel integral inequality provides another approach to further reduce conservatism, and improve the performance of the system. From the discussion above, we have got the sufficient condition for the existence of fuzzy H∞ filter. Next we will focus on the fuzzy H∞ filter design for system (3).
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Theorem 1: Given constants h, ρ, ω and γ > 0, the system (6) is asymptotically stable with w(t) ≡ 0, and satisfies the prescribed H∞ performance requirement (7), if there exist matrices P˜ P P˜ = P˜ T = 11 ˜22 , (20) ∗ P22 Y˜ = Y˜ T ∈ R2n×2n , Z˜ = Z˜ T ∈ R2n×2n , such that the following LMIs (21) are feasible. Υij + Υji < 0,
i ≤ j,
i, j = 1, 2, ..., p,
(21)
T T , , P22 = P22 where P11 , P12 are assumed to satisfy: P11 = P11 and P12 is invertible by invoking small perturbation if it is necessary. Let I 0 (27) F = −T T , P12 ∗ P22
and G = diag{F, F, F, F, 1}. Using diag{G, F, 1} and its transpose to post multiply and pre multiply (25), then we can get
where
√ T ˜ ˜T ˜ ij hΓ Γ Ξ 1ij 2ij Υij = ∗ −2ω P˜ + ω 2 Z˜ 0 , ∗ ∗ −1 ˜ ij = Λ ˜ +Ξ ˜ 3ij , Ξ ˜ −3Z Z˜ 12Z˜ −10Z˜ 0 ∗ −3Z˜ −8Z˜ 10Z˜ 0 ˜= 3 ∗ ˜ 60Z˜ 0 , Λ ∗ −64 Z h ∗ ∗ ∗ −60Z˜ 0 ∗ ∗ ∗ ∗ 0 ˜ Sym{λ1ij } + Y λ2ij 0 0 λ3ij ∗ −Y˜ 0 0 0 ˜ Ξ3ij = ∗ ∗ 0 0 0 , ∗ ∗ ∗ 0 0 ∗ ∗ ∗ ∗ −γ 2 ˜ 1ij = λ1ij λ2ij 0 0 λ3ij , Γ ˜ 2ij = Ei −Cj Edi 0 0 0 0 , Γ P11 Ai + Bj Ci Aj λ1ij = ˜ , P22 Ai + Bj Ci Aj P11 Bi + Bj Di P11 Adi + Bj Cdi 0 , , λ3ij = ˜ λ2ij = ˜ P22 Adi + Bj Cdi 0 P22 Bi + Bj Di
−1 Bˆ0 j = P˜22 Bj ,
Cˆ0 j = Cj .
P11 P˜22 T ˜ , P = F PF = ∗ P˜22 −T T P12 , Y˜ = F T Y F, Z˜ = F T ZF, Aj = P12 Aˆj P22 −T T ˆj , Cj = Cˆj P P , Bj = P12 B 12 22 −1 T P˜22 = P12 P22 P12 ,
−T T −1 P22 , Aj P12 Aˆj = P12 −T T ˆ Cj = Cj P12 P22 .
P ≤ −2ωP + ω Z.
P12 P22
(24)
,
(30)
−T −1 −T T −T T −1 Aˆ0 j = P12 P22 (P12 Aj P12 P22 )P22 P12 = P˜22 Aj , −T −1 −1 0 ˆ ˜ (31) B j = P12 P22 (P12 Bj ) = P22 Bj , −T T −T T 0 ˆ C j = (Cj P P22 )P P12 = Cj . 12
(26)
22
Besides, inequality (28) can also be denoted as
(22)
As a result, if the following inequality (25) holds, the inequality (9) is true. √ T Ξ(t) ΓT2 hΓ1 P ∗ (25) −2ωP + ω 2 Z 0 < 0. ∗ ∗ −1 Then introduce a partition as P11 P = ∗
ˆj = P −1 Bj , B 12
−1 T As P˜22 = P12 P22 P12 , through an equivalent transformation −T P12 P22 x ˆ(t), we can obtain an admissible fuzzy H∞ realization as:
which can also be presented as −P Z
(29)
where Υij is defined in (21). From (29), we can obtain:
˜ Φ(t) =
p p X X
υi (ψ(t))υj (ψ(t))Υij
i=1 j=1
=
p X i=1
2
(28)
with the changes of variables as
Proof: For arbitrary constant ω, the following inequality holds: (ωZ − P )Z −1 (ωZ − P ) ≥ 0, (23)
−1
υi (ψ(t))υj (ψ(t))Υij < 0,
i=1 j=1
and in this case, the parameters of the fuzzy H∞ filter can be presented as −1 Aˆ0 j = P˜22 Aj ,
p X p X
˜ Φ(t) =
υi (ψ(t))2 Υii +
p X p X
υi (ψ(t))υj (ψ(t))(Υij + Υji ),
i=1 i 0, the system (6) is asymptotically stable with w(t) ≡ 0, and satisfies the prescribed H∞ performance requirement (7), if there exist matrices P T ˜ ˜ P = P = 11 ∗
(33)
i ≤ j,
i, j = 1, 2, ..., p,
p X p X
p X p X
(34)
where Ωij = Υij − Jij + Kij +
m ¯ ab Jab −
a=1 b=1
m ¯ kl Kkl ,
k=1 l=1
Υij is defined in (21), and in this case, the parameters of the fuzzy H∞ filter can be expressed as −1 Aˆ0 j = P˜22 Aj ,
−1 Bˆ0 j = P˜22 Bj ,
Cˆ0 j = Cj .
− mij )µT (t)Jij µ(t) + =
p p X X
+
=
p X p X
(m ¯ ij
(mij − mij )µT (t)Kij µ(t)
mij µT (t)(Υij − Jij + Kij )µ(t)
i=1 j=1 p X p X
m ¯ ij µT (t)Jij µ(t) −
i=1 j=1 p p XX
p X p X
−
V˙ (t) + eT (t)e(t) − γ 2 wT (t)w(t) < 0,
(39)
which means that the filtering system (6) is asymptotically stable and the H∞ performance condition (7) can be satisfied, thus, we accomplish the proof of Theorem 2. Remark 2: In Theorem 2, the information of the membership functions is considered in the criterion, consequently, less conservative result can be obtained. However, compared with Theorem 1, the criterion presented in Theorem 2 is more complex, which implies that it will be more difficult to realize in engineering applications. Therefore, both Theorem 1 and Theorem 2 are meaningful. IV. N UMERICAL E XAMPLE In this section, we will use one numerical example to illustrate the effectiveness of the proposed approach. A. Example 1
υ1 (x1 (t)) = 1 −
mij µT (t)Kij µ(t)
i=1 j=1
mij µT (t)(Υij − Jij + Kij +
i=1 j=1 p X p X
(38) Consequently, if (34) holds, we can derive that the following inequality holds
and the membership functions are defined as
mij µT (t)Υij µ(t) +
i=1 j=1
i=1 i
