H
1
optimal singular and normal filter design
for uncertain singular systems
C.-M. Lee and I.K. Fong
Abstract: The robust H1 optimal filtering problem for singular systems with norm-bounded
uncertainties in all system matrices is considered. On the basis of the admissibility assumption of the uncertain singular systems, one singular and two normal filter design methods are proposed under the linear-matrix inequality framework. A numerical example is provided to illustrate the application of all three proposed methods.
1 Introduction
In the past decade, the H1 optimal filtering problem for
singular systems has been an important research topic. This is due, not only to the theoretical interest, but also to
the relevance of the topic in various engineering
applications. Many works, such as Xu et al. [1] and Yue
and Han[2], consider the filters with the same structure as the concerned singular system in the derivative term. Indeed, this is often a natural and convenient choice. However, it is more suitable for the cases in which the coef-ficient matrix for the derivative term has no uncertainty. In practical applications, more flexibility will be gained if all system matrices are allowed to contain uncertainties.
In this paper, the robust H1 optimal filtering problem is studied for singular systems, which contain norm-bounded uncertainties in all system matrices. First of all, a filter design method that generates singular filters is proposed as an extension of the method by Xu et al.[1]. Then, two kinds of normal filters (i.e. those with the system matrix for the derivative term being the identity matrix) [3] are considered. For singular as well as normal filters, the goal is to minimise the H1performance level of the correspond-ing filtercorrespond-ing error dynamics. The consideration of normal filters is beneficial when sometimes the physical realisations of singular filters are not easy [3, 4]. In order to realise a
singular system, often it needs special algorithms [5] to
convert a singular system model into a normal state-space form.
To handle systems with uncertainties in the system matrix for the derivative term, it is assumed that the uncer-tain systems are admissible and the concept of the restricted system equivalence (r.s.e.)[3]is applied. In addition, some preliminary results in Lin et al. [6], which considers the stabilisation problem for singular systems using the alge-braic Riccati equation method, provide the further assump-tions for the theoretical development of this paper. For easier applications, all proposed filter design methods are formulated within the linear-matrix inequality (LMI) framework.
Some notations to be used subsequently are introduced here. The inequality X 0 means that the matrix X is
symmetric and positive semi-definite, and X Y
means X 2 Y 0. Similar definitions apply to symmetric positive/negative definite matrices. For a matrix M, kMk denotes the spectral norm of M, and for a stable
continuous-time transfer function matrix G(s), kGk1¼
supv[[0,1)kG( jv)k is its H1norm. Iris the identity matrix with dimension r, the superscript T represents 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
First, consider the following nominal singular system Sn: E _xðtÞ ¼ AxðtÞ þ BuðtÞ
zðtÞ ¼ LxðtÞ
ð1Þ
where x(t) [Rnand rankE ¼ r , n. The unforced singular
system pair (E, A) of (1) with u ¼ 0 is regular if det(sE 2 A) is not identically zero. If deg(det(sE 2 A)) ¼ rank E, then (E, A) is said to be impulse free. The pair (E, A) is stable if all the roots of det(sE 2 A) ¼ 0 have negative real parts. Finally, (E, A) is admissible if it is regular, impulse free and stable. For Sn, its transfer function matrix from u(t) to z(t) is G(s) ¼ L(sE 2 A)21B.
Definition 1[3]: Suppose Snin (1) is regular. Let P and Q
be two n n non-singular matrices and Er¼ PEQ,
Ar¼ PAQ, Br¼ PB, Lr¼ LQ. The system
Sr: Er_xrðtÞ ¼ ArxrðtÞ þ BruðtÞ
zðtÞ ¼ LrxrðtÞ
ð2Þ
with xr(t) ¼ Q21x(t) is r.s.e. to Sn.
For any given regular Sn, there exist [3] non-singular
matrices P and Q such that
Er¼ Ir 0 0 0 ; Ar ¼ A1 A2 A3 A4 ; Br¼ B1 B2 ; Lr¼ ½L1 L2 ð3Þ
#The Institution of Engineering and Technology 2006
doi:10.1049/iet-cta:20060042 Paper first received 16th June 2005
The authors are with the Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan, Republic of China
Now consider the following uncertain singular system Su: E _xðtÞ ¼ ðA þdAÞxðtÞ þ ðB þdBÞuðtÞ
zðtÞ ¼ ðL þdLÞxðtÞ þ ðJ þdJ ÞuðtÞ
ð4Þ wheredA,dB,dL and dJ are constant uncertainties. Definition 2[6, 7]: The unforced pair (E, A þdA) in Suis said to be quadratically admissible for all uncertaintydA if there exists a matrix X such that
ETX ¼ XTE 0 ð5Þ
ðA þdAÞTX þ XTðA þdAÞ , 0 ð6Þ
The quadratic admissibility is also called the generalised quadratic stability by Xu and Yang [8] and Xu et al. [9], and it implies[7]the admissibility of a system.
Definition 3[7]: Given am. 0, the system Suis said to be quadratically admissible with disturbance attenuationmfor all uncertaintiesdA,dB,dL anddJ if there exists a matrix X such that ETX ¼ XTE 0 ð7Þ ðA þdAÞTX þXTðA þdAÞ ðB þdBÞTX m2I L þdL J þdJ I 2 6 6 6 4 3 7 7 7 5, 0 ð8Þ
It is known [7] that quadratically admissible with
disturb-ance attenuationmmeans that the H1norm of the transfer
function matrix from u(t) to z(t) is less thanm.
2.1 Problem formulation
The uncertain singular system to be discussed is S:
ðE0þdEÞ_xðtÞ ¼ ðA0þdAÞxðtÞ þ ðB0þdBÞuðtÞ yðtÞ ¼ ðC0þdCÞxðtÞ þ ðD0þdDÞuðtÞ zðtÞ ¼ ðL0þdLÞxðtÞ þ ðJ0þdJ ÞuðtÞ 8 < : ð9Þ where x(t) [Rn
is the state vector, y(t) [Rp
the measured
output vector, z(t) [Rq the vector to be estimated and
u(t) [Rm
the disturbance input vector. The matrices E0, A0, B0, C0, D0, L0and J0are known real constant matrices with appropriate dimensions. The constant uncertainty matrices satisfy dA dB dC dD dL dJ 2 4 3 5 ¼ Mx My Mz 2 4 3 5D½NxNu ð10Þ
dE ¼ MDN andDTDI. Suppose that the pair (E0þdE,
A0þdA) is admissible and rank(E0þdE) ¼ rank
E0¼ r , n for allDunder discussion. Let the non-singular P and Q be such that
PE0Q ¼ Ir 0 0 0 ; PA0Q ¼ Ar1 Ar2 Ar3 Ar4 PM ¼ M1 M2 ; NQ ¼ ½N1 N2 ð11Þ
By Lin et al.[6], it can be assumed without loss of general-ity that either M2¼ 0 or N2¼ 0. Here, it is further assumed that kN1M1k, 1.
To estimate z(t), the following filter is adopted Sf: Ef_xfðtÞ ¼ AfxfðtÞ þ BfyðtÞ
zfðtÞ ¼ CfxfðtÞ þ DfyðtÞ
ð12Þ where xf(t) [Rnf
and zf(t) [Rq. The matrices Ef, Af, Bf, Cfand Dfare to be determined. From S in (9) and Sfin (12), the filtering error dynamics may be written as
Se: Ee_xeðtÞ ¼ AexeðtÞ þ BeuðtÞ
eðtÞ ¼ CexeðtÞ þ DeuðtÞ
ð13Þ where e(t) ¼ z(t) 2 zf(t), xeT(t) ¼ [xT(t) xfT(t)] and
Ee¼ E0þdE 0 0 Ef ; Ae¼ A0þdA 0 BfðC0þdCÞ Af Be¼ B0þdB BfðD0þdDÞ Ce¼ ½ðL0þdLÞ DfðC0þdCÞ Cf De¼ ðJ0þdJ Þ DfðD0þdDÞ ð14Þ
The purpose here is to design the filter Sfsuch that the pair (Ef, Af) is admissible and
sup
D
kCeðsEeAeÞ1BeþDek1,me ð15Þ
for a prescribed H1-norm boundme. 0.
2.2 Restricted system equivalence
Under the assumption that the pair (E0þdE, A0þdA) is admissible, there exist[6] non-singular matrices P, Q, PD and QDsuch that the system S in (9) is r.s.e. to the system
~ Sr: ~ Er_~xðtÞ ¼ ~Ar~xðtÞ þ ~BruðtÞ yðtÞ ¼ ~Cr~xðtÞ þ ~DruðtÞ zðtÞ ¼ ~Lr~xðtÞ þ ~JruðtÞ 8 < : ð16Þ where ~ Er¼E0r¼PE0Q; A~r¼PDðA0rþMxrDNxrÞQD ~ Br¼PDðB0rþMxrDNurÞ; C~r¼ ðC0rþMyrDNxrÞQD ~ Dr¼ ðD0rþMyrDNurÞ; L~r¼ ðL0rþMzrDNxrÞQD ~Jr¼ ðJ0rþMzrDNurÞ ð17Þ and A0r¼PA0Q; B0r¼PB0; C0r¼C0Q D0r¼D0; L0r¼L0Q; J0r¼J0 Mxr¼PMx; Myr¼My; Mzr¼Mz Nxr¼NxQ; Nur¼Nu ð18Þ
More specifically, for N2¼ 0 in (11), PD¼ In2 M0rD˜ N0r
and QD¼ In, and for M2¼ 0 in (11), PD¼ In
and QD¼ In2 M0rD˜ N0r, where M0r¼ PM, N0r¼ NQ and
D˜ ¼D(I þ N1M1D)21.
Thus, the error dynamics S˜e from S˜rin (16) and Sfin (12) is ~ Se: E~e_~xeðtÞ ¼ ~Ae~xeðtÞ þ ~BeuðtÞ eðtÞ ¼ ~Ce~xeðtÞ þ ~DeuðtÞ ( ð19Þ
IET Control Theory Appl., Vol. 1, No. 1, January 2007 120
where ~xeT(t) ¼ [~xT(t) xfT(t)] and ~ Ee¼ E0r 0 0 Ef ; A~e¼ A~r 0 BfC~r Af " # ; B~e¼ B~r BfD~r " # ; ~ Ce¼ ½ ~LrDfC~r Cf; D~e¼ ~JrDfD~r ð20Þ
Lemma 1: Suppose there exist non-singular matrices Psand
Qssuch that the regular systems Ss1and Ss2are r.s.e., and
there exist non-singular matrices Pf and Qf such that the
filters Sf1and Sf 2are r.s.e., where for i ¼ 1, 2
Ssi: Esi_xsiðtÞ ¼ AsixsiðtÞ þ BsiusðtÞ ysðtÞ ¼ CsixsiðtÞ þ DsusðtÞ zsðtÞ ¼ LsixsiðtÞ þ JsusðtÞ 8 > < > : ð21Þ Sf i: Ef i˙xf iðtÞ ¼ Af ixf iðtÞ þ Bf iysðtÞ ^zfðtÞ ¼ Cf ixf iðtÞ þ DfysðtÞ ð22Þ
Then, the corresponding error dynamics Se1 and Se2 are
also r.s.e., where for i ¼ 1, 2
Sei: Eei_xeiðtÞ ¼ AeixeiðtÞ þ BeiusðtÞ
^eðtÞ ¼ CeixeiðtÞ þ DeusðtÞ
ð23Þ ^e(t) ¼ zs(t) 2 ^zf(t), xeiT(t) ¼ [xsiT(t) xfiT(t)] and
Eei¼ Esi 0 0 Ef i ; Aei¼ Asi 0 Bf iCsi Af i Bei¼ Bsi Bf iDs ; Cei¼ ½LsiDf iCsi Cf i De¼JsDfDs ð24Þ
Moreover, the transfer function matrices Ge1(s) and Ge2(s) of Se1and Se2, respectively, are the same.
Proof: By the assumptions, there exist non-singular matrices Ps, Qs, Pfand Qfsuch that
Ej2¼PjEj1Qj; Aj 2¼PjAj1Qj; Bj 2¼PjBj1;
Cj 2¼Cj1Qj; Lj 2¼Lj1Qj ð25Þ
for j¼ s and f. It is easy to see that the matrices in (24) satisfy
Ee2¼ ^PeEe1Q^e; Ae2¼ ^PeAe1Q^e
Be2¼ ^PeBe1; Ce2¼Ce1Q^e ð26Þ
where ^Pe¼ diag(Ps, Pf) and ^Qe¼ diag(Qs, Qf). Therefore Se1and Se2are also r.s.e. By (26)
Ge2ðsÞ ¼ Ce2ðsEe2Ae2Þ1Be2þDe
¼Ce1Q^eðs ^PeEe1Q^e ^PeAe1Q^eÞ1P^eBe1þDe ¼Ce1ðsEe1Ae1Þ1Be1þDe¼Ge1ðsÞ ð27Þ
A On Lemma 1, the filtering problem formulated in the last section can be reformulated as follows: design the filter Sf in (12) such that the pair (Ef, Af) is admissible and
sup
D
k ~Ceðs ~Ee ~AeÞ1B~eþ ~Dek1,me ð28Þ
2.3 Two useful lemmas
Lemma 2 [10]: Let I 2 JTJ . 0 and define the set Y ¼ fDðI ¯JDÞ1; DTDI g
Then, Y ¼ f JT(I 2 J JT)21þPT(I 2 J JT)21/2,
PTP(I 2 JTJ ) 21g.
Lemma 3 [11]: Let V, H, F and R . 0 be real matrices
with appropriate dimensions, and the matrix P satisfies
PTPR. Then, for allPTPR, the matrix inequality
VþHPF þ FTPTHT, 0
holds if and only if there exists a scalar 1 . 0 such that
V H HT 0 þ1 F TRF 0 0 I , 0
3 Robust filter design
Two kinds of robust filters are discussed here: the singular and the normal filters. For the sake of brevity, only the results for the case with N2¼ 0[6]will be presented.
3.1 Singular filter design
The filter Sfis called a singular filter when rank Ef, nfin (12). In the literature, for example, Xu et al.[1]and Yue and Han [2], the choice Ef¼ E is the most discussed, as it is often dictated by the derivation process. This case is studied here for comparison with the normal filters. Theorem 1 is developed as an extension of Theorem 1 in Xu et al.[1], which concerns nominal singular systems. Theorem 1: If and only if there exist feasible solutionsr, 1, Y [Rnn, Z [Rnp, M [Rnn, N [Rqn, Pc[Rnn and Df[Rqp to the LMIs ET0rPc¼PTcE0r0; ET0rY ¼ YTE0r0 ET0rðPcY Þ 0 ð29Þ AT0rQTY þ YTQA0r AT0rQTY þ PTcQA0r þZC0rþM ! PTcQA0rþAT0rQTPc þZC0rþCT0rZT ! BT0rQTY BT0rQTPcþDT0rZT L0rN L0r 1 1J2N0rA0r 1 1J2N0rA0r MT0rY MT0rPc MTxrQTY MTxrQTPcþMTyrZT rNxr rNxr 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 m2eI J0rDfD0r I 1 1J2N0rB0r 0 1 1 I 0 0 0 0 MTzrMTyrDTf 1 1M T xrN T 0rJ T 2 rNur 0 0
1 1 J3 0 rI 0 0 rI 3 7 7 7 7 7 7 7 7 7 7 7 5 , 0 ð30Þ where Q¼I þ M0rJ1N0r; J1¼MT1NT1ðI N1M1MT1NT1Þ1 J2¼ ðI N1M1MT1NT1Þ1=2; J3 ¼I MT1NT1N1M1 ð31Þ then the filter Sfin (12) is admissible with the filter gains
Ef ¼E0r; Af ¼UTM Y1W1
Bf ¼UTZ; Cf ¼N Y1W1
ð32Þ
where U and W are non-singular matrices satisfying ET0rV ¼ UTE0r; E0rW ¼ ~WTET0r PcY1¼I VW ; Y1Pc¼I ~W U
ð33Þ
for any non-singular matrices V and ~W and the filtering
error dynamics S˜e in (19) is quadratically admissible as well as satisfying (28).
Proof: (Sufficiency) The proof may be divided into three parts. The first part is to show that there exist non-singular
matrices U, V, W and ~W such that (33) holds. The second
part involves showing that
Pe¼ Pc V
U UY1W1
ð34Þ is non-singular, and for the error dynamics S˜ein (19) with
~
Ee¼ diag(E0r, E0r) in (20) ~
ETePe¼PTeE~e0 ð35Þ
is true. These two parts may be proved by exactly following the corresponding parts of the proofs for Theorem 1 in Xu et al.[1].
The third and only part that must be proved here is that if
the LMI (30) holds, then the error dynamics S˜e in (19)
satisfies the constraint (8). To start, denote
HT1 ¼ ½J2N0rA0r J2N0rA0r J2N0rB0r 0 F1¼ ½MT0rY MT0rPc 0 0
HTr ¼ ½MTxrQTY MxrTQTPcþMTyrZT 0 MTzrMTyrDTf 11MTxrNT0rJT2 0
Fr¼ ½Nxr Nxr Nur 0 0 0
Inequality (30) can be rewritten as
Vr¼ V1 HTr rI rFr 0 rI 2 6 4 3 7 5 , 0 ð36Þ where V1¼ V0 11HT1 11I F1 0 11J3 2 4 3 5 and V0¼ AT0rQTY þ YTQA0r AT0rQTY þ PTcQA0r þZC0rþM ! BT0rQTY L0rN 2 6 6 6 6 6 6 4 PTcQA0rþAT0rQTPc þZC0rþCT0rZT ! BT0rQTPcþDT0rZT m2eI L0r J0rDfD0r I 3 7 7 7 7 7 7 5
By the Schur complement[12]and Lemma 3, (36) with a
r. 0 is equivalent to
~
V1¼V1þHrDFrþFTrDTHTr , 0 ð37Þ
for allDTDI. The inequality (37) may be more explicitly written as AT0rQTY þ YTQA0r AT0rQTY þ PTcQA0r þZ C0rþM ! PTcQA0rþ AT0rQTPc þZ C0rþ CT0rZT ! BT0rQTY BT0rQTPcþ DT0rZT L0rN L0r 1 1J2N0rA0r 1 1J2N0rA0r MT0rY MT0rPc 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 m2eI J0rDfD0r I 1 1J2N0rB0r 0 1 1I 0 0 0 1 1J3 3 7 7 7 7 7 7 7 7 7 7 7 5 , 0 ð38Þ with A0r B0r C0r D0r L0r J0r 2 4 3 5 ¼ CA0r0r BD0r0r L0r J0r 2 4 3 5 þ MMxryr Mzr 2 4 3 5D½Nxr Nur ð39Þ
IET Control Theory Appl., Vol. 1, No. 1, January 2007 122
Letting V0¼ AT0rQTY þ YTQA0r AT0rQTY þ PTcQA0r þZ C0rþM ! BT0rQTY L0rN 2 6 6 6 6 6 6 4 PTcQA0rþ AT0rQTPc þZC0rþ CT0rZT ! BT0rQTPcþ DT0rZT m2eI L0r J0rDfD0r I 3 7 7 7 7 7 7 5 and HT1 ¼ ½J2N0rA0r J2N0rA0r J2N0rB0r 0 one sees that (38) with an 121. 0 is equivalent to
V0þ H1PF1þFT1PTHT1, 0 ð40Þ for allPTPJ
321by the Schur complement and Lemma
3. With PD¼ In2 M0r(2J1þPTJ2)N0r, the inequality
(40) may be rewritten as AT0rPTDY þ YTPDA0r AT0rPTDY þ PTcPDA0r þZ C0rþM ! PTcPDA0rþ AT0rPTDPc þZC0rþ CT0rZT ! BT0rPTDY BT0rPTDPcþ DT0rZT L0rN L0r 2 6 6 6 6 6 6 4 m2eI J0rDfD0r I 3 7 7 7 5, 0 ð41Þ Note that PD¼ In2 M0rD(I þ N1M1D) 21 N0r for D T DI
by Lemma 2 and the assumption kN1M1k, 1.
Pre-and post-multiplying (41) by diag(Y2T, I, I, I) and
diag(Y21, I, I, I), respectively, show that (41) is equivalent to PTlA~TePePlþPTlPTeA~ePl ~ BTePePl m2eI ~ CePl D~e I 2 6 4 3 7 5 , 0 ð42Þ where Peis defined in (34) Pl¼ Y 1 I W 0
are non-singular, and ~Ae, ~Be, ~Ceand ~Deare given in (20)
with Af, Bf and Cf from (32). Pre- and post-multiplying
(42) by diag(Pl2T, I, I) and diag(Pl21, I, I), respectively, one finally obtains
~ ATePeþPTeA~e ~ BTePe m2eI ~ Ce D~e I 2 6 4 3 7 5 , 0 ð43Þ
By Definition 3, the conditions in (35) and (43) with the Pe in (34) imply that S˜ein (19) is quadratically admissible with disturbance attenuationme. The filter Sfin (12) is therefore admissible.
(Necessity) As the filtering error dynamics S˜ein (19) and (20) is quadratically admissible and has a transfer function matrix satisfying (28), there exists a matrix Xesuch that
~ ETeXe¼XTeE~e0 ð44Þ ~ ATeXeþXTeA~e ~ BTeXe m2eI ~ Ce D~e I 2 6 4 3 7 5 , 0 ð45Þ
It is easy to see that (45) implies ~
ATeXeþXTeA~e, 0 ð46Þ
which implies Xeis non-singular. Partition Xeand Xe21as
Xe¼ Xe1 Xe2 U Xe3 ; X1e ¼ Ye1 Ye2 W Ye3 ð47Þ in accordance with the partition of ~Ae. The 2 – 2 block of (46) gives AfTXe3þXe3TAf, 0, which implies Xe3 is non-singular. By (46)
XTe A~Teþ ~AeX1e , 0 ð48Þ
The 1 – 1 block of (48) gives ~ArYe1þYe1TA~rT, 0, which implies Ye1is also non-singular. Note that from (44)
ET0rXe1¼XTe1E0r0; ETfXe3¼XTe3Ef 0 ð49Þ and Xe2TE~eT¼ ~EeXe210, whose 1 – 1 block implies
ET0rY1e1 ¼YTe1 E0r0 ð50Þ
As U and W can be chosen to have full-column rank[1], it is possible to define two full-column-rank matrices
^ T ¼ Ye1 I W 0 ; T ¼ I Xe1 0 U ð51Þ
so that XeT ¼ ^ T. Pre- and post-multiplying (45) by
diag( ^TT, I, I) and diag( ^T, I, I), respectively, and then
pre-and post-multiplying the resulting inequality by
diag(Ye12T, I, I, I) and diag(Ye121, I, I, I), respectively. Then, (30) is obtained by Lemmas 2 and 3 when
Pc¼Xe1; Y ¼ Y1e1; Af ¼UTM Ye1W1;
Bf ¼UTZ; Cf ¼N Ye1W1 ð52Þ
are substituted. Also, the first inequality of (49) provides the first inequality of (29) and (50) provides the second inequal-ity of (29). As Ye1¼ (Xe12 Xe2Xe321U)21, by (44) and the second inequality of (49), one has
ET0rðXe1Y1e1Þ ¼ET0rXe2X1e3U ¼ UTEfX1e3U 0 ð53Þ
which provides the last inequality of (29). A
Remark 1: On the basis of Theorem 1, the following convex optimisation problem may be formulated to find the H1 optimal filter of the form (12) such that (28) is satisfied with the minimalme2
min
m2
e;r;11;Y ;Z;M;N ;Pc;Df
m2e ð54Þ
subject to the LMIs (29) and (30). Note that by Theorem 1, the choice of the non-singular matrices U, V, W and ~W does not affect the result of the optimalme.
3.2 Normal filter design method I
The filter Sfin (12) is called a normal filter if Ef¼ Infin (12)
[3]. In some cases, the realisation of singular filters is not easy
[3, 4]. It may even be necessary to convert the derived singu-lar filters into the equivalent normal filters by using additional algorithms[5]. For such cases, it is reasonable to find the normal filter directly. In this section, the first of two normal filter design methods is introduced.
Theorem 2: If there exist feasible solutions 1,r, Pc[Rnn,
F[Rnp , C[Rqn , G[Rnn , Q [^ Rnn and Df[Rqp to the LMIs PcþPTc ðET0rþI ÞG GðE0rþI Þ 2G " # . 0; E T 0rPc ET0rG GE0r G " # 0 ð55Þ AT0rQTPcþPTcQA0r þCT0rFTE0rþET0rFC0r ! GQA0rþFC0rþ ^QTE0r Q þ ^^ QT BT0rQTPcþDT0rFTE0r BT0rQTGþDT0rFT L0rDfC0r C 1 1J2N0rA0r 0 MT0rPc MT0rG MTxrQTPcþMTyrFTE0r MTxrQTGþMTyrFT rNxr 0 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 m2eI J0rDfD0r I 1 1J2N0rB0r 0 1 1 I 0 0 0 0 MTzrMTyrDTf 1 1M T xrNT0rJ T 2 rNur 0 0 1 1 J3 0 rI 0 0 rI 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 , 0 ð56Þ
whereQ,J1,J2andJ3are given in (31), then any normal filter Sfwith Ef¼ Inand
Af ¼ ^QG1; Bf ¼F; Cf ¼CG1; Df ð57Þ
is stable and makes (28) hold for S˜ein (19). Proof: Let
Pe¼ Pc I
E0r G1
ð58Þ
With the normal filter, ~Eein (20) is diag(E0r, In), so inequal-ities in (55) guarantee that Peis non-singular and Pesatisfies
~ ETePe¼PTeE~e0 ð59Þ Next, define H1¼ PTcQMxrþET0rFMyr GQMxrþFMyr 0 MzrDfMyr 11J2N0rMxr 0 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 ; FT1 ¼ NTxr 0 NTur 0 0 0 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 ð60Þ V1¼ AT0rQTPcþPTcQA0r þCT0rFTE0rþET0rFC0r ! GQA0rþFC0rþ ^QTE0r Q þ ^^ QT BT0rQTPcþDT0rFTE0r BT0rQTGþDT0rFT L0rDfC0r C 1 1J2N0rA0r 0 MT0rPc MT0rG 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 m2eI J0rDfD0r I 1 1J2N0rB0r 0 1 1 I 0 0 0 1 1 J3 3 7 7 7 7 7 7 7 7 7 7 7 5 ð61Þ
By Lemma 3, (56) with ar. 0 is equivalent to
V1þH1DF1þFT1DTHT1 , 0 ð62Þ for allDTDI. Denote
HT2 ¼ ½J2N0rA0r 0 J2N0rB0r 0 F2¼ ½MT0rPc MT0rG 0 0 and V2¼ AT0rQTPcþPTcQA0r þ CT0rFTE0rþET0rFC0r ! GQ A0rþFC0rþ ^QTE0r Q þ ^^ QT BT0rQTPcþ DT0rFTE0r BT0rQTGþ DT0rFT L0rDfC0r C 2 6 6 6 6 6 6 4 m2eI J0rDfD0r I 3 7 7 7 5
in the left side of (62), where A0r, B0r, C0r, D0r, L0rand J0r
are defined in (39). By Lemma 3, (62) with an 121. 0 is
equivalent to
V2þH2PF2þFT2PTHT2 , 0 ð63Þ
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for all PTPJ321. With PD¼ In2 M0r(2J1þ
PTJ
2)N0r, the inequality (63) may be rewritten as PTcPDA0rþET0rFC0r þ AT0rPTDPcþ CT0rFTE0r ! GPDA0rþFC0rþ ^QE0r Q þ ^^ QT BT0rPTDPcþ DT0rFTE0r BT0rPTDGþ DT0rFT L0rDfC0r C 2 6 6 6 6 6 6 4 m2eI J0rDfD0r I 3 7 7 7 5, 0 ð64Þ
Note that PD¼ In2 M0rD(I þ N1M1D)21N0r for DTDI
by Lemma 2 and the assumption kN1M1k, 1. Pre- and
post-multiplying (64) by diag(I, PT, I, I), one obtains ~ ATePeþPTeA~e ~ BTePe m2eI ~ Ce D~e I 2 6 4 3 7 5 , 0 ð65Þ
where Peis given in (58) and ~Ae, ~Be, ~Ceand ~Deare given in (20) with Af, Bfand Cffrom (57). Expressions (59) and (65) together imply that S˜e in (19) is quadratically admissible
with disturbance attenuation me. The normal filter Sf in
(12) thus is stable. A
Remark 2: On the basis of Theorem 2, the following convex optimisation problem may be formulated to find the H1 optimal normal filter of order n such that (28) is satisfied with the minimalme
2
min
m2
e;r;11;Pc;F;C;G; ^Q;Df
m2e ð66Þ
subject to the LMIs (55) and (56).
3.3 Normal filter design method II
In this section, a method for designing normal filters of lower order than those obtainable by the first method is developed.
Theorem 3: If there exist feasible solutions 1,r, Pc[Rnn,
Pf[Rrr, F˜ [ Rrp, Q [~ Rrr, Cf[ Rqr and Df[Rqp to the LMIs Pf . 0; ET0rPc¼PTcE0r0 ð67Þ AT0rQTPcþPTcQA0r ~ FC0r Q þ ~~ QTþ2Pf BT0rQTPc D0rTF~T m2eI L0rDfC0r Cf J0rDfD0r 1 1J2N0rA0r 0 1 1J2N0rB0r MT0rPc 0 0 MTxrQTPc MTyrF~T 0 rNxr 0 rNur 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 I 0 1 1 I 0 0 1 1 J3 MTzrMTyrDTf 1 1 M T xrNT0rJ T 2 0 rI 0 0 0 0 rI 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 , 0 ð68Þ whereQ,J1,J2andJ3are given in (31), then any normal filter Sfwith Ef¼ Irand
Af ¼P1f Q þ I~ r; Bf ¼P1f F~; Cf; Df ð69Þ
is stable and makes (28) hold for S˜ein (19).
Proof: The proof is similar to that for Theorem 2, except Pe is set to diag(Pc, Pf). The remaining steps are the same and
omitted here for the sake of brevity. A
Remark 3: On the basis of Theorem 3, the following convex optimisation problem may be formulated to find the H1 optimal normal filter of order r such that (28) is satisfied with the minimalme2
min
m2
e;r;11;Pc;Pf; ~F; ~Q;Cf;Df
m2e ð70Þ
subject to the LMIs (67) and (68).
4 Numerical example
In this section, an example is worked out to illustrate the proposed filter design methods. Suppose that the system matrices of the system S in (9) are as follows
E0¼ 1 0 0 0 1 0 0 0 0 2 6 4 3 7 5; A0¼ 2 1 0 0:5 1 1 0 0 1 2 6 4 3 7 5 B0¼ 1 1 0 2 6 4 3 7 5; C0¼1 1 0 L0¼ 2 1 0:5; D0¼0:5; J0¼0 ð71Þ
The uncertainty matrices in (10) are assumed to be
M ¼ 0:1 0:1 0:1 2 6 4 3 7 5; Mx¼ 0:5 1 0:5 2 6 4 3 7 5; My¼ 1; Mz¼2 N ¼ 0:1 0:5 0; Nx¼0:1 0 0:1; Nu¼1
and jDj 1. It is easy to verify that (E0þMDN,
A0þMxDNx) is an admissible pair. Let P ¼ Q ¼ I3 in
(11) and PD¼ I32 MD(1 2 0.04D)21N as well as
QD¼ I3. Then, the considered system S is r.s.e. to the
Three H1 optimal filters are designed by solving the convex optimisation problems listed in Remarks 1, 2 and 3, respectively, which are implemented by the MATLAB
LMI Control Toolbox [13]. Because in this example
E0r¼ PE0Q ¼ diag(1, 1, 0), the non-strict LMIs in (29), (55) and (67) are manually adjusted to strict ones when applying the MATLAB LMI Control Toolbox. The
result-ing optimal me’s are 2.7591, 2.1415 and 2.4739,
respect-ively, for the singular, third-order normal and
second-order normal filters. Obviously, in this case, normal filters have better me’s than the singular filter. Also, the third-order normal filter, having larger degrees of freedom, owns a bettermethan that for the second-order normal filter.
As a check, for the H1 optimal singular filter, the pair (Ef, Af) corresponding to U ¼ V ¼ Pcin (33) is
1 0 0 0 1 0 0 0 0 2 4 3 5; 20:338816:2319 15:910512:4829 6:903020:3279 0:6263 0:4655 3:4626 2 4 3 5 0 @ 1 A which is admissible. 5 Conclusion
The H1optimal filter design problem has been considered
for uncertain singular systems, in which uncertainties appear in all system matrices. For designing singular filters, the method proposed in Yu et al.[1] was extended, but for designing normal filters, two new methods were pro-posed, one giving higher-order filters with more degrees of freedom and one giving lower-order filters. By the numeri-cal example used to illustrate the applications of the pro-posed methods, it is seen that singular filters should not be the only choice when considering uncertain singular systems.
6 Acknowledgment
This research was supported by the National Science Council of the Republic of China under grant NSC 93-2213-E-002-020.
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