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Lyapunov-Krasovskii method and strong delay-independent stability of linear delay systems
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Copyright © IFAC Linear Time Delay Systems, Ancona , Italy, 2000
C:Oc> Publications www.elsevier.comdocate--ifac
LYAPUNOV-KRASOVSKII METHOD AND STRONG DELAY-INDEPENDENT STABILITY OF LINEAR DELAY SYSTEMS Pierre-Alexandre Bliman .,1
• National Technical University, Dept of Mathematics, Zografou Campus, 157 80 Athens, Greece
Abstract: The issue of stability of linear systems with commensurate delays is considered. A precise link is established between a LMI condition obtained by use of quadratic Lyapunov-Krasovskii functionals, and the frequency domain condition of strong delay-independent stability. This comparison is achieved using an uncertainty analysis approach proposed by Chen and Latchman (1995) . Among other benefits, this permits to exhibit some classes of systems for which Lyapunov-Krasovskii method is nonconservative. Copyright ©2000 IFAC Keywords: Linear systems, Delay analysis, Stability analysis, Robust stability, Lyapunov methods, Frequency domains
1. INTRODUCTION
(Kharitonov, 1998; Niculescu et al., 1998). This property is expressed as:
Some issues on the asymptotic stability of linear delay systems with commensurate delays are considered here. Define the system: ;i;
= Aox + A 1 x(t -
h) ,
Vh
~
O,Vs E C with Res
~
0,
det(sI - Ao - e- sh Ad :I 0 .
(1)
Hale et al. have provided necessary and sufficient condition of asymptotic stability independent of the delay (Hale et al., 1985, Theorem 2.4), namely:
where A o, Al E IRNxN. As is well-known, for any given h ~ 0, the asymptotic stability of (1) is equivalent to VsECwith Res~O, det(sI-A o -e- sh A 1 ):l0 .
Rea(Ao + Ad < 0 ; V(s,z) E jIR x C with s:l 0, Izl = 1, det(sI - Ao - zAd :I 0 ,
Both for reasons of computational simplicity and stability robustness, the issue of stability of (1) independent of the delay h has been considered, see (Kamen, 1982; Kamen, 1983) and the surveys
(2a) (2b)
where a(Ao) designates the set consisting of the eigenvalues of A o , and Rea(Ao) the set of their real parts (in the sequel , p(Ao) = max la(Ao)1 denotes the spectral radius of Ao).
1 On leave from I.l'\.R.I.A., with the support of European Commission's Training and Mobility of Researchers Programme ERB FMRXCT-970137 "Breakthrough in the control of nonlinear systems". The author is indebted to the network coordinator Fran~oise Lamnabhi-Lagarrigue, and the coordinator of the greek team Ioannis Tsinias, for their help and support. Permanent and corresponding address: I.N.R.I.A., Rocquencourt B.P. 105, 78153 Le Chesnay Cedex, France. Phone: (33) 1 39635568, Fax: (33) 1 39 63 57 86, Email: pierre-a.lexancire.blimanClinria.fr
On the other hand, sufficient condition for asymptotic stability independent of the delay has been given using Lyapunov-Krasovskii functional method , see (Hale and Verduyn Lunel , 1993, 132134), (Verriest and Ivanov, 1994). More precisely, for P, Q E IRNxN, positive definite, denoting
33
V(cI»
d~f cl>T(O)pcl>(O) + [Oh cl>T(T)QcI>(T) dT forcl>EC([-h , O];I!~N) ,
Then,
°
(i) {::} Rea(Ao) < and min maxllM(sI - AO)-lAlM-llI < 1,
(3)
inve~ible
the quadratic functional V (cl» is bounded from below by IlcI>(O)1I 2 /I1P- l ll , and one has d ( x(t) ) "It ~ 0, dt V(xt} ~ x(t _ h) R
~f
(Ar P
T
R
(
8E]R
inve~ible
(8)
(4)
-Q
Afp
(7)
°
(ii) {::} Re a(Ao) < and max min IIM(sI - AO)-l AlM-llI < 1 .
x(t) ) x(t - h) ,
+ P Ao + Q PAl)
8E]R
In particular, (i) implies (ii) , and implies that the functional V defined in (3) is positive definite for any solution P, Q of (5) , and verifies, for any solution x of (1): "It ~ 0,
along the trajectories of system (1) (where as usual Xt(T) = x(t + T) , T E [-h, OJ) . Asymptotic stability of system (1) is then deduced when R < 0, using Krasovskii's time-domain techniques (Krasovskii, 1963) .
d dt [V(Xt)]
~ (X(t -
x(t) h) )
T
x(t) ) R (X(t - h) .
•
We attempt here to clarify and compare the two previous approaches . In a previous paper (Bliman, 2000), it was proved that the success of the Lyapunov-Krasovskii method (more precisely, the existence of positive matrices P, Q providing negative R in (4)) implies a certain frequency domain condition closely related to (2) , and called strong delay-independent stability by Niculescu et al. (Niculescu et al., 1996). In the present paper, the same result is repeated and expanded (Theorem 1 below), with a proof notably simpler. Also, the link with uncertainty analysis approach (Chen and Latchman, 1995; Niculescu, 1999; Huang and Zhou , 1999) (J.l analysis) is made. This permits to exhibit some classes of systems for which Lyapunov-Krasovskii approach is nonconservative (Corollary 5). The result is then generalized to systems with multiple , commensurate delays (Theorem 6) . Further results on linear delay systems may be found in (Bliman and Niculescu , 2000) . Results of absolute stability for non linear delay systems, obtained with the same mathematical techniques , are given in (Bliman, 1999a; Bliman, 1999b).
Condition (6) denotes strong delay-independent stability (according to the terminology in Niculescu et al. (Niculescu et al., 1996)), and should be linked with (2) . The contents of Theorem 1 should be compared to the classical stability result for rational systems: one may equivalently characterize the stability of the linear system :i; = Aox, either by the existence of P, Q, positive definite, such that A6' P + P Ao + Q < 0, or by a frequency domain property, namely Rea(Ao) < O. Incidentally, the delay-free case is obtained in Theorem 1 when taking Al = 0. The sufficiency of (7) for delayindependent stability has been shown in (Verriest et al., 1993).
2. SYSTEMS WITH ONE DELAY
In the right-hand side of (8), min{IIM(sI M invertible} = J.ltl«sIAO)-l AlM-11I AO)-l Ad , where J.ltl represents the structured singular value associated to the uncertainty structure ~ = {UN : 0 E Cl , see (Zhou et al., 1996, Chapter 11) , and (Chen and Latchman, 1995; Niculescu , 1999; Huang and Zhou , 1999) in the context of delay systems. The result of Theorem 1 clearly underlines the conservatism inherent to the use of quadratic Lyapunov-Krasovskii functionals of type (3).
Theorem 1. Consider the two following statements.
Remark 2. Condition (6) is equivalent (Hertz et al. , 1984, Remarks 1 and 2) to
(i) There exist P, Q E
P
= pT > 0,
R
~f
(Ar P
Q
jRN x N
such that
V(s , z) E 1(:2 with Res ~ 0, Izl ~ 1, det(sI - Ao - zAd
= QT > 0,
+ P Ao + Q PAl) < 0 Afp
-Q
.
(5)
Analogously, when (6b) holds, maximum modulus principle (see e.g. (Zhou et al. , 1996)) implies that condition (6a) may equivalently be replaced by Rea(Ao) < 0. 0
(ii) The following property holds:
Rea(Ao + Ad <
°; °.
Izl = 1, det(sI - Ao - zAd f:-
V(s , z) E jJR x I(: with
f:- 0 .
(6a)
Proof: By Kalman-Yakubovich-Popov lemma (Rantzer, 1996), (i) is equivalent to
(6b)
34
3. SYSTEMS WITH COMMENSURATE DELAYS
Rea(Ao) < 0, and 3Q = QT > 0, Vs E jIR, Ar(s" I - An-IQ(sI - AO)-I Al - Q
< 0,
(9)
which is itself equivalent to the right-hand side of (7): this provides (7).
The previous result is here generalized to the following class of systems: L
When Ao is Hurwitz, one has: Vs E C with Res ~ 0, min{lzl : z E C, det(sI - Ao - zAd} =max{lzl : zEiC, det(zI-(sI-Ao)-IAd}
=p«sI -
:i;
=L
(10)
Alx(t -lh) ,
1=0
where L E N and AI E JRNxN ,I = O,L. The idea to handle system (10), is to transform it into a new, augmented, system having a unique delay. More precisely, defining
Ao)-I Ad x(tx(t) - h)
mm IIM(sI-Ao)-IAIM-III, M invertible (
using (Zhou et al., 1996, p. 282). Using Remark 2, one deduces (8). The rest of the proof is straight0 forward. Remark 3. The equivalence (8) is proved in (Chen and Latchman, 1995, (3.4)-(3.5)) . 0 Remark 4. The implication (i) => (ii) may be proved more directly. One may apply discretetime version of Kalman-Yakubovich-Popov lemma (Rantzer, 1996) to (9), leading directly to (ii), see (Bliman, 2000) . A particularity of this method is to enlighten the fact that matrix P (resp. Q) in (i) is linked with the continuous-time (resp. discrete-time) frequency domain variable s (resp. z) in (ii).
Alternatively, R being defined in (5), one may remark that
(;;:)* R (;;:) = (Ar +z" ADp+P(Ao+zAd+(l-z"z)Q < O.
x(t -
X = AoX + AIX(t - Lh) ,
(11)
where Ao d~f
L
L
JI
181 AI , Al d~f
1=0
L
L
J(L-I)T
181 AI,
1=0
(12a)
nLxL , iJ j def . J E "" = 1 I'f t' + 1 = J,. 0 ot h erWlse. (12b) It is clear that the trajectories of system (10) may be obtained as projections of some trajectories of system (l1)i asymptotic stability of (11) hence implies asymptotic stability of (10). It turns out from the proof of Theorem 6, that strong delayindependent stability of (11) is indeed equivalent to strong delay-independent stability of (10), see Proposition 9 below. A natural Lyapunov-Krasovskii functional candidate for (11) is now: V(r/»
~f r/>T(O)pr/>(O) +
l:h
r/>T(T) Qr/>(T) dT
for r/>E C([-Lh,OjiJRLN) . (13) Theorem 6. Consider the two following statements.
(i) There exist P , Q E JRLNxLN such that P
Proof: When Ao and Al commute and have distinct eigenvaiues, they may be diagonalized in a common basis (Gantmakher, 1959) . Taking for M in (7) the corresponding similarity transformation, one sees that both expressions in (7) and (8) are equal to max{p«sI - Ao)-I Ad : s E jJR}, whence the result . When either Ao or Al has nondistinct eigenvalues, one may find arbitrarily small matrices B such that Ao + B and Al + B commute (i.e. B and Ao - Al commute) and have distinct eigenvalues. It then suffices to apply the previous argument, and to make B -t O. 0
(l- l)h)
one may "rewrite" system (10) as:
For any Izl ~ 1 and any nonzero eigenvector v of Ao + zAI' with eigenvalue SEC, the previous identity implies that (s + s")v" Pv < 0: any eigenvaiue of Ao + zAI has negative real part. 0 Corollary 5. Suppose that Ao and Al commute. Then, conditions (i) and (ii) of Theorem 1 are equivalent. •
)
= pT > 0,
R~f
Q = QT
> 0,
(A6P+PAo+Q PAl) 0 Arp -Q < ,
where A o , Al are given by (12) . (ii) The following property holds: L Rea(L At} < 0 i
(14)
(15a)
0
V(s, z) E jJR x C with Izl = I , L det(sI - Lz1A1):f: O . 1=0
35
(15b)
Then,
Proposition 9. Let Ao, A1 being defined by (12).
°
(i) {::} Rea(Ao) < and min maxllM(sI -.4o)-lA1M-11I inve~ible
BE}R
\:I(s , z) E C2 with Re s ~ 0, Izl ~ 1,
< 1,
L
°
det(sI - L zl At}
(ii) {::} Re a(Ao) < and max min IIM(sI BEjR
Then,
.40)-1 A1M-11I < 1 .
inve~ible
)T R (X(tX(t) ) - Lh)
o
Remark 7. Same commentary than in Remark 2 holds; (15) may be globally replaced by: \:I(s , z) E C2 with Re s ~ 0, Iz l ~ 1, L
o Proof: After applying Theorem 1, it only remains to compare det( sI - '£~ Zl At} and det( sI - Ao zL A1)' One has : L
L(JI + zL J(L-I)T) 181 AI . 1=0
Now, for any z E C, denote r
N
= r(L) ~f e-2p- ,
= N(z , L) E jRL XL,
Reasoning as for Corollary 5, one gets:
Corollary 10. Suppose that Ao and A1 commute. Then, conditions (i) and (ii) of Theorem 6 are equivalent. • As an example, when L = 2, the assumption of Corollary 10 is fulfilled iff AoA1 = A1Ao , AoA2 = A 2A o , A1A2 = A2A1 and Af = O.
det(sI - I>IA I ) i= 0 . o
= sI -
• This permits to conclude the proof of Theorem 6.
•
sI - Ao - zL A1
°
~ \:I(s, z) E C2 with Re s ~ 0, Izl ~ 1, det(sI - .40 - zL Ad i= 0 .
In particular, (i) implies (ii), and implies that the functional V defined in (13) is positive definite for any solution P , Q of (14) , and verifies, for any solution X of (11) : \:It ~ 0,
d ( X(t) dt[V(Xdl ~ X(t - Lh)
i=
o
N ij ~f (r j - 1 Z)i-1 .
The next lemma is proved easily:
4. REFERENCES
P.-A. Bliman, Stability of nonlinear delay systems: delay-independent small gain theorem and frequency domain interpretation of the Lyapunov-Krasovskii method, (submitted 1999) P .-A. Bliman, Lyapunov-Krasovskii functionals and frequency domain: delay-independent absolute stability criteria for delay systems (submitted 1999) P.-A. Bliman, Stability of linear delay systems. A note on frequency domain interpretation of Lyapunov-Krasovskii method, Proc. of 14th
Int . Symp . on Mathematical Theory of Networks and Syst. MTNS 2000, Perpignan,
Lemma 8. The following formula is true , for any 1= O,L:
France (2000) P .-A. Bliman, S.-I. Niculescu , A note on fre= N diag{zl , (rz)I , ... , (r L- 1z)I}N-1 . quency domain interpretation of LyapunovKrasovskii approach for linear delay systems , (submitted 2000) J . Chen , H.A. Latchman , Frequency sweeping tests for stability independent of delay, IEEE One deduces from Lemma 8, that Trans . Automat. Control 40 no 9 (1995) 1640-1645 det(sI - .40 - zL Ad F .R. Gantmakher , Theory of matrices, Chelsea L Publishing co (1959) L 1 = det(sI - L diag{zIAI ' (rz)1 AI , . .. , (r - z)1 Ad) J .K. Hale, E.F . Infante, F.S.P. Tsen , Stability in 1=0 linear delay equations, J. Math. Anal. Appl. L-1 L 115 (1985) 533-555 l = det(sI - L(r ' z/ At} . J.K. Hale , S.M. Verduyn Lunel , Introduction 1'=0 1=0 to functional-differential equations, Applied As Irl = 1, t he preceding formula yields the Mathematical Sciences 99 (Springer-Veriag, following result. New York , 1993)
JI + zL jtL-I)T
•
IT
36
D. Hertz, E.!. Jury, E. Zeheb, Stability independent and dependent of delay for delay differential systems, J. F'ranklin Institute 318 no 3 (1984) 143-150 Y.-P. Huang, K. Zhou, Robust control of uncertain time delay systems, Proc. of 38th Con/. on Decision and Control, Phoenix, Arizona (1999) 1130-1135 E .W . Kamen, Linear systems with commensurate time delays: stability and stabilization independent of delay. IEEE 'lhlns . A utomat. Control 27 no 2 (1982) 367-375 E.W . Kamen , Correction to "Linear systems with commensurate time delays: stability and stabilization independent of delay" , IEEE Trans. Automat. Control 28 no 2 (1983) 248-249 V.L. Kharitonov, Robust stability analysis of time delay systems: a survey, Proc. IFAC Syst. Struct. Contr. (1998) N.N. Krasovskii , Stability of motion, (Stanford University Press, Stanford, 1963) S.-1. Niculescu, J .-M. Dion, L. Dugard, H. Li , Asymptotic stability sets for linear systems with commensurable delays: a matrix pencil approach, IEEE/IMACS CESA '96, Lille, France (1996) S.-1. Niculescu, E.1. Verriest, L. Dugard, J.-M. Dion, Stability and robust stability of timedelay systems: a guided tour, Stability and control of time-delay systems, Lecture Notes in Control and Inform. Sci. 228, (Springer, London, 1998) 1-71 S.-1. Niculescu, On some frequency sweeping tests for delay-dependent stability: a model transformation case study, Proc. of 5th European Control Conference, Karlsruhe, Germany (1999) A. Rantzer, On the Kalman-Yakubovich-Popov lemma, Syst. Contr. Lett. 28 no 1 (1996) 7-10 E.I. Verriest, M.K.H . Fan, J. Kullstam , Frequency domain robust stability criteria for linear delay systems, Proc. of 32nd IEEE CDC, San Antonio (Texas) (1993) 3473-3478 E.!. Verriest, A.F. Ivanov, Robust stability of systems with delayed feedback, Circuits Systems Signal Process . 13 no 2-3 (1994) 213-222 K . Zhou , with J .C. Doyle, K. Glover, Robust and optimal control, Prentice Hall (1996)
37
Copyright © IFAC Linear Time Delay Systems, Ancona , Italy, 2000
C:Oc> Publications www.elsevier.comdocate--ifac
LYAPUNOV-KRASOVSKII METHOD AND STRONG DELAY-INDEPENDENT STABILITY OF LINEAR DELAY SYSTEMS Pierre-Alexandre Bliman .,1
• National Technical University, Dept of Mathematics, Zografou Campus, 157 80 Athens, Greece
Abstract: The issue of stability of linear systems with commensurate delays is considered. A precise link is established between a LMI condition obtained by use of quadratic Lyapunov-Krasovskii functionals, and the frequency domain condition of strong delay-independent stability. This comparison is achieved using an uncertainty analysis approach proposed by Chen and Latchman (1995) . Among other benefits, this permits to exhibit some classes of systems for which Lyapunov-Krasovskii method is nonconservative. Copyright ©2000 IFAC Keywords: Linear systems, Delay analysis, Stability analysis, Robust stability, Lyapunov methods, Frequency domains
1. INTRODUCTION
(Kharitonov, 1998; Niculescu et al., 1998). This property is expressed as:
Some issues on the asymptotic stability of linear delay systems with commensurate delays are considered here. Define the system: ;i;
= Aox + A 1 x(t -
h) ,
Vh
~
O,Vs E C with Res
~
0,
det(sI - Ao - e- sh Ad :I 0 .
(1)
Hale et al. have provided necessary and sufficient condition of asymptotic stability independent of the delay (Hale et al., 1985, Theorem 2.4), namely:
where A o, Al E IRNxN. As is well-known, for any given h ~ 0, the asymptotic stability of (1) is equivalent to VsECwith Res~O, det(sI-A o -e- sh A 1 ):l0 .
Rea(Ao + Ad < 0 ; V(s,z) E jIR x C with s:l 0, Izl = 1, det(sI - Ao - zAd :I 0 ,
Both for reasons of computational simplicity and stability robustness, the issue of stability of (1) independent of the delay h has been considered, see (Kamen, 1982; Kamen, 1983) and the surveys
(2a) (2b)
where a(Ao) designates the set consisting of the eigenvalues of A o , and Rea(Ao) the set of their real parts (in the sequel , p(Ao) = max la(Ao)1 denotes the spectral radius of Ao).
1 On leave from I.l'\.R.I.A., with the support of European Commission's Training and Mobility of Researchers Programme ERB FMRXCT-970137 "Breakthrough in the control of nonlinear systems". The author is indebted to the network coordinator Fran~oise Lamnabhi-Lagarrigue, and the coordinator of the greek team Ioannis Tsinias, for their help and support. Permanent and corresponding address: I.N.R.I.A., Rocquencourt B.P. 105, 78153 Le Chesnay Cedex, France. Phone: (33) 1 39635568, Fax: (33) 1 39 63 57 86, Email: pierre-a.lexancire.blimanClinria.fr
On the other hand, sufficient condition for asymptotic stability independent of the delay has been given using Lyapunov-Krasovskii functional method , see (Hale and Verduyn Lunel , 1993, 132134), (Verriest and Ivanov, 1994). More precisely, for P, Q E IRNxN, positive definite, denoting
33
V(cI»
d~f cl>T(O)pcl>(O) + [Oh cl>T(T)QcI>(T) dT forcl>EC([-h , O];I!~N) ,
Then,
°
(i) {::} Rea(Ao) < and min maxllM(sI - AO)-lAlM-llI < 1,
(3)
inve~ible
the quadratic functional V (cl» is bounded from below by IlcI>(O)1I 2 /I1P- l ll , and one has d ( x(t) ) "It ~ 0, dt V(xt} ~ x(t _ h) R
~f
(Ar P
T
R
(
8E]R
inve~ible
(8)
(4)
-Q
Afp
(7)
°
(ii) {::} Re a(Ao) < and max min IIM(sI - AO)-l AlM-llI < 1 .
x(t) ) x(t - h) ,
+ P Ao + Q PAl)
8E]R
In particular, (i) implies (ii) , and implies that the functional V defined in (3) is positive definite for any solution P, Q of (5) , and verifies, for any solution x of (1): "It ~ 0,
along the trajectories of system (1) (where as usual Xt(T) = x(t + T) , T E [-h, OJ) . Asymptotic stability of system (1) is then deduced when R < 0, using Krasovskii's time-domain techniques (Krasovskii, 1963) .
d dt [V(Xt)]
~ (X(t -
x(t) h) )
T
x(t) ) R (X(t - h) .
•
We attempt here to clarify and compare the two previous approaches . In a previous paper (Bliman, 2000), it was proved that the success of the Lyapunov-Krasovskii method (more precisely, the existence of positive matrices P, Q providing negative R in (4)) implies a certain frequency domain condition closely related to (2) , and called strong delay-independent stability by Niculescu et al. (Niculescu et al., 1996). In the present paper, the same result is repeated and expanded (Theorem 1 below), with a proof notably simpler. Also, the link with uncertainty analysis approach (Chen and Latchman, 1995; Niculescu, 1999; Huang and Zhou , 1999) (J.l analysis) is made. This permits to exhibit some classes of systems for which Lyapunov-Krasovskii approach is nonconservative (Corollary 5). The result is then generalized to systems with multiple , commensurate delays (Theorem 6) . Further results on linear delay systems may be found in (Bliman and Niculescu , 2000) . Results of absolute stability for non linear delay systems, obtained with the same mathematical techniques , are given in (Bliman, 1999a; Bliman, 1999b).
Condition (6) denotes strong delay-independent stability (according to the terminology in Niculescu et al. (Niculescu et al., 1996)), and should be linked with (2) . The contents of Theorem 1 should be compared to the classical stability result for rational systems: one may equivalently characterize the stability of the linear system :i; = Aox, either by the existence of P, Q, positive definite, such that A6' P + P Ao + Q < 0, or by a frequency domain property, namely Rea(Ao) < O. Incidentally, the delay-free case is obtained in Theorem 1 when taking Al = 0. The sufficiency of (7) for delayindependent stability has been shown in (Verriest et al., 1993).
2. SYSTEMS WITH ONE DELAY
In the right-hand side of (8), min{IIM(sI M invertible} = J.ltl«sIAO)-l AlM-11I AO)-l Ad , where J.ltl represents the structured singular value associated to the uncertainty structure ~ = {UN : 0 E Cl , see (Zhou et al., 1996, Chapter 11) , and (Chen and Latchman, 1995; Niculescu , 1999; Huang and Zhou , 1999) in the context of delay systems. The result of Theorem 1 clearly underlines the conservatism inherent to the use of quadratic Lyapunov-Krasovskii functionals of type (3).
Theorem 1. Consider the two following statements.
Remark 2. Condition (6) is equivalent (Hertz et al. , 1984, Remarks 1 and 2) to
(i) There exist P, Q E
P
= pT > 0,
R
~f
(Ar P
Q
jRN x N
such that
V(s , z) E 1(:2 with Res ~ 0, Izl ~ 1, det(sI - Ao - zAd
= QT > 0,
+ P Ao + Q PAl) < 0 Afp
-Q
.
(5)
Analogously, when (6b) holds, maximum modulus principle (see e.g. (Zhou et al. , 1996)) implies that condition (6a) may equivalently be replaced by Rea(Ao) < 0. 0
(ii) The following property holds:
Rea(Ao + Ad <
°; °.
Izl = 1, det(sI - Ao - zAd f:-
V(s , z) E jJR x I(: with
f:- 0 .
(6a)
Proof: By Kalman-Yakubovich-Popov lemma (Rantzer, 1996), (i) is equivalent to
(6b)
34
3. SYSTEMS WITH COMMENSURATE DELAYS
Rea(Ao) < 0, and 3Q = QT > 0, Vs E jIR, Ar(s" I - An-IQ(sI - AO)-I Al - Q
< 0,
(9)
which is itself equivalent to the right-hand side of (7): this provides (7).
The previous result is here generalized to the following class of systems: L
When Ao is Hurwitz, one has: Vs E C with Res ~ 0, min{lzl : z E C, det(sI - Ao - zAd} =max{lzl : zEiC, det(zI-(sI-Ao)-IAd}
=p«sI -
:i;
=L
(10)
Alx(t -lh) ,
1=0
where L E N and AI E JRNxN ,I = O,L. The idea to handle system (10), is to transform it into a new, augmented, system having a unique delay. More precisely, defining
Ao)-I Ad x(tx(t) - h)
mm IIM(sI-Ao)-IAIM-III, M invertible (
using (Zhou et al., 1996, p. 282). Using Remark 2, one deduces (8). The rest of the proof is straight0 forward. Remark 3. The equivalence (8) is proved in (Chen and Latchman, 1995, (3.4)-(3.5)) . 0 Remark 4. The implication (i) => (ii) may be proved more directly. One may apply discretetime version of Kalman-Yakubovich-Popov lemma (Rantzer, 1996) to (9), leading directly to (ii), see (Bliman, 2000) . A particularity of this method is to enlighten the fact that matrix P (resp. Q) in (i) is linked with the continuous-time (resp. discrete-time) frequency domain variable s (resp. z) in (ii).
Alternatively, R being defined in (5), one may remark that
(;;:)* R (;;:) = (Ar +z" ADp+P(Ao+zAd+(l-z"z)Q < O.
x(t -
X = AoX + AIX(t - Lh) ,
(11)
where Ao d~f
L
L
JI
181 AI , Al d~f
1=0
L
L
J(L-I)T
181 AI,
1=0
(12a)
nLxL , iJ j def . J E "" = 1 I'f t' + 1 = J,. 0 ot h erWlse. (12b) It is clear that the trajectories of system (10) may be obtained as projections of some trajectories of system (l1)i asymptotic stability of (11) hence implies asymptotic stability of (10). It turns out from the proof of Theorem 6, that strong delayindependent stability of (11) is indeed equivalent to strong delay-independent stability of (10), see Proposition 9 below. A natural Lyapunov-Krasovskii functional candidate for (11) is now: V(r/»
~f r/>T(O)pr/>(O) +
l:h
r/>T(T) Qr/>(T) dT
for r/>E C([-Lh,OjiJRLN) . (13) Theorem 6. Consider the two following statements.
(i) There exist P , Q E JRLNxLN such that P
Proof: When Ao and Al commute and have distinct eigenvaiues, they may be diagonalized in a common basis (Gantmakher, 1959) . Taking for M in (7) the corresponding similarity transformation, one sees that both expressions in (7) and (8) are equal to max{p«sI - Ao)-I Ad : s E jJR}, whence the result . When either Ao or Al has nondistinct eigenvalues, one may find arbitrarily small matrices B such that Ao + B and Al + B commute (i.e. B and Ao - Al commute) and have distinct eigenvalues. It then suffices to apply the previous argument, and to make B -t O. 0
(l- l)h)
one may "rewrite" system (10) as:
For any Izl ~ 1 and any nonzero eigenvector v of Ao + zAI' with eigenvalue SEC, the previous identity implies that (s + s")v" Pv < 0: any eigenvaiue of Ao + zAI has negative real part. 0 Corollary 5. Suppose that Ao and Al commute. Then, conditions (i) and (ii) of Theorem 1 are equivalent. •
)
= pT > 0,
R~f
Q = QT
> 0,
(A6P+PAo+Q PAl) 0 Arp -Q < ,
where A o , Al are given by (12) . (ii) The following property holds: L Rea(L At} < 0 i
(14)
(15a)
0
V(s, z) E jJR x C with Izl = I , L det(sI - Lz1A1):f: O . 1=0
35
(15b)
Then,
Proposition 9. Let Ao, A1 being defined by (12).
°
(i) {::} Rea(Ao) < and min maxllM(sI -.4o)-lA1M-11I inve~ible
BE}R
\:I(s , z) E C2 with Re s ~ 0, Izl ~ 1,
< 1,
L
°
det(sI - L zl At}
(ii) {::} Re a(Ao) < and max min IIM(sI BEjR
Then,
.40)-1 A1M-11I < 1 .
inve~ible
)T R (X(tX(t) ) - Lh)
o
Remark 7. Same commentary than in Remark 2 holds; (15) may be globally replaced by: \:I(s , z) E C2 with Re s ~ 0, Iz l ~ 1, L
o Proof: After applying Theorem 1, it only remains to compare det( sI - '£~ Zl At} and det( sI - Ao zL A1)' One has : L
L(JI + zL J(L-I)T) 181 AI . 1=0
Now, for any z E C, denote r
N
= r(L) ~f e-2p- ,
= N(z , L) E jRL XL,
Reasoning as for Corollary 5, one gets:
Corollary 10. Suppose that Ao and A1 commute. Then, conditions (i) and (ii) of Theorem 6 are equivalent. • As an example, when L = 2, the assumption of Corollary 10 is fulfilled iff AoA1 = A1Ao , AoA2 = A 2A o , A1A2 = A2A1 and Af = O.
det(sI - I>IA I ) i= 0 . o
= sI -
• This permits to conclude the proof of Theorem 6.
•
sI - Ao - zL A1
°
~ \:I(s, z) E C2 with Re s ~ 0, Izl ~ 1, det(sI - .40 - zL Ad i= 0 .
In particular, (i) implies (ii), and implies that the functional V defined in (13) is positive definite for any solution P , Q of (14) , and verifies, for any solution X of (11) : \:It ~ 0,
d ( X(t) dt[V(Xdl ~ X(t - Lh)
i=
o
N ij ~f (r j - 1 Z)i-1 .
The next lemma is proved easily:
4. REFERENCES
P.-A. Bliman, Stability of nonlinear delay systems: delay-independent small gain theorem and frequency domain interpretation of the Lyapunov-Krasovskii method, (submitted 1999) P .-A. Bliman, Lyapunov-Krasovskii functionals and frequency domain: delay-independent absolute stability criteria for delay systems (submitted 1999) P.-A. Bliman, Stability of linear delay systems. A note on frequency domain interpretation of Lyapunov-Krasovskii method, Proc. of 14th
Int . Symp . on Mathematical Theory of Networks and Syst. MTNS 2000, Perpignan,
Lemma 8. The following formula is true , for any 1= O,L:
France (2000) P .-A. Bliman, S.-I. Niculescu , A note on fre= N diag{zl , (rz)I , ... , (r L- 1z)I}N-1 . quency domain interpretation of LyapunovKrasovskii approach for linear delay systems , (submitted 2000) J . Chen , H.A. Latchman , Frequency sweeping tests for stability independent of delay, IEEE One deduces from Lemma 8, that Trans . Automat. Control 40 no 9 (1995) 1640-1645 det(sI - .40 - zL Ad F .R. Gantmakher , Theory of matrices, Chelsea L Publishing co (1959) L 1 = det(sI - L diag{zIAI ' (rz)1 AI , . .. , (r - z)1 Ad) J .K. Hale, E.F . Infante, F.S.P. Tsen , Stability in 1=0 linear delay equations, J. Math. Anal. Appl. L-1 L 115 (1985) 533-555 l = det(sI - L(r ' z/ At} . J.K. Hale , S.M. Verduyn Lunel , Introduction 1'=0 1=0 to functional-differential equations, Applied As Irl = 1, t he preceding formula yields the Mathematical Sciences 99 (Springer-Veriag, following result. New York , 1993)
JI + zL jtL-I)T
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