- Email: [email protected]
Robust H∞ Filtering for Linear Continuous-Time Uncertain Systems With Multiple Delays: An LMI Approach
Copyright ® IF AC Robust Control Design. Prague. Czech Republic. 2000
ROBUST Ho< FILTERING FOR LINEAR CONTINUOUS·TIME UNCERTAIN SYSTEMS WITH MULTIPLE DELAYS: AN LMI APPROACH
Reinaldo M. Palhares •. 1 Pedro L. D. Peres •• ,2 Carlos E. de Souza •••. 3
• Pontifical Catholic University of Minas Gerais, Graduate Program in Electrical Engineering, Av. Dom Jose Gaspar 500, Predio 4. 30535-610 Belo Horizonte, MG, Brazil. EmaiI.·[email protected] •• School of Electrical and Computer Engineering, University of Campinas, CP 6101, 13081-970 Campinas, Sp, Brazil. Email.· [email protected] ••• Department of Systems and Control, lAboratorio Nacional de ComputlJfiio Cient(fica -LNCC, Av. Genilio Vargas 333,25651-070 Petr6poIis, RJ, Brazil. Email: [email protected]
Abstract: The problem of robust JL filter design for uncertain continuous-time linear systems with multiple time-delayed states is addressed in this paper. The uncertainties are assumed to belong to convex bounded domains (polytope type uncertainty) and the time delays are supposed to be constant. Delay-independent as well as delay-dependent stability conditions assuring robust stability and a prescribed JL disturbance attenuation for the filtering error system are established. in both cases. in tenns of linear matrix inequalities. which can be efficiently solved by standard optimization procedures with global convergence assured. Two illustrative examples are analyzed. Copyright © 2000 lFAC Keywords: Time-delay systems. Delay-dependent stability. Delay-inde~ndent stability. Robust JL filtering, Linear matrix inequalities.
1. INTRODUCflON
ness against unmodeled dynamics in both control and filtering designs.
The literature on JL filtering design for linear continuous-time systems is very extensive nowadays; see, e.g., (Grigoriadis and Watson, 1997), (park and Kailath, 1997), (palhares and Peres, 1999) and the references therein. As is well known, the !JL criterion is largely used to deal with noise sources considered as arbitrary signals with bounded energy. providing robust-
I
2
Supponed in pan by FAPEMIG. TECJ027198 - Brazil. Supponed in pan by CNPq (304604189-5/PQ) and CAPES -
Brazil. 3
Supponed in pan by CNPq (301653/96-8/PQ) - Brazil.
249
The problem of robust !JL filter design. that is. the problem of finding an estimate of a linear combination of the states of an uncertain system through the design of a linear stable filter which assures a guaranteed !JL attenuation level to the filtering error system has been addressed more recently by many authors. When norm-bounded uncertainties are considered. sufficient conditions for the existence of a robust !JL filter are provided in de Souza et al. (1995), based on Riccati algebraic equations and in Li and Fu (1997) in tenns of Linear Matrix Inequalities (LMIs). For systems with polytope type uncertainty. very recent results based on
LMls can be found in Geromel and de Oliveira (1998) and Palhares and Peres (1999).
q
i(t)
= Aox(t) + 2, Ajx(t -
'tj) + Bw(t)
(1)
j=!
However, these approaches do not take into account the existence of time-delays. As is well known, timedelays are frequently encountered in engineering systems, being a common source of instability and poor performance (Dugard and Verriest, 1997). The literature on JL filter design for continuous-time systems with time-delays presents some recent results, mainly based on the Riccati algebraic equation, which could be divided into two classes: delay-dependent (Fattouh et. al., 1998), (Pila et aI., 1999) and delay-independent conditions (Shi, 1998) (see de Souza and Li (1999), Kolmanovskii et al. (1999) and references therein for details concerning delay-independent and delaydependent stability conditions). In Shi (1998), normbounded uncertainties are also taken into account. As discussed in de Souza and Li (1999), JL methods based on delay-independent conditions (i.e. the timedelays are allowed to be arbitrarily large) can be somehow conservative, especially in situations where the existing time-delays are small. In this sense, a delaydependent filtering design, i.e., which takes into account information on the delay size, constitutes an alternative to provide less conservative results. On the other hand, the strategies based on delay-dependent conditions show to be very conservative whenever systems allowing unlimited size delays are considered.
= Cx(t) + Dw(t) z(t) = Lx(t) + Tw(t)
(3)
x(t) = (t), 'Vt E [-'t,0),
(4)
y(t)
(2)
where x(t) E lR" is the state vector, y(t) E 1R' is the measurements vector, w(t) E IRm is the noise signal vector (including process and measurement noises), z(t) E ]RP is the signal to be estimated, (t) is a given initial vector function which is continuous on the segment [-'t,O], and 'tAt), j = 1, ... ,q, are constant bounded time-delays. The system matri~es are assumed to be unknown (uncertain) but belonging to a known convex compact set of poly topic type, namely (Ao, . .. ,Aq,B,C,D,L, T) E '1J
(5)
where '1J is a given convex bounded polyhedral domain described by f. vertices as follows: '1J ~ { (Ao, . .. , T) I (Ao, ... , T)
l
= 2,
(Xi
(AOi, . . . ,Aqi,
i=!
l
Bi ,Ci ,Di,Li,7;);
The design of robust JL filters for continuous-time linear systems with polytope type uncertainty and multiple constant state delays is investigated in this paper. To the authors' knowledge, polytope type uncertainties have not been investigated in the context of robust JL filtering design for systems with delays and, in the case of delay-dependent conditions, only precisely known systems have been addressed.
(Xi
2: 0;
2, (Xi =
I}
(6)
i=!
This kind of convex bounded parameter uncertainty has been widely used in the context of robust control (see, for instance, (Palhares et aI. , 1997) and references therein). In this paper, the following notion of robust stability for uncertain state-delayed systems is used (de Souza and Li, 1999):
The aim is to obtain a full-order stable linear filter assuring robust stability and a guaranteed JL performance for the filtering error system. For that, delaydependent as well as delay-independent sufficient conditions are established in terms of LMIs, allowing the problem to be solved by means of efficient numerical methods with global convergence assured in both cases.
Definition 1. The system (1)-(6) with w(t) == 0, is said to be robustly stable if the trivial solution x(t) == 0 of the functional differential equation (1) is globally uniformly asymptotically stable for all admissible uncertainties.
The robust filtering problem addressed in this paper consists on obtaining an estimate z(t) of the signal z(t) such that a guaranteed performance criterion is minimized in a filtering error sense. Attention is focused on the design of a linear time-invariant, asymptotically stable filter of order nf = n with state-space realization of the form:
Throughout the paper, the boldface characters I and o stand for the identity and the null matrices of convenient sizes, respectively. ~ denotes the space of square integrable vector functions on [0,(0) with norm 1I·lb ~ (Io-II· Wdt)!/2, where 11 ·11 denotes the Euclidean vector norm. For a symmetric block matrix, the symbol * denotes the symmetric blocks outside the main diagonal block.
i(t) = Afx(t) + Bfy(t) , z(t)
x(O) =0
=Cfx(t) + Dfy(t).
(7) (8)
2. PROBLEM FORMULATION Defining the augmented state i(t) ~ [x(t) the filtering error dynamics is given by
Consider the following linear system with multiple delayed states: 250
.i'(t)]',
q
i(t)
= Aoi(t) + LAjEi(t-'tj) + Hw(t)
j=l z(t) =Ci(t) + Dw(t) i(t) = [CP'(t)
(10)
~tE[-'t,O)
0]',
then the filtering error system is robustly stable with y noise attenuation level.
(9)
Based on the above delay-independent robust stability criterion with y noise attenuation level , the following theorem provides a sufficient condition for the existence of a robust linear stable filter assuring an 1L bound for uncertain systems with multiple state delays.
(11)
where the filtering error signal is denoted by z(t) ~ z(t) -z(t) and
Ao =
[~~
:j]' A
=
j
[~] , j = 1, ... , q
- [B] B= BjD ,C=[L-DjC
(12)
Theorem 1. Let y > 0 be a given scalar. If there exist matrices R = R' E IRnxn positive definite, S = Si E llM X n , P j-rj - nJ E TlIln x n , j• - 1, ... ,q, M E TlIln xn N E l'"l'"l'"-,
(13)
-Cj]
D=T-DjD,E=[I 0]
IRP x n , Z E IRn x r and Dj E
(14)
[ ~; ~;
e,
where
(P,;,) Robust 1L filtering problem with multiple time-delayed states: Given a scalar y > 0, determine a stable linear filter of the form (7)-(8) such that the filtering error system (9)-( JJ) is robustly stable and ensures a prescribed level y of 1L noise attenuation, namely, under zero-initial conditions and for any non-zero wE L2 ~ (Ao , .. . , T) E 'lJ
~i=
1, . ..
,e
(16)
'14;
S-R>O
The filtering problem addressed in this paper is as follows:
IIzl12 :::; 'YIlwIl2,
satisfying
~il <0,
1Ui
Throughout the paper, Ao; , Aj;, H;, C; and D;, i = 1, . . . , denote the matrices Ao, Aj, H, C and D evaluated at each of the vertices of the polytope 'lJ
JRP x r
(17)
'TJ and '14; are defined in Lemma 1, and
1U . = [AOiR + RAOi + I
±
j=l
Pj
* RAOi + AOiS + C;Z' + M' + L Pj q
j
AOiS + SAoi + ZCi + C;Z' + ji Pj j=l
(15)
In this situation, the filtering error system is said to be robustly stable with y noise attenuation level.
!RJi
= [RAli SAli RBi
'R.3i = [ SBi + ZDi
3. DELAY-INDEPENDENT ROBUST 1L FILTERING SYNTHESIS
... ...
RA.] SA;;
L~-CD' I I j
-NI]
L;-C;DJ
then the problem (P,;,) is solvable independent of delays. Moreover, under the above conditions, the transfer function matrix of a suitable robust filter is given by
Before presenting the robust filtering synthesis, a delay-independent sufficient condition for the robust stability for the filtering error system with time delays (9)-(11) is established. For the sake of space limitation, all the proofs are omitted.
with
Lemma 1. Consider the filtering error system (9)(11) and let y > 0 be a given scalar. If there exist symmetric positive definite matrices P E ]R2n x 2n and Pj E IRnxn , j = 1, ... ,q, such that
AO;P + PAo; + ~ E' PjE
'li;
%.;]
7;'i
'TJ
o
tJ:f.i
0
[ ~i
With the results of Theorem 1, the problem (p,;,) can be solved (independent of delay) by testing the feasibility of the LMIs (16)-(17). The minimum y attenuation level can be obtained by means of the following LMI optimization procedure:
<0
'14;
min
= 1, ... , e, where 'lii=[PAli rr _ .L3 -
R,s,Pj,M,N ,z,D,,fJ
...
d·Jag { - Pl,···, -
Cn
PAqi], '12i=[PH;
D} ,
rq
rr.. _ L41 -
[-rD;
I
0
subjectto (16) and (17)
_iY;I ]
whereo~r . 251
4. DELAY-DEPENDENT ROBUST JL FILTERING SYNTHESIS
Q4j E lRnxn. and matrices M E lRnxr andDf E JRPxr satisfying
Firstly, a delay-dependent bounded real criterion for the robust stability of the uncertain filtering error system with multiple time-delays is presented in this section. Similar results can be found in Cao et al. (1998) and de Souza and Li (1999). Lemma 2. Consider the filtering error system (9)(11) with (5)-(6) and let y > 0 be a given scalar. If there exist symmetric positive definite matrices P, Pj, Pj~ E lll>2n x 2n .~r ~ 1 + q. Pjl,"" Pjq E lI" lll>nxn , j. - 1, ... ,q. lI" satisfying pt~
Jl;
R;
'tl r/i
'tqrq;
R: tiP
-yZl
D;
-I
0 0
0 0
'tl r'/i
0
0
-'tI'l'1
'tq~;
0
0 'tl R;
0 0
0
'tqR;
0
1
D';
0
A2;
A3;
'tIL/i
A~; A~;
-yZl
D;
D: -I
0 0
'tIL'li
0
0
-'tln l
'tqL~;
0
0
't1~1;
0 0
0
'tq/!.q;
0
IRPxn,
ZE
'tqLq; 0 0
0
0 -'tqnq 0 0
0
0
'tl~;i
'tq~~i
0
0
o
o
<0
(20)
0
S-R>O
0 -, 'tqB; 0
0 'tl R: 0
A/i
0 -'tq'I'q 0
NE
lRnxn .
0
(21)
'V i = 1, ... , f. where
D; =
<0
T; - DfD;; AIi =
[:Ii =-2;
31i = AOiS+ SAo; + (fAji)' S )=1
+ S (tAj;)
'Vi = 1, ... ,f, where E is defined in (14) and Jl;
= Ao;P+PAo;+ (fAj;E) P+P (tEAj;)' J=I
rj;
= [Aj;EPj
Aj;Pjl
PA o; PE'A'liE'
+ZCi+C;Z';
J=I
32i
J=I
= RAo;+AoiS + R (fAj;) J=I
. . . Aj;Pjq Aj;EPj~ + (fAji)' S+C;Z' +M';
... PE'A~;E'l
)=1
'I'j=diag{Pj,Pj l, "',Pjq, Pj~, Pj, Pjl, ... , Pjq} 33i
then the uncertain filtering error system is robustly stable with y noise attenuation level.
= Ao;R+RAoi+ (tAji)' R+R (tAji); )=1
\)=1
A . - [SB;+ZD;] . A.- [ 21 -
Considering Lemma 2. the next theorem provides a sufficient condition for the existence of a linear stable filter solving problem (P!).
RB;
noj;
Theorem 2. Consider the system (1 )-( 6) and let y > 0 be a given scalar. If there exist symmetric matrices R E ]R"xn positive definite, S E lRnxn. Ylj E ]R"xn, Y2j E ]R"xn, Wlj E lRnxn. W2j E IRnxn. Gjk E lRnxn , k = l, . .. ,q, Qlj E lRnxn. Q2j E lRnxn , Q3j E IRnxn ,
njO; =
252
,3. -
Ojli
'"
L;-C';D'f
].
L!; - C';D'f - N' ,
njq;);
Oj~i = [:~: ~];
[
S]
QI J' +Q2 ' S J
Wlj
, Gjl , ... , Gjq
} ;
~ ' i = [SBi+ZDi] . e . = [Q3 j +Q4j fljBi'
J
O" r---~--~--"""'-~--""---"""
S] ;
S
J
Figure 1 (time delays considered as 'tl = 0.4 and 't2 = 0.5) and Figure 2 ('tl = 5, 't2 = 25) show the maximum singular value diagrams for the (delay-independent) robust filter (25) connected to the uncertain system given by (22)-(24) at the 4 vertices of the uncertain domain 'D. The effectiveness of the '1 = 0.7843 guaranteed bound obtained from the delay-independent approach is apparent in both different situations. y" = 0.7843
flj
OJ
0 .7
for j ,k = I , . .. ,q, then the problem (P!) is solvable. Moreover, under the above conditions, a suitable filter is given by (18)-(19).
..,'" "
~
0.'
la
-=ccc: Vi
With the results of Theorem 2, the problem (P!) can be solved by testing the feasibility of the LMls (20)-(21). The minimum y attenuation level can be obtained by means of the following LMI optimization procedure:
0.'
0.'
0.2
0.'
0 10"
10"
v
min
.0'
.0'
.0'
.0'
Frequency (rad/s)
S , R , Yl j ,Y2j , Wlj , W2j G jk,Ql j, Q2j , Q3} ,Q4j ,M ,N ,Z , D,
Fig. 1. Singular value diagram (4 vertices) for the robust filter (25) connected to the uncertain system (22)-(24) with time delays 'tl = 0.4 and 't2 = 0.5.
subjectto (20) - (21) where v g,
0.'
r. O .'r---~--~--"""'----""---"""
y" = 0.7843 0.'
5. EXAMPLES As the first example, consider the uncertain continuoustime delay system given by (1)-(6) with q = 2 delays and
Ao = [~3 A2
2 _4 +P]'
= [-~.2
C = [I
AI = [~~il -0.~+4>] (22)
-0~31+ 4>]'
0], D
= I,
B=
L = [I
[~]
(23)
2]
(24)
Frequency (rad/s)
The uncertain parameters satisfy Ip I ~ 2 and 14> I ~ 0.1, defining a 4 vertices uncertain domain.
Fig. 2. Singular value diagram (4 vertices) for the robust filter (25) connected to the uncertain system (22)-(24) with time delays 'tl 5 and't2 25.
=
Using the delay-independent approach presented in Theorem 1, one can obtain the following JL filtering noise attenuation level '1 = 0.7843 with A = [-0.3293 J -5.2502 CJ
= [0.9904
1.2408] B -4.0512 ' J 1.9854]' DJ
For this system, with 'tl = 0.4 and 't2 = 0.5, the delay-dependent approach from Theorem 2 yields the JL guaranteed filtering cost given by 'ft 1=0.4 t2=0.5 = 0.9725 and the associate filter matrices
= [0.3649] 2.1083
= 0.0078
=
(25)
-0.6454
= [ -6.2433 CJ = [0.6184
AJ
Actually, since the delay-independent approach yields a feasible solution, one can conclude that the system (22)-(24) is delay-independent, that is, stability and guaranteed JL cost are both assured for any delay values.
0.9559] B -4.6511 ' J
= [0.4497] 2.2446
1.4732], DJ = 0.2556
(26)
Smaller JL guaranteed costs can be obtained if 'tl and 't2 assume very small values; for instance. with 253
=
=
terns with time-varying delays. lEE Proceedings - Control Theory and Applications 145(3), 338344. de Souza, C. E. and X. Li (1999). Delay-dependent robust J£,., control of uncertain linear state-delayed systems. Automatica, 35(7), 1313-1321. de Souza, C. E., U. Shaked and M. Fu (1995). Robust J£,., filtering for continuous time varying
'tJ 0.1 and't2 0.2, the delay-dependent approach from Theorem 2 yields the JL guaranteed filtering 0.4126. As a general cost given by Yt1=O.1 T2=O.2 rule, y increases as the delays increase.
=
Feasible solutions are obtained through the delaydependent approach, with higher JL guaranteed costs, till the limiting situations: 'tl = 0, 't2 ~ 1.8184; 't2 = 0, 'tl ~ 1.6177 and 'tl = 't2 ~ 0.92. To conclude, one may say that the delay dependent approach tends to produce conservative results whenever the system presents a delay-independent behavior.
uncertain systems with deterministic input signals. IEEE Transactions on Signal Processing,
43(3),709-719. Dugard, L. and E. I. Verriest (1997). Stability and Control of Time-delay Systems. Springer-Verlag, Berlin, Germany Fattouh, A., O. Sename and 1. M. Dion (1998). JL observer design for time-delay systems. Proceed-
As the second example, consider the uncertain continuous-time delay system given by (1)-(6) with q 2 delays and
=
Ao
= [~3
2 _4 +P]' Al
A2 = [-g.9 C=[1
0]'
= [~~85
_0~56+4>]' D=I,
ings of the 37th , IEEE Conference on Decision and Control, December, Tampa, FL, 4545-4546. Geromel, 1. c., and M . C. de Oliveira (1998). :J6 and J£,., robust filtering for convex bounded uncertain systems. Proceedings of the 37th IEEE Conference on Decision and Contro, December, Tampa, FL, 146-151. Grigoriadis, K. M . and J. T. Watson (1997). Reducedorder J£,., and L2-I- filtering via linear matrix inequalities. IEEE Transactions on Aerospace and Electronic Systems 33(4), 1326-1338. Kolmanovskii, V. B. , and S. I. Niculescu and J. P. Richard (1999). On the Liapunov-Krasovskii
0 _1 +4>] (27)
B=
[~]
(28)
L=[l
2]
(29)
The uncertain parameters are such that Ipl :S 2 and -0.6 :S 4> :S 0.4, defining a 4 vertices uncertain domain . For this system, the delay-independent approach fails to find a feasible filter.
functionals for stability analysis of linear delay systems. International Journal of Control
However, the delay-dependent approach (Theorem 2) achieves, for 'tl 0.1 and 't2 0.2, the guaranteed JL filtering cost Yt1=O.1 T2=D.2 = 0.9257 and the filter matrices
=
A f
= [-1.5067
Cf
-8.3521
= [0.7318
=
0.3429] -8.6605' 1.5881],
B f
72(4),374-384. Li, H . and M. Fu (1997). A linear matrix inequality approach to robust JL filtering. IEEE Transactions
= [1.0583]
on Signal Processing 45(9), 2338-2350.
3.8307
Df=0.1831
Palhares, R. M. and P. L. D. Peres (1999). Robust JL filtering design with pole placement constraint via LMIs. Journal oJ Optimization Theory and
(30)
Applications 102(2), 239-261 .
The delay-dependent approach provides a feasible solution until values of delay around 'tl = 0 and 't2 ~ 0.644; or 'tl ~ 0.3671 and't2 0, or'tl 't2 ~ 0.24, but the associated filter matrices A f and Bf exhibit very large norms.
=
Palhares, R. M., R. H. C. Takahashi and P. L. D. Peres (1997). JL and :J:h. guaranteed costs computation for uncertain linear systems. International Jour-
=
nal of Systems Science 28(2), 183-188. Park, P. and T. Kailath (1997). JL filtering via convex optimization. International Journal of Control
66( 1), 15-22.
6. CONCLUSION
Pila, A. w., U. Shaked and C. E. de Souza (1999). JL filtering for continuous-time linear systems with delay. IEEE Transactions on Automatic Control
Delay-independent as well as delay-dependent LMI conditions assuring the existence of a robust JL filter for uncertain linear continuous-time systems with multiple time-delays and convex bounded uncertainties have been proposed in this paper. With these conditions, the robust filtering design can be performed through standard convex programming tools with global convergence assured.
44(7),1412-1417. Shi, P. (1998). Filtering on sampled-data systems with parametric uncertainty. IEEE Transactions
on Automatic Control, 43(7), 1022-1027.
7. REFERENCES Cao, Y. Y., Y. X. Sun and J. Lam (1998). Delaydependent robust JL control for uncertain sys254
ROBUST Ho< FILTERING FOR LINEAR CONTINUOUS·TIME UNCERTAIN SYSTEMS WITH MULTIPLE DELAYS: AN LMI APPROACH
Reinaldo M. Palhares •. 1 Pedro L. D. Peres •• ,2 Carlos E. de Souza •••. 3
• Pontifical Catholic University of Minas Gerais, Graduate Program in Electrical Engineering, Av. Dom Jose Gaspar 500, Predio 4. 30535-610 Belo Horizonte, MG, Brazil. EmaiI.·[email protected] •• School of Electrical and Computer Engineering, University of Campinas, CP 6101, 13081-970 Campinas, Sp, Brazil. Email.· [email protected] ••• Department of Systems and Control, lAboratorio Nacional de ComputlJfiio Cient(fica -LNCC, Av. Genilio Vargas 333,25651-070 Petr6poIis, RJ, Brazil. Email: [email protected]
Abstract: The problem of robust JL filter design for uncertain continuous-time linear systems with multiple time-delayed states is addressed in this paper. The uncertainties are assumed to belong to convex bounded domains (polytope type uncertainty) and the time delays are supposed to be constant. Delay-independent as well as delay-dependent stability conditions assuring robust stability and a prescribed JL disturbance attenuation for the filtering error system are established. in both cases. in tenns of linear matrix inequalities. which can be efficiently solved by standard optimization procedures with global convergence assured. Two illustrative examples are analyzed. Copyright © 2000 lFAC Keywords: Time-delay systems. Delay-dependent stability. Delay-inde~ndent stability. Robust JL filtering, Linear matrix inequalities.
1. INTRODUCflON
ness against unmodeled dynamics in both control and filtering designs.
The literature on JL filtering design for linear continuous-time systems is very extensive nowadays; see, e.g., (Grigoriadis and Watson, 1997), (park and Kailath, 1997), (palhares and Peres, 1999) and the references therein. As is well known, the !JL criterion is largely used to deal with noise sources considered as arbitrary signals with bounded energy. providing robust-
I
2
Supponed in pan by FAPEMIG. TECJ027198 - Brazil. Supponed in pan by CNPq (304604189-5/PQ) and CAPES -
Brazil. 3
Supponed in pan by CNPq (301653/96-8/PQ) - Brazil.
249
The problem of robust !JL filter design. that is. the problem of finding an estimate of a linear combination of the states of an uncertain system through the design of a linear stable filter which assures a guaranteed !JL attenuation level to the filtering error system has been addressed more recently by many authors. When norm-bounded uncertainties are considered. sufficient conditions for the existence of a robust !JL filter are provided in de Souza et al. (1995), based on Riccati algebraic equations and in Li and Fu (1997) in tenns of Linear Matrix Inequalities (LMIs). For systems with polytope type uncertainty. very recent results based on
LMls can be found in Geromel and de Oliveira (1998) and Palhares and Peres (1999).
q
i(t)
= Aox(t) + 2, Ajx(t -
'tj) + Bw(t)
(1)
j=!
However, these approaches do not take into account the existence of time-delays. As is well known, timedelays are frequently encountered in engineering systems, being a common source of instability and poor performance (Dugard and Verriest, 1997). The literature on JL filter design for continuous-time systems with time-delays presents some recent results, mainly based on the Riccati algebraic equation, which could be divided into two classes: delay-dependent (Fattouh et. al., 1998), (Pila et aI., 1999) and delay-independent conditions (Shi, 1998) (see de Souza and Li (1999), Kolmanovskii et al. (1999) and references therein for details concerning delay-independent and delaydependent stability conditions). In Shi (1998), normbounded uncertainties are also taken into account. As discussed in de Souza and Li (1999), JL methods based on delay-independent conditions (i.e. the timedelays are allowed to be arbitrarily large) can be somehow conservative, especially in situations where the existing time-delays are small. In this sense, a delaydependent filtering design, i.e., which takes into account information on the delay size, constitutes an alternative to provide less conservative results. On the other hand, the strategies based on delay-dependent conditions show to be very conservative whenever systems allowing unlimited size delays are considered.
= Cx(t) + Dw(t) z(t) = Lx(t) + Tw(t)
(3)
x(t) = (t), 'Vt E [-'t,0),
(4)
y(t)
(2)
where x(t) E lR" is the state vector, y(t) E 1R' is the measurements vector, w(t) E IRm is the noise signal vector (including process and measurement noises), z(t) E ]RP is the signal to be estimated, (t) is a given initial vector function which is continuous on the segment [-'t,O], and 'tAt), j = 1, ... ,q, are constant bounded time-delays. The system matri~es are assumed to be unknown (uncertain) but belonging to a known convex compact set of poly topic type, namely (Ao, . .. ,Aq,B,C,D,L, T) E '1J
(5)
where '1J is a given convex bounded polyhedral domain described by f. vertices as follows: '1J ~ { (Ao, . .. , T) I (Ao, ... , T)
l
= 2,
(Xi
(AOi, . . . ,Aqi,
i=!
l
Bi ,Ci ,Di,Li,7;);
The design of robust JL filters for continuous-time linear systems with polytope type uncertainty and multiple constant state delays is investigated in this paper. To the authors' knowledge, polytope type uncertainties have not been investigated in the context of robust JL filtering design for systems with delays and, in the case of delay-dependent conditions, only precisely known systems have been addressed.
(Xi
2: 0;
2, (Xi =
I}
(6)
i=!
This kind of convex bounded parameter uncertainty has been widely used in the context of robust control (see, for instance, (Palhares et aI. , 1997) and references therein). In this paper, the following notion of robust stability for uncertain state-delayed systems is used (de Souza and Li, 1999):
The aim is to obtain a full-order stable linear filter assuring robust stability and a guaranteed JL performance for the filtering error system. For that, delaydependent as well as delay-independent sufficient conditions are established in terms of LMIs, allowing the problem to be solved by means of efficient numerical methods with global convergence assured in both cases.
Definition 1. The system (1)-(6) with w(t) == 0, is said to be robustly stable if the trivial solution x(t) == 0 of the functional differential equation (1) is globally uniformly asymptotically stable for all admissible uncertainties.
The robust filtering problem addressed in this paper consists on obtaining an estimate z(t) of the signal z(t) such that a guaranteed performance criterion is minimized in a filtering error sense. Attention is focused on the design of a linear time-invariant, asymptotically stable filter of order nf = n with state-space realization of the form:
Throughout the paper, the boldface characters I and o stand for the identity and the null matrices of convenient sizes, respectively. ~ denotes the space of square integrable vector functions on [0,(0) with norm 1I·lb ~ (Io-II· Wdt)!/2, where 11 ·11 denotes the Euclidean vector norm. For a symmetric block matrix, the symbol * denotes the symmetric blocks outside the main diagonal block.
i(t) = Afx(t) + Bfy(t) , z(t)
x(O) =0
=Cfx(t) + Dfy(t).
(7) (8)
2. PROBLEM FORMULATION Defining the augmented state i(t) ~ [x(t) the filtering error dynamics is given by
Consider the following linear system with multiple delayed states: 250
.i'(t)]',
q
i(t)
= Aoi(t) + LAjEi(t-'tj) + Hw(t)
j=l z(t) =Ci(t) + Dw(t) i(t) = [CP'(t)
(10)
~tE[-'t,O)
0]',
then the filtering error system is robustly stable with y noise attenuation level.
(9)
Based on the above delay-independent robust stability criterion with y noise attenuation level , the following theorem provides a sufficient condition for the existence of a robust linear stable filter assuring an 1L bound for uncertain systems with multiple state delays.
(11)
where the filtering error signal is denoted by z(t) ~ z(t) -z(t) and
Ao =
[~~
:j]' A
=
j
[~] , j = 1, ... , q
- [B] B= BjD ,C=[L-DjC
(12)
Theorem 1. Let y > 0 be a given scalar. If there exist matrices R = R' E IRnxn positive definite, S = Si E llM X n , P j-rj - nJ E TlIln x n , j• - 1, ... ,q, M E TlIln xn N E l'"l'"l'"-,
(13)
-Cj]
D=T-DjD,E=[I 0]
IRP x n , Z E IRn x r and Dj E
(14)
[ ~; ~;
e,
where
(P,;,) Robust 1L filtering problem with multiple time-delayed states: Given a scalar y > 0, determine a stable linear filter of the form (7)-(8) such that the filtering error system (9)-( JJ) is robustly stable and ensures a prescribed level y of 1L noise attenuation, namely, under zero-initial conditions and for any non-zero wE L2 ~ (Ao , .. . , T) E 'lJ
~i=
1, . ..
,e
(16)
'14;
S-R>O
The filtering problem addressed in this paper is as follows:
IIzl12 :::; 'YIlwIl2,
satisfying
~il <0,
1Ui
Throughout the paper, Ao; , Aj;, H;, C; and D;, i = 1, . . . , denote the matrices Ao, Aj, H, C and D evaluated at each of the vertices of the polytope 'lJ
JRP x r
(17)
'TJ and '14; are defined in Lemma 1, and
1U . = [AOiR + RAOi + I
±
j=l
Pj
* RAOi + AOiS + C;Z' + M' + L Pj q
j
AOiS + SAoi + ZCi + C;Z' + ji Pj j=l
(15)
In this situation, the filtering error system is said to be robustly stable with y noise attenuation level.
!RJi
= [RAli SAli RBi
'R.3i = [ SBi + ZDi
3. DELAY-INDEPENDENT ROBUST 1L FILTERING SYNTHESIS
... ...
RA.] SA;;
L~-CD' I I j
-NI]
L;-C;DJ
then the problem (P,;,) is solvable independent of delays. Moreover, under the above conditions, the transfer function matrix of a suitable robust filter is given by
Before presenting the robust filtering synthesis, a delay-independent sufficient condition for the robust stability for the filtering error system with time delays (9)-(11) is established. For the sake of space limitation, all the proofs are omitted.
with
Lemma 1. Consider the filtering error system (9)(11) and let y > 0 be a given scalar. If there exist symmetric positive definite matrices P E ]R2n x 2n and Pj E IRnxn , j = 1, ... ,q, such that
AO;P + PAo; + ~ E' PjE
'li;
%.;]
7;'i
'TJ
o
tJ:f.i
0
[ ~i
With the results of Theorem 1, the problem (p,;,) can be solved (independent of delay) by testing the feasibility of the LMIs (16)-(17). The minimum y attenuation level can be obtained by means of the following LMI optimization procedure:
<0
'14;
min
= 1, ... , e, where 'lii=[PAli rr _ .L3 -
R,s,Pj,M,N ,z,D,,fJ
...
d·Jag { - Pl,···, -
Cn
PAqi], '12i=[PH;
D} ,
rq
rr.. _ L41 -
[-rD;
I
0
subjectto (16) and (17)
_iY;I ]
whereo~r . 251
4. DELAY-DEPENDENT ROBUST JL FILTERING SYNTHESIS
Q4j E lRnxn. and matrices M E lRnxr andDf E JRPxr satisfying
Firstly, a delay-dependent bounded real criterion for the robust stability of the uncertain filtering error system with multiple time-delays is presented in this section. Similar results can be found in Cao et al. (1998) and de Souza and Li (1999). Lemma 2. Consider the filtering error system (9)(11) with (5)-(6) and let y > 0 be a given scalar. If there exist symmetric positive definite matrices P, Pj, Pj~ E lll>2n x 2n .~r ~ 1 + q. Pjl,"" Pjq E lI" lll>nxn , j. - 1, ... ,q. lI" satisfying pt~
Jl;
R;
'tl r/i
'tqrq;
R: tiP
-yZl
D;
-I
0 0
0 0
'tl r'/i
0
0
-'tI'l'1
'tq~;
0
0 'tl R;
0 0
0
'tqR;
0
1
D';
0
A2;
A3;
'tIL/i
A~; A~;
-yZl
D;
D: -I
0 0
'tIL'li
0
0
-'tln l
'tqL~;
0
0
't1~1;
0 0
0
'tq/!.q;
0
IRPxn,
ZE
'tqLq; 0 0
0
0 -'tqnq 0 0
0
0
'tl~;i
'tq~~i
0
0
o
o
<0
(20)
0
S-R>O
0 -, 'tqB; 0
0 'tl R: 0
A/i
0 -'tq'I'q 0
NE
lRnxn .
0
(21)
'V i = 1, ... , f. where
D; =
<0
T; - DfD;; AIi =
[:Ii =-2;
31i = AOiS+ SAo; + (fAji)' S )=1
+ S (tAj;)
'Vi = 1, ... ,f, where E is defined in (14) and Jl;
= Ao;P+PAo;+ (fAj;E) P+P (tEAj;)' J=I
rj;
= [Aj;EPj
Aj;Pjl
PA o; PE'A'liE'
+ZCi+C;Z';
J=I
32i
J=I
= RAo;+AoiS + R (fAj;) J=I
. . . Aj;Pjq Aj;EPj~ + (fAji)' S+C;Z' +M';
... PE'A~;E'l
)=1
'I'j=diag{Pj,Pj l, "',Pjq, Pj~, Pj, Pjl, ... , Pjq} 33i
then the uncertain filtering error system is robustly stable with y noise attenuation level.
= Ao;R+RAoi+ (tAji)' R+R (tAji); )=1
\)=1
A . - [SB;+ZD;] . A.- [ 21 -
Considering Lemma 2. the next theorem provides a sufficient condition for the existence of a linear stable filter solving problem (P!).
RB;
noj;
Theorem 2. Consider the system (1 )-( 6) and let y > 0 be a given scalar. If there exist symmetric matrices R E ]R"xn positive definite, S E lRnxn. Ylj E ]R"xn, Y2j E ]R"xn, Wlj E lRnxn. W2j E IRnxn. Gjk E lRnxn , k = l, . .. ,q, Qlj E lRnxn. Q2j E lRnxn , Q3j E IRnxn ,
njO; =
252
,3. -
Ojli
'"
L;-C';D'f
].
L!; - C';D'f - N' ,
njq;);
Oj~i = [:~: ~];
[
S]
QI J' +Q2 ' S J
Wlj
, Gjl , ... , Gjq
} ;
~ ' i = [SBi+ZDi] . e . = [Q3 j +Q4j fljBi'
J
O" r---~--~--"""'-~--""---"""
S] ;
S
J
Figure 1 (time delays considered as 'tl = 0.4 and 't2 = 0.5) and Figure 2 ('tl = 5, 't2 = 25) show the maximum singular value diagrams for the (delay-independent) robust filter (25) connected to the uncertain system given by (22)-(24) at the 4 vertices of the uncertain domain 'D. The effectiveness of the '1 = 0.7843 guaranteed bound obtained from the delay-independent approach is apparent in both different situations. y" = 0.7843
flj
OJ
0 .7
for j ,k = I , . .. ,q, then the problem (P!) is solvable. Moreover, under the above conditions, a suitable filter is given by (18)-(19).
..,'" "
~
0.'
la
-=ccc: Vi
With the results of Theorem 2, the problem (P!) can be solved by testing the feasibility of the LMls (20)-(21). The minimum y attenuation level can be obtained by means of the following LMI optimization procedure:
0.'
0.'
0.2
0.'
0 10"
10"
v
min
.0'
.0'
.0'
.0'
Frequency (rad/s)
S , R , Yl j ,Y2j , Wlj , W2j G jk,Ql j, Q2j , Q3} ,Q4j ,M ,N ,Z , D,
Fig. 1. Singular value diagram (4 vertices) for the robust filter (25) connected to the uncertain system (22)-(24) with time delays 'tl = 0.4 and 't2 = 0.5.
subjectto (20) - (21) where v g,
0.'
r. O .'r---~--~--"""'----""---"""
y" = 0.7843 0.'
5. EXAMPLES As the first example, consider the uncertain continuoustime delay system given by (1)-(6) with q = 2 delays and
Ao = [~3 A2
2 _4 +P]'
= [-~.2
C = [I
AI = [~~il -0.~+4>] (22)
-0~31+ 4>]'
0], D
= I,
B=
L = [I
[~]
(23)
2]
(24)
Frequency (rad/s)
The uncertain parameters satisfy Ip I ~ 2 and 14> I ~ 0.1, defining a 4 vertices uncertain domain.
Fig. 2. Singular value diagram (4 vertices) for the robust filter (25) connected to the uncertain system (22)-(24) with time delays 'tl 5 and't2 25.
=
Using the delay-independent approach presented in Theorem 1, one can obtain the following JL filtering noise attenuation level '1 = 0.7843 with A = [-0.3293 J -5.2502 CJ
= [0.9904
1.2408] B -4.0512 ' J 1.9854]' DJ
For this system, with 'tl = 0.4 and 't2 = 0.5, the delay-dependent approach from Theorem 2 yields the JL guaranteed filtering cost given by 'ft 1=0.4 t2=0.5 = 0.9725 and the associate filter matrices
= [0.3649] 2.1083
= 0.0078
=
(25)
-0.6454
= [ -6.2433 CJ = [0.6184
AJ
Actually, since the delay-independent approach yields a feasible solution, one can conclude that the system (22)-(24) is delay-independent, that is, stability and guaranteed JL cost are both assured for any delay values.
0.9559] B -4.6511 ' J
= [0.4497] 2.2446
1.4732], DJ = 0.2556
(26)
Smaller JL guaranteed costs can be obtained if 'tl and 't2 assume very small values; for instance. with 253
=
=
terns with time-varying delays. lEE Proceedings - Control Theory and Applications 145(3), 338344. de Souza, C. E. and X. Li (1999). Delay-dependent robust J£,., control of uncertain linear state-delayed systems. Automatica, 35(7), 1313-1321. de Souza, C. E., U. Shaked and M. Fu (1995). Robust J£,., filtering for continuous time varying
'tJ 0.1 and't2 0.2, the delay-dependent approach from Theorem 2 yields the JL guaranteed filtering 0.4126. As a general cost given by Yt1=O.1 T2=O.2 rule, y increases as the delays increase.
=
Feasible solutions are obtained through the delaydependent approach, with higher JL guaranteed costs, till the limiting situations: 'tl = 0, 't2 ~ 1.8184; 't2 = 0, 'tl ~ 1.6177 and 'tl = 't2 ~ 0.92. To conclude, one may say that the delay dependent approach tends to produce conservative results whenever the system presents a delay-independent behavior.
uncertain systems with deterministic input signals. IEEE Transactions on Signal Processing,
43(3),709-719. Dugard, L. and E. I. Verriest (1997). Stability and Control of Time-delay Systems. Springer-Verlag, Berlin, Germany Fattouh, A., O. Sename and 1. M. Dion (1998). JL observer design for time-delay systems. Proceed-
As the second example, consider the uncertain continuous-time delay system given by (1)-(6) with q 2 delays and
=
Ao
= [~3
2 _4 +P]' Al
A2 = [-g.9 C=[1
0]'
= [~~85
_0~56+4>]' D=I,
ings of the 37th , IEEE Conference on Decision and Control, December, Tampa, FL, 4545-4546. Geromel, 1. c., and M . C. de Oliveira (1998). :J6 and J£,., robust filtering for convex bounded uncertain systems. Proceedings of the 37th IEEE Conference on Decision and Contro, December, Tampa, FL, 146-151. Grigoriadis, K. M . and J. T. Watson (1997). Reducedorder J£,., and L2-I- filtering via linear matrix inequalities. IEEE Transactions on Aerospace and Electronic Systems 33(4), 1326-1338. Kolmanovskii, V. B. , and S. I. Niculescu and J. P. Richard (1999). On the Liapunov-Krasovskii
0 _1 +4>] (27)
B=
[~]
(28)
L=[l
2]
(29)
The uncertain parameters are such that Ipl :S 2 and -0.6 :S 4> :S 0.4, defining a 4 vertices uncertain domain . For this system, the delay-independent approach fails to find a feasible filter.
functionals for stability analysis of linear delay systems. International Journal of Control
However, the delay-dependent approach (Theorem 2) achieves, for 'tl 0.1 and 't2 0.2, the guaranteed JL filtering cost Yt1=O.1 T2=D.2 = 0.9257 and the filter matrices
=
A f
= [-1.5067
Cf
-8.3521
= [0.7318
=
0.3429] -8.6605' 1.5881],
B f
72(4),374-384. Li, H . and M. Fu (1997). A linear matrix inequality approach to robust JL filtering. IEEE Transactions
= [1.0583]
on Signal Processing 45(9), 2338-2350.
3.8307
Df=0.1831
Palhares, R. M. and P. L. D. Peres (1999). Robust JL filtering design with pole placement constraint via LMIs. Journal oJ Optimization Theory and
(30)
Applications 102(2), 239-261 .
The delay-dependent approach provides a feasible solution until values of delay around 'tl = 0 and 't2 ~ 0.644; or 'tl ~ 0.3671 and't2 0, or'tl 't2 ~ 0.24, but the associated filter matrices A f and Bf exhibit very large norms.
=
Palhares, R. M., R. H. C. Takahashi and P. L. D. Peres (1997). JL and :J:h. guaranteed costs computation for uncertain linear systems. International Jour-
=
nal of Systems Science 28(2), 183-188. Park, P. and T. Kailath (1997). JL filtering via convex optimization. International Journal of Control
66( 1), 15-22.
6. CONCLUSION
Pila, A. w., U. Shaked and C. E. de Souza (1999). JL filtering for continuous-time linear systems with delay. IEEE Transactions on Automatic Control
Delay-independent as well as delay-dependent LMI conditions assuring the existence of a robust JL filter for uncertain linear continuous-time systems with multiple time-delays and convex bounded uncertainties have been proposed in this paper. With these conditions, the robust filtering design can be performed through standard convex programming tools with global convergence assured.
44(7),1412-1417. Shi, P. (1998). Filtering on sampled-data systems with parametric uncertainty. IEEE Transactions
on Automatic Control, 43(7), 1022-1027.
7. REFERENCES Cao, Y. Y., Y. X. Sun and J. Lam (1998). Delaydependent robust JL control for uncertain sys254