It can be determined by solving the resulting equations simultaneously and we obtain
(13)a=(n-1ui+)n-11nn.
3.1. Asymptotic Stability for Certain Neutral System
Theorem 6.
The neutral system with time-delay and actuator saturation as described in (1) and (3) is asymptotic stability if ∥h|B|+D∥<1 and there exist scalars γi>0, i=1,2,3, P>0, P^>0, Q=[Qij]2×2, Qij>0, R=[Rij]2×2, Rij>0, Ti>0 (i=1,2,3,4,5,6), and T~i>0 (i=3,4) such that the following symmetric linear matrix inequality holds:
(19)Φ=[Φ11Φ120Φ14Φ15Φ160⋆Φ22Φ230000⋆⋆Φ330000⋆⋆⋆Φ44Φ4500⋆⋆⋆⋆Φ5500⋆⋆⋆⋆⋆Φ660⋆⋆⋆⋆⋆⋆Φ77]<0,
where
(20)Wn(ρ)={∑i=1m((n-1)ρnnn/(n-1)ui+)2n-2(KiTKi)nif ∥Ki∥ρn>ui+,0otherwise,Φ11=A^cTP+PA^c+1γ1P^+(γ1+γ2+γ3)∥CTC∥Wi(ρi)+Q11+R11+T1+T2+h22(T~3+T5)+τ22(T~4+T6),Φ22=Q22-Q11,Φ33=-Q22-T1,Φ44=R22-R11,Φ55=1γ2DTP^D-R22-T2,Φ66=1γ3BTP^B-T3,Φ77=-T4,Φ12=Q12,Φ14=R12,Φ15=-A^cTPD,Φ16=A^cTPB,Φ23=-Q12,Φ45=-R12.
Proof.
Define a legitimate Lyapunov functional candidate as follows:
(21)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t))+V6(x(t)),
where
(22)V1(x(t))=ℒT(x(t))Pℒ(x(t)),V2(x(t))=∫t-(h/2)t[x(s)x(s-h2)]T[Q11Q12⋆Q22]ghhhhh×[x(s)x(s-h2)]ds+∫t-(τ/2)t[x(s)x(s-τ2)]Tghhhhhh×[R11R12⋆R22][x(s)x(s-τ2)]ds,V3(x(t))=∫t-htxT(s)T1x(s)ds+∫t-τtxT(s)T2x(s)ds,V4(x(t))=∫t-ht[∫stxT(s1)ds1]T3[∫stx(s2)ds2]ds,V5(x(t))=∫t-τt[∫stxT(s1)ds1]T4[∫stxT(s2)ds2]ds,(23)V6(x(t))=12∫t-ht(s-t+h)2xT(s)T5x(s)ds+12∫t-τt(s-t+τ)2xT(s)T6x(s)ds,
where P>0, Q=[Qij]2×2, Qij>0, R=[Rij]2×2, Rij>0, and Ti>0, i=1,2,3,4,5,6.
Then
(24)V˙(x(t))=V˙1(x(t))+V˙2(x(t))+V˙3(x(t))+V˙4(x(t))+V˙5(x(t))+V˙6(x(t)).
By (19)-(22), we obtain that
(25)V˙1(x(t))=2{xT(t)-xT(t-τ)DT+[∫t-htxT(s)ds]BT}×P[A^cx(t)+CDz(Kx)]=2xTPA^cx+2xTPCDz(Kx)-2xTA^cTPDx(t-τ)-2xT(t-τ)DTPCDz(Kx)+2xT(t)A^cTPB[∫t-htx(s)ds]+2[∫t-htxT(s)ds]BTPCDz(Kx)≤xT[A^cTP+PA^c+1γ1P^ghh+(γ1+γ2+γ3)∥CTC∥Wi(ρi)1γ1]x+1γ2xT(t-τ)DTP^Dx(t-τ)+1γ3[∫t-htxT(s)ds]BTP^B[∫t-htx(s)ds]-2xT(t)A^cTPDx(t-τ)+2xT(t)A^cTPB[∫t-htx(s)ds],
where P^>0, since P^=P2. Consider
(26)V˙2(x(t))=[x(t)x(t-h2)]T[Q11Q12⋆Q22][x(t)x(t-h2)]-[x(t-h2)x(t-h)]T[Q11Q12⋆Q22][x(t-h2)x(t-h)]+[x(t)x(t-τ2)]T[R11R12⋆R22][x(t)x(t-τ2)]-[x(t-τ2)x(t-τ)]T[R11R12⋆R22][x(t-τ2)x(t-τ)],V˙3(x(t))=xT(t)T1x(t)-xT(t-h)T1x(t-h)+xT(t)T2x(t)-xT(t-τ)T2x(t-τ),V˙4(x(t))=-[∫t-htxT(s1)ds1]T3[∫t-htx(s2)ds2]+∫t-htxT(t)T3[∫stx(s2)ds2]ds+∫t-ht[∫stx(s2)ds2]T3x(t)ds=-[∫t-htxT(s1)ds1]T3[∫t-htx(s2)ds2]+∫t-ht(s-t+h)xT(t)T3x(s)ds+∫t-ht(s-t+h)xT(s)T3x(t)ds=-[∫t-htxT(s1)ds1]T3[∫t-htx(s2)ds2]-∫t-ht(s-t+h)(T3x(t)-T5x(s))Tghhhh×T5-1(T3x(t)-T5x(s))ds+h22xT(t)T3T5-1T3x(t)+∫t-ht(s-t+h)xT(s)T5x(s)ds≤-[∫t-htxT(s1)ds1]T3[∫t-htx(s2)ds2]+h22xT(t)T~3x(t)+∫t-ht(s-t+h)xT(s)T5x(s)ds,
where T~3>0, since T~3=T3T5-1T3.
Similarly we have
(27)V˙5(x(t))≤-[∫t-τtxT(s1)ds1]T4[∫t-τtx(s2)ds2]+τ22xT(t)T~4x(t)+∫t-τt(s-t+τ)xT(s)T6x(s)ds,
where T~4>0, since T~4=T4T6-1T4. Consider
(28)V˙6(x(t)) =12h2xT(t)T5x(t)-∫t-ht(s-t+h)xT(s)T5x(s)ds +12τ2xT(t)T6x(t)-∫t-τt(s-t+τ)xT(s)T6x(s)ds.
Substituting these into (24), the time-derivative of V has new upper bound as follows:
(29)V˙(x(t))≤ξT(t)Φξ(t),
where(30)ξT(t)=[xT(t)xT(t-h2)xT(t-h)xT(t-τ2)xT(t-τ)∫t-htxT(s)ds∫t-τtxT(s)ds];Φ is defined as stated in (19).
If linear matrix inequality (19) is feasible, then we can get V˙(x(t))<0, for all x∈B(ρ). Therefore, if constant scalar h, constant parameter matrices B,D such that ∥h|B|+D∥<1 and there exist P>0, P^>0, Q=[Qij]2×2, Qij>0, R=[Rij]2×2, Rij>0, Ti>0 (i=1,2,3,4,5,6), and T~i>0 (i=3,4) satisfying (19) for real scalars γi>0 (i=1,2,3), from Hale and Verduyn Lunel [24], we can draw the neutral system which can be described by (1) and (3) is asymptotic stability. This completes the proof.
Remark 7.
W
i
(
ρ
i
)
(
i
=
1,2
,
…
,
n
)
are created by the parameter ρi which is a measure tool for domain of attraction. With these functions, we obtain the novel stability criterion. However, in Theorem 6 we need to look for the largest value of ρi with the optimal Wi(ρi). These can be seen in Section 3.3 below.
Remark 8.
Theorem 6 gives a delay-dependent stability criterion for neutral system with (1) and (3) using a delay-dividing approach. The delay differential conditions in other works, such as in [26], are usually more strict. These facts mean that our result is less conservative than some previous approaches.
The delay-dependent stability criterion for system (1) with τ≡h is presented in the following corollary.
Corollary 9.
The neutral systems (1) and (3) with τ≡h are asymptotic stability if ∥h|B|+D∥<1 and there exist P>0, P^>0, Q=[Qij]2×2, Qij>0, Ti>0 (i=1,2,3), and T~2>0 such that the following symmetric linear matrix inequality holds for real constant scalars γi>0, i=1,2,3:
(31)Φ¯=[Φ¯11Φ¯12Φ¯13Φ¯14⋆Φ¯22Φ¯230⋆⋆Φ¯330⋆⋆⋆Φ¯44]<0,
where Wi(ρi), i=1,2,3,…,n, are defined as before. Consider
(32)Φ¯11=A^cTP+PA^c+1γ1P^+(γ1+γ2+γ3)∥CTC∥Wi(ρi)+Q11+T1+h22(T~2+T3),Φ¯22=Q22-Q11,Φ¯33=1γ2DTP^D-Q22-T1,Φ¯44=1γ3BTP^B-T2,Φ¯12=Q12,Φ¯13=-A^cTPD,Φ¯14=A^cTPB,Φ¯23=-Q12.
Proof.
Define a legitimate Lyapunov functional candidate as
(33)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t)),
where
(34)V1(x(t))=ℒT(x(t))Pℒ(x(t)),V2(x(t)) =∫t-(h/2)t[x(s)x(s-h2)]T[Q11Q12⋆Q22][x(s)x(s-h2)]ds,V3(x(t))=∫t-htxT(s)T1x(s)ds,V4(x(t))=∫t-ht[∫stxT(s1)ds1]T2[∫stx(s2)ds2]ds,V5(x(t))=12∫t-ht(s-t+h)2xT(s)T3x(s)ds,
where P>0, Q=[Qij]2×2, Qij>0, and Ti>0, i=1,2,3.
According to (34) we obtain
(35)V˙1(x(t))≤xT[A^cTP+PA^c+1γ1P^ghh+(γ1+γ2+γ3)∥CTC∥Wi(ρi)1γ1]x+1γ2xT(t-h)DTP^Dx(t-h)+1γ3[∫t-htxT(s)ds]BTP^B[∫t-htx(s)ds]-2xT(t)A^cTPDx(t-h)+2xT(t)A^cTPB[∫t-htx(s)ds],
where P^>0, since P^=P2. Consider
(36)V˙2(x(t))=[x(t)x(t-h2)]T[Q11Q12⋆Q22][x(t)x(t-h2)]-[x(t-h2)x(t-h)]T[Q11Q12⋆Q22][x(t-h2)x(t-h)],V˙3(x(t))=xT(t)T1x(t)-xT(t-h)T1x(t-h),V˙4(x(t))≤-[∫t-htxT(s1)ds1]T2[∫t-htx(s2)ds2]+h22xT(t)T~2x(t)+∫t-ht(s-t+h)xT(s)T3x(s)ds,
where T~2>0, since T~2=T2T3-1T2. Consider
(37)V˙5(x(t))=12h2xT(t)T3x(t)-∫t-ht(s-t+h)xT(s)T3x(s)ds.
Then, the time-derivative of V has new upper bound as follows:
(38)V˙(x(t))=V˙1(x(t))+V˙2(x(t))+V˙3(x(t))+V˙4(x(t))+V˙5(x(t))≤ξ¯T(t)Φ¯ξ¯(t),
where Φ¯ is defined as stated in (31) and
(39)ξ¯T(t)=[xT(t)xT(t-h2)xT(t-h)∫t-htxT(s)ds].
The corollary can then be proved following [24].
3.2. Asymptotic Stability for Uncertain Neutral System
Consider the following uncertain neutral system with time-delay and actuator saturation:
(40)x˙(t)-Dx˙(t-τ)=(A+ΔA(t))x(t)+Bx(t-h)+(C+ΔC(t))Sat(u(t)),
where ΔA(t) and ΔC(t) stand for the uncertainties. For simplicity, the constant parameter matrices A, B, C, and D are square matrices. The spectral norm bound of the unknown uncertainties is
(41)∥ΔA(t)∥≤α, ∥ΔC(t)∥≤β, ∀t≥0.
Using the nonlinear function Dz(·), rewrite the uncertain neutral system as follows:
(42)x˙(t)-Dx˙(t-τ)=(Ac+ΔA(t)-ΔC(t)K)x(t)+Bx(t-h)+(C+ΔC(t))Dz(Kx(t)),
where Ac is defined as the same with certain neutral system with time-delay and actuator saturation. In particular, when ∥ΔA(t)∥=0 and ∥ΔC(t)∥=0, the uncertain neutral system becomes the certain case.
Similarly, we employ the operator ℒ:C0→ℝn with
(43)ℒ(x(t))=x(t)+B∫t-htx(s)ds-Dx(t-τ).
The following transformed system is then obtained:
(44)ddtℒ(x(t))=A¯cx(t)+(C+ΔC(t))Dz(Kx(t)),
where A¯c=A^c+ΔA(t)-ΔC(t)K=A+B-CK+ΔA(t)-ΔC(t)K.
Theorem 10.
The uncertain neutral system in (40) with feedback control (3) is asymptotic stability if ∥h|B|+D∥<1 and there exist scalars γi>0, i=1,2,3,4,5,6, P>0, P^>0, Q=[Qij]2×2, Qij>0, R=[Rij]2×2, Rij>0, Ti>0 (i=1,2,3,4,5,6), and T~i>0(i=3,4) such that the following symmetric linear matrix inequality holds:
(45)Ξ=[Ξ11Ξ120Ξ14Ξ15Ξ160⋆Ξ22Ξ230000⋆⋆Ξ330000⋆⋆⋆Ξ44Ξ4500⋆⋆⋆⋆Ξ5500⋆⋆⋆⋆⋆Ξ660⋆⋆⋆⋆⋆⋆Ξ77]<0,
where
(46)Ξ11=A^cTP+PA^c+1γ1P^+1γ2P^+(γ1+γ3+γ5)(α2+2αβ∥K∥+β2∥K∥2)I+(γ2+γ4+γ6)(β2+2β∥C∥+∥C∥2)Wi(ρi)+Q11+R11+T1+T2+h22(T~3+T5)+τ22(T~4+T6),Ξ22=Q22-Q11,Ξ33=-Q22-T1,Ξ44=R22-R11,Ξ55=(1γ3+1γ4)DTP^D-R22-T2,Ξ66=(1γ5+1γ6)BTP^B-T3,Ξ77=-T4,Ξ12=Q12,Ξ14=R12,Ξ15=-A^cTPD,Ξ16=A^cTPB,Ξ23=-Q12,Ξ45=-R12.
Proof.
Define the legitimate Lyapunov functional candidate as
(47)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t))+V6(x(t)),
where V1(x(t)), V2(x(t)), V3(x(t)), V4(x(t)), V5(x(t)), and V6(x(t)) are the same as in Theorem 6.
The time-derivative of V(x(t)) along the trajectories of closed system (44) is given by the following:
(48)V˙(x(t))=V˙1(x(t))+V˙2(x(t))+V˙3(x(t))+V˙4(x(t))+V˙5(x(t))+V˙6(x(t)).
Then
(49)V˙1(x(t))=2{[∫t-htxT(s)ds]xT(t)-xT(t-τ)DT h+[∫t-htxT(s)ds]BT}×P[A¯cx(t)+(C+ΔC)Dz(Kx)]=2xTPA¯cx+2xTP(C+ΔC)Dz(Kx)-2xTA¯cTPDx(t-τ)-2xT(t-τ)DT×P(C+ΔC)Dz(Kx)+2xT(t)A¯cTPB[∫t-htx(s)ds]+2[∫t-htxT(s)ds]BTP(C+ΔC)Dz(Kx)=2xTPA^cx+2xTP(ΔA-ΔCK)x+2xTP(C+ΔC)Dz(Kx)-2xT(t-τ)DTPA^cx(t)-2xT(t-τ)DTP(ΔA-ΔCK)x-2xT(t-τ)DTP(C+ΔC)Dz(Kx)+2[∫t-htxT(s)ds]BTPA^cx+2[∫t-htxT(s)ds]BTP(ΔA-ΔCK)x+2[∫t-htxT(s)ds]BTP(C+ΔC)Dz(Kx)≤xT[A^cTP+PA^c]x-2xT(t)A^cTPDx(t-τ)+2xT(t)A^cTPB[∫t-htx(s)ds]+1γ1xTP^x+γ1xT(ΔA-ΔCK)T(ΔA-ΔCK)x+1γ2xTP^x+γ2DzT(Kx)(C+ΔC)T×(C+ΔC)Dz(Kx)+1γ3xT(t-τ)DTP^Dx(t-τ)+γ3xT(ΔA-ΔCK)T(ΔA-ΔCK)x+1γ4xT(t-τ)DTP^Dx(t-τ)+γ4DzT(Kx)(C+ΔC)T(C+ΔC)Dz(Kx)+1γ5[∫t-htxT(s)ds]BTP^B[∫t-htx(s)ds]+γ5xT(ΔA-ΔCK)T(ΔA-ΔCK)x+1γ6[∫t-htxT(s)ds]BTP^B[∫t-htx(s)ds]+γ6DzT(Kx)(C+ΔC)T(C+ΔC)Dz(Kx)≤xT{(A^cTP+PA^c)+[1γ1P^+γ1(α2+2αβ∥K∥+β2∥K∥2)I](A^cTP+PA^c) +[1γ1P^+γ1(α2+2αβ∥K∥+β2∥K∥2)I] +[1γ2P^+γ2(β2+2β∥C∥+∥C∥2)Wi(ρ)] +γ3(α2+2αβ∥K∥+β2∥K∥2)I +γ4(β2+2β∥C∥+∥C∥2)Wi(ρi) +γ5(α2+2αβ∥K∥+β2∥K∥2)I +γ6(β2+2β∥C∥+∥C∥2)Wi(ρi)(A^cTP+PA^c)+[1γ1P^+γ1(α2+2αβ∥K∥+β2∥K∥2)I]}x+xT(t-τ)(1γ3DTP^D+1γ4DTP^D)x(t-τ)+[∫t-htxT(s)ds][1γ5BTP^B+1γ6BTP^B]×[∫t-htx(s)ds]-2xT(t)A^cTPDx(t-τ)+2xT(t)A^cTPB[∫t-htx(s)ds],
where P^>0, since P^=P2.
V
˙
2
(
x
(
t
)
)
, V˙3(x(t)), V˙4(x(t)), V˙5(x(t)), and V˙6(x(t)) are obtained similarly as in Theorem 6. Substituting these into (48), the time-derivative of V has new upper bound as follows:
(50)V˙(x(t))≤ξT(t)Ξξ(t),
where(51)ξT(t)=[xT(t)xT(t-h2)xT(t-h)xT(t-τ2)xT(t-τ)∫t-htxT(s)ds∫t-τtxT(s)ds];Ξ is defined as stated in (45).
If linear matrix inequality (45) is feasible, then V˙(x(t))<0, for all x∈B(ρ). The theorem can then be proved following [24].
Theorem 10 provides new asymptotic stability conditions for the uncertain neutral systems in (40) and (3). The following corollary is presented as a special case of the theorem.
Corollary 11.
The uncertain neutral system in (40) and (3) with τ≡h is asymptotic stability if ∥h|B|+D∥<1 and there exist P>0, P^>0, Q=[Qij]2×2, Qij>0, Ti>0(i=1,3), and T~2>0 such that the following symmetric linear matrix inequality holds for real constant scalars γi>0, i=1,2,3,4,5,6:
(52)Ξ¯=[Ξ¯11Ξ¯12Ξ¯13Ξ¯14⋆Ξ¯22Ξ¯230⋆⋆Ξ¯330⋆⋆⋆Ξ¯44]<0,
where Wi(ρ), i=1,2,3,…,n are defined as before and
(53)Ξ¯11=A^cTP+PA^c+1γ1P^+1γ2P^+(γ1+γ3+γ5)(α2+2αβ∥K∥+β2∥K∥2)I+(γ2+γ4+γ6)(β2+2β∥C∥+∥C∥2)Wi(ρi)+Q11+T1+h22(T~2+T3),Ξ¯22=Q22-Q11,Ξ¯33=(1γ3+1γ4)DTP^D-Q22-T1,Ξ¯44=(1γ5+1γ6)BTP^B-T2,Ξ¯12=Q12,Ξ¯13=-A^cTPD,Ξ¯14=A^cTPB,Ξ¯23=-Q12.
Proof.
Choose a legitimate Lyapunov functional candidate as
(54)V(x(t))=V1(x(t))+V2(x(t))+V3(x(t))+V4(x(t))+V5(x(t))
which are the same as (34).
V
˙
1
(
x
(
t
)
)
can be evaluated similarly as in Theorem 10 and Corollary 9. The proof can be readily obtained.
3.3. The Algorithm with W2(ρ2) and the Algorithm to Solve the Optimal Wi(ρi)
From Definition 4, it is seen that W1(ρ1) are different from W2(ρ2), W3(ρ3), and W4(ρ4),…,Wn(ρn). We compare between W2(ρ2), W3(ρ3), and W4(ρ4),…,Wn(ρn) and intend to reduce the conservativeness of the result. To that end, we should obtain ρ2 in the first place. In what follows we present the stability analysis algorithm with W2(ρ2) to solve ρ2.
Step 1.
Give α,β,K.
Step 2.
Set positive values γi (i=1,2,3,4,5,6).
Step 3.
Initialize ρ2.
Step 4.
With W2(ρ2), solve linear matrix inequality (45) by Matlab LMI Toolbox.
Step 5.
If the solution satisfies the stability condition, go to Step 6; otherwise, reduce ρ2 and go to Step 3.
Step 6.
Increase ρ2 and go to Step 3.
Step 7.
End.
Remark 12.
The above algorithm is stated with respect to uncertain systems (40) and (3). Other cases can be dealt with similarly.
After defining ρ2, we replace W2(ρ2) by Wi(ρi) to reduce the conservativeness. We analyse these functions W2(ρ2),W3(ρ3),W4(ρ4),…,Wn(ρn) and find the optimal Wi(ρi) to obtain the maximum ρi among ρ2,ρ3,…,ρn.
Theorem 13.
Given W2(ρ2),W3(ρ3),W4(ρ4),…,Wn(ρn), if there exists j≤n, such that ρj-1≥ρk and ∥Wj(ρj-1)∥≤∥Wk(ρk)∥, for all k=2,3,…,j-1, then we have
(55)∥Ki∥ρj-1ui+≤fk(j), ∀k=2,3,…,j-1,
where ρj is the domain attraction obtained by Wj(ρj) and
(56)fk(j)=(k-1)(k-1)/(j-k)kk/(j-k)×jj/(j-k)(j-1)(j-1)/(j-k).
Proof.
Recall the definition of Wj(ρj), we know it can be expressed in the following equality:
(57)Wj(ρj)=∑i=1m((j-1)ρjjj/(j-1)ui+)2j-2(KiTKi)j.
If we have ρj-1≥ρk, for all k=2,3,…,j-1, then from the above equality we obtain
(58)∥Wk(ρk)∥≤∥Wk(ρj-1)∥, ∀k=2,3,…,j-1.
With regard to Wj(ρj-1), we have
(59)Wj(ρj-1)=∑i=1m((j-1)ρj-1jj/(j-1)ui+)2j-2(KiTKi)k(KiTKi)j-k=∑i=1m[(k-1)ρj-1kk/(k-1)ui+]2k-2(KiTKi)k(KiTKi)j-kgh×(ρj-1ui+)2(j-k)k2k(k-1)2k-2(j-1)2j-2j2j=∥Ki∥2(j-k)(ρj-1ui+)2(j-k)×k2k(k-1)2k-2(j-1)2j-2j2jWk(ρj-1).
If we have ∥Wj(ρj-1)∥≤∥Wk(ρk)∥, then we can obtain
(60)∥Wj(ρj-1)∥=∥Ki∥2(j-k)(ρj-1ui+)2(j-k)k2k(k-1)2k-2×(j-1)2j-2j2j∥Wk(ρj-1)∥≤∥Wk(ρk)∥.
By (58), we have
(61)(∥Ki∥ρj-1ui+)2(j-k)k2k(k-1)2k-2(j-1)2j-2j2j≤1,
where j=3,4,5,…, and k=2,3,…,j-1.
Then we easily obtain
(62)∥Ki∥ρj-1ui+≤fk(j)=(k-1)(k-1)/(j-k)kk/(j-k)×jj/(j-k)(j-1)(j-1)/(j-k)
which completes the proof.
Remark 14.
Theorem 13 provides us a criterion to find the optimal Wj(ρj). That is, if condition (55) cannot be satisfied, the procedure to find the optimal one should be ended. For example, when ρ4 is obtained, we compute ∥Ki∥ρ4/ui+ and the following inequalities need to be satisfied:
(63)∥Ki∥ρ4ui+≤f2(5), ∥Ki∥ρ4ui+≤f3(5),∥Ki∥ρ4ui+≤f4(5).
Otherwise, the algorithm terminates with ρ4.
The following optimality algorithm is used to obtain the optimal Wn(ρn).
Step 1.
Set positive value δ (small value, for example, 10-5).
Step 2.
Solve ρ2 based on the above stability analysis algorithm with W2(ρ2).
Step 3.
Compute ∥Ki∥ρ2/ui+, if (∥Ki∥ρ2/ui+)-fk(j)≤f2(3), then go to Step 4; otherwise, go to Step 7.
Step 4.
Set ρj+1=ρj, repeat the stability analysis algorithm with Wj+1(ρj) then obtain ρj+1 iteratively.
Step 5.
Compute ∥Ki∥ρj+1/ui+, if
(64)∥Ki∥ρj+1ui+≤fk(j+2), |∥Ki∥ρj+1ui+-fk(j+2)|>δ, ∀k=2,3,…,j+1
then go to Step 6; otherwise go to Step 7.
Step 6.
Set j=j+1 and go to Step 4.
Step 7.
End.