- Email: [email protected]
Frequency-domain robustness of time-lag systems
FREQUENCY-DOMAIN ROBUSTNESS OF TIME-LAG SYSTEMS...
14th World Congress ofIFAC
Copyright to 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
G-2e-IO-4
FREQUENCY-DOMAIN ROBUSTNESS OF TIME-LAG SYSTEMS Magdi S. Mallnloud Depal~tnlent of Elec:trical and COmp\lter Engineering Kll\Vait TJniversity, PO Box 5969., Safat-13060
KUWAIT Email: magdi(geng.kllniv.ed11.kw
Abstract: This paper investigates the robustness of linear tilne-lag systenls with respect to real parameteric uncertainties. A linear quadratic tirne-lag regulator (LQTLR) is derived in the frequency-domain and its relation to the Kalman equation, the circle eondit.ion and the Hamiltonian s,ysterll is disclosed. Frequency-dolllahl robustness properties are provided. These include performance improvement and closed-loop eigenvalne diRtribut.ion. Copyright © 1999 IFAC Keywords: Tilue-Iag systeIllS, Linear regulators, Performance
1
INTRODUCTION
Linear quadratic Gaussian (LQC) control has been \vell studied for Inore than three decades C_A..ndcrson and ivloore, 1990) _ As a design tool~ it has been based all the assu111ption that th.e dYllamic model of the system under consideratioIl is exactly kno""vn and its disturbances are Gaussian white noise processes with kno\vn statistics. The early result of Doyle (1978) hovvever has sho\vn that the LQG control design cannot guarantee stabilit.y robustness and perfoTInance against modelling uncertainties. This has brought into focu:;; the importance of robust control; which has subsequently attracted significant research interests over the past decade. Several robust control methodologies have been developed in~luding loop transfer recovery (Doyle and Stein~ 1981) , quadratic stabilization (Khargonekar ~ et aI., 1990) and robust linear quadratic design (Douglas and Athans, 1994). all another front of research, control problems of systems with state and/oT input delay have under investigation during the past tVlenty years. Tinle delay often arise in connection \vith system lueasuremcnts or nue to phy~ieal properties such as hydraulic systenls and rolling mills (11alek-Zavarei and Jalnshidi, 1987). Other sources of delays ll"lay occur in the transnlission of inforrnation bet\veen different parts of dynarnical systelTIS including chemical proce,ssing systems, comIIlunjcations and pO~Ter systems. l\/Iost of the rescent research efforts on tirne-delay systeIlls have been directed to robust stability and stabihzation in the tinle-domain; see (Loll and Van Den Bosch, 1997; :vIahmoud, 1994) and their references. L,ittle attention has been paid to robust measures of UI1CeI'tain tirrle-lag sy~teIllS in the frequecy-doHlain. The purpose of thi::> paper is to bridge this gap by et TO bust regulator design of a class
robustness~
Parameteric uncertainties.
of uncertain linear time-lag systenls with quadratic performance measures. It extends the results of (Douglas and Athans, 1994) to the class of time-lag systems with norm-bounded llncertaintie~. Based on the optiJnality conditions, a robust frequency-domain equality is de-
rived which exhibits a frequency-domain relation for the actual return-difference transfer function Inatrix of the time-lag systenl. Its relation to the Kalman equathe circle condition and the Hamiltonian system is disclosed. Frequency-donlain robustness properties are then provided. These include performance improvement and closed-loop eigenvalue distribution. tion~
Notations and Facts In the scquel~ wc denote by )'(lifl ) the transpose} the inverse , the complex conjugate transpose , the determinant and the eigenvalues of any square lnatrix RT. lV' > 0 (lV < 0) stands for a posit.ive- (negative-) definite nlatrix {tiT and C- represents the open left-half of the conlplex plane. SOInetiIIles, the arguluents of a function will be omitted when no confusion can arise. v~lt, W- 1 .' W*, det(W) and
Fact 1: Given matrices El , 2:2 and 1:3 of appropriate dillieIl~iolls \vith L: 1 == ~i. ~1hen ~1
+
L3~(t)E2
where .6,t (t)Ll(t) S I ~1
2
\j
+ 'E~~t(t):Et <
0
t , if and only if for
+ /-L-I :E~E3 + ft:E2I;~ <
SOlllC
J-t
>0
0
PROBLEM DESCRIPTION
In this section, we consider a claRs of uncertain time-lag systems represented by a state space il1.odel of the form:
eJosely exalniniIlg
(Eb.:) x(t)
3395
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FREQUENCY-DOMAIN ROBUSTNESS OF TIME-LAG SYSTEMS...
-+-
lJ(t)
preliminary results. It is v-rell-kno\vn (Flagbedzi and Pe-arson, 1990) that \Vhell systeIll (Eo) is stabiliz;able in the manner of (8), then the unique optimal control is given by:
Dox(t -- T) [~4o
+
14th World Congress of IFAC
+
~A]x(t)
+ EouCt)
Dox(t - T)
(1)
Cox(t)
(2)
where t E ~R is the tirrle~ :.c( t) E :1(-n is the st.ate, ~RTn is the control input and T is a constant scalar representing the arnount of delay in the state . The matrices A o E ~Rnxn, Eo E 2R nxm and Do E :Rnxn are real constant representing the norninal plant \vith the triplet (Aa' B o' Co) being controllable-observable. T'he uncertain nlatrix LiA == _~ A(t)AJ "Htith ~(t) being bounded in the farnl
u,(t) E
where 0 < P
=: pt E ~Rnxn, 0 S VvT E ~Rnxn and !Jt7lxn are the solution of the Riccati Equations
o ~ L, E (RE~s):
o
PA o
x(t)
:)
y(t)
Aox(t) Cax(t)
+ Bnu(t) + Dox(t -
(11)
- [d~~n)] + A~W{n)+D~L{O,a)
o and 1\.1 and N are real constant IllatriceH with appropriate dimensions. Distinct from system(1)-(2) is the nominal system:
(E u
+ A~P + Vf/~(O)Do + D~~V(O)
P B oR;;l B~P + Qo
p BoR~;; 1 B~ ~~/-( 0:)
(12)
8 [a !J + aGO'] L{{J, a)
o
T) (3) (4)
(13)
L(j3, O)B o R;;l B;L(O, a:) along ,vith boundary conditions
The transfer function Toes) from input u to output Y is
P
given by:
A o - e-
ST
DoJ
;::=
In a sinlilar way, we have for system (2:.6,.): (:0
Hll(s)-l B o ~
. .4~
~
e-- ST Do]
p
(5)
v{here H o ( s) is the quasi-characteristic Inatrix. Define the set (6) j\l == {s: det [1/0 ( s ) 1 O}
{sI
Til ( -7); lV(a)
::::=
L( ~T, ex); 0: E {-r, 0]
(7)
==
pt; L(p, Q)
L· t (O!.,j3);o:
=
E [-T~OL;3 E
[-T10J (15)
\~.rhen
the controller (10) is applied to systerll (1)-(2), it produces a stable closed-loop tilne--lag (CLTL,) systern. In the case of systelns \vithout delay, propcr~ ties of the closed-loop system are well-known (.A..rlderson and ~1oore~ 1990) in the tinle-domain as well as the freqllency-doruain. Our task no\v is to obtairl sinlilar properties fOT the CI.TL system. It follows from ell) that
Definition l(Flagbedzi and Pearson~ 1990): System (Eo) is stabilizable if and only if
~P(sI - Aa) - (-sI - A~)P
+ W(O)D o + D~W(O)
~P BoR;;! B~P + Qo == 0
(8)
3
Postnlultiplying (12) by Doe:S over {-r,O] using (14), wc get:
LINEAR QUADRATIC TIMELAG REGULATOR (LQTLR)
P Doe- ST
1=
{:ct(t) Qo x{t)
+
u/(t) R
o
u(t) dt} dt
~
(
-sI -
(16) ct
and integrating wrt a
A~) LOr W( O')DoesC
+ [0.,- D;,L(O, o:)D" (eS"')da -
""Ve associate \vith systeul (2:.6.) the quadratic perfoTn1.ance measure:
.J(u) =
( 14)
and symlnetry conditions
Co H;;l(s) .B a
[sI -
==-
.fr
-PBoR- 1 B t
W(O)D"
W(o:)Do(esC<)do: = 0
(17)
In a sin1ilar "'lay, "re can \"\l'ite:
(9) 'Arhere 0 < Qo :::::: Q~ E ~Rnxn and 0 < R'O = R~ E ~Rmxrn arc state and control weighting matrices. System (ED,) plus perforrnance J (u) forms a linear quadratic tirnelag regualtor (LQTLR). Our ohjective is to exarrline its frequency-domain robustness properties vlith respect to real par ameter unccrt aintics. 'To achieve our go al~ ,ve fo~ eus attention on the IloIninal LQTLR to develop SOlne
-D~lvt(O)
+ eSTD~P
+ LOT e-s,B D~L{(3, O)Dod,B
- JOT e- ,13 D~Wt(p)da(-sI s
- Aa)
_e-s,B D~ l1Jr t (f3)d;3B o R- 1 B~P
== 0
(18)
3396
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FREQUENCY-DOMAIN ROBUSTNESS OF TIME-LAG SYSTEMS...
14th World Congress ofIFAC
Nov{~ Vole prernultiply (13) by e- s /3 D~ , postmultiply the result by DoP.- sa and integrate "vrt Q: , .8 over [-7,0]
This is the first luain result in tIllS paper. Before exploring the properties of the RFDE (23) or (24), we provide some results relevant for the noulinal system (Eo).
using (14)~ we reach:
-1:
D~L(O, n)Do(eSb)da
-L: [Or L: e-
sB
+eST D~
+
-i:
Relllark 2: It is important to observe that the RFDE for system (~o) has the forIn
D~L(f3, O)Do df3
[I
W(a)Do(eSO<)da
R' D
e-sf'lD~wt(;J)dae-ST
c- sB
=
(19)
0
Define
res)
+
p
+ vr(s~ T)
[1 + R;;l B~r(jw)Go(jw)Bu]*Rn [1 + R;l B~r(j4-')Go(jw)BoJ ~ Ra Vw 2:: 0(26)
p+.Lor D~wt(a)e-sadQ
rt(-s)
p
+ Vt(-S,T) (20)
Inequality (26) provides a characterization of the circle condition for tilne-laR systclns where [1 + li;;"1 B~r(j~))Go(jw)BoJ corresponds to the return difference matrixRelDark 4: Frolll (Nlalek-Zavarci and Jamshidi) the Ifamiltonian matrix associated with systClll (b o ) is given by:
By combining (16)-(19) and using (20), we get:
r t (--8 )Ho(s) + Ii~( - 8 )r(s) + r t ( -s)BoR;;-1 B~r(s) - Qo ==
(25)
ReITIark 3: By further substituting s == jw into (25) and noting that the matrix B~G;(-s)QoGo(s)Bo ~ o Vw 2: 0) it fo1101V8 that
LOT vV(Cl)DocSClda
P
+ B;G~(-s)QoGo(s)Bo
"vhere Go(s) ::::::: [.f{o(.t])]-l. Equation (25) is analogous to the well-known Kalman equation in optimal control (Allderson and J\tloorc, 1990). rrhe basic difference lies in the dependence of r(s) OIl the delay factors T and Do,
D~wt(f3)df3BoR:;l
B; lOr W(a)Do(eS")da
+ B~G~(-s)rt(-s)BoR~l]Ro
[1 + R;;lB~r(s)Go(s)Bo) :;:::::
1987)~
(21)
0
which, in view of (5) and (7), can be rewitten as:
(27)
r ( -s)Hb,.(s) + H1 (-/i)f(s) + r t ( -s)Bo R-;;l B;r(s) ~ Q r t ( -S)(AA - Aa) + (A6. - _4. o )tr(s) == 0 (22) t
On usirlg
(21)~
it follows that 11. o (s) satisfies the inden-
tity
By letting G(s) =-= lH~(.~)J-l postmultiplying; (21) by B~Gi, ( -8)~ premultiply by G( s )Bo and rearranging terms, v..re arrive at: j
[I + (I
[rt(-s) _ [ITo(S)
B~Gt(-.9)rt( -s)BoR-;;l]R o
+ R -;l B~r(s)G(s)Bo] = + B~Gt.( -8)[00 + r t ( -S)(.J4~
-
IIo(s)
l
Ro
[1 + I{oG(s)Bo)t Rn[I + KoG(s)BoJ ~ rt.(-s)(A~ - Aa)
(AA - Ao)tr(s)]G(s)B o
!!I]
BaR;;! B~ ] II(s);(-s)
Ho(s)
+ BaR;;! B~r(s)
(28)
det[sI - '"It 0 (s)] ~ (- )11.det[Ho (s) + BaR;;l B~r(s)] x
(~6.).
Rem.ark 1: In (23), the quantity G(s)B o represents the open-loop t.ransfer rnatrix ,"vhen all the states are measurable. Also, from (10) a.nd (20) the Inatrix R;;lB~r(s) == Ko is the optinlal gain. l:sing this gain, we express (23) a..~:
+
[rfs)
It is quite cleaT froJn (28) that
(23)
which describes a robust frequency-dornain equality
R o + B~Gt(-s)lQo
==
- 1-£o(s)]
- A o ) -+-
(A~ - Ao)tr(s)]G(s)Bo
(R:F'DE) for the time-lag system
0
~I] [sI
+ (24)
det[H~(-s) +rt(-s)BoR;;lB~]
(29)
"\vhich implies that the (2n) roots of Ho(s) are sylnnletTic about the irnaginary axis and the stable roots are those of det[Ho(s) + BoR;;t B~r(s)J == o. rrhe results achieved in (25), (26) and (29) I'epresellt the second main result as they reveal qualitative properties of the CLTL system.
3397
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FREQUENCY-DOMAIN ROBUSTNESS OF TIME-LAG SYSTEMS...
14th World Congress ofIFAC
ROBUSTNESS PROPERTIES
4
for SOllle Q ~ Qt > O. An interpretation of (37) i~ that for all admissible uncertainties satisfying At ( t).6. (t ) ~ I} Q is a lo\ver-bound on the nominal LQTLR design characterized by Q 0 ~ P and l' (s 7).
An ilnportant special case of the perforulance Ineasure J(u) is when R == p I. In this case the RFDE (24) reduces to
[1 + KoG(s)Bo]t[l + ](-oG(s)DoJ === I + p-l B~Gt(-8){(Jo + ri'(-s)(A~ ~ Aa) (Ad - Ao)tr(s)]G(s)Bo
1
By virtue of (34) if "re guarantee Aj ' [1 + J(s)] 2: 1, it iluplies that wc guarantee that the singular values of the sensitivity functiol1 are le-ss tha!l unity, that is Aj [I + J(S)]-l ~ 1. This in turn guarantees perfOTlnance robustness for the class of uncertain time-Jag systems satisfying ~ t(t.)~(t) S; I. I\1ore illlportantly, by letting Jo(s) ::::: I\oGo(s)B n and subtracting (25) from (30) with R === pI~ ~re can sho\v :
+ (30)
For systeIll (E.a.) and the class of uncertainties satisfying (3), it is readily seen that
[Qo +
r t (-s)(A 6
-
A o)
(A~ -
+
Ao)tr(s)l ? 0
(31)
(I + J(s)]t[I + J(s)] - [1 + J o (s)]t[1
[Qo
+ rt(-8)(A~
=::
p-l B~Gt(- 8 )[r t (-5 )1'1 ~(t)1l1 -r-
~ A o ) + (Ab. - Ao)tr(s)]
f2 t (-s) 0(8)
=
+ Jo(s)]
p-l B~IGt(-s)QoG(s) - G~( -s)QoGo(s)JB o +
\.vhich implies that
l~ft~t(t)lvtr(s)JG(s)Bo
(32)
for sorne ~)(8). By letting J(s) == KoG(s)B o , then froll1 (30)-(31) \ve get:
If + J(s)]t[I + J(s)] J + fJ-1fn(s)G(8)Bor~!2(s)G(s)Bo]
~ 0
(38)
This inlplies
=::
(33)
\vhich implies that
Aj [I
+ J(s)]
=:=
{I + p--l A;[D(s)G(s)Bo ]} 1/2
(34)
In line of (Douglas and Athans, 1994), it is significant to no te that (34) guaT antees the ¥lell-kno-vvl1 ro bustness properties of the LQR design of systems ""vithout delay. Tllese include phase and gain margins. For singleinput single-output systenlS and using COITlplex plan argUluents, expression (34) states that the Nyquist plot of the uncertain tiIne-delay systeln remains outside the unit disk centered at the critical point. In this ,vay, a certain level of robustness to unstructured uncertainty as well as stability and performance robustness to paraluetric uIlcertainty \vill be acquired.
Eqlla tion (39) represents the third rnain result of this paper. It states that the lllaxin1-Unl singular value of the sensitivity function of the nominal LQR design is greater than tl1at of the. R,LQR design at every frequency. This quantifies the iluprovement of performance robustness in the RLQR design based on the sensitivity transfer function. Finally, by evaluating the eingenvalue distribution of the uncertain tiIne-lag systenl (E~ :), vile get
(40) where 1il::t. (s) satisfies t.he indentity
[rt/-s)
To examine the robustness properties of the RFDE (30), \Vc start by substituting (AA - A o ) =: J.\F 6 (t).l\l and keeping first-order uncertaillties. Thus G(s)
[H~(s)J-l
-
-
[I1~(S) 0
~I]
+ 2V' 6,(t)A1GoCs)J
det(sI -
(35) 1~ Vle
+ [P + ~/t ( -s, T )]NL\(t).l\1 + t M ~t(t)Nt[P + V(s, T)J :=:; t t (-20 + .-\-1 Plv'-1\r P + (T~lVt( - 8 , r).lV1V +(A + rr)iWt;.\f 2: 0
'l
r
(s, r) (36)
det[IT 6 (s)] det[II~ (--o)J
(J -ll:T t =::
0
(-
S)
1-C~(s)1 ~
detl-H~ (-s)
sufficient condition for (36) is
+ A ~ 1 P N l\r/, P -t+ cr)I\:f t NI - Q
+(A
(41)
+
+ B o R-;;l B~r(s)]
r
iJ
(
-s)BoR-;:l B~]
x (42)
which repre..~eIlt the (2n) roots of systenl (EA): n-stable roots obtained from det[fI.L3.(s)J == 0 and the relnaining n-roots are unst.able corresponding to det[II~(-s)) :::::: 0 Using (5) ,(7) and (28) into (41) yields
Qo
Q0
~
+ B o R-;;l B~r(s)
(-~)ndettH.Q.(8)
get:
> 0) rr > O. A
!!I]
It is quite clear from (41) that
lJsing (20) and (35) into (31) v.rith the aid of Fact
for some .A that
II~( -s)
rr~(s) :::;: H~(s)
[r(s)
J~
BoR;;} B~
[Ho(s) - 1\F.6.(t).ivI]-l G{)(s)~I
[sI - 'HL>(s)J
t
T) l\l1V 1/( s, -r ) (37)
det~IIo(s) ~
tlA]
det[IIo(s)]det[I ~ II~l(8)~Al d et [IT; (--- s) - A t. A] det[II~(-.s)]det[I - n~t( -s)l\tA] (43)
3398
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FREQUENCY-DOMAIN ROBUSTNESS OF TIME-LAG SYSTEMS...
14th World Congress of IFAC
Olle interpretation of (43) is that, for a given realization of the uncertainties 6...14, the polyno111ial W( s) ::= det[I -n;-l(s)D.A] is the source of the shift in the eigenvalue clustpr in the proper left-half plane. Another interpretation is that the shift in eigcnvalue cluster is symmetric about the inlaginary axis. Note that this result ~rill not be valid if the uncertainties enter the input matrix.
tainty~lEEE
Trans.
~4utomatic
Control, 39, 107-
111.
Doyle , J. C. (1978). Guaranteed 1\-1argins for LQG RegulatoTs,IEEE Trans. Automatic G"ontrol, 26, 756-757.
Doyle ,J. C. and G. Stein (1981). I\r1ultivariable Feedback Design:
Concepts
Synthesis~IEEE
fOT
a
Classical/~'Iodern
Trans . .4utomatic Control, 26, 4-
16.
5
CONCLUSIONS
Flagbedzi , Y. A. and A. E. Pearson (1990)~ Output Feedback stabilizatioIl of Delay SystemR via Generalization of the Transformation Method, 1nl. Journal of Contror', 51, 801-822.
For a class of linear, uncertain tiule-lag systeulS, this paper has: 1) derived a rohust frequency-domain equality exhibits a relation for the actual return-difference transfer function 111atrix of the tinle-lag systelu. 2) provided relevant properties of the closed-loop timelag system \vith regards to the Kalman equation, the circle condition and the Hanliltonian SyStC111. 3) presented a Illeasure of perforrnanee robustness in RLQR design based on the sensitivity trnasfer function.
REFERENCES Anderson , B. D. O. and J. B~ l\1oore (1990)~ Optimal Control: L'inear Quadratic l'vlethods PrcnticeHall, New York~ j
Douglas ) J, and 1\.1..AthallS (1994). Robust Linear Quadratic Designs v~rith R.eal Pal'anleteT Uncer-
Khargonekar , P. P., I. R. Petcrsen and K. Zhou (1990). R.obu::it Stability of lTncertain Systems and Hoc Optimal Control ,IEEE Trans. Automatic Control, 35 , 356-361.
J.S. and P. P. J. Van Den Bosch (1997). Independent-of-Delay Stability Criteria for lJncertain Linear State Space l\1odels, Autom.atica, 33, 1'71-179.
LO"ll,
Mahtnoud ,:Lvi. S. (1994). Output Feedback Stabilization of (Jncertain Systenls with State-Delay'~, in Control and Dynarnic SyBte'm,s, VaL 63~pp. 197257, Acaden1.ic Press, Nevl Y'ork.
Malek-Zavarei , M. and NI. Jalllshidi (1987) TimeDelay Systems: Analysis, Optimization and ..t1pplications~ ~orth-Holland,
Aluestrdam.
3399
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
Copyright to 1999 IFAC 14th Triennial World Congress, Beijing, P.R. China
G-2e-IO-4
FREQUENCY-DOMAIN ROBUSTNESS OF TIME-LAG SYSTEMS Magdi S. Mallnloud Depal~tnlent of Elec:trical and COmp\lter Engineering Kll\Vait TJniversity, PO Box 5969., Safat-13060
KUWAIT Email: magdi(geng.kllniv.ed11.kw
Abstract: This paper investigates the robustness of linear tilne-lag systenls with respect to real parameteric uncertainties. A linear quadratic tirne-lag regulator (LQTLR) is derived in the frequency-domain and its relation to the Kalman equation, the circle eondit.ion and the Hamiltonian s,ysterll is disclosed. Frequency-dolllahl robustness properties are provided. These include performance improvement and closed-loop eigenvalne diRtribut.ion. Copyright © 1999 IFAC Keywords: Tilue-Iag systeIllS, Linear regulators, Performance
1
INTRODUCTION
Linear quadratic Gaussian (LQC) control has been \vell studied for Inore than three decades C_A..ndcrson and ivloore, 1990) _ As a design tool~ it has been based all the assu111ption that th.e dYllamic model of the system under consideratioIl is exactly kno""vn and its disturbances are Gaussian white noise processes with kno\vn statistics. The early result of Doyle (1978) hovvever has sho\vn that the LQG control design cannot guarantee stabilit.y robustness and perfoTInance against modelling uncertainties. This has brought into focu:;; the importance of robust control; which has subsequently attracted significant research interests over the past decade. Several robust control methodologies have been developed in~luding loop transfer recovery (Doyle and Stein~ 1981) , quadratic stabilization (Khargonekar ~ et aI., 1990) and robust linear quadratic design (Douglas and Athans, 1994). all another front of research, control problems of systems with state and/oT input delay have under investigation during the past tVlenty years. Tinle delay often arise in connection \vith system lueasuremcnts or nue to phy~ieal properties such as hydraulic systenls and rolling mills (11alek-Zavarei and Jalnshidi, 1987). Other sources of delays ll"lay occur in the transnlission of inforrnation bet\veen different parts of dynarnical systelTIS including chemical proce,ssing systems, comIIlunjcations and pO~Ter systems. l\/Iost of the rescent research efforts on tirne-delay systeIlls have been directed to robust stability and stabihzation in the tinle-domain; see (Loll and Van Den Bosch, 1997; :vIahmoud, 1994) and their references. L,ittle attention has been paid to robust measures of UI1CeI'tain tirrle-lag sy~teIllS in the frequecy-doHlain. The purpose of thi::> paper is to bridge this gap by et TO bust regulator design of a class
robustness~
Parameteric uncertainties.
of uncertain linear time-lag systenls with quadratic performance measures. It extends the results of (Douglas and Athans, 1994) to the class of time-lag systems with norm-bounded llncertaintie~. Based on the optiJnality conditions, a robust frequency-domain equality is de-
rived which exhibits a frequency-domain relation for the actual return-difference transfer function Inatrix of the time-lag systenl. Its relation to the Kalman equathe circle condition and the Hamiltonian system is disclosed. Frequency-donlain robustness properties are then provided. These include performance improvement and closed-loop eigenvalue distribution. tion~
Notations and Facts In the scquel~ wc denote by )'(lifl ) the transpose} the inverse , the complex conjugate transpose , the determinant and the eigenvalues of any square lnatrix RT. lV' > 0 (lV < 0) stands for a posit.ive- (negative-) definite nlatrix {tiT and C- represents the open left-half of the conlplex plane. SOInetiIIles, the arguluents of a function will be omitted when no confusion can arise. v~lt, W- 1 .' W*, det(W) and
Fact 1: Given matrices El , 2:2 and 1:3 of appropriate dillieIl~iolls \vith L: 1 == ~i. ~1hen ~1
+
L3~(t)E2
where .6,t (t)Ll(t) S I ~1
2
\j
+ 'E~~t(t):Et <
0
t , if and only if for
+ /-L-I :E~E3 + ft:E2I;~ <
SOlllC
J-t
>0
0
PROBLEM DESCRIPTION
In this section, we consider a claRs of uncertain time-lag systems represented by a state space il1.odel of the form:
eJosely exalniniIlg
(Eb.:) x(t)
3395
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FREQUENCY-DOMAIN ROBUSTNESS OF TIME-LAG SYSTEMS...
-+-
lJ(t)
preliminary results. It is v-rell-kno\vn (Flagbedzi and Pe-arson, 1990) that \Vhell systeIll (Eo) is stabiliz;able in the manner of (8), then the unique optimal control is given by:
Dox(t -- T) [~4o
+
14th World Congress of IFAC
+
~A]x(t)
+ EouCt)
Dox(t - T)
(1)
Cox(t)
(2)
where t E ~R is the tirrle~ :.c( t) E :1(-n is the st.ate, ~RTn is the control input and T is a constant scalar representing the arnount of delay in the state . The matrices A o E ~Rnxn, Eo E 2R nxm and Do E :Rnxn are real constant representing the norninal plant \vith the triplet (Aa' B o' Co) being controllable-observable. T'he uncertain nlatrix LiA == _~ A(t)AJ "Htith ~(t) being bounded in the farnl
u,(t) E
where 0 < P
=: pt E ~Rnxn, 0 S VvT E ~Rnxn and !Jt7lxn are the solution of the Riccati Equations
o ~ L, E (RE~s):
o
PA o
x(t)
:)
y(t)
Aox(t) Cax(t)
+ Bnu(t) + Dox(t -
(11)
- [d~~n)] + A~W{n)+D~L{O,a)
o and 1\.1 and N are real constant IllatriceH with appropriate dimensions. Distinct from system(1)-(2) is the nominal system:
(E u
+ A~P + Vf/~(O)Do + D~~V(O)
P B oR;;l B~P + Qo
p BoR~;; 1 B~ ~~/-( 0:)
(12)
8 [a !J + aGO'] L{{J, a)
o
T) (3) (4)
(13)
L(j3, O)B o R;;l B;L(O, a:) along ,vith boundary conditions
The transfer function Toes) from input u to output Y is
P
given by:
A o - e-
ST
DoJ
;::=
In a sinlilar way, we have for system (2:.6,.): (:0
Hll(s)-l B o ~
. .4~
~
e-- ST Do]
p
(5)
v{here H o ( s) is the quasi-characteristic Inatrix. Define the set (6) j\l == {s: det [1/0 ( s ) 1 O}
{sI
Til ( -7); lV(a)
::::=
L( ~T, ex); 0: E {-r, 0]
(7)
==
pt; L(p, Q)
L· t (O!.,j3);o:
=
E [-T~OL;3 E
[-T10J (15)
\~.rhen
the controller (10) is applied to systerll (1)-(2), it produces a stable closed-loop tilne--lag (CLTL,) systern. In the case of systelns \vithout delay, propcr~ ties of the closed-loop system are well-known (.A..rlderson and ~1oore~ 1990) in the tinle-domain as well as the freqllency-doruain. Our task no\v is to obtairl sinlilar properties fOT the CI.TL system. It follows from ell) that
Definition l(Flagbedzi and Pearson~ 1990): System (Eo) is stabilizable if and only if
~P(sI - Aa) - (-sI - A~)P
+ W(O)D o + D~W(O)
~P BoR;;! B~P + Qo == 0
(8)
3
Postnlultiplying (12) by Doe:S over {-r,O] using (14), wc get:
LINEAR QUADRATIC TIMELAG REGULATOR (LQTLR)
P Doe- ST
1=
{:ct(t) Qo x{t)
+
u/(t) R
o
u(t) dt} dt
~
(
-sI -
(16) ct
and integrating wrt a
A~) LOr W( O')DoesC
+ [0.,- D;,L(O, o:)D" (eS"')da -
""Ve associate \vith systeul (2:.6.) the quadratic perfoTn1.ance measure:
.J(u) =
( 14)
and symlnetry conditions
Co H;;l(s) .B a
[sI -
==-
.fr
-PBoR- 1 B t
W(O)D"
W(o:)Do(esC<)do: = 0
(17)
In a sin1ilar "'lay, "re can \"\l'ite:
(9) 'Arhere 0 < Qo :::::: Q~ E ~Rnxn and 0 < R'O = R~ E ~Rmxrn arc state and control weighting matrices. System (ED,) plus perforrnance J (u) forms a linear quadratic tirnelag regualtor (LQTLR). Our ohjective is to exarrline its frequency-domain robustness properties vlith respect to real par ameter unccrt aintics. 'To achieve our go al~ ,ve fo~ eus attention on the IloIninal LQTLR to develop SOlne
-D~lvt(O)
+ eSTD~P
+ LOT e-s,B D~L{(3, O)Dod,B
- JOT e- ,13 D~Wt(p)da(-sI s
- Aa)
_e-s,B D~ l1Jr t (f3)d;3B o R- 1 B~P
== 0
(18)
3396
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FREQUENCY-DOMAIN ROBUSTNESS OF TIME-LAG SYSTEMS...
14th World Congress ofIFAC
Nov{~ Vole prernultiply (13) by e- s /3 D~ , postmultiply the result by DoP.- sa and integrate "vrt Q: , .8 over [-7,0]
This is the first luain result in tIllS paper. Before exploring the properties of the RFDE (23) or (24), we provide some results relevant for the noulinal system (Eo).
using (14)~ we reach:
-1:
D~L(O, n)Do(eSb)da
-L: [Or L: e-
sB
+eST D~
+
-i:
Relllark 2: It is important to observe that the RFDE for system (~o) has the forIn
D~L(f3, O)Do df3
[I
W(a)Do(eSO<)da
R' D
e-sf'lD~wt(;J)dae-ST
c- sB
=
(19)
0
Define
res)
+
p
+ vr(s~ T)
[1 + R;;l B~r(jw)Go(jw)Bu]*Rn [1 + R;l B~r(j4-')Go(jw)BoJ ~ Ra Vw 2:: 0(26)
p+.Lor D~wt(a)e-sadQ
rt(-s)
p
+ Vt(-S,T) (20)
Inequality (26) provides a characterization of the circle condition for tilne-laR systclns where [1 + li;;"1 B~r(j~))Go(jw)BoJ corresponds to the return difference matrixRelDark 4: Frolll (Nlalek-Zavarci and Jamshidi) the Ifamiltonian matrix associated with systClll (b o ) is given by:
By combining (16)-(19) and using (20), we get:
r t (--8 )Ho(s) + Ii~( - 8 )r(s) + r t ( -s)BoR;;-1 B~r(s) - Qo ==
(25)
ReITIark 3: By further substituting s == jw into (25) and noting that the matrix B~G;(-s)QoGo(s)Bo ~ o Vw 2: 0) it fo1101V8 that
LOT vV(Cl)DocSClda
P
+ B;G~(-s)QoGo(s)Bo
"vhere Go(s) ::::::: [.f{o(.t])]-l. Equation (25) is analogous to the well-known Kalman equation in optimal control (Allderson and J\tloorc, 1990). rrhe basic difference lies in the dependence of r(s) OIl the delay factors T and Do,
D~wt(f3)df3BoR:;l
B; lOr W(a)Do(eS")da
+ B~G~(-s)rt(-s)BoR~l]Ro
[1 + R;;lB~r(s)Go(s)Bo) :;:::::
1987)~
(21)
0
which, in view of (5) and (7), can be rewitten as:
(27)
r ( -s)Hb,.(s) + H1 (-/i)f(s) + r t ( -s)Bo R-;;l B;r(s) ~ Q r t ( -S)(AA - Aa) + (A6. - _4. o )tr(s) == 0 (22) t
On usirlg
(21)~
it follows that 11. o (s) satisfies the inden-
tity
By letting G(s) =-= lH~(.~)J-l postmultiplying; (21) by B~Gi, ( -8)~ premultiply by G( s )Bo and rearranging terms, v..re arrive at: j
[I + (I
[rt(-s) _ [ITo(S)
B~Gt(-.9)rt( -s)BoR-;;l]R o
+ R -;l B~r(s)G(s)Bo] = + B~Gt.( -8)[00 + r t ( -S)(.J4~
-
IIo(s)
l
Ro
[1 + I{oG(s)Bo)t Rn[I + KoG(s)BoJ ~ rt.(-s)(A~ - Aa)
(AA - Ao)tr(s)]G(s)B o
!!I]
BaR;;! B~ ] II(s);(-s)
Ho(s)
+ BaR;;! B~r(s)
(28)
det[sI - '"It 0 (s)] ~ (- )11.det[Ho (s) + BaR;;l B~r(s)] x
(~6.).
Rem.ark 1: In (23), the quantity G(s)B o represents the open-loop t.ransfer rnatrix ,"vhen all the states are measurable. Also, from (10) a.nd (20) the Inatrix R;;lB~r(s) == Ko is the optinlal gain. l:sing this gain, we express (23) a..~:
+
[rfs)
It is quite cleaT froJn (28) that
(23)
which describes a robust frequency-dornain equality
R o + B~Gt(-s)lQo
==
- 1-£o(s)]
- A o ) -+-
(A~ - Ao)tr(s)]G(s)Bo
(R:F'DE) for the time-lag system
0
~I] [sI
+ (24)
det[H~(-s) +rt(-s)BoR;;lB~]
(29)
"\vhich implies that the (2n) roots of Ho(s) are sylnnletTic about the irnaginary axis and the stable roots are those of det[Ho(s) + BoR;;t B~r(s)J == o. rrhe results achieved in (25), (26) and (29) I'epresellt the second main result as they reveal qualitative properties of the CLTL system.
3397
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FREQUENCY-DOMAIN ROBUSTNESS OF TIME-LAG SYSTEMS...
14th World Congress ofIFAC
ROBUSTNESS PROPERTIES
4
for SOllle Q ~ Qt > O. An interpretation of (37) i~ that for all admissible uncertainties satisfying At ( t).6. (t ) ~ I} Q is a lo\ver-bound on the nominal LQTLR design characterized by Q 0 ~ P and l' (s 7).
An ilnportant special case of the perforulance Ineasure J(u) is when R == p I. In this case the RFDE (24) reduces to
[1 + KoG(s)Bo]t[l + ](-oG(s)DoJ === I + p-l B~Gt(-8){(Jo + ri'(-s)(A~ ~ Aa) (Ad - Ao)tr(s)]G(s)Bo
1
By virtue of (34) if "re guarantee Aj ' [1 + J(s)] 2: 1, it iluplies that wc guarantee that the singular values of the sensitivity functiol1 are le-ss tha!l unity, that is Aj [I + J(S)]-l ~ 1. This in turn guarantees perfOTlnance robustness for the class of uncertain time-Jag systems satisfying ~ t(t.)~(t) S; I. I\1ore illlportantly, by letting Jo(s) ::::: I\oGo(s)B n and subtracting (25) from (30) with R === pI~ ~re can sho\v :
+ (30)
For systeIll (E.a.) and the class of uncertainties satisfying (3), it is readily seen that
[Qo +
r t (-s)(A 6
-
A o)
(A~ -
+
Ao)tr(s)l ? 0
(31)
(I + J(s)]t[I + J(s)] - [1 + J o (s)]t[1
[Qo
+ rt(-8)(A~
=::
p-l B~Gt(- 8 )[r t (-5 )1'1 ~(t)1l1 -r-
~ A o ) + (Ab. - Ao)tr(s)]
f2 t (-s) 0(8)
=
+ Jo(s)]
p-l B~IGt(-s)QoG(s) - G~( -s)QoGo(s)JB o +
\.vhich implies that
l~ft~t(t)lvtr(s)JG(s)Bo
(32)
for sorne ~)(8). By letting J(s) == KoG(s)B o , then froll1 (30)-(31) \ve get:
If + J(s)]t[I + J(s)] J + fJ-1fn(s)G(8)Bor~!2(s)G(s)Bo]
~ 0
(38)
This inlplies
=::
(33)
\vhich implies that
Aj [I
+ J(s)]
=:=
{I + p--l A;[D(s)G(s)Bo ]} 1/2
(34)
In line of (Douglas and Athans, 1994), it is significant to no te that (34) guaT antees the ¥lell-kno-vvl1 ro bustness properties of the LQR design of systems ""vithout delay. Tllese include phase and gain margins. For singleinput single-output systenlS and using COITlplex plan argUluents, expression (34) states that the Nyquist plot of the uncertain tiIne-delay systeln remains outside the unit disk centered at the critical point. In this ,vay, a certain level of robustness to unstructured uncertainty as well as stability and performance robustness to paraluetric uIlcertainty \vill be acquired.
Eqlla tion (39) represents the third rnain result of this paper. It states that the lllaxin1-Unl singular value of the sensitivity function of the nominal LQR design is greater than tl1at of the. R,LQR design at every frequency. This quantifies the iluprovement of performance robustness in the RLQR design based on the sensitivity transfer function. Finally, by evaluating the eingenvalue distribution of the uncertain tiIne-lag systenl (E~ :), vile get
(40) where 1il::t. (s) satisfies t.he indentity
[rt/-s)
To examine the robustness properties of the RFDE (30), \Vc start by substituting (AA - A o ) =: J.\F 6 (t).l\l and keeping first-order uncertaillties. Thus G(s)
[H~(s)J-l
-
-
[I1~(S) 0
~I]
+ 2V' 6,(t)A1GoCs)J
det(sI -
(35) 1~ Vle
+ [P + ~/t ( -s, T )]NL\(t).l\1 + t M ~t(t)Nt[P + V(s, T)J :=:; t t (-20 + .-\-1 Plv'-1\r P + (T~lVt( - 8 , r).lV1V +(A + rr)iWt;.\f 2: 0
'l
r
(s, r) (36)
det[IT 6 (s)] det[II~ (--o)J
(J -ll:T t =::
0
(-
S)
1-C~(s)1 ~
detl-H~ (-s)
sufficient condition for (36) is
+ A ~ 1 P N l\r/, P -t+ cr)I\:f t NI - Q
+(A
(41)
+
+ B o R-;;l B~r(s)]
r
iJ
(
-s)BoR-;:l B~]
x (42)
which repre..~eIlt the (2n) roots of systenl (EA): n-stable roots obtained from det[fI.L3.(s)J == 0 and the relnaining n-roots are unst.able corresponding to det[II~(-s)) :::::: 0 Using (5) ,(7) and (28) into (41) yields
Qo
Q0
~
+ B o R-;;l B~r(s)
(-~)ndettH.Q.(8)
get:
> 0) rr > O. A
!!I]
It is quite clear from (41) that
lJsing (20) and (35) into (31) v.rith the aid of Fact
for some .A that
II~( -s)
rr~(s) :::;: H~(s)
[r(s)
J~
BoR;;} B~
[Ho(s) - 1\F.6.(t).ivI]-l G{)(s)~I
[sI - 'HL>(s)J
t
T) l\l1V 1/( s, -r ) (37)
det~IIo(s) ~
tlA]
det[IIo(s)]det[I ~ II~l(8)~Al d et [IT; (--- s) - A t. A] det[II~(-.s)]det[I - n~t( -s)l\tA] (43)
3398
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FREQUENCY-DOMAIN ROBUSTNESS OF TIME-LAG SYSTEMS...
14th World Congress of IFAC
Olle interpretation of (43) is that, for a given realization of the uncertainties 6...14, the polyno111ial W( s) ::= det[I -n;-l(s)D.A] is the source of the shift in the eigenvalue clustpr in the proper left-half plane. Another interpretation is that the shift in eigcnvalue cluster is symmetric about the inlaginary axis. Note that this result ~rill not be valid if the uncertainties enter the input matrix.
tainty~lEEE
Trans.
~4utomatic
Control, 39, 107-
111.
Doyle , J. C. (1978). Guaranteed 1\-1argins for LQG RegulatoTs,IEEE Trans. Automatic G"ontrol, 26, 756-757.
Doyle ,J. C. and G. Stein (1981). I\r1ultivariable Feedback Design:
Concepts
Synthesis~IEEE
fOT
a
Classical/~'Iodern
Trans . .4utomatic Control, 26, 4-
16.
5
CONCLUSIONS
Flagbedzi , Y. A. and A. E. Pearson (1990)~ Output Feedback stabilizatioIl of Delay SystemR via Generalization of the Transformation Method, 1nl. Journal of Contror', 51, 801-822.
For a class of linear, uncertain tiule-lag systeulS, this paper has: 1) derived a rohust frequency-domain equality exhibits a relation for the actual return-difference transfer function 111atrix of the tinle-lag systelu. 2) provided relevant properties of the closed-loop timelag system \vith regards to the Kalman equation, the circle condition and the Hanliltonian SyStC111. 3) presented a Illeasure of perforrnanee robustness in RLQR design based on the sensitivity trnasfer function.
REFERENCES Anderson , B. D. O. and J. B~ l\1oore (1990)~ Optimal Control: L'inear Quadratic l'vlethods PrcnticeHall, New York~ j
Douglas ) J, and 1\.1..AthallS (1994). Robust Linear Quadratic Designs v~rith R.eal Pal'anleteT Uncer-
Khargonekar , P. P., I. R. Petcrsen and K. Zhou (1990). R.obu::it Stability of lTncertain Systems and Hoc Optimal Control ,IEEE Trans. Automatic Control, 35 , 356-361.
J.S. and P. P. J. Van Den Bosch (1997). Independent-of-Delay Stability Criteria for lJncertain Linear State Space l\1odels, Autom.atica, 33, 1'71-179.
LO"ll,
Mahtnoud ,:Lvi. S. (1994). Output Feedback Stabilization of (Jncertain Systenls with State-Delay'~, in Control and Dynarnic SyBte'm,s, VaL 63~pp. 197257, Acaden1.ic Press, Nevl Y'ork.
Malek-Zavarei , M. and NI. Jalllshidi (1987) TimeDelay Systems: Analysis, Optimization and ..t1pplications~ ~orth-Holland,
Aluestrdam.
3399
Copyright 1999 IFAC
ISBN: 0 08 043248 4