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Stall control for axial compressors*
STALL CONTROL FOR AXIAL COMPRESSORS...
14th World Congress ofIFAC
E-2c-11-6
Copyright © 1999 IFA C 14th Triennial Wodd Congress, Beijing, P. R. China
STALL CONTROL FOR AXIAL COMPRESSORS * Calin Belta§, Guoxiang Gu§,l, Andrew Sparks t , and Siva Bandat
§Department of Electrical and Computer Engineering Louisiana State University, Baton Rouge, LA 70803-5901, U.S.A. tFlight Dynamics Directorate, Wright Laboratory \Vright-Patterson Air Force Base, Dayton, OH 45433-7531, V.S.A.
Abstract: Rotating stall is a primary constraint fol' the performance of flow compressors. This paper establishes a necessary and sufficient condition for a feedback controller to locally stabilize the critical equilibrium of the uniform flOVl at the inception of rotating stall. The explicit condition obtained in this paper provides an effective synthesis tool for the stall control. Copyright © 1999 IFAC Keywords:
Hopf Bifurcation, Rotating stall, local stability, stabilization.
1. INTRODUCTION ~~xial
This paper focuses on rotating sta.ll control based on the multi-mode rvloore-Greitzer model [16], which in the limiting case converges to the full PDE (partial differential equation) model of ivloore-Greitzer [18]. A feedback control law, consisting of both linear and quadratic feedback terms, will be investigated which is a generalization of the feedback control laws in [15, 20, 5] for the third order (or single-mode) MooreGreitzer model. It will be shown that the critical operating point a.t the peak pressure rise of the performance characteristic curve is locally stabilizable with the proposed feedback control law_ .A. necessary and sufficient condition will be derived for local stabiliza.tion of the underlying feedback compression system. This condition holds, and results in lumped feedback controllers for the full PDE Moore-Greitzer model in the limiting case. In a special case, the proposed feedback control law reduces to the linear feedback control law proposed in [5] Vv"hich was also studied in [14] recently. In contrast to the global stabilization results in [4], our local stabilization conditions are explicit, and provide an effective synthesis tool for rotating stall control "9lithout distributed sensors. Hence our result compliments those in [4].
flow compressors are subject to two distinct
aerodynamic instabilities, rotating stall and surge, which can severely limit the compressor performances.
Both these inst abilities are disruptions of the normal operating condition which is designed for steady and
axisymmetric flow. Rotating stall is a severely nonaxisymmetric distribution of axial flow velocity, tak-
ing the forlll of a wave or "stall cell" , that propagates steadily in the circumferential direction at a fraction of the rotor speed. Surge~ on the other hand, is an axisymmetric oscillation of the m.ass How along the axial length of the compressor. Both can result in catastrophic consequences. Feedback control v..~ proposed to improve compressor performance in [6], and has since received great attention in recent years. The existing results include linear control law [19], bifurcation stabilization [15, 20], and backstepping lllethod [13, 4].
*
This research was supported in part by A.FOSR and ARO.
The corresponding author with email: ggu'~ee.lsu.edu, and fax: (225) 388-5200. 1
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Copyright 1999 IFAC
ISBN: 0 08 043248 4
STALL CONTROL FOR AXIAL COMPRESSORS...
14th World Congress of IFAC
2. MULTI-MODE COMPRESSOR MODEL J.~
The above multi-mode model converge to the PDE model described in (1) and (2) uniformly as lV -+
schematic axial compressor is sho,vn in Figure 1. Throttle Y
TeN
Ps
= eNV2N + 1,
TTe,V
= ~. 2~r + 1
The performance characteristic curve
Wc (.)
is assumed
to be a cubic polynomial [18], and given by Plenum Vp
'l.jJc(r/JN) Figure 1
Schematic of compressor.
= H [coeN + Clr:PN + C3tPJ]
•
(8)
where x· 2 ;;;;;:: x . x denotes Hadmard product [9], and x· k raising each element of x to the poviter of k. The uniform flow can be represented by
A post-stall model for compression systems was develGreitzer~ and its full PDE form is
oped by iv100re and described by [18]
=
Proposition 2.1 The steady equilibria (r.PN, '1') (l/JeeN, We) for the multi-mode compressor model in (4) and (6) are dependent on the throttle parameter r, and
governed by;
>it
-1'~),
4;210 (
rjJ
= + 'P1'7=o'
(2)
The first equality in (9) is obtained by setting
(PN =
0
in (4), while the second equality in (9) by setting q, == o in (6). The proof is thus omitted. ~4.. local model around equilibria (rPeeN, We) has the form
All notations axe consistent with [18}. The incompressible and irrotational assumption the gas flow implies the existence of disturbance flow potential that satisfies Laplace's equation ~rith zero boundary condition at 11 == -ip ::; -le) 'w'hich in turn implies that the disturbance flow at station 0 has the form:
on
:i:
= f('i x)
= Lx
+
Q[x,x]
+ C[x, x,x] +".,
(10)
N
CPI"l=O
= L- (An cos(n9) + B n sin(n8)) ,
(3)
n=l
where !'v" -+ 00. Let {e j9n }~~o be a set of 2N + 1 uniformly distributed samples on the unit circle. Denote rjJ(8 z•v +d
],
1
1
.,j2
--./2 cos 92 N+l sin ()2N+l
COS
e1
sin (}l
cos 1V02N+l sin NfJ 2 1\l+1
cos lV(h, sin N(J2
COS N(}l
sin j\"T 81
diag (lel m 1 I 21 m212,··· ~ m N I 2),
E
m cosh(nl nsinh (nlp
1
F )
mn F
diag (0, F l
a ) + -,
I
,FN
),
,'"
1
TT E- FT, M2
2 =:
[10 10] '
Fn =
[~
\Vhile Ko is independent of the throttle parameter, T and 0 are functions of l' By the form of eN, the linear matrix can be written as:
-01;],
L = S-lDS~
= TT E~lT~
Do =
where t N
f2
= V2N+1'
T- 1 =TT.
K.]
[T
a/le
-K T
for 1 $ n :::; lv
•
)
D.rjJN ,j,
M 1cPlv
~
+ M27/Jc(~ifJN) -
- eN E R
Dn --
[n
nIb
-nib] -1 a mn
The eigenvalues of L can now be easily
ZN
M 2en I
+1 ,
A1,2
(4)
==
),2,,+1,2>1.+2
(5)
a.5rc ± 0.5j
v"4(K
2
= (Cl! ± j ~) m;;-l,
+ Ta/le) - Tl,
(17)
= T + et/le
(18)
Te
dZle (:J~1 -1'#) ,
(6)
where n =:; 1; 2; ... J N. functions of '""'I.
[1
(7)
Assume S-shape for the performance characteristic curve tf;i;('), and Cl + 3C3 = 0 as in [18J. A schematic
1
1
cOlllputed as
Then the full PDE model can be approximated by [16]
rPN
D == diag(Do,D 1 , · · · ,DN))
1 ] .
Hence eigenvalues are also
2335
Copyright 1999 IFAC
ISBN: 0 08 043248 4
STALL CONTROL FOR AXIAL COMPRESSORS...
14th World Congress of IFAC
compressor characteristic is shown in Figure 2. The designed operating point is uniquely determined by
the intersection of the throttle line (dashed lines A-B or C-D for two different values of the throttle position) with the compressor performance curve 1fC(')i according to (9). The maximum pressure rise takes place at a = 0 (point A) because that is "\vhere the derivative of "-Pc. C·) equals to zero. Since a < 0 on the right side of the maximum pressure rise, the lv pairs of eigenvalues T
of L Inatrix in (18) are stable. By T < 0 the pair of eigenvalues of (17) are also stable. Hence any point on right side of the peak of 1/Jc (.) is a stable operating point, and the uniform flow is a stable equilibrium, as highlighted by the solid line. However if the throttle value , decreases~ then the flow rate intensity
Al
"Ye
VW; e=ifJcl W"e='1'c
=
1Jc -Ji/Jr;(f/lc)
l
stalled flo,v for the compression system. In the next section, some of the existing results on Hopf bifurcations Vi,rill be reviewed which will be used in Section 4 for the synthesis of a stabilizing feedback control law.
3. STABILITY FOR HOPF BIFURCATIONS This section reviews briefly the projection and Lyapunov methods for local stability of Hopf bifurcations. For stability analysis of Hopf bifurcations, the system under consideration is the following nth order parametrized nonlinear system: x=!("X),
!("Y,xo) =0 \f--yER, xoESca
n ,
(20) where x E Rn, ! is a real-valued parameter, and S a linear subspace of Rn l to be clarified later. It is assumed that f (', .) is sufficiently smooth such that the equilibrium solution X e , satisfying f("Y, x e ) == 0, is a smooth function of ,. The smoothness of f(·,·) implies the existence of a Taylor series expansion near the origin of Rn in the forID of
x = fe"~ x) = L(-y)x + Q(')') [x, x] + C(,)[x, x, x] + ... (21) where L(,)x, Q(l")[x,x], and C("y) [x, x,x] can each be expanded into
(19)
c=2,"V
2W
L(,)x Q(,) [x, x]
and A2n+l , 2n+2 == ±jwn V\o;'th W n = %m;;l for 1 ::; n :5 .IV. .A.t the criticality 'r = "Ye, Hopf bifurcations occur for the multi-mode model of the compression sy·stem in (4) and (6) that induce rotating stalL Furthermore if the underlying bifurcation is subcritical, or unstable, the ~l\-C portion of the stall curve is unstable which is sketched for 1'1 = 1. Stall cells will be born at point i\., and gl'o",r which will throttle the operating point frOIn _I\. to B quickly which is a stable operating point., although undesirable. There is a tremendous drop in both the pressure rise, and floV\o" rate. Moreover increasing the throttle position and flow rate at point B does not increase the pressure rise. Rather the pressure rise becOIll€S even IOVv"er before it comes to point C at which it jUIIlPS back to (due to again loss of stability) the performance curve Wc(·). The hysteresis loop A-B-C-D is the main cause for loss of compressor performance, and the potential damage to aeroengines.
C(j)[x, x, x)
Lox
+ 6'JL l x + (01')2 L 2 x + ... ,
Qo[x,x]
+O~Ql[X,X]
Co[x, x, xl
+"',
+ o,C1 [x, x, x] + ... ,
with 0'1 = r -le, and Lo, L l , £2 constant matrices of size n x n. Suppose that L( "y) possesses m « n/2) pairs of complex eigenvalues Ak(....,,) = ak CT) ± j!3k (7) , dependent smoothly on ,. It is assumed that for 1 ~ k~m,
l.l
= 0,
/3('Ye) = Wk
d;;
¥- 0, O:~('Ye) =
(-Ye)
¥- 0,
(22) while all other eigenvalues of L( 1') are stable at, and in a neighborhood of J == Ic. That is, m pairs of eigenvalues cross the imaginary axis simultaneously. . Then each Ak(,) is a critical eigenvalue, so is its conjugate. The center space, or the eigen-space for the m pairs of critical eigenvalues is denoted by Se, and is assumed to be orthogonally complement to S in the sense that S E& Sr; = R 11. Since projection of Xo E S to Se is zero, the equilibria Xo satisfying (20) will be called zero solution. The strict crossing assumption implies that the zero solution changes its stability as ~ crosses rc' For instance Ct~ (le) > 0 implies that the zero solution is locally stable for J < re) and becomes unstable for { > 'Ye. The crucial problem is the determination of local stability near the critical parameter le at \vhich Hopf bifurcations occur, and the periodic solutions are born.
'Y
= YC~I"""" ~-
3.1 The Projection fv1ethod
"{ > 'Ye
......,;,-=-':.~::;....~-_~----------------'--------:=-
o
Figure 2
2W
4le
The center space Se, spanned by all the critica.l eigenvectors, is complete. IT all the critical eigenvalues are distinct, Se = SC1 ffi SC2 EB . - . ffi SC
Compressor characteristic, and stall.
Tn
The above discussion indicates that stabilization of the critical operating point, and A-C portion of the stall curve is a key issue for rotating stall control ,vhich will eliminate the hysteresis loop, and allow smooth change of the operating point between axisymmetric flow and
,
where SCk denotes the eigen-space spanned by kth pair of the critical eigenvectors. The projection method in [11] projects the nonlinear dynamics into the subspace of SCk so that stability of the projected dynamics can be analyzed4 Associated with each pair of the critical
2336
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
STALL CONTROL FOR AXIAL COMPRESSORS...
eigenvalues (.Ak, A/c), there can define a SCV (stability characteristic value) X~k) whose sign determines local
for k, p
= 1, ... ~ m
11Jith k
f:. p,
where
stability of the projected dynamics. See [11, 10~ 1] for details. Let the Taylor series of f (1' i x) be of the form in (21) where L o L(O). ..~n algorithm to compute >..~k) is outlined next.
and
Vk
'2Qo[rk) rk].
(23) (24)
Due to unpredictability, it is difficult to design axial flow compressors such that the critical operating condition corresponding to rotating stall is stable. Thus feedback control is proposed to suppress rotating stall [6J. This paper considers the use of bleedvalve as actuator for the compression system. It is noted that the throttle parameter is a composition of uncertainty 1 0 ; synthesized disturbance from inlet and combustion chamber~ and the bleed valve position o--y that can be employed as actuator. Assume that W is measurable. Then the throttle parameter ca.n be expressed as
+ ikQo[rk, Vk] +~fhCoITk,rh,fh]}. (25)
2Re {2lkQo[rk> lik]
Theorem 3.1 Suppose that £0 has m pairs of nonzero critical eigenvalues on the imaginary axis with the rest on open left half plane. Then the projected dynarnics along the kth pair of the eigenvectors is stable, if ~2 < 0, and unstable if)..2 > O. For the case of l'l = 1, the associated Hop! bifurcation is supercriticalor stable, if;\2 < 0, and subcritica~ or unstable, if .\2 > O.
"Y
x=
z
16Re {lk (Qo[rk ~ jlp]
+Qo{rp, I/k p ]
p ]
= Lx + Q[x,x] + G[x
j
x, x]
+ ... + ru, (31)
Dz +
sru, sr = [ e~ 0
] K.,j(2N
+ l)lc,
0
D
o Thus the state variables corresponding to the subsystem of Do are deviations jn pressure rise Jw = W - 'lie, and in average flo,v rate ~ 1» = ep - 4>e respectivelY4 The two eigenvalues of Do as in (17) are associated with surge dynamics, and are linearly controllable. The state variables corresponding to subsystems Dl DN represent amplitudes of harmonics for disturbance flow rate in (3). Thus the N pairs of eigenvalues for D 1 ~ DN are associated with the rotating stall dynamics, and are linearly uncontrollable. While linear feedback law can be used for surge control, the uncontrollability for the critical modes of D 1 ~ D .v implies that quadratic feedback law is more appropriate for rotating stall control [11. The above consideration suggests f'.I
(26)
+ Qo[rp~ JJk + ~Co[Tk,rPlrp])}
fe"~ x)
Do
Theorem 3.3 Suppose that L o admits m > 1 pairs of eigenllalues on the imaginary U:I:is with the 1'est on open left half plane, and the nonlinear system (21) is locally center-symmetric in the sense of Lyapunov. Then there exists a Lyapunov function that guarantees local stability of (21) at the 'criticality, if
Xkp
(30)
=
Definition 3.2 Under the condition that L o admits lV > 1 pairs of eigenvalues on the imaginary axis with the rest on open left half plane, the nonlinear system (21) is said to be locally center-symmetric in the sense of Lyap'unov if (i) Wi =1= qwp for q = 1,2,3, whenever 'fn ~ 2, (ii) qWi =f:. w p +Wk, Wi #- qw p +Wk for q = 1,2, whenever m ~ 3, and (iii) Wi ±wp i= Wk +WI, whenever m 2: 4 where ail indices are distinct.
= 1, ... ,rn,
u
Vii'
vlhere r = [ 0 0 1] T ~2-lc, and Lx, Q[x,::cl, and G[x, x, x] are linear, quadratic, and cubic terms respectively as given in (11)-(15) with [ replaced by 1 0 ' Applying similarity transform z Sx yields the linearized system
The Lyapunov method for multiple critical modes is developed by Fu [7]. It gives sufficient conditions on the existence of a Lyapunov function that guarantees local stability of the bifurcated system. The following notion is needed.
for k
-
ten as
3.2 The Lyapunov Method
8X~k) < 0,
== ')'() + Or = 10
where u is the feedback control law to be designed. Our objective is local (asymptotic) stabilization around the critical equilibrium of the uniform flow (4)ceN) wc) for the closed-loop cOIllpression system. The compression control system can now be writ-
It should be clear that if £0 admits only a single pair of imaginary eigenvalues, then local stability of the bifurcated system is equivalent to X~k) < O. However if L o admits m.ore than one pair of iInaginary eigenvalues, local stability of each projected dynamics does not guarantee local stability of the bifurcated systeIns. In this case, local stability of the bifurcated systems can be tackled by the Lyapunov method [7].
Xkk
(29)
4. LOCAL FEEDBACK STABILIZATION
• Step 3: The coefficient 5,.~k) is given by
j~k)
(28)
Since Xkk == 8X~k). local stability of Hopf bifurcation with multiple pairs of critical modes requires more than local stability of each projected dynamics. Roughly speaking, the nonpositivity of Xkp for k 1= p takes the coupling of different pairs of critical modes into account for local stability of the bifurcated system in (21).
from the
1
satisfy
"2 Qo [r k ' T p ].
equations
~Qolrh' rh],
Vk p
1
• Step 1: Compute left row eigenvector ik and right column eigenvector Tk of £0 corresponding to the kth critical eigenvalue of Ak(D) = jWk- Normalize by setting lkrk = 1. J.tk
J
iQolrk, rp ],
=
• Step 2: Solve column vectors
j..tk p
(27)
:5 0,
2337
Copyright 1999 IFAC
ISBN: 0 08 043248 4
STALL CONTROL FOR AXIAL COMPRESSORS...
14th World Congress ofIFAC
the employment of the following feedback control law: (32)
The linear term needs be designed to take care of the surge dynamics, and the quadratic terms need be designed to stabilize the rotating stall dynamics locally. The linear feedback term involving the average disturbance flo~v is intentiously skipped because of the difficulty in measuring Xl ~ X2N+1 which requires distributed sensors as N ~ CXJ. For the same reason, more general quadratic form is avoided in the proposed feed back control law. Define discrete Fourier transform
o ::; n It
Vl.~il1
:s 2N.
(33)
cornpressor parameters and linear feedback gain K'li. As a result) distributed sensing for the flow rate rP1V is unnecessary in the limiting case N -+ 00. 11oreover precise information on the cubic performance characteristic curve, and on the critical operating condition is not required, provided that unpper and lower bounds on C3~ H, W~ epc, and rc/~ are available. RelDark 4.3 In f4], backstepping method was used to derive a global stabilization condition on a more general state feedback control law than (32). It was shown tha.t any given operating point (or set point) 1 0 2: "Ye can be globally stabilized. It should be clear that if the set point is chosen as at the criticality) local stability will be ensured for ro = 'c. However it requires distributed sensing as N ---t 00. Moreover the stabilizability condition in [4] is less explicit than the one in (35). The stabilization condition (35) also indicates that the critical operating point can be locally stabilized by using linear feedback control law, i.e., by setting KQ = o in (32). The next result follows easily from Theorem 4.2 1 and thus the proof is skipped.
be shown later that local stabilization requires
i
0, rather than KQ;. :f:. 0 for all i > O. Hence distributed sensors are unnecessary in the Hmi ting case l\l --+ 00. For the feedback control law as in (32), the closedthat
Assume that C3 < DJ rPc > W; and the condition (34) is satisfied~ Moreover suppose that the imaginary part of the eigenvalue.t; in (18) are locally center-symmetric in the sense of Lyapunov. Then the rotating stall dynamics is locally asymptotically stable at the criticality 1'0 = re under feedback control u K'l-8w; if and only if
Corollary 4.4
loop compression system can now be expanded into the
follO"~ving
X L*
Taylor series: L*x
+
Q*[x,
8- 1 D*S,
[ [
D~
Q*[X,X]
KQ
T
xl + C*[x~x) xl +.'., = diag(D(;, D~ ~ ... ) DAr),
+ ",2l c K..:v -K
3~2~
=
D*
~ le
]
(tv
~ 1) kf2 2 K l c KQ
[ KQt
D~
1
KQ2N+l
= Dn ,
02N+l T
..
]
X ·2
,
By the fact that satisfies (34).
-4W e ],
Do
Under the feedback control law (32}) the surge dyn.amics is locally stable at the criticality '"Yo = le, if and only if
K,
lc
(34)
're
'YQ="1c
= 2~'
C3
< 0,
( ~c ) W- 1
2
-
(36)
stabilizing gain K'I! exists that
measurement of pressure rise was considered in [8, 5] for the third order !vloore-Greitzer model which is a crude approximation. Corollary 4.4 is surprising ~"hich indicates that linear control law is also effective in stabilization of the critical operating point for the multimode model, which in the limiting case converges to the PDE Moore-Greitzer model. It should be pointed that the same feedback control law was studied in [14] as well. Using eqllilibria dependent feedback control, [14] was able to obtain a global stabilization result for ""to > ""'/c 1. In combination with the result in Corollary
LelTIm.a 4.1
< - ~I 2
VV
ReD'lark 4.5 Feedback stabilization using only the
and O*[x,x,x] = C[x,x,x]. It is noted that is associated with the surge dynamics, and n:, with the rotating stall dynamics for n ?: 1. The follo\ving result can be easily shown.
K\l!
"Ye
K~>2~+12Hc3
The main result of this section is the following.
Its proof is based on the projection method, and Lyapunov method in Section 3. Due to the space limit, the proof is omitted.
4.4, global stabilization can be claiIned for "'to
~
re·
Ass'Urne that C3 < 0, W and the condition (34) is satisfied. Moreover suppose that the imaginary part of the eigenvalues in (18) are locally center-symmetric in the sense of Lyapunov. Then the
TheorelTI 4.2
J
rotating stall dynamic.s is locally a.symptotically stable at the criticality (0 = le for the feedback compression
system described in (34) '" (34), if and only if
The significance of Theorelll 4.2 lies in the fact t.hat
lIn [14)1 glQbal stabilization condition was claimed for 1'0 ~ le. which is incorrect. In light of Corollary 4.4, condition (36)
the stabilizability condition (35) requires only the s~m of all KO t ~s to satisfy a bound determined by various
needs be enforced at
"{o
:=;;
"'/c
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ISBN: 0 08 043248 4
14th World Congress ofIFAC
STALL CONTROL FOR AXIAL COMPRESSORS...
References [1] E.H. Abed and Jyun-Horng Fu, ~(Local feedback stabilization and bifurcation control) 1. Hopf bifurcation," Systems and Control Letters, 7 (1986) i 11-17.
[12] "\Vei Kang, "Bifurcation control via state feedback for single input nonlinear systems: Part I and Part 11" , preprint, 1996. Krstic~ J.M. Protz, J.D. Paduano~ and P.V. Kokotovic, UBackstepping designs for jet engine stall and surge control," Proc. of 34th IEEE Conf. Dec~ and Contr., 3049-3055, 1995.
[13] M.
[2J R.A. Adomaitis and E.H. Abed, "Bifurcation analysis of nonuniform flow patterns in axial-flow gas compressors," in 1st World Congress of Nonlinear Analysis, Aug. 1992.
[14] A. Leonessa, 'l.-S. Chellaboina, and W.M. Had-
[3] 0.0. Badmus, S. ChowdhurYt and C.K. Nett, "Nonlinear control of surge in axial compression systems," Automatica,'VoL 32,59-70,1996. [4J A. Banaszuk, H.A. Hauksson
j
dad, HGlobally stabilizing controllers for multimode axial flow compressors via equilibriadependent Lyapunov functions)" Proc. 1997 A mer. Contr. Conf.) 993-1002, 1997.
and I. Mezic
(1996), "A Backstepping controller for Moore-
[15] D.-C. Liaw and E.H. Abed, "Active control of compressor stall inception: A bifurcationtheoretical approach," A'Utomatica~ voL 32, 109-
Greitzer PDE model describing stall and surge in compressors", prcprint.
116, 1996.
[5] X. Cbeli, G. Gu, P. 11artin, and K. Zhou, "Rotating stall control via bifurcation stabilization," Automatica: vol 34, 437-443, l\pril 1998.
[16] C.A. Mansoux, J.D. Setiav~. . a n, D.L. Gysling) and J.D. Paduano~ "Distributed nonlinear modeling a.nd stability analysis of ~"'Cial compressor stall and surge," Proc. 1994 Amer. Contr. Con!., 23052316, 1994.
[61 A.H. Epstein, J.E. FfO-"TCS Williams, and E.1vI. Greitzer, "Active suppression of aerodynamic instabilities in turbomachinery," J. Propulsion) vol. 5, 204~211) 1989.
[17] F .E. McCaughan, "Bifurcation analysis ofaxial flow compressor stability/ SIAM J. Applied Mathematics, vol. 20, 1232-1253, 1990.
[7] J.-H Fu) aLyapunov functions and stability cri-
teria for nonlinear systems "rjth multiple critical modesH'l Mathematics of Contr. , Sig., and Syst.)
[18] F.K. Moore and E.M. Gieitzer (1986), "A theory of post-stall transients in axial compressors: Part. I - development of the equations," ASME J. of Engr. for Gas Turbines and Power, voL 108, pp. 68-76.
7, 255-278, 1994. [8] G. Gu) A. Sparks, and C. Belta (1997), "Stability analysis for rotating stall and surge in axial flow cornpressors", preprint.
[19] J.D. Paduano, A.H. Epstein, L. Valavani, J.P.
[9] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
Longley~
E.J\,L Greitzer,
and G.R. Guenette
(1993), 'tActive control of rotating stall in a lowspeed axial compressor/' J. Turbomachinery, vo1. 115, 48-56.
[10] LaN. Howard) "Nonlinear oscillations," in: F.G. Hoppensteadt, Ed' J Nonlinear Oscillations in Biology, (Amer. Math. Soc., Providence, RI, 1979), pp. 1-68.
[20} H.O. Wang, R.A. Adomaitis and E.H. Abed, "Nonlinear analysis and control of rotating stall in axial flow compressors/' in American Control
[11] G. Iooss and D.D. Joseph, Elementary Stabil-
Conference~
ity and Bifurcation Theory, Springer-Verlag, New
2317-2321, 1994.
York, 1980.
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Copyright 1999 IFAC
ISBN: 008 0432484
14th World Congress ofIFAC
E-2c-11-6
Copyright © 1999 IFA C 14th Triennial Wodd Congress, Beijing, P. R. China
STALL CONTROL FOR AXIAL COMPRESSORS * Calin Belta§, Guoxiang Gu§,l, Andrew Sparks t , and Siva Bandat
§Department of Electrical and Computer Engineering Louisiana State University, Baton Rouge, LA 70803-5901, U.S.A. tFlight Dynamics Directorate, Wright Laboratory \Vright-Patterson Air Force Base, Dayton, OH 45433-7531, V.S.A.
Abstract: Rotating stall is a primary constraint fol' the performance of flow compressors. This paper establishes a necessary and sufficient condition for a feedback controller to locally stabilize the critical equilibrium of the uniform flOVl at the inception of rotating stall. The explicit condition obtained in this paper provides an effective synthesis tool for the stall control. Copyright © 1999 IFAC Keywords:
Hopf Bifurcation, Rotating stall, local stability, stabilization.
1. INTRODUCTION ~~xial
This paper focuses on rotating sta.ll control based on the multi-mode rvloore-Greitzer model [16], which in the limiting case converges to the full PDE (partial differential equation) model of ivloore-Greitzer [18]. A feedback control law, consisting of both linear and quadratic feedback terms, will be investigated which is a generalization of the feedback control laws in [15, 20, 5] for the third order (or single-mode) MooreGreitzer model. It will be shown that the critical operating point a.t the peak pressure rise of the performance characteristic curve is locally stabilizable with the proposed feedback control law_ .A. necessary and sufficient condition will be derived for local stabiliza.tion of the underlying feedback compression system. This condition holds, and results in lumped feedback controllers for the full PDE Moore-Greitzer model in the limiting case. In a special case, the proposed feedback control law reduces to the linear feedback control law proposed in [5] Vv"hich was also studied in [14] recently. In contrast to the global stabilization results in [4], our local stabilization conditions are explicit, and provide an effective synthesis tool for rotating stall control "9lithout distributed sensors. Hence our result compliments those in [4].
flow compressors are subject to two distinct
aerodynamic instabilities, rotating stall and surge, which can severely limit the compressor performances.
Both these inst abilities are disruptions of the normal operating condition which is designed for steady and
axisymmetric flow. Rotating stall is a severely nonaxisymmetric distribution of axial flow velocity, tak-
ing the forlll of a wave or "stall cell" , that propagates steadily in the circumferential direction at a fraction of the rotor speed. Surge~ on the other hand, is an axisymmetric oscillation of the m.ass How along the axial length of the compressor. Both can result in catastrophic consequences. Feedback control v..~ proposed to improve compressor performance in [6], and has since received great attention in recent years. The existing results include linear control law [19], bifurcation stabilization [15, 20], and backstepping lllethod [13, 4].
*
This research was supported in part by A.FOSR and ARO.
The corresponding author with email: ggu'~ee.lsu.edu, and fax: (225) 388-5200. 1
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14th World Congress of IFAC
2. MULTI-MODE COMPRESSOR MODEL J.~
The above multi-mode model converge to the PDE model described in (1) and (2) uniformly as lV -+
schematic axial compressor is sho,vn in Figure 1. Throttle Y
TeN
Ps
= eNV2N + 1,
TTe,V
= ~. 2~r + 1
The performance characteristic curve
Wc (.)
is assumed
to be a cubic polynomial [18], and given by Plenum Vp
'l.jJc(r/JN) Figure 1
Schematic of compressor.
= H [coeN + Clr:PN + C3tPJ]
•
(8)
where x· 2 ;;;;;:: x . x denotes Hadmard product [9], and x· k raising each element of x to the poviter of k. The uniform flow can be represented by
A post-stall model for compression systems was develGreitzer~ and its full PDE form is
oped by iv100re and described by [18]
=
Proposition 2.1 The steady equilibria (r.PN, '1') (l/JeeN, We) for the multi-mode compressor model in (4) and (6) are dependent on the throttle parameter r, and
governed by;
>it
-1'~),
4;210 (
rjJ
= + 'P1'7=o'
(2)
The first equality in (9) is obtained by setting
(PN =
0
in (4), while the second equality in (9) by setting q, == o in (6). The proof is thus omitted. ~4.. local model around equilibria (rPeeN, We) has the form
All notations axe consistent with [18}. The incompressible and irrotational assumption the gas flow implies the existence of disturbance flow potential that satisfies Laplace's equation ~rith zero boundary condition at 11 == -ip ::; -le) 'w'hich in turn implies that the disturbance flow at station 0 has the form:
on
:i:
= f('i x)
= Lx
+
Q[x,x]
+ C[x, x,x] +".,
(10)
N
CPI"l=O
= L- (An cos(n9) + B n sin(n8)) ,
(3)
n=l
where !'v" -+ 00. Let {e j9n }~~o be a set of 2N + 1 uniformly distributed samples on the unit circle. Denote rjJ(8 z•v +d
],
1
1
.,j2
--./2 cos 92 N+l sin ()2N+l
COS
e1
sin (}l
cos 1V02N+l sin NfJ 2 1\l+1
cos lV(h, sin N(J2
COS N(}l
sin j\"T 81
diag (lel m 1 I 21 m212,··· ~ m N I 2),
E
m cosh(nl nsinh (nlp
1
F )
mn F
diag (0, F l
a ) + -,
I
,FN
),
,'"
1
TT E- FT, M2
2 =:
[10 10] '
Fn =
[~
\Vhile Ko is independent of the throttle parameter, T and 0 are functions of l' By the form of eN, the linear matrix can be written as:
-01;],
L = S-lDS~
= TT E~lT~
Do =
where t N
f2
= V2N+1'
T- 1 =TT.
K.]
[T
a/le
-K T
for 1 $ n :::; lv
•
)
D.rjJN ,j,
M 1cPlv
~
+ M27/Jc(~ifJN) -
- eN E R
Dn --
[n
nIb
-nib] -1 a mn
The eigenvalues of L can now be easily
ZN
M 2en I
+1 ,
A1,2
(4)
==
),2,,+1,2>1.+2
(5)
a.5rc ± 0.5j
v"4(K
2
= (Cl! ± j ~) m;;-l,
+ Ta/le) - Tl,
(17)
= T + et/le
(18)
Te
dZle (:J~1 -1'#) ,
(6)
where n =:; 1; 2; ... J N. functions of '""'I.
[1
(7)
Assume S-shape for the performance characteristic curve tf;i;('), and Cl + 3C3 = 0 as in [18J. A schematic
1
1
cOlllputed as
Then the full PDE model can be approximated by [16]
rPN
D == diag(Do,D 1 , · · · ,DN))
1 ] .
Hence eigenvalues are also
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STALL CONTROL FOR AXIAL COMPRESSORS...
14th World Congress of IFAC
compressor characteristic is shown in Figure 2. The designed operating point is uniquely determined by
the intersection of the throttle line (dashed lines A-B or C-D for two different values of the throttle position) with the compressor performance curve 1fC(')i according to (9). The maximum pressure rise takes place at a = 0 (point A) because that is "\vhere the derivative of "-Pc. C·) equals to zero. Since a < 0 on the right side of the maximum pressure rise, the lv pairs of eigenvalues T
of L Inatrix in (18) are stable. By T < 0 the pair of eigenvalues of (17) are also stable. Hence any point on right side of the peak of 1/Jc (.) is a stable operating point, and the uniform flow is a stable equilibrium, as highlighted by the solid line. However if the throttle value , decreases~ then the flow rate intensity
Al
"Ye
VW; e=ifJcl W"e='1'c
=
1Jc -Ji/Jr;(f/lc)
l
stalled flo,v for the compression system. In the next section, some of the existing results on Hopf bifurcations Vi,rill be reviewed which will be used in Section 4 for the synthesis of a stabilizing feedback control law.
3. STABILITY FOR HOPF BIFURCATIONS This section reviews briefly the projection and Lyapunov methods for local stability of Hopf bifurcations. For stability analysis of Hopf bifurcations, the system under consideration is the following nth order parametrized nonlinear system: x=!("X),
!("Y,xo) =0 \f--yER, xoESca
n ,
(20) where x E Rn, ! is a real-valued parameter, and S a linear subspace of Rn l to be clarified later. It is assumed that f (', .) is sufficiently smooth such that the equilibrium solution X e , satisfying f("Y, x e ) == 0, is a smooth function of ,. The smoothness of f(·,·) implies the existence of a Taylor series expansion near the origin of Rn in the forID of
x = fe"~ x) = L(-y)x + Q(')') [x, x] + C(,)[x, x, x] + ... (21) where L(,)x, Q(l")[x,x], and C("y) [x, x,x] can each be expanded into
(19)
c=2,"V
2W
L(,)x Q(,) [x, x]
and A2n+l , 2n+2 == ±jwn V\o;'th W n = %m;;l for 1 ::; n :5 .IV. .A.t the criticality 'r = "Ye, Hopf bifurcations occur for the multi-mode model of the compression sy·stem in (4) and (6) that induce rotating stalL Furthermore if the underlying bifurcation is subcritical, or unstable, the ~l\-C portion of the stall curve is unstable which is sketched for 1'1 = 1. Stall cells will be born at point i\., and gl'o",r which will throttle the operating point frOIn _I\. to B quickly which is a stable operating point., although undesirable. There is a tremendous drop in both the pressure rise, and floV\o" rate. Moreover increasing the throttle position and flow rate at point B does not increase the pressure rise. Rather the pressure rise becOIll€S even IOVv"er before it comes to point C at which it jUIIlPS back to (due to again loss of stability) the performance curve Wc(·). The hysteresis loop A-B-C-D is the main cause for loss of compressor performance, and the potential damage to aeroengines.
C(j)[x, x, x)
Lox
+ 6'JL l x + (01')2 L 2 x + ... ,
Qo[x,x]
+O~Ql[X,X]
Co[x, x, xl
+"',
+ o,C1 [x, x, x] + ... ,
with 0'1 = r -le, and Lo, L l , £2 constant matrices of size n x n. Suppose that L( "y) possesses m « n/2) pairs of complex eigenvalues Ak(....,,) = ak CT) ± j!3k (7) , dependent smoothly on ,. It is assumed that for 1 ~ k~m,
l.l
= 0,
/3('Ye) = Wk
d;;
¥- 0, O:~('Ye) =
(-Ye)
¥- 0,
(22) while all other eigenvalues of L( 1') are stable at, and in a neighborhood of J == Ic. That is, m pairs of eigenvalues cross the imaginary axis simultaneously. . Then each Ak(,) is a critical eigenvalue, so is its conjugate. The center space, or the eigen-space for the m pairs of critical eigenvalues is denoted by Se, and is assumed to be orthogonally complement to S in the sense that S E& Sr; = R 11. Since projection of Xo E S to Se is zero, the equilibria Xo satisfying (20) will be called zero solution. The strict crossing assumption implies that the zero solution changes its stability as ~ crosses rc' For instance Ct~ (le) > 0 implies that the zero solution is locally stable for J < re) and becomes unstable for { > 'Ye. The crucial problem is the determination of local stability near the critical parameter le at \vhich Hopf bifurcations occur, and the periodic solutions are born.
'Y
= YC~I"""" ~-
3.1 The Projection fv1ethod
"{ > 'Ye
......,;,-=-':.~::;....~-_~----------------'--------:=-
o
Figure 2
2W
4le
The center space Se, spanned by all the critica.l eigenvectors, is complete. IT all the critical eigenvalues are distinct, Se = SC1 ffi SC2 EB . - . ffi SC
Compressor characteristic, and stall.
Tn
The above discussion indicates that stabilization of the critical operating point, and A-C portion of the stall curve is a key issue for rotating stall control ,vhich will eliminate the hysteresis loop, and allow smooth change of the operating point between axisymmetric flow and
,
where SCk denotes the eigen-space spanned by kth pair of the critical eigenvectors. The projection method in [11] projects the nonlinear dynamics into the subspace of SCk so that stability of the projected dynamics can be analyzed4 Associated with each pair of the critical
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STALL CONTROL FOR AXIAL COMPRESSORS...
eigenvalues (.Ak, A/c), there can define a SCV (stability characteristic value) X~k) whose sign determines local
for k, p
= 1, ... ~ m
11Jith k
f:. p,
where
stability of the projected dynamics. See [11, 10~ 1] for details. Let the Taylor series of f (1' i x) be of the form in (21) where L o L(O). ..~n algorithm to compute >..~k) is outlined next.
and
Vk
'2Qo[rk) rk].
(23) (24)
Due to unpredictability, it is difficult to design axial flow compressors such that the critical operating condition corresponding to rotating stall is stable. Thus feedback control is proposed to suppress rotating stall [6J. This paper considers the use of bleedvalve as actuator for the compression system. It is noted that the throttle parameter is a composition of uncertainty 1 0 ; synthesized disturbance from inlet and combustion chamber~ and the bleed valve position o--y that can be employed as actuator. Assume that W is measurable. Then the throttle parameter ca.n be expressed as
+ ikQo[rk, Vk] +~fhCoITk,rh,fh]}. (25)
2Re {2lkQo[rk> lik]
Theorem 3.1 Suppose that £0 has m pairs of nonzero critical eigenvalues on the imaginary axis with the rest on open left half plane. Then the projected dynarnics along the kth pair of the eigenvectors is stable, if ~2 < 0, and unstable if)..2 > O. For the case of l'l = 1, the associated Hop! bifurcation is supercriticalor stable, if;\2 < 0, and subcritica~ or unstable, if .\2 > O.
"Y
x=
z
16Re {lk (Qo[rk ~ jlp]
+Qo{rp, I/k p ]
p ]
= Lx + Q[x,x] + G[x
j
x, x]
+ ... + ru, (31)
Dz +
sru, sr = [ e~ 0
] K.,j(2N
+ l)lc,
0
D
o Thus the state variables corresponding to the subsystem of Do are deviations jn pressure rise Jw = W - 'lie, and in average flo,v rate ~ 1» = ep - 4>e respectivelY4 The two eigenvalues of Do as in (17) are associated with surge dynamics, and are linearly controllable. The state variables corresponding to subsystems Dl DN represent amplitudes of harmonics for disturbance flow rate in (3). Thus the N pairs of eigenvalues for D 1 ~ DN are associated with the rotating stall dynamics, and are linearly uncontrollable. While linear feedback law can be used for surge control, the uncontrollability for the critical modes of D 1 ~ D .v implies that quadratic feedback law is more appropriate for rotating stall control [11. The above consideration suggests f'.I
(26)
+ Qo[rp~ JJk + ~Co[Tk,rPlrp])}
fe"~ x)
Do
Theorem 3.3 Suppose that L o admits m > 1 pairs of eigenllalues on the imaginary U:I:is with the 1'est on open left half plane, and the nonlinear system (21) is locally center-symmetric in the sense of Lyapunov. Then there exists a Lyapunov function that guarantees local stability of (21) at the 'criticality, if
Xkp
(30)
=
Definition 3.2 Under the condition that L o admits lV > 1 pairs of eigenvalues on the imaginary axis with the rest on open left half plane, the nonlinear system (21) is said to be locally center-symmetric in the sense of Lyap'unov if (i) Wi =1= qwp for q = 1,2,3, whenever 'fn ~ 2, (ii) qWi =f:. w p +Wk, Wi #- qw p +Wk for q = 1,2, whenever m ~ 3, and (iii) Wi ±wp i= Wk +WI, whenever m 2: 4 where ail indices are distinct.
= 1, ... ,rn,
u
Vii'
vlhere r = [ 0 0 1] T ~2-lc, and Lx, Q[x,::cl, and G[x, x, x] are linear, quadratic, and cubic terms respectively as given in (11)-(15) with [ replaced by 1 0 ' Applying similarity transform z Sx yields the linearized system
The Lyapunov method for multiple critical modes is developed by Fu [7]. It gives sufficient conditions on the existence of a Lyapunov function that guarantees local stability of the bifurcated system. The following notion is needed.
for k
-
ten as
3.2 The Lyapunov Method
8X~k) < 0,
== ')'() + Or = 10
where u is the feedback control law to be designed. Our objective is local (asymptotic) stabilization around the critical equilibrium of the uniform flow (4)ceN) wc) for the closed-loop cOIllpression system. The compression control system can now be writ-
It should be clear that if £0 admits only a single pair of imaginary eigenvalues, then local stability of the bifurcated system is equivalent to X~k) < O. However if L o admits m.ore than one pair of iInaginary eigenvalues, local stability of each projected dynamics does not guarantee local stability of the bifurcated systeIns. In this case, local stability of the bifurcated systems can be tackled by the Lyapunov method [7].
Xkk
(29)
4. LOCAL FEEDBACK STABILIZATION
• Step 3: The coefficient 5,.~k) is given by
j~k)
(28)
Since Xkk == 8X~k). local stability of Hopf bifurcation with multiple pairs of critical modes requires more than local stability of each projected dynamics. Roughly speaking, the nonpositivity of Xkp for k 1= p takes the coupling of different pairs of critical modes into account for local stability of the bifurcated system in (21).
from the
1
satisfy
"2 Qo [r k ' T p ].
equations
~Qolrh' rh],
Vk p
1
• Step 1: Compute left row eigenvector ik and right column eigenvector Tk of £0 corresponding to the kth critical eigenvalue of Ak(D) = jWk- Normalize by setting lkrk = 1. J.tk
J
iQolrk, rp ],
=
• Step 2: Solve column vectors
j..tk p
(27)
:5 0,
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STALL CONTROL FOR AXIAL COMPRESSORS...
14th World Congress ofIFAC
the employment of the following feedback control law: (32)
The linear term needs be designed to take care of the surge dynamics, and the quadratic terms need be designed to stabilize the rotating stall dynamics locally. The linear feedback term involving the average disturbance flo~v is intentiously skipped because of the difficulty in measuring Xl ~ X2N+1 which requires distributed sensors as N ~ CXJ. For the same reason, more general quadratic form is avoided in the proposed feed back control law. Define discrete Fourier transform
o ::; n It
Vl.~il1
:s 2N.
(33)
cornpressor parameters and linear feedback gain K'li. As a result) distributed sensing for the flow rate rP1V is unnecessary in the limiting case N -+ 00. 11oreover precise information on the cubic performance characteristic curve, and on the critical operating condition is not required, provided that unpper and lower bounds on C3~ H, W~ epc, and rc/~ are available. RelDark 4.3 In f4], backstepping method was used to derive a global stabilization condition on a more general state feedback control law than (32). It was shown tha.t any given operating point (or set point) 1 0 2: "Ye can be globally stabilized. It should be clear that if the set point is chosen as at the criticality) local stability will be ensured for ro = 'c. However it requires distributed sensing as N ---t 00. Moreover the stabilizability condition in [4] is less explicit than the one in (35). The stabilization condition (35) also indicates that the critical operating point can be locally stabilized by using linear feedback control law, i.e., by setting KQ = o in (32). The next result follows easily from Theorem 4.2 1 and thus the proof is skipped.
be shown later that local stabilization requires
i
0, rather than KQ;. :f:. 0 for all i > O. Hence distributed sensors are unnecessary in the Hmi ting case l\l --+ 00. For the feedback control law as in (32), the closedthat
Assume that C3 < DJ rPc > W; and the condition (34) is satisfied~ Moreover suppose that the imaginary part of the eigenvalue.t; in (18) are locally center-symmetric in the sense of Lyapunov. Then the rotating stall dynamics is locally asymptotically stable at the criticality 1'0 = re under feedback control u K'l-8w; if and only if
Corollary 4.4
loop compression system can now be expanded into the
follO"~ving
X L*
Taylor series: L*x
+
Q*[x,
8- 1 D*S,
[ [
D~
Q*[X,X]
KQ
T
xl + C*[x~x) xl +.'., = diag(D(;, D~ ~ ... ) DAr),
+ ",2l c K..:v -K
3~2~
=
D*
~ le
]
(tv
~ 1) kf2 2 K l c KQ
[ KQt
D~
1
KQ2N+l
= Dn ,
02N+l T
..
]
X ·2
,
By the fact that satisfies (34).
-4W e ],
Do
Under the feedback control law (32}) the surge dyn.amics is locally stable at the criticality '"Yo = le, if and only if
K,
lc
(34)
're
'YQ="1c
= 2~'
C3
< 0,
( ~c ) W- 1
2
-
(36)
stabilizing gain K'I! exists that
measurement of pressure rise was considered in [8, 5] for the third order !vloore-Greitzer model which is a crude approximation. Corollary 4.4 is surprising ~"hich indicates that linear control law is also effective in stabilization of the critical operating point for the multimode model, which in the limiting case converges to the PDE Moore-Greitzer model. It should be pointed that the same feedback control law was studied in [14] as well. Using eqllilibria dependent feedback control, [14] was able to obtain a global stabilization result for ""to > ""'/c 1. In combination with the result in Corollary
LelTIm.a 4.1
< - ~I 2
VV
ReD'lark 4.5 Feedback stabilization using only the
and O*[x,x,x] = C[x,x,x]. It is noted that is associated with the surge dynamics, and n:, with the rotating stall dynamics for n ?: 1. The follo\ving result can be easily shown.
K\l!
"Ye
K~>2~+12Hc3
The main result of this section is the following.
Its proof is based on the projection method, and Lyapunov method in Section 3. Due to the space limit, the proof is omitted.
4.4, global stabilization can be claiIned for "'to
~
re·
Ass'Urne that C3 < 0,
TheorelTI 4.2
J
rotating stall dynamic.s is locally a.symptotically stable at the criticality (0 = le for the feedback compression
system described in (34) '" (34), if and only if
The significance of Theorelll 4.2 lies in the fact t.hat
lIn [14)1 glQbal stabilization condition was claimed for 1'0 ~ le. which is incorrect. In light of Corollary 4.4, condition (36)
the stabilizability condition (35) requires only the s~m of all KO t ~s to satisfy a bound determined by various
needs be enforced at
"{o
:=;;
"'/c
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STALL CONTROL FOR AXIAL COMPRESSORS...
References [1] E.H. Abed and Jyun-Horng Fu, ~(Local feedback stabilization and bifurcation control) 1. Hopf bifurcation," Systems and Control Letters, 7 (1986) i 11-17.
[12] "\Vei Kang, "Bifurcation control via state feedback for single input nonlinear systems: Part I and Part 11" , preprint, 1996. Krstic~ J.M. Protz, J.D. Paduano~ and P.V. Kokotovic, UBackstepping designs for jet engine stall and surge control," Proc. of 34th IEEE Conf. Dec~ and Contr., 3049-3055, 1995.
[13] M.
[2J R.A. Adomaitis and E.H. Abed, "Bifurcation analysis of nonuniform flow patterns in axial-flow gas compressors," in 1st World Congress of Nonlinear Analysis, Aug. 1992.
[14] A. Leonessa, 'l.-S. Chellaboina, and W.M. Had-
[3] 0.0. Badmus, S. ChowdhurYt and C.K. Nett, "Nonlinear control of surge in axial compression systems," Automatica,'VoL 32,59-70,1996. [4J A. Banaszuk, H.A. Hauksson
j
dad, HGlobally stabilizing controllers for multimode axial flow compressors via equilibriadependent Lyapunov functions)" Proc. 1997 A mer. Contr. Conf.) 993-1002, 1997.
and I. Mezic
(1996), "A Backstepping controller for Moore-
[15] D.-C. Liaw and E.H. Abed, "Active control of compressor stall inception: A bifurcationtheoretical approach," A'Utomatica~ voL 32, 109-
Greitzer PDE model describing stall and surge in compressors", prcprint.
116, 1996.
[5] X. Cbeli, G. Gu, P. 11artin, and K. Zhou, "Rotating stall control via bifurcation stabilization," Automatica: vol 34, 437-443, l\pril 1998.
[16] C.A. Mansoux, J.D. Setiav~. . a n, D.L. Gysling) and J.D. Paduano~ "Distributed nonlinear modeling a.nd stability analysis of ~"'Cial compressor stall and surge," Proc. 1994 Amer. Contr. Con!., 23052316, 1994.
[61 A.H. Epstein, J.E. FfO-"TCS Williams, and E.1vI. Greitzer, "Active suppression of aerodynamic instabilities in turbomachinery," J. Propulsion) vol. 5, 204~211) 1989.
[17] F .E. McCaughan, "Bifurcation analysis ofaxial flow compressor stability/ SIAM J. Applied Mathematics, vol. 20, 1232-1253, 1990.
[7] J.-H Fu) aLyapunov functions and stability cri-
teria for nonlinear systems "rjth multiple critical modesH'l Mathematics of Contr. , Sig., and Syst.)
[18] F.K. Moore and E.M. Gieitzer (1986), "A theory of post-stall transients in axial compressors: Part. I - development of the equations," ASME J. of Engr. for Gas Turbines and Power, voL 108, pp. 68-76.
7, 255-278, 1994. [8] G. Gu) A. Sparks, and C. Belta (1997), "Stability analysis for rotating stall and surge in axial flow cornpressors", preprint.
[19] J.D. Paduano, A.H. Epstein, L. Valavani, J.P.
[9] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
Longley~
E.J\,L Greitzer,
and G.R. Guenette
(1993), 'tActive control of rotating stall in a lowspeed axial compressor/' J. Turbomachinery, vo1. 115, 48-56.
[10] LaN. Howard) "Nonlinear oscillations," in: F.G. Hoppensteadt, Ed' J Nonlinear Oscillations in Biology, (Amer. Math. Soc., Providence, RI, 1979), pp. 1-68.
[20} H.O. Wang, R.A. Adomaitis and E.H. Abed, "Nonlinear analysis and control of rotating stall in axial flow compressors/' in American Control
[11] G. Iooss and D.D. Joseph, Elementary Stabil-
Conference~
ity and Bifurcation Theory, Springer-Verlag, New
2317-2321, 1994.
York, 1980.
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