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Robust H∞ Fir Filter for Uncertain Systems
Copyright © 1996 IFAC 13th Triennial World Congress, San Francisco. USA
2d-21 4
ROBUST H= FIR FILTER FOR UNCERTAIN SYSTEMS Oh-Kyu Kwon,!, Hee-Seob Ryllj and Kyung-Sang Yooj tLaooratory Jor Control and In~tr1J.men~a.tion Sll$tem~ Dept, of Eledrical Engineering, lnha UniVf.TSi'y, Inchon 40f!~751, [(orea Td: +82-:J2-86 0-7.'/91;, Fax:+82-S2-863_1;8f!2, E-mail;oHwon950 dragon. inha. a I'.kr tDept. of Electrical Engineering, Doowon Techmcal Col/ege, Ansung 456-890, Korea
Abstract. This paper deals with the issue of the robust Hoo FfR(Finit.e Impulse Response) filtering problem for linear systems with parameter uncertainty in the state-space model. Firstly, the H 00 FIR filter is presented for the system without the uncertainty, which corresponds to the Hoo filter proprn!ed by Nag-pal and Khargonekar (1991). The impulse response of the Hoo FIR filter is calculated by the equivalence relationship between the FIR filter and the recursive filter (Kwon et al., 1990; 1994), Secondly, the robust Hoo FIR is proposed for "he linear uncertain syst.em with the param.eter uncertainty. It is also derived from the relationship between the robust Hoo FIR filter and the robust H 00 filter proposed by Xie and de Souza (1991) and Fu et al. (1992). Estima.tion performances of the Hoo FIR filters proposed are compared with two other technique/o! by Kahnan filter and the robW!,t Hoo filter (Shaked and Theodor, 1992; Fu d al., 1992) via some simulations to exemplify the superiority of the pr,)polSed method over other techniques. Key Words.
l.
IT 00 estimation; robust estimation; FIR filter; uncertain linear systems.
INTRODUCTION
Response) filter structure.
There exists a vast literature on H co estimation) see for example (Nagpal and Khargonekar, 1991; Shaked, 1990; Shaked and Theodor, 1992; Fu et al., 1992; Xie and de Souza, 1992). However, the lloo filters proposed so far are mainly rest.rided t.o time-invariant systems. Nagpal and Khargonekar (1991) have proposed the Hoo filter for linear time-varying systems, but it does not work for the systems with parameter uncertainties. Even though Xie and de Souza (1991) have proposed the robust Hoo filter for time-varying systems with parameter uncertainties) it is restricted to the filtering problem on the finite horizon [0) T] or to the periodic systems. Therefore they can not be applied to general time-varying systems on the infinite horizon [0: 00) since one of two Riccati differential equations required to solve the problem can not be computed on the infinite horizon. How can we solve the robust Hco filtering problem for time-varying systems on the infinite horizon? This is the main issue of the current paper. The ba..<;ic idea of this paper is to formulate the robust H= filtering problem on the moving hori7.0n [t - T, tJ and 1.0 adopt the FIR(Finite Impulse • Author to whom
an correspondence should be addressed.
3386
FIR filters are widely used in the signal processing area, and they were utilized in the estimation problem as the optimal FIR filters by Kwon et al. (1987; 1989; 1990). Since the optimal FIR filters use the finite observations only over a finite preceding time interval, they overcome the divergence prob:iem and have the built-in BIIlO (Bounded Input/Bounded Output) stability and the robustness to the numerical problems such as coefficient quantization errors and roundoff errors, which are well known properties of the FIR structure in signal processing area (Lim, 1985). Also note that IIR(Infinite Impulse Response) or recurSlVe filter structure (e.g., Kalman filter) rcquires the i:litial conditions on each horizon, which is an impractical assumption) but that FIR filter structure does not :t'equires the initial conditions. The optimal FIR filtcf:i arc) however J presented not in the lloo setting but in the minimum variance formulation. The H co filter proposed here is to be called as the robust Ho<. FIR filter in thc scnsc that it is an Hoo filter with the FIR structure for uncertain systems. It is noted that this filter will work for the general time-varying systems
with param eter uncertainties, and that this point will be one of the. main contrihutions of the current paper. The layout of this paper is as follows: In Section 2, the robust R oo FIR filtering problem is formulated. The Hex; FIR. filtering problem is solved in Section 3, and th e robust 1100 FIR filtering problem is solved in Section 4. Section 5 exemplifies the estimation performances of the (robust) H oo FIR filters, which are compared to those of CXi!:iLiug Hoc; filt.ers. Conclusions are summarized in Section 6.
2. PROBLEM FORMULATION AND PRELIMINARIES
y(t)
= =
+ Ll.A(t)]x(t) + R(t)lII(t) [C(t) + Ll.C(t)Jx(t) + D(t)w(t)
(2)
L(t)x(t),
(~)
[A(t)
z(t)
(1)
n
where x(t) E H is the state, w(t) E Hq is the noise which belongs to L,[O, 00), y(t) E R m is the measured output : z(l) E RP is a linear combination of the state variables to be estimated, A, B, (./, D and L are known real piecewise continuous bounded matrices that describe the nominal system and that satisfies t he <:ondition aT(t) 0 (4) D(t ) [ VT(t) = 1 '
1 [ 1
and Ll.A(t) and Ll.C(t) represent the time-va.rying parameter uncertainties . These uncert ainties are assumed to be in the: following structure:
Ll.A(t) [ Ll.C(t)
1= [ H, H 1F(t)E 1
(5)
(6) where HI, H2 , and E are known real constan t matrices with appro priat.e dimensions. Her~, t.he: ~upe rscript. 'T' denotes the transpose , I denotes the identity matrix with a.ppropriate dimension , and the notation X ~ Y (X > Y) means that X - Y is positive semidefinite ( respcdivc1y , positive definite) .
H(t,r;T)y(r)dr
L(t)E(t IT),
xr
=
Provided that there are no paramet er uu c.ertainties in t.he system, i.e., Ll.A(t) 0 and Ll.C (t) 0 for all t in the above formulation, the pNblem reduces to the Hoc FIR filtering problem, which cJrresponds to the H 00 filtermg problem of Nagpal and Khargonekar (1991) and Shaked (1990). Let us solve the H"" FIR filtering prob{em first, and then the robust H00 FIR filtering problem.
=
=
H oo FIR FILTER
3.
In this section let us comider the system without the un certainty, I. e., the syste m (1)-(3) with Ll.A(t) = 0 and .t1.C(t) = O. The necessary artd sufficient condition for the Hco FIR filter to be an unbiased estimator 1S given as follows (Kwon et al., 1987; 1990) : 4>(I,s) =
l'
H(t,7 ;1')C4>(T,s)dT,
(10)
where ~(-,.) is the transition matrix of the system matrixAof(I). Thc~
Hoo FIR filter is obtained by constructing its impulse response from that of ohe lIoo filter (Fu et al., 1992; Xie and de Souza, 1992) o n a finite moving horizon [I - '1', I).
Theorem 1 There exists a s{·iution to the H DO FIR filtering problem, if and only if th ere exists a symmetric
matrix R(t,u)
~
0 for 0:::;
(7
:~
T satisfying the Rie""ti
diDerential equation
au R(t, (7 ) = AR(t , u; + R(t , u)AT
+ BBT
(7)
+ H(t, (7) [1';-' rT .~ - c T c )H(t, u)
(8)
R(t,O) = P(t - T, 1 -1'), 0 < u :::; T.
'-T
i (t I T)
the FlR .
a
In t his paper, t he FIR filter is defined by t he form
l'
Gi1 1en the 8y,,;tem (1)-(3) and a prescribed level of noise attenuation 1"1 > 0, find arJ. es timation for z(t) of
'-T
with F( ·) : R - Rixj being an unknown matrix fUllction cmd ~atisfying
x(t IT)
Then the robust Hoo FTR filte. -ing problem is formulated as follows:
=
Let us consider uncertain linear time-varying systems with st.ate-space model of the form
i:(t)
where H(t, ·; T) is the fini te impulse response with the finite duration T. The estim {~tion error is here defined by e(t) = z(t) - i(t IT ). (9)
3387
(11)
The impuls e response of the Hoo FIR filt er ( 7) is th en del ermin ed as follows:
%17 H(t , S;,,) = [A - fl(t, ,,)CT C]11 (I , S; (7)
(12)
to assume that the initial conditions are given on each moving horizon. Even if the initial state x(t - T) is completely unknown, i.e., P(t - T , t - T) 001, the impulse response of the H 00 :?IR filter is calculated as follows:
=
O:O;T-t+s
i.~
asymptot-
Proof: If the Hoo filter of Nagpal and Khargonekar (1 991) is applied t.o t.he H ~ filtering problem on the horizon [t - T, t], it is given as follows:
:<, (1) = [A - K(I)CJx,(I) wh ere
KO
+ K(I)y(I) ,
Corollary 1 Assume that Ihe system of (I) and (2) is uniformly completely observable, and that B is uniformly bounded. Then there e:r.ists a solution to the lloo FIR filtering probh:m with uuknown initial covariancc if and only if there exist a b.nmdcd symmetric matrix S (t , T) > 0 for all T 2': to , wh ere l o is the ob .. rvabi/ity index, satisfying the Riccati d·ifferenlial equation : {)
(13)
-1~'LTL - S(I,o-)BBTS(t , ,,) (16)
POCT and PO satisfies the differential
:=
S(I, 0)
Riccati equation
P(I7)
-S(t,I7)A- ATS(t , o-) + CTC
{)o- S(t, ,,)
=
AP(,,)
PtO)
= P(t -
- -y~' LT L] Pt,,)
T, t - T),
H(t,s;T)
0 < " :0; T
= S-l(I,T)L(I,s;T),
t-T :O; s~t
a" L(t, s; 0-) = _[AT + S(I, ,,)BBTjL(t, s; 0-)
(17) (18)
L(I,s;1'-I+s) = C T , 0 :0; T-t+. < 0- :O;T. Then the estimation error ('f the Hoo FIR filter is asymptotically stable.
w(t, t - T)x , (t - T)
+
l'
1/I(t , r)K(r)y(r)dr,
(14)
'-T
wh ere W(-, .) is the transition matrix of A-K(-) C . It can be shown that 'I!(-,.) satisfieB the following relationship (Kwon, et al. , 1990; 1993; 1994):
1/I(t, r)K(r)
= H(t, r; T),
1- T:O;
r:O; I
'I!(t,r) =
(15)
From the properties (la) and (1,5) of the FIR filter, it can be shown that 1/I(t, t - T) = 0 and that (14) redu ce.. to (7). The asymptotic stability of the H 00 filter guarante~ the asy mptotic. stability of the H 00 FIR filter. Hence the proof is completed. ODD It. is n ot~d that R(i, T) is the estimation error covarian ce of the Hoo FIR filter in the worst case , i .e. ,
= E[x (t) -
< 0- :0; T .
I)
And th e t!stimation error of the Hoo filter is asymptotically sta ble. If the filter (13) is applied on the moving hori ?On [t - T , t] , t.hen t.he Hoo FIR filter is to be derived as follows:
i'(t I T)
(I
And the impulse response of tile Hoo FIR filter is determined Q,,, follows:
+ P(,,)A T + BBT
- Pt,,) [C T c
= 0,
=
Proof: Let us define S(t,o-) R - '(t,I7), and L(t,s : := S(t,I7)H(I , s;o-) . Th"n (17) i. obtained , and (18) and (16) aTe come from (12) and (11), respectively. The uniform complete c,bservability of the system and the uniform boundedness of B (and D) guarantee Sit, T) > 0 for T > and the asymptotic stability of the estimation error of the f'IF: filter (K won et al., 1993; 1994). Hence the pToof is completed. ODD
0-)
e,
Noie that the prescribed leve :~ It of the Hoc FIR filter obtained by Corollary 1 is always larger than that by Theorem 1, which means the fNmer is more conservative than the latter. It is also noted that in case the system of (1) - (3) is time-invariant , the iioo FIR filter of Corollary 1 i. always time-invariant, i.e. , H(t , s ;T) = H(! - 8;1')
and
x(t IT)
= LT H(t -
r ;T)y(r)dr.
(19)
In this case, the impulse resp ) nBe H(r ;T) of the filter
Theorem 1 requirp.B that. the initial Ht.at.p. c:ovari ance is
is calculated on the interval 0 ~ T ~ T once and for all, and the filter algorithm becomes very simple, which is one of some advantages of thl~ FIR filter (Kwon et al.,
bound ed and given. It would be) however, impractical
1987; 1989).
P(t)
i(t I T)][x(t) - i(t I T)f :0; R(t , T ).
3388
4.
ROBUST Hoo FIR FILTER
The robust 11 co FlU filtering problem is to be solved in this section. Using the matching condition (5), the parameterized system which corresponds to the system (1)-(3) is derived a.s follows (~'u et al., 1992; Xie and de Souza, 1991),
x(t) yet) z(l)
+ iJw(t) Cx(t) + Dw(t)
(20)
Lx(l),
(22)
Ax(t)
=
(21)
biJT = 0, and that the FIR filter can be applied to this system, The robust HOC) FIR filter is then obtained as follows: Theorem 2 Given a prescribed leve.l of noise attenuation f! > 0, a solution to He robust HOC) FIR filterin.q problem exists if for sornl': scalar ft > 0 the following conditions are satisfied: (1) There exists a solution Q(t, u) ? 0 to the differential Riceat; equation (23) for all u E [0, T); (b) There exist, a solution R(t, u) ::.: 0 to the differential Riccati equation
:" R(t, u)
where
= Am R(I, ut + R(I, ,,)A;';; + Bw B~
A= A+'Y,'iJiJTQ(t,O)
C=
B=[B
1"[1,1,
+R(t, u)
c + 'Y,' bi3T Q(t, 0) b=[D 'Y'II,I,
"
R(t, 0) = pet - T, I - f),
R=bb T ,
"
and Q(t,O) is the solution of the following differential Riccati equation iJ
--Q(t, u)
Du
AT Q(t, u)
Q(t,T) = 0,
o
ou H(t, s; u)
=
fX)
= Amx(t) + Bww(t) + Byy(t),
(24)
= [Am -
-j
-
f{1 t, tT)C]H(t, s; u)
T) are
(27)
O::;T-t+s
achieved for the robust HO<..> estimation problem.
In case bfF 0, the system of (20) and (21) is uncorrelated, and then the robust H 00 FIR. filter can be derived similarly as the H FlU filter by substituting B and eT k- 1 for B and eT in Theorem 1. In case, however, the system of (20) and (21) is correlated, the state equation (20) should be modified in order to apply the FIR filter to the system since it requires the system be uncorrelated. Using a technique proposed by Middleton and Goodwin (1990) and Kim (1993), the modified state equation is obtained as follows:
(26)
= L(t)i(t IT),
whcre the impulse responses H(t) '; T) and Hy(t, determined as follows:
0::; tT::; T
such that Q(l, tT) ::.: 0 for all t. And w(l) E R,+i is a noise signal which belongs to L2[O, (0), it > 0 is a scaling parameter to be chosen on each horizon [t -T, tJ, and It > 0 is the disturbance attenuation level to be
x(t)
< u::; T.
1~T [Ii(t,r;T)+ Iiy(t, r; T)ly(r)dr i(1 IT)
(23)
0
(25)
Then the robust H= FIR jilt'r for the system of (20) and (£1) is derived as
x(t IT) =
+ AQ(t, u) + '~lF E
+'Y,'Q(t, ,,)BW Q(t, u)
ht' LT L -- eT kl cl R(t, u)
By(t, s; T)
=
l'
= K(t,T-t +s)
1I(t, r; T)C
(28)
'-T
where
and ~m(-,') is the state tronsliion matrix of Am. Proof; Since (20) is equivalEnt to (24), the filter for (20) is also equivalent to (24). Applying the optimal FIR filter algorithm with the control input presented by Kwon et al. (1993) to the system of (24) and (21), the above results come directly. ODD
where
-
-
Bw = B - ByD. The modified state equation (24) is equivalent to the state equation (20). Note that there is no correlation between Bww and lJ1..V in the system (24) and (21) since
3389
Note that Theorem 2 presents only the sufficient CODditions for the solution to the robust HOC) FIR filtering problem while Theorem 1 gives the necessary and sufficient conditions. Theorem 2 also requires that the initial state covariance is bounded and given. In case the initial state x(t - T) is completely unknown, i.e., P(t - T, t - T) = CX!I, impulse responses of the robust H 00 FIR filter are obtained as follows:
Corollary 2 Assume Ihat Ihe syslem of (20) and (21) is uniformly completely observable, and that lJ is uniformly bounded. Then there. exi.~t<; a ,'iolufion to the. Hco FIR filtering problem with unknown initial state cDvariance, if the following conditions arc satisfied: (a) There exi,<;ts a ,<;oilliion Q(t, t'T) 2:: 0 to the. differential Riceali equalion (2.1) for all er E [0, T); (b) There exists a bounded symmetric matrix Set, a) > 0 for all er which satisfies the differential Riccati equation
:er 8 (t,,,)
-8(t, er)Am - A;'S(t, er)
+ eT k'e
-'i~'LTL-8(I,er)BwB~8(t,er)
8(t,0)
=[~
l'
H,
= (I,
E
= [0.5
OA].
The discrctization interval has been taken a.'1 0.01 scc, and the finite duration has been chosen as T = O.02sec for the robust H = FIR filter. The disturbance signal w(·) is generated by pas.~jng a zer(}-ffican random noise with standard deviation tJ 0.5 through a low-pass filter with normalized cutoff frequency Wn = 0.4.
(29)
= 0,
= S-'(l,T)L(I.,,;T),
H,
=
And then the .mpulse responses H(t,'; T) and Hy(l, ·;T) of Ihe robust H 00 FIR jiller (26) for the syslem are determined as follows: H(t,8;T)
The nominal system to be estimated is stable and nonminimum-phase. The parameter uncertainty is taken as follows:
1- T :0: s :0: t (30)
=
The design parameters are cl:osen as f 0.6551, l' 0.5275 for the robust Hoo filter (Fu el al., 1992), and 0.5561, 1 = 0.2176 for the robust Hoo FIR filter proposed here. The unknown function FC) is taken as F(t) 1 in this simulation to represent the model uncertainty.
,=
=
The simulation results are sho'Nn in Fig. 1, which shows the t.rajectories of the residual z(t) - i(l) of Kalman filter, the robust Hoo filter (Fu et al., 1992) and the robust H(X, FIR filter. It is shown in the simulation results that the performance of the robust H FIR filter proposed is robuster than that of Kalman fi1ter with respect to the parameter uncertainty, and that it is similar to that of the robust Hoo filter. It is, however, noted that the robust IIoo FIR filter can be applied directly to timevarying systems, but that the robust Hoo filter ofShaked and Theodor (1992) and Fu e;~ al. (1992) can not.
=
AT
~-1
L(I,s;T-t+s)=C R
,
and lly(t,·; T) IS calculated by (28). The eslimalion error of Ihe robusl Hoo FIR jiller (26) is asymptotically stable with P(t - T, t - T) 001 and T 2: R,m., where f. nm is the observability index of the pair {Am, C}.
=
Proof: Applying the optimal FIR filter algorithm with the control input presented by Kwon et al. (1993) to the system, the above results come directly. ODD
Note that Corollary 2 gives only t.he sufficient condit.ions. It. is also noted t.hat the robust Hoc FIR filter of Corollary 2 alway~ becorm~ time-invariant when the system of (20) - (22) is time-invariant, which is the same property as that of the Hoo FIR filter of Corollary 1 when t.he system of (1) - (3) is time-invariant.
5.
SIMl:LATION
In order t.o exemplify the performance of the robust H= FIR filter, it has been applied to the uncert.ain system taken by Shaked and Theodor (1992), which has the system matrices as follows:
A C
n
= [~l ~2l, B = [ = [-1 1], D = 0.1, L = [1
0].
3390
6.
CONCLUSIONS
In this paper the robust H 00 FIR filter has been proposed for linear time-varying systems with parameter uncertainty. Firstly, the H 00 FIR filter is obtained for the system without the uncertainty, and secondly the robust Hoo FIR filter for the uncertain system is derived by modifying the uncertain system. The performance of the filter proposed has been exemplified by some simulations. It is noted that the filter proposed works in general time-varying uncertai:rl systems. However, the strict performance analysis requires further research.
ACKNOWLEDGEMENT Thc~
first author acknowledge~; very helpful suggestions made by Dr. Carlos E. de Souza at Department of Electrical and Computer En~;ineering, The University of Newcastle, Australia. This work was supported in part by Inha University ReSEarch Grant, Inchon 402751, Korea in 1995.
REFERENCES
Fu, M., C.E. de Souza and L. Xie (1992). Hoo estimation for uncertain systems. Int. J. Robust and NonJincar Contro l, 2, 87- 105. Kim, J.H. (1993). Opllmal F'IR Filters for State.Space Models with Con-eiated Noises. M.E. Dissertati on, Dept . of Electrical Engineering, lnha Univ ., Jod oo, Korea. Kwon, O.K. , W .H. Kwon and K.S. Lee (1989). FIR filters and recursive forms for discrete-time state-space modell:l. Aui()matica, 25 , 715-728. Kwon, O.K., W .H. Kwon , K.S. Yoo and M..l. Kim ( 1993) . Receding huri w lI LQG controller using optimal FIR. filter with control input . Proc. of 32nd CV C, Orlando, Florida, 12n-1297. Kwoll, W .R. and O.K. Kwon (1987). FIR filters and rec ursive forms for continuous t ime-invariant statespace models. lEb'E Trans. Automatic CQntrol, 32,
O.05 --~··
~M~ ---~,oo~-~~~--~ ~' ---~~~--~--~ OD To_
352·31\6. Kwon , W .II., O.K. Kwon and K.S. Lee (1990) . Opti· mal PTR filters for time-varying state-space models. IEEE Trans. Aerospace and l!:'ledromc Syst ems, 26 , 1011·1021 Kwon, W.Il. , V.S. Suh , K.S. Lee and O.K. Kwon (1994). Equ ivalence of finite memory filters . IEEE Trans. Ae ros pace and Electronic Systems, 30 , 968-972 . eim, J .S. (1985). Fundamental of digital signal proce ... ing. In : Modern Signal Processing(T. Kailath , Ed.), 1-58) Hemisphere Publishing Co., \Vashington . Middlct on, R.H. and G .C. Goodwin (1990). Digi. tal Control and Estimation: A Unified Approach. Prentice-Hall , Inc., Englcwood Cliffs. Nagpal, K.M. and P.P. Khargonekar (1991) . Filtering a.nd smoothing in an Hoo setting. lEEE 1Tans. Automatic Control, 36 , 152-166. Shaked, U. (1990) . Hoo minimum error state estima· t.ion oflillcar stntionary processes. IEEE Trans. Automat. Cont r., 35, 554-558. Shaked, U. anti Y. Theodor (1992). H= optimal esti· mation: A tutorial. Proc. 31st IEEE CDC, Tucson, Arizoua, 2'278-2286. Xi., T.. an d C. B. de Souza (1991). Hoo filtering for linear periodic systems with parameter uncertainty. System and Con trol Letters , 17, 343-350.
3391
-0.05 - -- ,
o
'00
200
300
' 00
Time
Kalman F
I
~~ ~~------~-===:J o
Fig. I.
100
200
30CI
To"",
400
500
600
Simulation results: (a) Robust H~ FIR filter, (b) Robust H 00 filter, and (c) Kalman filter.
2d-21 4
ROBUST H= FIR FILTER FOR UNCERTAIN SYSTEMS Oh-Kyu Kwon,!, Hee-Seob Ryllj and Kyung-Sang Yooj tLaooratory Jor Control and In~tr1J.men~a.tion Sll$tem~ Dept, of Eledrical Engineering, lnha UniVf.TSi'y, Inchon 40f!~751, [(orea Td: +82-:J2-86 0-7.'/91;, Fax:+82-S2-863_1;8f!2, E-mail;oHwon950 dragon. inha. a I'.kr tDept. of Electrical Engineering, Doowon Techmcal Col/ege, Ansung 456-890, Korea
Abstract. This paper deals with the issue of the robust Hoo FfR(Finit.e Impulse Response) filtering problem for linear systems with parameter uncertainty in the state-space model. Firstly, the H 00 FIR filter is presented for the system without the uncertainty, which corresponds to the Hoo filter proprn!ed by Nag-pal and Khargonekar (1991). The impulse response of the Hoo FIR filter is calculated by the equivalence relationship between the FIR filter and the recursive filter (Kwon et al., 1990; 1994), Secondly, the robust Hoo FIR is proposed for "he linear uncertain syst.em with the param.eter uncertainty. It is also derived from the relationship between the robust Hoo FIR filter and the robust H 00 filter proposed by Xie and de Souza (1991) and Fu et al. (1992). Estima.tion performances of the Hoo FIR filters proposed are compared with two other technique/o! by Kahnan filter and the robW!,t Hoo filter (Shaked and Theodor, 1992; Fu d al., 1992) via some simulations to exemplify the superiority of the pr,)polSed method over other techniques. Key Words.
l.
IT 00 estimation; robust estimation; FIR filter; uncertain linear systems.
INTRODUCTION
Response) filter structure.
There exists a vast literature on H co estimation) see for example (Nagpal and Khargonekar, 1991; Shaked, 1990; Shaked and Theodor, 1992; Fu et al., 1992; Xie and de Souza, 1992). However, the lloo filters proposed so far are mainly rest.rided t.o time-invariant systems. Nagpal and Khargonekar (1991) have proposed the Hoo filter for linear time-varying systems, but it does not work for the systems with parameter uncertainties. Even though Xie and de Souza (1991) have proposed the robust Hoo filter for time-varying systems with parameter uncertainties) it is restricted to the filtering problem on the finite horizon [0) T] or to the periodic systems. Therefore they can not be applied to general time-varying systems on the infinite horizon [0: 00) since one of two Riccati differential equations required to solve the problem can not be computed on the infinite horizon. How can we solve the robust Hco filtering problem for time-varying systems on the infinite horizon? This is the main issue of the current paper. The ba..<;ic idea of this paper is to formulate the robust H= filtering problem on the moving hori7.0n [t - T, tJ and 1.0 adopt the FIR(Finite Impulse • Author to whom
an correspondence should be addressed.
3386
FIR filters are widely used in the signal processing area, and they were utilized in the estimation problem as the optimal FIR filters by Kwon et al. (1987; 1989; 1990). Since the optimal FIR filters use the finite observations only over a finite preceding time interval, they overcome the divergence prob:iem and have the built-in BIIlO (Bounded Input/Bounded Output) stability and the robustness to the numerical problems such as coefficient quantization errors and roundoff errors, which are well known properties of the FIR structure in signal processing area (Lim, 1985). Also note that IIR(Infinite Impulse Response) or recurSlVe filter structure (e.g., Kalman filter) rcquires the i:litial conditions on each horizon, which is an impractical assumption) but that FIR filter structure does not :t'equires the initial conditions. The optimal FIR filtcf:i arc) however J presented not in the lloo setting but in the minimum variance formulation. The H co filter proposed here is to be called as the robust Ho<. FIR filter in thc scnsc that it is an Hoo filter with the FIR structure for uncertain systems. It is noted that this filter will work for the general time-varying systems
with param eter uncertainties, and that this point will be one of the. main contrihutions of the current paper. The layout of this paper is as follows: In Section 2, the robust R oo FIR filtering problem is formulated. The Hex; FIR. filtering problem is solved in Section 3, and th e robust 1100 FIR filtering problem is solved in Section 4. Section 5 exemplifies the estimation performances of the (robust) H oo FIR filters, which are compared to those of CXi!:iLiug Hoc; filt.ers. Conclusions are summarized in Section 6.
2. PROBLEM FORMULATION AND PRELIMINARIES
y(t)
= =
+ Ll.A(t)]x(t) + R(t)lII(t) [C(t) + Ll.C(t)Jx(t) + D(t)w(t)
(2)
L(t)x(t),
(~)
[A(t)
z(t)
(1)
n
where x(t) E H is the state, w(t) E Hq is the noise which belongs to L,[O, 00), y(t) E R m is the measured output : z(l) E RP is a linear combination of the state variables to be estimated, A, B, (./, D and L are known real piecewise continuous bounded matrices that describe the nominal system and that satisfies t he <:ondition aT(t) 0 (4) D(t ) [ VT(t) = 1 '
1 [ 1
and Ll.A(t) and Ll.C(t) represent the time-va.rying parameter uncertainties . These uncert ainties are assumed to be in the: following structure:
Ll.A(t) [ Ll.C(t)
1= [ H, H 1F(t)E 1
(5)
(6) where HI, H2 , and E are known real constan t matrices with appro priat.e dimensions. Her~, t.he: ~upe rscript. 'T' denotes the transpose , I denotes the identity matrix with a.ppropriate dimension , and the notation X ~ Y (X > Y) means that X - Y is positive semidefinite ( respcdivc1y , positive definite) .
H(t,r;T)y(r)dr
L(t)E(t IT),
xr
=
Provided that there are no paramet er uu c.ertainties in t.he system, i.e., Ll.A(t) 0 and Ll.C (t) 0 for all t in the above formulation, the pNblem reduces to the Hoc FIR filtering problem, which cJrresponds to the H 00 filtermg problem of Nagpal and Khargonekar (1991) and Shaked (1990). Let us solve the H"" FIR filtering prob{em first, and then the robust H00 FIR filtering problem.
=
=
H oo FIR FILTER
3.
In this section let us comider the system without the un certainty, I. e., the syste m (1)-(3) with Ll.A(t) = 0 and .t1.C(t) = O. The necessary artd sufficient condition for the Hco FIR filter to be an unbiased estimator 1S given as follows (Kwon et al., 1987; 1990) : 4>(I,s) =
l'
H(t,7 ;1')C4>(T,s)dT,
(10)
where ~(-,.) is the transition matrix of the system matrixAof(I). Thc~
Hoo FIR filter is obtained by constructing its impulse response from that of ohe lIoo filter (Fu et al., 1992; Xie and de Souza, 1992) o n a finite moving horizon [I - '1', I).
Theorem 1 There exists a s{·iution to the H DO FIR filtering problem, if and only if th ere exists a symmetric
matrix R(t,u)
~
0 for 0:::;
(7
:~
T satisfying the Rie""ti
diDerential equation
au R(t, (7 ) = AR(t , u; + R(t , u)AT
+ BBT
(7)
+ H(t, (7) [1';-' rT .~ - c T c )H(t, u)
(8)
R(t,O) = P(t - T, 1 -1'), 0 < u :::; T.
'-T
i (t I T)
the FlR .
a
In t his paper, t he FIR filter is defined by t he form
l'
Gi1 1en the 8y,,;tem (1)-(3) and a prescribed level of noise attenuation 1"1 > 0, find arJ. es timation for z(t) of
'-T
with F( ·) : R - Rixj being an unknown matrix fUllction cmd ~atisfying
x(t IT)
Then the robust Hoo FTR filte. -ing problem is formulated as follows:
=
Let us consider uncertain linear time-varying systems with st.ate-space model of the form
i:(t)
where H(t, ·; T) is the fini te impulse response with the finite duration T. The estim {~tion error is here defined by e(t) = z(t) - i(t IT ). (9)
3387
(11)
The impuls e response of the Hoo FIR filt er ( 7) is th en del ermin ed as follows:
%17 H(t , S;,,) = [A - fl(t, ,,)CT C]11 (I , S; (7)
(12)
to assume that the initial conditions are given on each moving horizon. Even if the initial state x(t - T) is completely unknown, i.e., P(t - T , t - T) 001, the impulse response of the H 00 :?IR filter is calculated as follows:
=
O:O;T-t+s
i.~
asymptot-
Proof: If the Hoo filter of Nagpal and Khargonekar (1 991) is applied t.o t.he H ~ filtering problem on the horizon [t - T, t], it is given as follows:
:<, (1) = [A - K(I)CJx,(I) wh ere
KO
+ K(I)y(I) ,
Corollary 1 Assume that Ihe system of (I) and (2) is uniformly completely observable, and that B is uniformly bounded. Then there e:r.ists a solution to the lloo FIR filtering probh:m with uuknown initial covariancc if and only if there exist a b.nmdcd symmetric matrix S (t , T) > 0 for all T 2': to , wh ere l o is the ob .. rvabi/ity index, satisfying the Riccati d·ifferenlial equation : {)
(13)
-1~'LTL - S(I,o-)BBTS(t , ,,) (16)
POCT and PO satisfies the differential
:=
S(I, 0)
Riccati equation
P(I7)
-S(t,I7)A- ATS(t , o-) + CTC
{)o- S(t, ,,)
=
AP(,,)
PtO)
= P(t -
- -y~' LT L] Pt,,)
T, t - T),
H(t,s;T)
0 < " :0; T
= S-l(I,T)L(I,s;T),
t-T :O; s~t
a" L(t, s; 0-) = _[AT + S(I, ,,)BBTjL(t, s; 0-)
(17) (18)
L(I,s;1'-I+s) = C T , 0 :0; T-t+. < 0- :O;T. Then the estimation error ('f the Hoo FIR filter is asymptotically stable.
w(t, t - T)x , (t - T)
+
l'
1/I(t , r)K(r)y(r)dr,
(14)
'-T
wh ere W(-, .) is the transition matrix of A-K(-) C . It can be shown that 'I!(-,.) satisfieB the following relationship (Kwon, et al. , 1990; 1993; 1994):
1/I(t, r)K(r)
= H(t, r; T),
1- T:O;
r:O; I
'I!(t,r) =
(15)
From the properties (la) and (1,5) of the FIR filter, it can be shown that 1/I(t, t - T) = 0 and that (14) redu ce.. to (7). The asymptotic stability of the H 00 filter guarante~ the asy mptotic. stability of the H 00 FIR filter. Hence the proof is completed. ODD It. is n ot~d that R(i, T) is the estimation error covarian ce of the Hoo FIR filter in the worst case , i .e. ,
= E[x (t) -
< 0- :0; T .
I)
And th e t!stimation error of the Hoo filter is asymptotically sta ble. If the filter (13) is applied on the moving hori ?On [t - T , t] , t.hen t.he Hoo FIR filter is to be derived as follows:
i'(t I T)
(I
And the impulse response of tile Hoo FIR filter is determined Q,,, follows:
+ P(,,)A T + BBT
- Pt,,) [C T c
= 0,
=
Proof: Let us define S(t,o-) R - '(t,I7), and L(t,s : := S(t,I7)H(I , s;o-) . Th"n (17) i. obtained , and (18) and (16) aTe come from (12) and (11), respectively. The uniform complete c,bservability of the system and the uniform boundedness of B (and D) guarantee Sit, T) > 0 for T > and the asymptotic stability of the estimation error of the f'IF: filter (K won et al., 1993; 1994). Hence the pToof is completed. ODD
0-)
e,
Noie that the prescribed leve :~ It of the Hoc FIR filter obtained by Corollary 1 is always larger than that by Theorem 1, which means the fNmer is more conservative than the latter. It is also noted that in case the system of (1) - (3) is time-invariant , the iioo FIR filter of Corollary 1 i. always time-invariant, i.e. , H(t , s ;T) = H(! - 8;1')
and
x(t IT)
= LT H(t -
r ;T)y(r)dr.
(19)
In this case, the impulse resp ) nBe H(r ;T) of the filter
Theorem 1 requirp.B that. the initial Ht.at.p. c:ovari ance is
is calculated on the interval 0 ~ T ~ T once and for all, and the filter algorithm becomes very simple, which is one of some advantages of thl~ FIR filter (Kwon et al.,
bound ed and given. It would be) however, impractical
1987; 1989).
P(t)
i(t I T)][x(t) - i(t I T)f :0; R(t , T ).
3388
4.
ROBUST Hoo FIR FILTER
The robust 11 co FlU filtering problem is to be solved in this section. Using the matching condition (5), the parameterized system which corresponds to the system (1)-(3) is derived a.s follows (~'u et al., 1992; Xie and de Souza, 1991),
x(t) yet) z(l)
+ iJw(t) Cx(t) + Dw(t)
(20)
Lx(l),
(22)
Ax(t)
=
(21)
biJT = 0, and that the FIR filter can be applied to this system, The robust HOC) FIR filter is then obtained as follows: Theorem 2 Given a prescribed leve.l of noise attenuation f! > 0, a solution to He robust HOC) FIR filterin.q problem exists if for sornl': scalar ft > 0 the following conditions are satisfied: (1) There exists a solution Q(t, u) ? 0 to the differential Riceat; equation (23) for all u E [0, T); (b) There exist, a solution R(t, u) ::.: 0 to the differential Riccati equation
:" R(t, u)
where
= Am R(I, ut + R(I, ,,)A;';; + Bw B~
A= A+'Y,'iJiJTQ(t,O)
C=
B=[B
1"[1,1,
+R(t, u)
c + 'Y,' bi3T Q(t, 0) b=[D 'Y'II,I,
"
R(t, 0) = pet - T, I - f),
R=bb T ,
"
and Q(t,O) is the solution of the following differential Riccati equation iJ
--Q(t, u)
Du
AT Q(t, u)
Q(t,T) = 0,
o
ou H(t, s; u)
=
fX)
= Amx(t) + Bww(t) + Byy(t),
(24)
= [Am -
-j
-
f{1 t, tT)C]H(t, s; u)
T) are
(27)
O::;T-t+s
achieved for the robust HO<..> estimation problem.
In case bfF 0, the system of (20) and (21) is uncorrelated, and then the robust H 00 FIR. filter can be derived similarly as the H FlU filter by substituting B and eT k- 1 for B and eT in Theorem 1. In case, however, the system of (20) and (21) is correlated, the state equation (20) should be modified in order to apply the FIR filter to the system since it requires the system be uncorrelated. Using a technique proposed by Middleton and Goodwin (1990) and Kim (1993), the modified state equation is obtained as follows:
(26)
= L(t)i(t IT),
whcre the impulse responses H(t) '; T) and Hy(t, determined as follows:
0::; tT::; T
such that Q(l, tT) ::.: 0 for all t. And w(l) E R,+i is a noise signal which belongs to L2[O, (0), it > 0 is a scaling parameter to be chosen on each horizon [t -T, tJ, and It > 0 is the disturbance attenuation level to be
x(t)
< u::; T.
1~T [Ii(t,r;T)+ Iiy(t, r; T)ly(r)dr i(1 IT)
(23)
0
(25)
Then the robust H= FIR jilt'r for the system of (20) and (£1) is derived as
x(t IT) =
+ AQ(t, u) + '~lF E
+'Y,'Q(t, ,,)BW Q(t, u)
ht' LT L -- eT kl cl R(t, u)
By(t, s; T)
=
l'
= K(t,T-t +s)
1I(t, r; T)C
(28)
'-T
where
and ~m(-,') is the state tronsliion matrix of Am. Proof; Since (20) is equivalEnt to (24), the filter for (20) is also equivalent to (24). Applying the optimal FIR filter algorithm with the control input presented by Kwon et al. (1993) to the system of (24) and (21), the above results come directly. ODD
where
-
-
Bw = B - ByD. The modified state equation (24) is equivalent to the state equation (20). Note that there is no correlation between Bww and lJ1..V in the system (24) and (21) since
3389
Note that Theorem 2 presents only the sufficient CODditions for the solution to the robust HOC) FIR filtering problem while Theorem 1 gives the necessary and sufficient conditions. Theorem 2 also requires that the initial state covariance is bounded and given. In case the initial state x(t - T) is completely unknown, i.e., P(t - T, t - T) = CX!I, impulse responses of the robust H 00 FIR filter are obtained as follows:
Corollary 2 Assume Ihat Ihe syslem of (20) and (21) is uniformly completely observable, and that lJ is uniformly bounded. Then there. exi.~t<; a ,'iolufion to the. Hco FIR filtering problem with unknown initial state cDvariance, if the following conditions arc satisfied: (a) There exi,<;ts a ,<;oilliion Q(t, t'T) 2:: 0 to the. differential Riceali equalion (2.1) for all er E [0, T); (b) There exists a bounded symmetric matrix Set, a) > 0 for all er which satisfies the differential Riccati equation
:er 8 (t,,,)
-8(t, er)Am - A;'S(t, er)
+ eT k'e
-'i~'LTL-8(I,er)BwB~8(t,er)
8(t,0)
=[~
l'
H,
= (I,
E
= [0.5
OA].
The discrctization interval has been taken a.'1 0.01 scc, and the finite duration has been chosen as T = O.02sec for the robust H = FIR filter. The disturbance signal w(·) is generated by pas.~jng a zer(}-ffican random noise with standard deviation tJ 0.5 through a low-pass filter with normalized cutoff frequency Wn = 0.4.
(29)
= 0,
= S-'(l,T)L(I.,,;T),
H,
=
And then the .mpulse responses H(t,'; T) and Hy(l, ·;T) of Ihe robust H 00 FIR jiller (26) for the syslem are determined as follows: H(t,8;T)
The nominal system to be estimated is stable and nonminimum-phase. The parameter uncertainty is taken as follows:
1- T :0: s :0: t (30)
=
The design parameters are cl:osen as f 0.6551, l' 0.5275 for the robust Hoo filter (Fu el al., 1992), and 0.5561, 1 = 0.2176 for the robust Hoo FIR filter proposed here. The unknown function FC) is taken as F(t) 1 in this simulation to represent the model uncertainty.
,=
=
The simulation results are sho'Nn in Fig. 1, which shows the t.rajectories of the residual z(t) - i(l) of Kalman filter, the robust Hoo filter (Fu et al., 1992) and the robust H(X, FIR filter. It is shown in the simulation results that the performance of the robust H FIR filter proposed is robuster than that of Kalman fi1ter with respect to the parameter uncertainty, and that it is similar to that of the robust Hoo filter. It is, however, noted that the robust IIoo FIR filter can be applied directly to timevarying systems, but that the robust Hoo filter ofShaked and Theodor (1992) and Fu e;~ al. (1992) can not.
=
AT
~-1
L(I,s;T-t+s)=C R
,
and lly(t,·; T) IS calculated by (28). The eslimalion error of Ihe robusl Hoo FIR jiller (26) is asymptotically stable with P(t - T, t - T) 001 and T 2: R,m., where f. nm is the observability index of the pair {Am, C}.
=
Proof: Applying the optimal FIR filter algorithm with the control input presented by Kwon et al. (1993) to the system, the above results come directly. ODD
Note that Corollary 2 gives only t.he sufficient condit.ions. It. is also noted t.hat the robust Hoc FIR filter of Corollary 2 alway~ becorm~ time-invariant when the system of (20) - (22) is time-invariant, which is the same property as that of the Hoo FIR filter of Corollary 1 when t.he system of (1) - (3) is time-invariant.
5.
SIMl:LATION
In order t.o exemplify the performance of the robust H= FIR filter, it has been applied to the uncert.ain system taken by Shaked and Theodor (1992), which has the system matrices as follows:
A C
n
= [~l ~2l, B = [ = [-1 1], D = 0.1, L = [1
0].
3390
6.
CONCLUSIONS
In this paper the robust H 00 FIR filter has been proposed for linear time-varying systems with parameter uncertainty. Firstly, the H 00 FIR filter is obtained for the system without the uncertainty, and secondly the robust Hoo FIR filter for the uncertain system is derived by modifying the uncertain system. The performance of the filter proposed has been exemplified by some simulations. It is noted that the filter proposed works in general time-varying uncertai:rl systems. However, the strict performance analysis requires further research.
ACKNOWLEDGEMENT Thc~
first author acknowledge~; very helpful suggestions made by Dr. Carlos E. de Souza at Department of Electrical and Computer En~;ineering, The University of Newcastle, Australia. This work was supported in part by Inha University ReSEarch Grant, Inchon 402751, Korea in 1995.
REFERENCES
Fu, M., C.E. de Souza and L. Xie (1992). Hoo estimation for uncertain systems. Int. J. Robust and NonJincar Contro l, 2, 87- 105. Kim, J.H. (1993). Opllmal F'IR Filters for State.Space Models with Con-eiated Noises. M.E. Dissertati on, Dept . of Electrical Engineering, lnha Univ ., Jod oo, Korea. Kwon, O.K. , W .H. Kwon and K.S. Lee (1989). FIR filters and recursive forms for discrete-time state-space modell:l. Aui()matica, 25 , 715-728. Kwon, O.K., W .H. Kwon , K.S. Yoo and M..l. Kim ( 1993) . Receding huri w lI LQG controller using optimal FIR. filter with control input . Proc. of 32nd CV C, Orlando, Florida, 12n-1297. Kwoll, W .R. and O.K. Kwon (1987). FIR filters and rec ursive forms for continuous t ime-invariant statespace models. lEb'E Trans. Automatic CQntrol, 32,
O.05 --~··
~M~ ---~,oo~-~~~--~ ~' ---~~~--~--~ OD To_
352·31\6. Kwon , W .II., O.K. Kwon and K.S. Lee (1990) . Opti· mal PTR filters for time-varying state-space models. IEEE Trans. Aerospace and l!:'ledromc Syst ems, 26 , 1011·1021 Kwon, W.Il. , V.S. Suh , K.S. Lee and O.K. Kwon (1994). Equ ivalence of finite memory filters . IEEE Trans. Ae ros pace and Electronic Systems, 30 , 968-972 . eim, J .S. (1985). Fundamental of digital signal proce ... ing. In : Modern Signal Processing(T. Kailath , Ed.), 1-58) Hemisphere Publishing Co., \Vashington . Middlct on, R.H. and G .C. Goodwin (1990). Digi. tal Control and Estimation: A Unified Approach. Prentice-Hall , Inc., Englcwood Cliffs. Nagpal, K.M. and P.P. Khargonekar (1991) . Filtering a.nd smoothing in an Hoo setting. lEEE 1Tans. Automatic Control, 36 , 152-166. Shaked, U. (1990) . Hoo minimum error state estima· t.ion oflillcar stntionary processes. IEEE Trans. Automat. Cont r., 35, 554-558. Shaked, U. anti Y. Theodor (1992). H= optimal esti· mation: A tutorial. Proc. 31st IEEE CDC, Tucson, Arizoua, 2'278-2286. Xi., T.. an d C. B. de Souza (1991). Hoo filtering for linear periodic systems with parameter uncertainty. System and Con trol Letters , 17, 343-350.
3391
-0.05 - -- ,
o
'00
200
300
' 00
Time
Kalman F
I
~~ ~~------~-===:J o
Fig. I.
100
200
30CI
To"",
400
500
600
Simulation results: (a) Robust H~ FIR filter, (b) Robust H 00 filter, and (c) Kalman filter.