3.1. Krein Space Model Design
Before we proceed, we would like to propose the following lemma to construct an auxiliary stochastic system in Krein space.
Lemma 3.
Given a scalar γ>0 and an integer l>0, then the H∞ performance (4) is fulfilled if and only if there exists a fault estimator fˇ(k-l∣k) such that the following inequality holds:
(7)J=x0TP0-1x0+∑k=0NfT(k)f(k)+∑k=0Nv0T(k)v0(k)+∑k=0N-1dT(k)d(k)+∑k=0NvzT(k)vz(k)-γ-2∑k=lNvsT(k)vs(k)>0,
subject to the following dynamic constraints:
(8)x(k+1)=A(k)x(k)+Bf(k)f(k)+D(k)d(k),y0(k)=ρC(k)x(k)+v0(k),yz(k)=ρ(1-ρ) C(k)x(k)+vz(k),fˇ(k-l∣k)=f(k-l)+vs(k),x(0)=x0,
where y0(k) and yz(k) are the fictitious observations with their corresponding observation noises v0(k) and vz(k), respectively. The instantaneous value of y0 at each time instant k is equal to y(k) along with yz(k)≡0.
Proof.
Consider the following.
Necessity. From (1), the state transition matrix Φ is defined as
(9)Φ(k,j)={A(k-1)⋯A(j),0<k<j,I,k=j;
hence, we have
(10)x(k)=Φ(k,0)x0+∑i=0k-1Φ(k,i+1)Bf(i)f(i)+∑i=0k-1Φ(k,i+1)D(i)d(i).
Define
(11)yk=[yT(0)⋯yT(k)]T,vs,k=[vsT(0)⋯vsT(k)]T,fˇk=[fˇT(0∣l)⋯fˇT(k-l∣k)]T.
Then, in view of (10), we have
(12)yN=Ξ(k)Gxx0+Ξ(k)GffN+Ξ(k)GddN+vN,fˇN=fN-l+vs,N,
where
(13)Ξ(k)=diag{θ(1),…,θ(k)},Gf(k,i)=C(k)Φ(k,i+1)Bf(i),Gd(k,i)=C(k)Φ(k,i+1)D(i),Gx=[C(0)Φ(0,0)C(1)Φ(1,0)⋮C(N)Φ(N,0)],Gf=[0⋯⋯0Gf(1,0)0⋯0⋮⋱⋱Gf(N,0)Gf(N,1)⋯0],Gd=[0⋯⋯0Gd(1,0)0⋯0⋮⋱⋱Gd(N,0)Gd(N,1)⋯0].
Thus, by substituting (12) into (4) and taking (2) into consideration, we have
(14)J0=E{x0TP0-1x0+∑k=0NfT(k)f(k)+∑k=0N-1dT(k)d(k) -(yN-Ξ(k)Gxx0-Ξ(k)GffN-Ξ(k)GddN)T ×(yN-Ξ(k)Gxx0-Ξ(k)GffN-Ξ(k)GddN) -γ-2∑k=lN(fˇ(k-l∣k)-f(k-l))T ×(fˇ(k-l∣k)-f(k-l))∑k=0N}=x0TP0-1x0+∑k=0NfT(k)f(k)+∑k=0N-1dT(k)d(k) +(y0,N-Ξ-Gxx0-Ξ-GffN-Ξ-GddN)T ×(y0,N-Ξ-Gxx0-Ξ-GffN-Ξ-GddN) +(yz,N-Ξ~Gxx0-Ξ~GffN-Ξ~GddN)T ×(yz,N-Ξ~Gxx0-Ξ~GffN-Ξ~GddN) -γ-2∑k=lN(fˇ(k-l∣k)-f(k-l))T ×(fˇ(k-l∣k)-f(k-l)),
where
(15)y0,k=[y0T(0)⋯y0T(k)]T, yz,k=[yzT(0)⋯yzT(k)]T,y0(i)=y(i), yz(i)=0, (i=0,…,k),Ξ-=ρI, Ξ~=ρ(1-ρ) I.
Therefore, if the H∞ performance index (4) is satisfied, then, following the same line with the correlation between (1) and (4), we have J>0 subject to the dynamics (8) over x0, fk, and dk.
Sufficiency. For (8), since the value of y0(k) is equivalent to y(k) and yz(k)≡0, in light of (14), it is easy to find out that for a given constant γ>0 and an integer l>0, J0=J, which indicates that if J>0 holds, then the H∞ performance (4) is satisfied. Combing the sufficiency and necessity part, the proof is complete.
In virtue of Lemma 3, the auxiliary performance index J in (7) can be converted into the following compact form:
(16)J=[x0dNfNva,N]T[I0000I0000I0000Qa,N]-1[x0dNfNva,N],
where
(17)va(k)={v1(k)=[v0(k)vz(k)],0≤k<l,v2(k)=[v0(k)vz(k)vs(k)],k≥l,va,N=[vaT(0)⋯vaT(N)]T,(18)Qa(k)={Qv1(k)=diag{I,I},0≤k<l,Qv2(k)=diag{I,I,-γ2I},k≥l,Qa,N=diag{Qa(0),…,Qa(N)}.
From (8) and (17), we have
(19)yf(k)=[y(k)yz(k)]=C1(k)x(k)+v1(k),ya(k)={yf(k),0≤k<l,[yf(k)fˇ(k-l∣k)] =C2(k)x(k) +Hf(k-l)+v2(k),k≥l,
where
(20)C1(k)=[ρC(k)ρ(1-ρ) C(k)], C2(k)=[C1(k)0],H=[00I]T.
Thus, according to [20, 21], we introduce the following Krein space system associated with (8), (16), (18), and (19):
(21)x(k+1)=A(k)x(k)+Bf(k)f(k)+D(k)d(k),ya(k)={yf(k)=C1(k)x(k)+v1(k),0≤k<l,[yf(k)fˇ(k-l∣k)] =C2(k)x(k)+Hf(k-l)+v2(k),k≥l,x(0)=x0,
where x0(i), d(i), f(i), v1(i), and v2(i) are uncorrelated white noises in Krein space satisfying
(22)〈[x0d(i)f(i)va(i)],[x0d(i)f(i)va(i)]〉=[Iδij0000Iδij0000Iδij0000Qa(i)δij],(23)va(k)={v1(k)=[v0(k)vz(k)],0≤k<l,v2(k)=[v0(k)vz(k)vs(k)],k≥l,
with v0(k), vz(k), and vs(k) being fictitious noise in Krein space corresponding to (17).
Consequently, on the basis of Lemma 4.2.1 in [20], we have the following lemma.
Lemma 4.
For (8), given a scalar γ>0 and an integer l>0, then the H∞ performance (7) has a minimum over x0, f, d if and only if Qa(k) and Qw(k) have the same inertia, where Qw(k)=〈w(k),w(k)〉 is the covariance matrix of innovation sequence w(k) given by
(24)w(k)=ya(k)-y^a(k),
where y^a(k) is the projection of ya(k) onto L{{ya(j)}j=0k-1}. Furthermore, the minimum value of J is
(25)Jmin=∑k=0l-1[yf(k)-C1(k)x^(k)]T ×Qw-1(k)[yf(k)-C1(k)x^(k)]+∑k=lN[yf(k)-C1(k)x^(k)fˇ(k-l∣k)-fˇ(k-l∣k-1)]T ×Qw-1(k)[yf(k)-C1(k)x^(k)fˇ(k-l∣k)-f^(k-l∣k-1)],
where x^(k) and f^(k-l∣k-1) are, respectively, calculated from the Krein space projections of x(k) and f(k-l) onto L{{ya(j)}j=0k-1}.
Remark 5.
According to Lemmas 3 and 4, the purpose of establishing the dynamic model (8) associated with (7) is to derive a positive minimum of the cost function (4) by applying the projection theory in Krein space. Notice that although the measurement {y(k)}k=0N is a substantially stochastic sequence, the instantaneous values of y(k) and fˇ(k-l∣k) at each instant are available for the estimator. Thus, the equivalent cost function (7) and its corresponding dynamic constraint are constructed in a “conditional expectation” sense by gathering up {y(k)}k=0N (cf. (14) in the proof of Lemma 3).
3.2. Kalman Filtering in Krein Space
From the analysis above, the key step to achieve our goal is to find a suitable x^(k) and f^(k-l∣k-1). To this end, let
(26)y1(k)=yf(k), y2(k)=[yf(k)fˇ(k∣k+l)];
then
(27)y1(k-l+i)=C1(k-l+i)x(k-l+i)+v~1(k-l+i), i=1,…,l,y2(i)=C2(i)x(i)+Hf(i)+v~2(i), i=0,…,k-l,
where v~1(k)=v1(k) and v~2=[v1T(k)vsT(k+l)]T are zero-mean white noises with the following covariance matrices, respectively:
(28)Qv~1(k)=diag{I,I}, Qv~2(k)=diag{I,I,-γ2I}.
It is easy to check out that {y2(0),…,y2(k-l);y1(k-l+1),…,y1(k)} span the same linear space as L{{ya(j)}j=0k}.
To proceed, the following definition is introduced.
Definition 6 (see [32]).
For t>k-l, the estimator η^(t,1) is the optimal estimation of η(t) on the observation L{{y2(t)}t=0k-l-1;{y1(t)}t=k-lt=k-1}. For 0<t≤k-l, the estimator η^(t,2) is the optimal estimation of η(t) on the observation L{{y2(t)}t=0t=k-1}.
In accordance with (24), the innovation sequence is defined as follows:
(29)w1(k-l+i)=C1(k-l+i)e1(k-l+i)+v~1(k-l+i), i=0,…,l,w2(i)=C2(i)e2(i)+Hf(i)+v~2(i), i=0,…,k-l,
where
(30)e1(k-l+i)=x(k-l+i)-x^(k-l+i,1), i=0,…,l,e2(i)=x(i)-x^(i,2), i=0,…,k-l,
with the corresponding covariance matrices given as
(31)P1(k-l+i)=〈e1(k-l+i),e1(k-l+i)〉, i=0,…,l,P2(i)=〈e2(i),e2(i)〉, i=0,…,k-l.
In light of Lemma 2.2.1 in [20], the innovation sequences L{{w2(t)}t=0k-l-1;{w1(t)}t=k-lt=k-1} are uncorrelated white noises and span the same linear space as L{{ya(j)}j=0k}.
For deriving x^(k-l,2) (k=l+1,l+2,…), applying the Krein space based projection formula in [21] by taking (21) and (22) into account, we have that
(32)x^(k-l,2)=A(k-l-1)x^(k-l-1,2) +〈x(k-l),w2(k-l-1)〉 ×〈w2(k-l-1),w2(k-l-1)〉-1 ×w2(k-l-1)=A(k-l-1)x^(k-l-1,2) +K2(k-l-1)w2(k-l-1),x^(0)=0,
where
(33)K2(k-l-1)=(A(k-l-1)P2(k-l-1)C2T(k-l-1) +Bf(k-l-1)H)Q2-1(k-l-1),
with Q2(k-l-1)=C2(k-l-1)P2(k-l-1)C2T(k-l-1)+HHT+Qv~2(k-l-1).
In addition, following the definition of P2(i) and (32), P2(i) (i=0,1,…,k-l-1) is the solution to the following standard Riccati equation:
(34)P2(i+1)=A(i)P2(i)AT(i)+Bf(i)BfT(i)+D(i)DT(i)-K2(i)Q2-1(i)K2T(i),P2(0)=P0.
For calculating x^(k-l+i,1) (i=1,…,l) with the initial condition x^(k-l,1)=x^(k-l,2), we apply the projection formula once again such that
(35)x^(k-l+i+1,1) =A(k-l+i)x^(k-l+i,1) +A(k-l+i)〈x(k-l+i),w1(k-l+i)〉 ×〈w1(k-l+i),w1(k-l+i)〉-1w1(k-l+i) =A(k-l+i)x^(k-l+i,1)+K1(k-l+i)w1(k-l+i),
where
(36)K1(k-l-1)=A(k-l+i)P1(k-l+i)×C1T(k-l+i)Q1-1(k-l+i),
with Q1(k-l+i)=C1(k-l+i)P1(k-l+i)C1T(k-l+i)+Qv~1(k-l+i), and P1(k-l+i) is computed recursively in the following form:
(37)P1(k-l+i+1) =A(k-l+i)P1(k-l+i)AT(k-l+i) +Bf(k-l+i)BfT(k-l+i) +D(k-l+i)DT(k-l+i) -K1(k-l+i)Q2-1(k-l+i)K1T(k-l+i),P1(k-l)=P2(k-l).
Similarly, the projection formula is reutilized to compute f^(k-l∣k-1); that is,
(38)f^(k-l∣k-1) =∑i=0l-1〈f(k-l),w1(k-l+i)〉Q1-1(k-l+i)w1(k-l+i) =∑i=0l-1Ωk-l+ik-lC1T(k-l+i)Q1-1(k-l+i)w1(k-l+i), i=1,…,l-1,
where Ωk-l+ik-l, i=1,…,l-1 is obtained recursively in terms of
(39)Ωk-l+ik-l=Ωk-l+i-1k-l[C1(k-l+i-1)A(k-l+i-1)-K1(k-l+i-1) ×C1(k-l+i-1)]T,Ωk-l+1k-l=BfT(k-l).
Finally, in order to calculate Qw(k) which is associated with Jmin and fˇ(k-l)∣k, define f~(k-l)=f(k-l)-f^(k-l∣k-1), and then, from (38), we know that
(40)〈f~(k-l),f~(k-l)〉 =I-∑i=0l-1Ωk-l+ik-lC1T(k-l+i)Q1-1(k-l+i) ×(Ωk-l+ik-lC1T(k-l+i))T.
Combining (29) and (40), we have (41)Qw(k)={C1(k)P1(k)C1T(k)+I,0<k<l,[C1(k)P1(k)C1T(k)+IC1(k)(Ωkk-l)TΩkk-lC1T(k)-γ2I+I-〈f~(k-l),f~(k-l)〉],k≥l,where P1(k) and Ωl-l+ik-l are the same as in (37) and (39).