3.1. Passivity Analysis and ε-Bound Estimation
In this subsection, using a novel storage function depending on the singular perturbation parameter, a sufficient condition for the system to be passive is proposed. Furthermore, bisectional search algorithms are formulated to get the best estimate of the ε-bound and the optimal dissipation η, respectively.
Theorem 4.
Given η and ε->0, system (1) with u(t)=0 is passive with dissipation η for all ε∈(0,ε-], if there exist scalars μ1>0, μ2>0, and matrices Zi (i=1,…,5) with Zi=ZiT (i=1,2,3,4) satisfying the following LMIs:
(11)Z1>0,[Z1+ε-Z3ε-Z5Tε-Z5ε-Z2]>0,[Z1+ε-Z3ε-Z5Tε-Z5ε-Z2+ε-2Z4]>0,(12)[μ1α2MTM+μ2β2NTN+Δ(0)ZT(0)Bω-CTZT(0)Bf0*2ηI-DωT-Dω0-Dg**-μ1I0***-μ2I] <0, [μ1α2MTM+μ2β2NTN+Δ(ε-)ZT(ε-)Bω-CTZT(ε-)Bf0*2ηI-DωT-Dω0-Dg**-μ1I0***-μ2I] <0,
where Z(ε)=[Z1+εZ3εZ5TZ5Z2+εZ4], Δ(ε)=ATZ(ε)+ZT(ε)A.
Proof.
Define a storage function as follows:
(13)V(x)=xT(t)E(ε)Z(ε)x(t).
By Lemma 3, LMIs (11) imply that
(14)E(ε)Z(ε)=ZT(ε)E(ε)>0, ∀ε∈(0,ε-].
Thus, V(x) defined by (13) is positive definite for any ε∈(0,ε-].
By Lemma 2, it follows from (12) that, for any ε∈(0,ε-],
(15)[ATZ(ε)+ZT(ε)A+Δ(ε)ZT(ε)Bω-CTZT(ε)Bf0*2ηI-DT-D0-Dg**-μ1I0***-μ2I]<0.
Taking the derivative of V(x) along the trajectories of system (1) with u(t)=0 and using constraints (2) and (3), for any scalars μ1>0, μ2>0, and ε∈(0,ε-], we have
(16)V˙(x)|(1)=x˙T(t)E(ε)Z(ε)x(t)+xT(t)E(ε)Z(ε)x˙(t)=(Ax(t)+Bff(t,x)+Bωω(t))T×Z(ε)x(t)+xT(t)ZT(ε)×(Ax(t)+Bff(t,x)+Bωω(t))=xT(t)(ATZ(ε)+ZT(ε)A)x(t)+2xT(t)ZT(ε)Bff(t,x)+2xT(t)ZT(ε)Bωω(t)≤xT(t)(ATZ(ε)+ZT(ε)A)x(t)+2xT(t)ZT(ε)Bff(t,x)+2xT(t)ZT(ε)Bωω(t)+μ1(α2xT(t)MTMx(t)-fTf)+μ2(β2xT(t)NTNx(t)-gTg).
Therefore, (17)V˙(x)|(1)-2ωT(t)z(t)+2ηωT(t)ω(t) =[xωfg]T[ATZ(ε)+ZT(ε)A+Δ(ε)ZT(ε)Bω-CTZT(ε)Bf0*2ηI-DT-D0-Dg**-μ1I0***-μ2I][xωfg].
Taking into account (15) and (17), we have
(18)V˙(x)|(1)-2ωT(t)z(t)+2ηωT(t)ω(t)≤0.
Taking the integral on the two sides of (18) from 0 to T, we obtain
(19)∫0TV˙(x)-2ωT(t)z(t)+2ηωT(t)ω(t)≤0,
which implies
(20)∫0T2ωT(t)z(t)-2ηωT(t)ω(t)≥V(x(T))-V(x(0)).
Therefore,
(21)∫0TωT(t)z(t)-ηωT(t)ω(t)dt≥0, ∀T>0
holds for all trajectories with zero initial condition x(0)=0.
Hence, system (1) is passive with dissipation η for all ε∈(0,ε-].
Remark 5.
In [17], passivity of fuzzy SPSs was investigated based on decomposing the original system into fast and slow subsystems. Since the system decomposition requires the singular perturbation to be small enough, the proposed results are only sufficient conditions for the existence of an ε-bound for the SPSs to preserve passivity but cannot produce an estimate of the ε-bound. In this paper, the newly developed method implicitly employs the singular perturbation structure of the SPSs rather than depends on system decomposition, which provides convenience for estimating the value of the ε-bound.
Remark 6.
In [19], a sufficient condition for system (1) to be passive with dissipation η=0 was proposed by using a storage function in the form of
(22)V(x)=[x1Tx2T][Z1εZ5TεZ5εZ2][x1x2],
which is a special case of (13) with Z3=0 and Z4=0. Thus, Theorem 4 is more general and less conservative than the existing results in [19], which will be illustrated by the example in the next section.
The upper bound ε- given by Theorem 4 is guessed. We now propose a bisectional search algorithm to get the best estimate of ε-.
Algorithm 7.
Bisectional search algorithm optimizing ε- for given dissipation η.
Step 1. Given positive scalars ρ,σ,γ,δ, where ρ,δ are sufficiently small, γ is sufficiently large, and ρ<σ<γ. Set τ_=τ-=σ.
Step 2. Check LMIs (11)-(12) with ε-=σ. If they are feasible, set τ_=σ and σ:=2σ; otherwise, set τ-=σ and σ:=0.5σ.
Step 3. If τ-=ρ or τ_>γ, go to Step 7. Else if τ_≥τ-, go to Step 2.
Step 4. Set ε*=0.5(τ_+τ-).
Step 5. Check LMIs (11)-(12) with ε-=ε*. If they are feasible, set τ_=ε*; otherwise, set τ-=ε*.
Step 6. Go to Step 4 if |τ_-τ-|>δ; otherwise, go to Step 7.
Step 7. If τ-<ρ, the proposed method cannot give an answer. If τ_>γ, the optimal upper bound ε- is larger than γ. Otherwise, the optimal upper bound produced by the proposed method is the value of ε-. End.
Remark 8.
In Algorithm 7, Step 1 presents the initial and terminal conditions. Steps 2-3 will show that the proposed method does not work or the optimal upper bound ε- is larger than the given γ or determine a search interval [τ_,τ-] for Steps 4–6 such that LMIs (11)-(12) are feasible with ε-=τ_ but not with ε-=τ-. Steps 4–6 are used to search the best estimate of the upper bound ε- in [τ_,τ-].
Similarly, we have the following algorithm to get the largest dissipation η for given ε-.
Algorithm 9.
Bisectional search algorithm optimizing η for given ε-.
Step 1. Given positive scalars ρ,σ,γ,δ, where ρ,δ are sufficiently small, γ is sufficiently large, and ρ<σ<γ. Set τ_=τ-=σ.
Step 2. Check LMIs (11)-(12) with η=σ. If they are feasible, set τ_=σ and σ:=2σ; otherwise, set τ-=σ and σ:=0.5σ.
Step 3. If τ-=ρ or τ_>γ, go to Step 7. Else if τ_≥τ-, go to Step 2.
Step 4. Set η*=0.5(τ_+τ-).
Step 5. Check LMIs (11)-(12) with η=η*. If they are feasible, set τ_=η*; otherwise, set τ-=η*.
Step 6. Go to Step 4 if |τ_-τ-|>δ; otherwise, go to Step 7.
Step 7. If τ-<ρ, the proposed method cannot give an answer. If τ_>γ, the optimal value of η is larger than γ. Otherwise, the optimal value produced by the proposed method is the value of η. End.
Remark 10.
Since the conditions proposed in Theorem 4 are sufficient but not necessary for the system to be passive, thus the best estimates of ε-bound and dissipation η obtained by Algorithms 7 and 9 are less than or equal to their true values. Thus, the conservatism of the algorithms results from Theorem 4. Although Theorem 4 is less conservative than the existing results in [19], there may be some room to further reduce the conservatism, as will be considered in our future work.
3.2. Passive Controller Design
In this subsection, we will design a state feedback controller for system (1) to achieve a given ε-bound. The controller under consideration is in the form of
(23)u(t)=F(ε)x(t),
where F(ε)∈Rl×n is the controller gain to be designed. Then, the closed-loop system is as follows:
(24)E(ε)x˙(t)=(A+BuF(ε))x(t)+Bff(t,x)+Bωω(t),z(t)=Cx(t)+Dgg(t,x)+Dωω(t).
Theorem 11.
Given η and ε->0, if there exist scalars μ1>0, μ2>0, μ-1>0, μ-2>0, and matrices Zi (i=1,…,5) with Zi=ZiT (i=1,2,3,4) and K satisfying(25)Z1>0,[Z1+ε-Z3ε-Z5Tε-Z5ε-Z2]>0,(26)[Z1+ε-Z3ε-Z5Tε-Z5ε-Z2+ε-2Z4]>0,(27)[Θ(0)Bω-ZT(0)CTBf0αZT(0)MTβZT(0)NT*2ηI-DωT-Dω0-Dg00**-μ1I000***-μ2I00****-μ-1I0*****-μ-2I]<0,(28)[Θ(ε-)Bω-ZT(ε-)CTBf0αZT(ε-)MTβZT(ε-)NT*2ηI-DωT-Dω0-Dg00**-μ1I000***-μ2I00****-μ-1I0*****-μ-2I]<0,(29)μ1μ-1=1, μ2μ-2=1,where Z(ε)=[Z1+εZ3εZ5TZ5Z2+εZ4], Θ(ε)=AZ(ε)+ZT(ε)AT+BK+KTBT, then, the closed-loop system (24) is passive with dissipation η for all ε∈(0,ε-], and the gain matrix can be designed as F(ε)=KZ-1(ε).
Proof.
By Lemma 3, LMIs (25) and (26) imply that
(30)E(ε)Z(ε)=ZT(ε)E(ε)>0, ∀ε∈(0,ε-],
which shows that
(31)Z-T(ε)E(ε)=E(ε)Z-1(ε)>0, ∀ε∈(0,ε-].
Define a storage function as follows:
(32)V(x)=xT(t)Z-T(ε)E(ε)x(t).
Thus, V(x) defined by (32) is positive definite for any ε∈(0,ε-].
By Lemma 2, it follows from (27), (28), and (29) that, for any ε∈(0,ε-], (33)[Θ(ε)Bω-ZT(ε)CTBf0αZT(ε)MTβZT(ε)NT*2ηI-DωT-Dω0-Dg00**-μ1I000***-μ2I00****-μ-1I0*****-μ-2I]<0,which is equivalent to (34)[Θ(ε)+μ1α2ZT(ε)MTMZ(ε)+μ2β2ZT(ε)NTNZ(ε)Bω-ZT(ε)CTBf0*2ηI-DωT-Dω0-Dg**-μ1I0***-μ2I]<0.
Pre- and postmultiplying (34) by diag{Z-T,I,I,I} and its transpose gives(35)[Θ-(ε)+μ1α2MTM+μ2β2NTNZ-T(ε)Bω-CTZ-T(ε)Bf0*2ηI-DωT-Dω0-Dg**-μ1I0***-μ2I]<0, ε∈(0,ε-],where Θ-(ε)=Z-T(ε)(A+BF(ε))+(A+BF(ε))TZ-1(ε), F(ε)=KZ-1(ε).
Taking the derivative of V(x) along the trajectories of system (24) and using constraints (2) and (3), for any scalars μ1>0, μ2>0, and ε∈(0,ε-], we have
(36)V˙(x)|(22)=2xT(t)Z-T(ε)E(ε)x˙(t)=2xT(t)Z-T(ε)×((A+BF(ε))x(t)+Bff(t,x)+Bωω(t))=xT(t)({(A+BF(ε))T}Z(ε)-T(A+BF(ε))xT(t) +(A+BF(ε))TZ-1(ε))x(t)+2xT(t)Z-T(ε)Bff(t,x)+2xT(t)Z-T(ε)Bωω(t)≤xT(t)({(A+BF(ε))TZ-1}Z(ε)-T(A+BF(ε))xT(t) +(A+BF(ε))TZ-1(ε))x(t)+2xT(t)Z-T(ε)Bff(t,x)+2xT(t)Z-T(ε)Bωω(t)+μ1(α2xT(t)MTMx(t)-fTf)+μ2(β2xT(t)NTNx(t)-gTg).
Therefore, (37)V˙(x)|(22)-2ωT(t)z(t)+2ηωT(t)ω(t) =[xωfg]T[Θ-(ε)+μ1α2MTM+μ2β2NTNZ-T(ε)Bω-CTZ-T(ε)Bf0*2ηI-DωT-Dω0-Dg**-μ1I0***-μ2I][xωfg].
Taking into account (35) and (37), we have
(38)V˙(x)|(22)-2ωT(t)z(t)+2ηωT(t)ω(t)≤0.
Taking the integral on the two sides of (38) from 0 to T, we obtain
(39)∫0TV˙(x)-2ωT(t)z(t)+2ηωT(t)ω(t)≤0,
which implies
(40)∫0T2ωT(t)z(t)-2ηωT(t)ω(t)≥V(x(T))-V(x(0)).
Therefore,
(41)∫0TωT(t)z(t)-ηωT(t)ω(t)dt≥0, ∀T>0
holds for all trajectories with zero initial condition x(0)=0.
Hence, system (24) is passive with dissipation η for all ε∈(0,ε-].
Remark 12.
It follows from LMIs (25) that Z1 and Z2 are nonsingular matrices, which guarantees that Z(0) is nonsingular. Then, F=KZ-1(ε) is well defined for all ε∈(0,ε-]. Furthermore, it can be seen that the controller reduces to an ε-independent one, that is, u=Fx=K[Z10Z5Z2]-1x, when ε is sufficiently small.
Using Theorem 4, we have the following corollary to design an ε-independent controller.
Corollary 13.
Given η and ε->0, if there exist scalars μ1>0, μ2>0, μ-1>0, μ-2>0, and matrices Zi (i=1,2,5) with Zi=ZiT>0 (i=1,2) and K satisfying (27) and (29), then the closed-loop system (24) is passive with dissipation η for all ε∈(0,ε-], and the gain matrix can be designed as F=K[Z10Z5Z2]-1.
Remark 14.
It is noted that the condition presented in Theorem 11 is not a convex set due to the equality constraints in (29). Several approaches have been proposed to solve such nonconvex feasibility problems, among which the cone complementarity linearization (CCL) method [22] is most commonly used. For computational simplicity, one can set μ1=μ-1=μ2=μ-2=1; then, the LMI conditions in Theorem 11 are easy to be solved by LMI toolbox. However, this may lead to conservatism.