H
∞
F
ILTER
D
ESIGN FOR
U
NCERTAIN
D
ISCRETE
-T
IME
S
INGULAR
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YSTEMS VIA
N
ORMAL
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RANSFORMATION
*
Ching-Min Lee
1and I-Kong Fong
1,2Abstract. This paper concerns the robust H∞filtering problem for discrete-time singular systems with norm-bounded uncertainties. Based on the admissibility assumption of sin-gular systems, a set of necessary and sufficient conditions for the existence of the desired filters is established, and a normal filter design method under the linear matrix inequality framework is developed. A numerical example is given to illustrate the application of the proposed method.
Key words: Singular system, restricted system equivalence, admissibility, robust filter,
LMI.
1. Introduction
In the past decades, the H∞ filtering problem for singular systems has been an important research topic. This is due not only to the theoretical interests but also to the relevance of the topic in various engineering applications. Many works [10], [17], [23], [26] consider robust filters for continuous-time singular systems, in which the filter design criteria are mainly based on the generalized Lyapunov theorem [12], [18] for singular systems, and the formulations are under the linear matrix inequality (LMI) framework for easier applications. Unlike the discrete-time singular system stabilization problem [20]–[22], [27], in the filtering problem for discrete-time singular systems, applications of the approaches parallel to those for the continuous-time systems are not often adopted. One possible reason is the difficulty in managing the resultant constraints related to the singular matrix in
∗Received August 15, 2005; accepted December 14, 2005; published online August 17, 2006.
This research was supported by the National Science Council of the Republic of China (Taiwan) under grant NSC 93-2213-E-002-020.
1Department of Electrical Engineering, National Taiwan University, Taipei, 10617, Taiwan.
E-mail for Lee: [email protected]; E-E-mail for Fong: [email protected]
2Present address: Room 211, Electrical Engineering Building II, 1 Roosevelt Rd., Sec. 4, Taipei,
the difference term of the state-space model, especially when the constraints need to be represented as LMIs.
In this paper, the robust H∞ filtering problem is discussed for discrete-time singular systems with norm-bounded uncertainties. The goal of the filter is to satisfy the H∞performance level requirement on the filtering error dynamics. The proposed filter design method is formulated under the LMI framework. Unlike [10], [23], [26], which directly handle singular systems by using the generalized Lyapunov theorem, here a “normal transformation” to obtain normal system mod-els (i.e., those with the system matrix for the difference term being the identity matrix) [3] from singular system models is applied first, and normal filters are found directly. Then, instead of using criteria such those in [20]–[22], [27], an easier-to-use criterion based on the direct Lyapunov theorem for normal systems is applied. It is believed that the consideration of normal filters is beneficial, because sometimes the physical realizations of singular filters are not easy [3], [4]. In order to realize a singular system, one often needs special algorithms [15] to convert a singular system model into a normal state-space form.
Some of the notation to be used subsequently is introduced here. The inequality
X ≥ 0 means that the matrix X is symmetric and positive semi-definite, and X≥ Y means X−Y ≥ 0. Similar definitions apply to symmetric positive/negative
definite matrices. For a matrix M,M denotes its spectral norm, and for a stable discrete-time transfer function matrix G(z), G∞= supω∈[0,2π)G(ejω) is its
H∞norm. Ir is the identity matrix with dimension r , the superscriptTrepresents
the transpose of a matrix, and diag(X, Y, . . . , Z) is the block diagonal matrix with diagonal elements X, Y,. . . , Z. Finally, ∗ is used to simplify the presentation of symmetric matrices.
2. Preliminaries and problem formulation
2.1. Preliminaries
First, consider the following nominal singular system:
0:
E0x(k + 1) = A0x(k) + B0u(k)
z(k) = L0x(k), (1)
where x(k) ∈ Rn and rank E0 = r < n. The unforced singular system pair
(E0, A0) of (1) with u(k) ≡ 0 is regular, if det(zE0− A0) is not identically zero.
If deg(det(zE0− A0)) = rank E0, then(E0, A0) is said to be causal. The pair
(E0, A0) is stable if all the roots of det(zE0− A0) = 0 have magnitudes less than
unity. Finally,(E0, A0) is admissible if it is regular, causal, and stable. For 0, its transfer function matrix from u(k) to z(k) is G(z) = L0(zE0− A0)−1B0.
Definition 1 [3]. Suppose0in (1) is regular. Let P0and Q0be two n× n non-singular matrices, and E0r = P0E0Q0, A0r = P0A0Q0, B0r = P0B0, L0r =
L0Q0. The system
0r:
E0rxr(k + 1) = A0rxr(k) + B0ru(k)
z(k) = L0rxr(k), (2)
with xr(k)=Q−10 x(k) is restricted system equivalent (r.s.e.) to 0.
For any given regular0, there exist [3] nonsingular matrices P0and Q0such that E0r = Ir 0 0 0 , A0r = A1 A2 A3 A4 , B0r = B1 B2 , L0r = L1 L2 . (3)
Lemma 1 [24]. Suppose0rin (2) is regular and has the system matrices in (3).
Then the pair(E0r, A0r) is causal and stable if and only if A4 ∈ R(n−r)×(n−r)
is invertible, and all the roots of det(zE0r− A0r) = 0 have magnitudes less than
unity.
Lemma 1 is the discrete-time version of the corresponding Lemma in [24], and can be proved similarly [3].
Lemma 2. Suppose0r in (2) is r.s.e. to0 in (1). The pair(E0, A0) in (1) is
admissible if and only if the pair(E0r, A0r) in (2) is admissible.
Proof. The pair(E0, A0) is admissible if and only if [20] there exists a nonsin-gular matrix X such that
ET0XE0≥ 0, AT0XA0− ET0XE0< 0. (4)
Since0and0rare r.s.e., there exist nonsingular matrices P0and Q0such that
E0= P−10 E0rQ−10 and A0= P−10 A0rQ−10 . Thus (4) is equivalent to
ET0rXrE0r ≥ 0, AT0rXrA0r− ET0rXrE0r < 0, (5)
with Xr = P−T0 XP−10 , which means exactly that(E0r, A0r) is admissible. 2
2.2. System transformation
The uncertain singular system to be discussed is
: Ex(k + 1) = (A + δA)x(k) + (B + δB)u(k) y(k) = (C + δC)x(k) + (D + δD)u(k) z(k) = (L + δL)x(k) + (J + δJ)u(k), (6) where x(k) ∈ Rnis the state vector, y(k) ∈ Rpis the measured output vector,
z(k) ∈ Rq is the vector to be estimated, and u(k) ∈ Rm is the disturbance input
B, C, D, L, and J are known real constant matrices with appropriate dimensions.
The constant uncertainty matrices satisfy δA δBδC δD δL δJ = HHxy Hz ∆Ex Eu (7)
with ∆T∆≤ I and ∆ ∈ Rd1×d2. Assume that the pair(E, A+δA) is admissible, so there exist [3] nonsingular matrices P and Q such that in (6) is r.s.e. to the system r: Er˜x(k + 1) = (Ar+ δAr)˜x(k) + (Br + δBr)u(k) y(k) = (Cr + δCr)˜x(k) + (D + δD)u(k) z(k) = (Lr + δLr)˜x(k) + (J + δJ)u(k), (8)
where˜x(k) = Q−1x(k) = ˜xT1(k) ˜xT2(k)T,˜x1(k) ∈ Rr,˜x2(k) ∈ Rn−r, and the constant uncertainty matrices satisfy
δArδCr δBrδD δLr δJ = HHxry Hz ∆Exr Eu (9)
with ∆T∆≤ I. The matrices Er = PEQ = Ir 0 0 0 , Ar = PAQ = A11 A12 A21 A22 , Br = PB = B1 B2 , Cr = CQ = C1 C2 , Lr = LQ = L1 L2 , Hxr = PHx = Hx1 Hx2 , Exr = ExQ= Ex1 Ex2 . (10)
The r.s.e. systemrin (8) may be more explicitly written as
˜x1(k + 1) = (A11+ Hx1∆Ex1)˜x1(k) + (A12+ Hx1∆Ex2)˜x2(k)
+(B1+ Hx1∆Eu)u(k), (11)
0= (A21+ Hx2Ex1)˜x1(k) + (A22+ Hx2∆Ex2)˜x2(k)
+(B2+ Hx2∆Eu)u(k), (12)
y(k) = (C1+ Hy∆Ex1)˜x1(k) + (C2+ Hy∆Ex2)˜x2(k)
+(D + Hy∆Eu)u(k), (13)
z(k) = (L1+ Hz∆Ex1)˜x1(k) + (L2+ HzEx2)˜x2(k)
+(J + Hz∆Eu)u(k). (14)
By Lemma 2, the pair(Er, Ar+δAr) of r with parameter matrices in (9) and
(10) is admissible. In addition, by Lemma 1, the term(A22+ Hx2∆Ex2) in (12)
nonsingular. Let the nonsingular matrices ¯P= diag(Ir, A−122) and ¯Q = In. Then rin (11)–(14) is, via ¯P and ¯Q, r.s.e. to
˜
r:
Er˜x(k + 1) = ¯P(Ar+ δAr)˜x(k) + ¯P(Br + δBr)u(k)
y(k) = (Cr + δCr)˜x(k) + (D + δD)u(k)
z(k) = (Lr + δLr)˜x(k) + (J + δJ)u(k),
(15) which can be represented more explicitly by (11), (13), (14), and
0= ( ¯A21+ ¯Hx2Ex1)˜x1(k) + (In−r+ ¯Hx2Ex2)˜x2(k)
+( ¯B2+ ¯Hx2∆Eu)u(k) (16)
with ¯A21= A−122A21, ¯B2= A−122B2, and ¯Hx2= A−122Hx2.
By Lemma 1, the term(In−r+ ¯Hx2∆Ex2) in (16) is also nonsingular, because
of the admissibility of ˜rmaintained by Lemma 2. Using the identity
(I + MN)−1= I − M(I + NM)−1N (17)
for any real matrices M and N with appropriate dimensions, one has
(In−r+ ¯Hx2∆Ex2)−1= In−r− ¯Hx2ˆ∆Ex2, (18)
where ˆ∆= ∆(Id2+ Ex2¯Hx2∆)−1. Therefore, (16) may be rearranged as
˜x2(k) = −( ¯A21+ ¯Hx2ˆ∆ ¯Ex1)˜x1(k) − ( ¯B2+ ¯Hx2ˆ∆ ¯Eu)u(k), (19)
where ¯Ex1 = Ex1− Ex2¯A21 and ¯Eu = Eu − Ex2¯B2. By substituting (19) into
(11), (13), and (14), the system ˜ris reduced to ˜r:
˜x1(k + 1) = ( ¯A11+ ¯Hx1ˆ∆ ¯Ex1)˜x1(k) + ( ¯B1+ ¯Hx1ˆ∆ ¯Eu)u(k)
y(k) = ( ¯C1+ ¯Hy ˆ∆ ¯Ex1)˜x1(k) + ( ¯D + ¯Hy ˆ∆ ¯Eu)u(k)
z(k) = ( ¯L1+ ¯Hz ˆ∆ ¯Ex1)˜x1(k) + (¯J + ¯Hz ˆ∆ ¯Eu)u(k),
(20)
where
¯A11 = A11− A12¯A21, ¯B1= B1− A12¯B2, ¯C1 = C1− C2¯A21,
¯D = D − C2¯B2, ¯L1= L1− L2¯A21, ¯J = J − L2¯B2, ¯Hx1= Hx1− A12¯Hx2, ¯Hy = Hy− C2¯Hx2, ¯Hz = Hz− L2¯Hx2,
¯Ex1= Ex1− Ex2¯A21, ¯Eu = Eu− Ex2¯B2.
(21) Note that ˜r in (20) is a normal system [3], and its stability is guaranteed by Lemma 2 with the r.s.e. relationship.
The transformation from singular to normal system models enables one to handle the robust filtering problem for uncertain singular systems more easily, because many existing filter design methods for normal systems can be applied. Besides, filters designed this way have fewer states than singular filters designed directly from the singular system models. Finally, sometimes the physical realiza-tions of singular filters are not easy [3], [4]. In order to realize singular filters, one often needs special algorithms [15] to convert to a normal state-space form.
However, it must be pointed out that in general the transformation is not unique, and for in (6), there may be more than one pair of nonsingular matrices {P, Q} capable of making PEQ = diag(Ir, 0). Among the various methods to find a
feasible pair{P, Q}, one is stated here. Let a singular value decomposition [9] of a given E in (6) be E = ¯U diag(Σ, 0) ¯VT, where ¯U, ¯V ∈ Rn×n are uni-tary, Σ = diag(σ1, . . . , σr), and σi > 0, i = 1, . . . , r, are the singular values of E. Thus, diag(Σ−1, In−r) ¯UTE ¯V = diag(Ir, 0), and a feasible pair {P, Q} is
{diag(Σ−1, In−r) ¯UT, ¯V}.
2.3. Problem statement
Consider the normal stable system ˜r in (20) subject to ˆ∆ = ∆(Id2 +
Ex2¯Hx2∆)−1and ∆T∆≤ I. To estimate z(k), the following filter:
f :
xf(k + 1) = Afxf(k) + Bfy(k)
zf(k) = Cfxf(k) + Dfy(k) (22)
is adopted, where xf(k) ∈ Rr and zf(k) ∈ Rq. The matrices Af, Bf, Cf,
and Df are to be determined. From ˜rin (20) andfin (22), the filtering error dynamics may be written as
e: xe(k + 1) = Aexe(k) + Beu(k) e(k) = Cexe(k) + Deu(k), (23) where e(k) = z(k) − zf(k), xTe(k) = [ ˜xT1(k) xTf(k) ], Ae = ˆA 0 Bf ˆC Af , Be= ˆB Bf ˆD , Ce= ˆL − Df ˆC −Cf , De= ˆJ − Df ˆD, (24) and
ˆA = ¯A11+ ¯Hx1ˆ∆ ¯Ex1, ˆB = ¯B1+ ¯Hx1ˆ∆ ¯Eu, ˆC = ¯C1+ ¯Hy ˆ∆ ¯Ex1,
ˆD = ¯D + ¯Hy ˆ∆ ¯Eu, ˆL = ¯L1+ ¯Hz ˆ∆ ¯Ex1, ˆJ = ¯J + ¯Hz ˆ∆ ¯Eu.
(25) The purpose here is to design a stable filterfsuch that
sup
∆
Ce(zI2r−Ae)−1Be+ De∞< µe (26)
for a prescribed H∞-norm boundµe> 0.
At this point an extra assumptionEx2¯Hx2 < 1 is added, which is solely for
enabling the LMI formulation in Theorem 2 to be developed in Section 3. Though this extra assumption limits the systems that may be handled, its validity is not affected by the choice of the transformation matrices P and Q, as can be proved in the following lemma.
Lemma 3. Consider the uncertain singular system in (6)–(7) with the
admissi-ble pair(E, A+δA). The value of Ex2¯Hx2 is independent of the transformation
matrix pair{P, Q}, making PEQ = diag(Ir, 0) in (10).
Proof. Suppose the nonsingular matrix pair{ ˜P, ˜Q} makes ˜PE ˜Q = diag(Ir, 0)
and ˜PA ˜Q= A11 A12 A−122A21 In−r , ˜PHx= Hx1 A−122Hx2 , Ex˜Q=Ex1 Ex2 . (27) By (16) and (27), the concerned norm isEx2A−122Hx2. Let ˆP, ˆQ ∈ Rn×nbe any
two nonsingular matrices satisfying ˆP ˜PE ˜Q ˆQ= diag(Ir, 0). Partition ˆP and ˆQ as
ˆP = ˆP11 ˆP12 ˆP21 ˆP22 , ˆQ = ˆQ11 ˆQ12 ˆQ21 ˆQ22 , (28)
where ˆP11, ˆQ11 ∈ Rr×r, and ˆP22, ˆQ22 ∈ R(n−r)×(n−r). Then ˆP11ˆQ11 = Ir,
ˆP21 = 0, and ˆQ12 = 0. From (27), the (2, 2) block of ˆP ˜P(A + Hx∆Ex) ˜Q ˆQ
is ˆP22ˆQ22+ ˆP22A−122Hx2∆Ex2ˆQ22. By Lemma 1, ˆP22ˆQ22is nonsingular, which implies that both ˆP22 and ˆQ22 are also nonsingular. Therefore, when{ ˆP ˜P, ˜Q ˆQ} is regarded as another transformation matrix pair, the corresponding new Ex2
and ¯Hx2are Ex2ˆQ22and( ˆP22ˆQ22)−1ˆP22A−122Hx2, respectively, and the concerned
norm isEx2ˆQ22( ˆP22ˆQ22)−1ˆP22A−122Hx2 = Ex2A−122Hx2. 2 2.4. Three useful lemmas
The following is a well-known lemma extended from the Bounded Real Lemma [7] for characterizing the H∞-norm constraint.
Lemma 4 [8], [25]. The error dynamic systemein (23) is quadratically stable
[1] and satisfies (26) for a givenµe> 0, if and only if there exists a Pe> 0 such
that −Pe 0 ATePe CTe 0 −µ2eI BTePe DTe PeAe PeBe −Pe 0 Ce De 0 −I < 0. (29)
It is known [1] that the quadratic stability of a system implies its asymptotic stability. Because ˜rin (20) is stable, the quadratic stability ofein (23) implies that the filterfin (22) is asymptotically stable.
The next two lemmas are useful for formulating the problem within the LMI framework.
Lemma 5 [19]. Let I− ΓTΓ> 0, and define the set
ϒ ={∆(I − Γ∆)−1, ∆T∆≤ I}.
Then,ϒ = {ΓT(I − ΓΓT)−1+ ΠT(I − ΓΓT)−1/2, ΠTΠ≤ (I − ΓTΓ)−1}.
Lemma 6 [11]. Let Ω, ¯M, ¯N, and R > 0 be real matrices with appropriate
dimensions, and let the matrix ¯Π satisfy ¯ΠT¯Π ≤ R. Then for all ¯ΠT¯Π ≤ R
the matrix inequality
Ω+ ¯M ¯Π ¯N + ¯NT¯ΠT¯MT< 0
holds if and only if there exists a scalarε > 0 such that
Ω ¯M ¯MT 0 + ε ¯NTR ¯N 0 0 −I < 0.
3. Robust filter design
In the literature, many authors [6], [13], [14], [25] have discussed normal robust filtering problems with various specifications, mainly based on Lemma 4. Here the method for proving Theorem 1 of [13] is modified to treat a different kind of uncertainty, and to derive the following preliminary theorem, which is the first step toward developing an LMI solution to the problem stated in Section 2.
Theorem 1. The filtering error dynamicse in (23) is quadratically stable and
satisfies (26) for all admissible uncertainties, if and only if there exist Φ∈ Rr×r,
X∈ Rr×r, Y∈ Rq×r, Z∈ Rr×q, W∈ Rr×r, and Df ∈ Rq×psuch that
−Φ ∗ ∗ ∗ ∗ ∗ −Φ −X ∗ ∗ ∗ ∗ 0 0 −µ2eI ∗ ∗ ∗ Φ ˆA Φ ˆA Φ ˆB −Φ ∗ ∗ X ˆA+ Z ˆC + W X ˆA + Z ˆC X ˆB + Z ˆD −Φ −X ∗ ˆL − Df ˆC − Y ˆL − Df ˆC ˆJ − Df ˆD 0 0 −I < 0, (30) Φ Φ Φ X > 0, (31)
where ˆA, ˆB, ˆC, ˆD, ˆL, and ˆJ are defined in (25). When the preceeding inequalities
hold, the filterfin (22) with filter gains
Af = −U−1WU−T, Bf = U−1Z, Cf = −YU−T, Df (32) is a solution to the considered robust filtering problem, where U is nonsingular
Proof. (Sufficiency) By the Schur complement [2] and the inequality (31), Φ> 0
and X− Φ > 0. Thus, I − XΦ−1 is nonsingular and there exist nonsingular matrices U and V such that I− XΦ−1= UVT. Let
ˆT = Φ−1 I VT 0 , ˇT = I X 0 UT , (33) where ˆT is nonsingular as ˆT−1 = 0 V−T I −Φ−1V−T . Define Pe = ˇT ˆT−1 = X U UT I
by letting U = −ΦV. Under this arrangement Pe > 0 because
X − UUT = X + UVTΦ = Φ > 0. Next, pre- and post-multiply (30)
by diag(Φ−1, I, I, Φ−1, I, I) at the same time. Substituting (24), (32), (33),
U = −ΦV, and Pe = ˇT ˆT−1 into the resulting inequality, as well as pre- and
post-multiplying by diag( ˆT−T, I, ˆT−T, I) and diag( ˆT−1, I, ˆT−1, I), respectively, give (29). By Lemma 4, the error dynamics in (23) is quadratically stable, which implies the filter in (22) with gains in (32) is asymptotically stable, and the H∞ performance requirement (26) is satisfied for all admissible uncertainties.
(Necessity) If the filtering error dynamicse is quadratically stable and has the H∞-norm boundµe, then by Lemma 4 there exists a Pe> 0 such that (29) is
satisfied. Let Peand its inverse P−1e be partitioned as
Pe= X U UT Ψ , P−1e = Φ−1 V VT , (34)
where X > 0, Φ > 0, and denotes the submatrix which is insignificant in this proof. From PeP−1e = I, it is seen that I − XΦ−1 = UVT with U, V
nonsingular [16], and U = −ΦVΨT. Form a nonsingular matrix ˆT as in (33).
Substitute Pe in (34) into (29), and pre- and post-multiply the resultant
inequal-ity by diag( ˆTT, I, ˆTT, I) and diag( ˆT, I, ˆT, I), respectively. Then (30) is obtained when
Af = U−1WΦ−1V−TΨ−1, Bf = U−1Z, Cf = YΦ−1V−TΨ−1(35)
are substituted, and the resultant inequality is pre- and post-multiplied by
diag(Φ, I, I, Φ, I, I) at the same time. A similar but much simpler procedure
applied to Pein (34) produces the inequality in (31). 2
Note that in addition to the filter gain matrices shown in the sufficiency part of Theorem 1, the following filter gains:
are also usable, because the transfer function matrix Gf(z) of the filter from y(k)
to zf(k) satisfies
Gf(z) = −YU−T(zI + U−1WU−T)−1U−1Z+ Df
= −Y[zI + (UUT)W]−1(UUT)−1
Z+ Df
= −Y[zI − (Φ − X)−1W]−1(X − Φ)−1Z+ Df. (37)
Next, in order to put the results of Theorem 1 under the LMI framework, the uncertainty ˆ∆is reformulated by the equivalent description
ˆ∆ = ΘT(Id
2− ΘΘ
T)−1+ ΠT(Id
2− ΘΘ
T)−1/2, (38)
by Lemma 5 and the assumptionEx2¯Hx2 < 1, where ΠTΠ≤ (Id1− ΘTΘ)−1
and Θ= −Ex2¯Hx2. Correspondingly, the matrices in (25) may be represented as
ˆA = ˜A + ¯Hx1ΠT˜Ex1, ˆB = ˜B + ¯Hx1ΠT˜Eu, ˆC = ˜C + ¯HyΠT˜Ex1,
ˆD = ˜D + ¯HyΠT˜Eu, ˆL = ˜L + ¯HzΠT˜Ex1, ˆJ = ˜J + ¯HzΠT˜Eu,
(39) where
˜A = ¯A11+ ¯Hx1ΘT(I − ΘΘT)−1¯Ex1, ˜B = ¯B1+ ¯Hx1ΘT(I − ΘΘT)−1¯Eu,
˜C = ¯C1+ ¯HyΘT(I − ΘΘT)−1¯Ex1, ˜D = ¯D + ¯HyΘT(I − ΘΘT)−1¯Eu,
˜L = ¯L1+ ¯HzΘT(I − ΘΘT)−1¯Ex1, ˜J = ¯J + ¯HzΘT(I − ΘΘT)−1¯Eu,
˜Ex1= (I − ΘΘT)−1/2¯Ex1, ˜Eu = (I − ΘΘT)−1/2¯Eu.
(40) Then Theorem 2 below is an LMI version of Theorem 1.
Theorem 2. Under the assumption ofEx2¯Hx2 < 1, the filtering error
dynam-icse in (23) is quadratically stable and satisfies (26) for a givenµe > 0 with
all considered uncertainties, if and only if there exist Φ ∈ Rr×r, X∈ Rr×r,
Y ∈ Rq×r, Z ∈ Rr×q, W ∈ Rr×r, Df ∈ Rq×p, and ε−1 > 0 such that the
LMIs in (31) and −Φ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −Φ −X ∗ ∗ ∗ ∗ ∗ ∗ 0 0 −µ2eIm ∗ ∗ ∗ ∗ ∗ Φ ˜A Φ ˜A Φ ˜B −Φ ∗ ∗ ∗ ∗ M51 M52 M53 −Φ −X ∗ ∗ ∗ M61 M62 M63 0 0 −Iq ∗ ∗ ε−1˜E x1 ε−1˜Ex1 ε−1˜Eu 0 0 0 −ε−1Id2 ∗ 0 0 0 ¯HTx1Φ M85 M86 0 M88 < 0, (41)
are satisfied, where
M51 = X ˜A+Z ˜C+W, M52 = X ˜A+Z ˜C, M53 = X ˜B+Z ˜D,
M61 = ˜L−Df ˜C−Y, M62 = ˜L−Df ˜C, M63 = ˜J−Df ˜D,
M85 = ¯HTx1X+ ¯HTyZT, M86 = ¯HTz− ¯HTyDTf, M88 = −ε−1(Id1−Θ TΘ).
When the above inequalities hold, the filterfin (22) with filter gains (32) or (36)
is a solution to the considered robust filtering problem.
Proof. It is enough to establish the equivalence of (30) and (41) with anε−1> 0.
By (38), (30) may be rewritten as ˜Ω + ˜MΠ ˜N + ˜NTΠT˜MT< 0, (43) with ΠTΠ≤ (Id1− Θ TΘ)−1, where ˜Ω = −Φ ∗ ∗ ∗ ∗ ∗ −Φ −X ∗ ∗ ∗ ∗ 0 0 −µ2eIm ∗ ∗ ∗ Φ ˜A Φ ˜A Φ ˜B −Φ ∗ ∗ X ˜A+Z ˜C+W X ˜A+Z ˜C X ˜B+Z ˜D −Φ −X ∗ ˜L−Df ˜C−Y ˜L−Df ˜C ˜J−Df ˜D 0 0 −Iq , (44) ˜MT= ˜E x1 ˜Ex1 ˜Eu 0 0 0 , (45) ˜N = 0 0 0 ¯HTx1Φ ¯HTx1X+ ¯HTyZT ¯HTz− ¯HTyDTf . (46) By Lemma 6 and the Schur complement, it is seen that (43) is equivalent to (41)
with anε−1> 0. 2
Remark 1. Based on Theorem 2, the following convex optimization problem
may be formulated with respect to a chosen pair{P, Q} in (10) to find the H∞ optimal filter of the form (22) such that (26) is satisfied with the minimalµe:
min
µ2
e, ε−1,Φ, W, X, Y, Z, Df
µ2e, (47)
subject to the LMIs (31), (41),ε−1> 0, and µ2e > 0.
4. A numerical example
In this section, an example is worked out to illustrate the proposed filter design method. Suppose that the system matrices of the system in (6) are as follows:
E= 10 22 11 1 0 0 , A= 00.1530.1560 00.0450.2520 00.0690.1560 0.1350 −0.1710 −0.6360 , B= 11 0.2 , C =0.1 0 0.5 , D= −0.5, L= −1 0.3 −0.5, J= 0. (48)
The uncertainty matrices in (7) are assumed to be HTx = 1.5 3 1.5 , Hy = −1, Hz = 2, Ex = 0.05 0 0.1 , Eu= 1, (49)
and|| ≤ 1. The prescribed H∞-norm boundµein (26) is 2. It is easy to verify
that(E, A + HxEx) is an admissible pair, and rank E = 2. By applying singular
value decomposition to E, one may choose
P= 00.2283.2850 −0.39770.2045 00.0238.6827 −0.5774 0.5774 0.5774 , Q= 00.2521.8655 −0.22550.9677 0.44720 0.4328 −0.1128 −0.8944 . (50)
BecauseEx2¯Hx2 = 0.5291 < 1, the assumption of Theorem 2 is satisfied. The
filterfin (22) is designed by solving the LMIs of Theorem 2, and the filter gains (36) are found to be Af = 0.0236 −0.0619 −0.1601 0.2558 , Bf = −0.7226 1.1036 , Cf = −0.0535 0.9527, Df = −0.9518, (51)
which is a second-order normal stable filter as desired. With respect to the chosen {P, Q} in (50), the corresponding H∞optimal filter is also designed by solving the convex optimization problem mentioned in Remark 1, which is implemented by the MATLAB LMI Control Toolbox [5]. The resulting optimalµe is 1.1761, and the filter gains (36) are found to be
Af = 0.0251 −0.0740 −0.1564 0.2626 , Bf = −0.6156 1.0481 , Cf = −0.0667 1.0134, Df = −0.8753. (52)
Of course, for the E in (48) there are other choices of{P, Q} capable of making
PEQ= diag(I2, 0), and an example is P= 10 −11 00 −1 1 1 , Q= 10 00.5 01 0 0 −2 . (53) Corresponding to this choice, the concerned normEx2¯Hx2 is still 0.5291 < 1.
are found to be Af = 0.1874 −0.0883 −0.2544 0.0968 , Bf = 0.8676 −1.9944 , Cf = 0.9448 −0.1419 , Df = −0.9202, (54)
which are clearly different from those in (51). Similarly, resolving the convex optimization problem mentioned in Remark 1 gives a different optimal µe =
1.8531 from the previous one.
5. Conclusion
The H∞ filter design problem has been considered for discrete-time singular systems with norm-bounded uncertainties. The algebraic equations in the singular system model are eliminated, and a normal dynamic system model is constructed with uncertainties in the linear fractional transformation form. For the H∞filter design problem, the normal system model allows one to utilize many existing methods to design normal filters directly, but the question of how to utilize the degrees of freedom in the choices of normal system models is worthy of further investigations. In this paper, a set of necessary and sufficient conditions is pro-vided in terms of LMIs for a normal filter design.
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