Bifurcation Stabilization with Local Output Feedback.
Local output feedback stabilization with smooth nonlinear controllers is studied for parameterized nonlinear systems for which the linearized system possesses either a simple zero eigenvalue or a pair of imaginary eigenvalues and the bifurcated solution is unstable at the critical value of the parameter. It is assumed that the unstable mode corresponding to the critical eigenvalue of the linearized system is not linearly controllable. Results are established for bifurcation stabilization using output feedback where the critical mode can be either linearly observable or linearly unobservable. The stabilizability conditions are characterized in explicit forms that can be used to synthesize stabilizing controllers. The results obtained in this paper are applied to rotating stall control for axial flow compressors as an application example.
Introduction
Stabilization of nonlinear control systems with smooth state feedback control has been studied by
a number of people [4, 2, 3, 9, 19]. An interesting situation for nonlinear stabilization is when the
linearized system has uncontrollable modes on imaginary axis with the rest of modes stable. This is
so called critical cases. for which the linearization theory is inadequate. It becomes more intricate if
the underlying nonlinear system involves a real-valued parameter. At critical values of the parameter,
linearized system has unstable modes corresponding eigenvalues on imaginary axis, and additional
This research was supported in part by grants from AFOSR, ARO, and LEQSF. A preliminary version of this paper
is to appear in 1997 American Control Conference.
y Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge, LA 70803-5901.
z Control Analysis Section, WL/FIGC, 2210 Eighth Street, Suite 21, WPAFB, OH 45433-7531
equilibrium solutions will be born. The bifurcated solutions may, or may not be stable. The instability of
the bifurcated solution may cause "hysteresis loop" in bifurcation diagram for both subcritical pitchfork
bifurcation and Hopf bifurcation [11], and induce undesirable physical phenomena. This is manifested
by rotating stall in axial flow compressors [1, 17, 18] that has received great attention for the past several
years [5, 6, 7, 15, 16, 20]. Thus bifurcation stabilization is an important topic in nonlinear control.
Abed and Fu studied bifurcation stabilization using smooth local state feedback control [2, 3]. For
Hopf bifurcation, stabilization conditions were obtained for both the case where the critical modes
of the linearized system are controllable and uncontrollable. For stationary bifurcation, stabilization
conditions were derived for the case where the critical mode of the linearized system is controllable.
The uncontrollable case was investigated in [13] where normal forms of the nonlinear system are used.
In this paper we study bifurcation stabilization via local output feedback control laws that are smooth.
Throughout the paper it is assumed that the critical mode of the linearized system is uncontrollable.
Moreover only output measurement is available for feedback. It should be clear that measurement of
all state variables is unrealistic in practice, especially when the underlying nonlinear system has a high
order. Often some of state variables are more expensive, or more difficult to measure than others. Hence
the bifurcation stabilization problem studied in this paper has more engineering significance.
Two stabilization issues will be investigated. The first one is concerned with bifurcation stabilization
where the critical mode of the linearized system is unobservable through linearized output measurement.
Stabilizability conditions are established for both stationary bifurcation and Hopf bifurcation. Roughly
speaking, it is shown that nonlinear controllers do not offer any advantage over the linear ones for
bifurcation stabilization. Stabilizing controllers, if exist, can be taken as linear ones. The second one is
concerned with bifurcation stabilization where the critical mode of the linearized system is observable
through linearized output measurement that includes state-feedback as a special case. Stabilizability
conditions are also obtained in this case. It is shown that linear controllers are adequate for stabilization
of transcritical bifurcation, and quadratic controllers are adequate for stabilization of pitchfork and
Hopf bifurcations, respectively. It should be pointed out that the stabilization conditions obtained in
this paper are characterized in explicit form that provide synthesis procedures for design of stabilizing
controllers, if they exist. Rotating stall control for axial flow compressors will be used as an application
example to demonstrate the use of the stabilization results established in this paper.
The notations in this paper are standard. The collections of real and complex numbers are denoted
by R and C, respectively. If c 2 C, its complex conjugate is denoted by - c. The collection of real and
complex vectors with size n are denoted by R n and C n , respectively. A matrix M of size p \Theta m can be
viewed as a linear map from C m to C p , and its transpose is denoted by M T . For
to be stable, if all its eigenvalues are in the open left half plane. Notions such as linear controllability
and linear observability can be found in [13].
Stability and Projection Method
This section considers the stability issue for bifurcated system using the projection method developed
in [11]. The system under consideration is the following nth order parametrized nonlinear system:
where x 2 R n , fl is a real-valued parameter, and ffi ? 0 is a sufficiently small real number. It is assumed
that f(\Delta; \Delta) is sufficiently smooth such that the equilibrium solution x e , satisfying f(fl; x e
smooth function of fl. Since f(fl; small neighborhood of is a fixed equilibrium
point locally which is called zero solution. The linearized system at the zero solution is given by
Two different types of bifurcation will be considered in this paper and the determination of their local
stability will be discussed in the following two subsections.
2.1 Local Stability for Stationary Bifurcation
For stationary bifurcation, it is assumed that L(fl) possesses a simple eigenvalue -(fl), dependent
smoothly on fl, satisfying
while all other eigenvalues are stable in a neighborhood of It implies that the zero solution
changes its stability as fl crosses 0. For instance, - implies that the zero solution is locally
stable for fl ? 0 and becomes unstable for additional equilibria x e 6= 0 will be
born at are smooth functions of fl by smoothness of f(\Delta; \Delta). Such bifurcated solutions
are independent of time t, and called stationary bifurcation. Thus is the critical value of the
parameter, and -(fl) is called critical eigenvalue. The nonlinear system (1) at is referred to as
critical system. The bifurcated solution of the nonlinear system born at may, or may not be
locally stable. For simplicity, only double points [11] will be considered in this paper. It should be
clear that stability of the bifurcated solution plays an important role for the dynamics of the nonlinear
system, and is hinged to stability of the critical system. A useful tool to determine local stability of the
bifurcated solution and of the critical system is the projection method developed in [11] and advocated
in [2, 11].
Let ' and r denote the left row and right column eigenvectors of L(0), corresponding to the critical
eigenvalue
also an equilibrium solution of (1), or bifurcated solution in a small neighborhood of
by [11] there exists a series expansion
x ek
Since f(fl; x) is sufficiently smooth, there exists Taylor expansion near the origin of R n of the form
where L(fl)x, Q(fl)[x; x], and C(fl)[x; x; x] are vector valued linear, quadratic, and cubic terms of f(fl; x)
respectively having symmetric form in each of their entry, and they can each be expanded into
are n \Theta n constant matrices.
Let ~
- be the critical eigenvalue for the linearized system matrix at the new (bifurcated) equilibria
close to the origin. Then ~
There exists a series expansion [11]
~
The computation of the first two coefficients of ~
- can proceed as follows [11, 2]:
is a function of fl.
ffl Step 2: Set x and calculate
ffl Step 3: Compute x e2 from equations 'x
Local stability of the bifurcated solution is given by the following theorem [3].
Theorem 2.1 Suppose that all eigenvalues of L 0 are stable, except one critical eigenvalue. For the
case the branch of the bifurcated equilibrium solution is locally stable for fl sufficiently close to
For the case the bifurcated solution is locally
stable for fl sufficiently close to 0 if ~
~
It should be clear that local bifurcation for the case fl 1 6= 0 is transcritical [21]. Thus the branch of
the bifurcated solution at " ? 0 has the opposite stability property as the one at " ! 0. We are interested
in stabilization of transcritical bifurcation for the branch of " ? 0, that has no loss of generality. On
the other hand, local bifurcation for the case
of the bifurcated solution share the same stability property.
2.2 Local Stability for Hopf Bifurcation
For Hopf bifurcation, it is assumed that L(fl) possesses a pair of complex eigenvalues -(fl); -
-(fl), dependent
smoothly on fl, while all other eigenvalues are stable in a neighborhood of
imaginary. It is assumed that
Thus -(fl) is a critical eigenvalue, so is its conjugate. It implies that the zero solution changes its
stability as fl crosses 0. For instance ff 0 (0) ? 0 implies that the zero solution is locally stable and
becomes unstable for fl ? 0. Furthermore Hopf Bifurcation Theorem asserts the existence of a one-parameter
family nonconstant periodic solutions of (1) emerging from the
zero solution at This is none stationary bifurcation. The positive real number " is a measure
of the amplitude of the periodic solution and " 0 is sufficiently small. The periodic solutions
period near 2-=! c and occur for parameter values fl given by a smooth function fl("). Exactly one of
the characteristic exponents of p " is near zero, and is given by
~
Local stability of Hopf bifurcation is hinged to the first nonzero coefficient of ~
-("), denoted by ~
N - 1. Generically
Let the Taylor series of f(fl; x) be of the form in (4) where L An algorithm to compute ~ - 2
is quoted from [2]. See also [10].
right column eigenvector r of L 0 corresponding to the
critical eigenvalue of . Normalize by setting
ffl Step 2: Solve column vectors - and - from the equations
\Gamma1:
ffl Step 3: The coefficient ~ - 2 is given by
~
ae
oe
Theorem 2.2 Suppose all eigenvalues of L 0 are stable in a neighborhood of except the critical
pair of complex eigenvalues. Then the Hopf bifurcation is stable, if ~ - 2 ! 0, and unstable if ~
To facilitate the development of our main results for bifurcation stabilization, Theorems 2.1 and 2.2
will be employed to serve for the definition of bifurcation stability throughout the rest of the paper,
although they differ from the conventional notion.
3 Output Feedback Stabilization for Stationary Bifurcation
We consider first feedback stabilization for stationary bifurcation. The nonlinear control system in
consideration has the form
where f(fl; x) is the same as in (4), and g(\Delta; \Delta) and h(\Delta) are also smooth functions satisfying
It is assumed that u 2 R, and y 2 R p with p - 1. Thus the nonlinear system (7) has only one control
input, but may have more than one output measurement. Its Taylor series expansion is given by
by assumptions in (8) where ~
are the linear and quadratic terms for the linear control
component of g(x; u), ~
is the linear term for the quadratic control component and ~
L[x; x] is the
quadratic term for the linear control component of of g(x; u), and are the coefficient vectors
of u, u 2 , and u 3 , respectively. It is assumed that L 0 has only one zero eigenvalue with rest of the
eigenvalues stable, and that the bifurcated solution born at locally stable. The assumption
on stability of the nonzero eigenvalues of L 0 has no loss of generality. If some of the nonzero eigenvalues
of L 0 are unstable, then linear control method, such as pole placement [12], can be employed to stabilize
those unstable modes corresponding to nonzero eigenvalues. It is the unstable mode corresponding to
the critical eigenvalue renders linear control methods inadequate because of bifurcation.
We seek a local output feedback control law of the form
that stabilizes the bifurcated solution, where K 2 [\Delta; \Delta]; K 3 [\Delta; \Delta; \Delta] are the quadratic, and cubic terms of K(y)
respectively, satisfying K 2 [0; The output has a Taylor series expansion
are the linear, quadratic and cubic terms of h(x) respectively, satisfying
loss of generality, it is assumed that the quadratic and cubic terms
of the feedback control law in (10) are of the form
~
~
~ ~
with K 2 and K 3 some constant matrices, and ~
Abed
and Fu studied the same problem in [3] for the case of state feedback where the critical mode of the
linearized system at We will consider the case of output feedback where the
critical mode of L 0 is linearly uncontrollable. It should be clear that in practice, measurement of all
state variables is unrealistic. Moreover some of the state variables are more expensive and more difficult
to measure than others. Often only partial, or a nonlinear function of state variables are measurable.
Under this circumstance, the critical mode of the linearized system may, or may not be observable based
on linearized output measurements. Hence the problem considered in this paper has more engineering
significance than that of [3]. Because y our results are also applicable to state
feedback by taking H We will establish stabilizability
conditions for bifurcated systems where the bifurcated solution is unstable near in the following
two subsections.
3.1 Unobservable Critical Mode
We consider first when the critical mode of the linearized system corresponding to the zero eigenvalue
at is not observable through linearized output measurement y Note that by assumption
the eigenvalue is invariant under feedback control because of both uncontrollability and
unobservability of the critical mode. Thus L
also possesses the critical zero eigenvalue as L 0 . Denote
'   and r   the left row and right column eigenvectors for L
0 corresponding to the critical eigenvalue.
Then it is easy to see that '  due to again the uncontrollability and unobservability of
the critical eigenvalue. This can be seen from PBH test [12]. Denote ~ -   as the critical eigenvalue of the
linearized feedback system at the bifurcated solution to be stabilized. It is a function of
in the form of
local feedback controller in (10) does not change the zero solution. Denote L   (fl) as linearized
system matrix for the closed-loop system at the zero solution. Its critical eigenvalue is denoted by -   (fl).
Since L
d-
is also invariant under feedback control. It follows that the bifurcated solution x of the closed-loop
system changes its stability and bifurcates at well. The problem is whether or not the
bifurcated solution can be stabilized with output feedback control. The next result is negative for the
transcritical bifurcation.
Theorem 3.1 Consider the nonlinear control system in (9) with output feedback control law in (10).
Suppose that the critical mode of L 0 is not linearly observable. Then for the case ~ - 1 ? 0, there does not
exist a feedback control law that stabilizes the branch of the bifurcated solution at " ? 0, and
~
is invariant under output feedback control in (10).
Proof: With smooth feedback controller in (10), the closed-loop system has the form
where the linear, quadratic, and cubic terms are given by
~
~
~
~
~
~ ~
The uncontrollability and unobservability of the critical mode for the linearized system at imply
that both 'B test [12]. Hence applying Theorem 2.1 to the nonlinear system
in (14) gives the first coefficient of the critical eigenvalue for the branch of the bifurcated solution at
~
by again 'B imply 'Q
It follows that the sign of ~ -
1 is the same
as ~ - 1 that can not be altered by feedback controller in (10).
Although the stability property of transcritical bifurcation can not be altered by output feedback,
the situation for pitchfork bifurcation is quite different. We have the following result.
Theorem 3.2 Consider the nonlinear control system in (9) with output feedback control law in (10)
under the same hypothesis as in Theorem 3.1. For simplicity, assume in addition that ~
Then for the case there exists a smooth feedback control law K(y) as in (10) that ensures
~
stabilizes the bifurcated solutions, if and only if there exists a linear feedback control law
that stabilizes the bifurcated solution. Moreover there exists a nonsingular matrix T 2 R n\Thetan
such that
where as the identity matrix
of size (n \Gamma 1) \Theta (n \Gamma 1), and
I
Then the existence of the stabilizing feedback control law is equivalent to that
and that L 00 +B 11 K 1 H 11 is stable for some K 1
Proof: Let x e2 and x
e2 be the unique solutions for
By the proof of Theorem 3.1, and the condition 'B
~
~
The formula governing the vector x
e2 as in (18) can be readily verified by noting that Q
due to ~
Applying Theorem 2.1 again yields that
~
~
~
~
It follows that the bifurcated solution is stabilized, if and only if there exists K 1 6= 0 such that ~
holds, and the nonzero eigenvalues of L lie in the open left half plane. It remains to show
that stabilizability of the bifurcated solution is equivalent to the existence of K 1 6= 0 such that (17)
holds, plus stability of L 00 +B 11 K 1 H 11 . From (18), the following equations
are obtained. Thus the solutions x e2 and x
e2 solved from (18) satisfy
The expression for ~ -
2 can now be written as
where ae is the same as in (16). Since the critical mode of the linearized system is uncontrollable
and unobservable, there exists a nonsingular matrix T 2 R n\Thetan such that (15) is true by Kalman
decomposition. Denoting L
the last two equations of (18) yield
that implies that the last element of the column vector
e2 is zero. Denote
I
I
where I n\Gamma1 is an identity matrix of size (n \Gamma 1) \Theta (n \Gamma 1). Then x
e2 is determined by ff and T as
ff
Applying (15) and (20) to the expression in (19) gives
by (20). Denote -
Using the identity
whenever the products AB and BA are square matrices and make sense, and the
following sequence of equalities is true:
I
~
~
I
~
Similarly there holds
Thus the expression of ~ -
2 in (21) can now be written as
~
Hence stabilization of pitchfork bifurcation is equivalent to the existence of K 1 6= 0 such that L 00
stable, and
~
~
because stability of L 00 , and L 00 +B 11 K 1 H 11 ensures that det(L 00 ) and det(L 00 +B 11 K 1 H 11 ) have the
same sign. This concludes the proof.
Theorem 3.2 is surprising as it indicates that even though the critical mode of the linearized system is
neither controllable, nor observable, there exists an output feedback control law that stabilizes pitchfork
bifurcation under some mild conditions. This is in sharp contrast to the linear control theory. Moreover,
nonlinear feedback control law does not offer any advantage over the linear ones as far as stabilization
of stationary bifurcation is concerned. This is due to the fact that linear feedback control, though has
no influence on stability of the linear term, changes quadratic terms of the state-space equation that
in turn determines the stability property of the pitchfork bifurcation. It is worth to point out that the
condition (17) in Theorem 3.2 is characterized in explicit form. It simplifies greatly the synthesis part
for bifurcation stabilization. Indeed, all K 1 satisfying (17) can be easily parameterized, and then
be substituted into L
to determine the right K 1 that ensures stability of L
00 . To
be specific for is a semi-infinite interval of the real line. Thus the stabilizing
value of K 1 can be easily determined through root locus of
where K 1 changes in the semi-infinite interval determined by inequality (17). If p ? 1, then all stabilizing
lie on one side of a hyper-plane in R p that does not pass through the origin, as K
stabilizing. In this case one needs search for the right K 1 on the given side of the hyper-plane in R p
to ensures the stability of L
parameterized root locus method can be
used for (22). The assumption ~
Theorem 3.2 is not very restrictive if the critical mode of
the linearized system is not observable through linearized output measurement. For instance it holds
for the case where output measurement consists of linear combination of state-variables. In the case
~
the quadratic gain K 2 plays a role as well for stabilization of pitchfork bifurcation that
will be illuminated further in the next subsection.
3.2 Observable Critical Mode
Suppose that the stabilizability condition in Theorem 3.2 does not hold. Then bifurcation stabilization
with smooth controllers is not possible. In this case we have to consider the case where the critical
mode is linearly observable based on output measurement. Extra sensor or sensors have to be deployed
so that H 1 r 6= 0 is valid. Clearly the nonlinear differential equation (14) holds with output feedback
controller in (10) for the case H 1 r 6= 0 where the linear, quadratic, and cubic terms, have the same
expressions as in the proof of Theorem 3.1.
Consider first transcritical bifurcation. Without loss of generality, the branch of " ? 0 is assumed
to be unstable for fl ? 0. This is equivalent to ~ Our goal is to seek a controller of the form
that stabilizes the bifurcated solution for changing the stability property of the zero
solution. It is noted that by assumption the eigenvalue is invariant under feedback control.
Thus L
also possesses the critical zero eigenvalue as L 0 . Denote '   and r   the left row and right column
eigenvectors for L
0 corresponding to the critical eigenvalue. Then '  due to the uncontrollability of
the critical eigenvalue. Denote ~ -   as the critical eigenvalue of L
under feedback. It has the same form
of the series expansion as in (13). However r   6= r in general due to H 1 r 6= 0 by the observability of the
critical mode. The next result concerns with stabilization for transcritical bifurcation.
Theorem 3.3 Consider the nonlinear control system as in (9) with output feedback control law in (10).
Suppose that the critical mode of the linearized system corresponding to the zero eigenvalue at
observable. Then for the case ~
there exists a nonlinear feedback control law
that stabilizes the given branch of the bifurcated solution at " ? 0, if and only if there exists a
linear feedback control law that stabilizes the given branch of the bifurcated solution at " ? 0. Moreover
there exists a nonsingular matrix T 2 R n\Thetan such that
where have the same sizes as in Theorem 3.2, respectively. Let ' i be the ith element
of ', and r T Q 0k r be the kth element of Q
Partition
The existence of stabilizing feedback control law, subject to the same stability
property for the zero solution, is equivalent to the existence of K 1 6= 0 such that
~
? 0; and
where
d
~
d ~
scalar, d
I
is the critical eigenvalue in (3).
Proof: From the proofs of Theorem 3.1, and of Theorem 2.1, the first non-zero coefficient of the
critical eigenvalue for the feedback system has the expression
~
due to 'B by the uncontrollability of the critical mode and the condition of the theorem. Because
only linear gain K 1 is present in the expression of ~
1 , the existence of stabilizing control law is equivalent
to the existence of linear stabilizing control law. It remains to show that conditions (i) - (iii) are
equivalent to the existence of the stabilizing feedback control law with the same stability property for
the zero solution retained. By Kalman decomposition a nonsingular matrix T exists such that (23)
holds where the lower triangular Schur form of used. The hypothesis on L 0 implies that
yields the form of 'T \Gamma1 as in (23). Since
'r
the right eigenvector of L
0 corresponding to the zero eigenvalue is found to be
r
By noticing that
the expression for can be simplified as
It is claimed that the condition (ii) is equivalent to the invariance of the stability property for the zero
solution, provided that L
stable. Indeed, denote -   as the critical eigenvalue of
the linearized system under feedback control law
d-
I n\Gamma1#
by the fact that if K reduces to -(fl). Substituting the expression of
j as in (25) yields that
It follows that (ii) is equivalent to that 'L 1 r   has the same sign as - 0 (0) due to stability assumption
for L
00 and L 00 , that is in turn equivalent to that the zero solution for the feedback system changes its
stability also at shares the same stability property as that of uncontrolled system. Thus the
stability property of the zero solution is retained. Similarly,
~
~
~
~
d ~
with ~
Combining with the expression in (26), there holds
~
~
For the term 'Q 0 [r   ; r   ], there holds
for some real symmetric matrix Q ' . To show that Q setting leads to
~
because the last element of T r is one, by ('T Therefore,
Combining (26)-(28) with rearrangement yields the expression
a
that can be simplified to (i). The necessity of the conditions (i)-(iii) are now clear. For the sufficiency,
(iii) implies that L
Hence
Condition (i) then
implies that ~
ensures stability of the given branch of the bifurcated solution, while condition
(ii) guarantees the invariance of the stability property of the zero solution under feedback.
It is noticed that if then (i) of Theorem 3.6 can be further simplified to as
by the fact that condition (iii) and stability of L 00 imply that
Theorem 3.6 indicates that nonlinear control does not offer any advantage over linear control for
stabilization of transcritical bifurcation either. More importantly the conditions (i)-(iii) also provide
explicit formulas for synthesis of stabilizing linear gain K 1 . Indeed, for the case of
The set of K 1 satisfying each of (i)-(iii) can be easily computed that are either finite intervals, or semi-infinite
intervals. In particular, the set of K 1 satisfying (iii) can be obtained through root locus. For
the case p ? 1, the conditions (i)-(iii) offer more freedoms for the synthesis of stabilizing linear gain
But on the other hand, the synthesis becomes more complex due to more than one element for K 1 .
Remark 3.4 Suppose that the linearized system has been transformed into the form of (23). Then the
last entry of the state vector is the critical state variable. Measurement of this critical state variable
corresponds to H 12 6= 0. Theorem 3.6 indicates that the feedback of the critical state variable is crucial.
This is due to the fact that if H the sign of ~
1 remains the same as that of ~ - 1 , and thus the given
branch of the bifurcated solution at " ? 0 is not stabilizable that is exactly the result of Theorem 3.1.
However this fact does not imply that the measurement of non-critical state-variables is unnecessary that
corresponds to H 11 6= 0. By condition (i) of Theorem 3.3, the measurement of non-critical state variables
becomes necessary, if a then the measurement of non-critical state
variables are redundant, and thus unnecessary for the purpose of bifurcation stabilization.
Theorem 3.3 has its implications to state feedback control:
with
the linear state feedback gain, and K 2 [\Delta; \Delta]; K 3 [\Delta; \Delta; \Delta] the quaratic and cubic
terms respectively. The next result is a direct consequence of Theorem 3.3.
Corollary 3.5 Suppose that all the hypotheses in Theorem 3.3 hold. Then there exists nonlinear state
feedback control law that stabilizes the given branch of the bifurcated solution, if and only if
there exists a linear state feedback control law that stabilizes the given branch of the bifurcated solution.
Moreover with the same notation as in Theorem 3.3, the existence of stabilizing state feedback control
law, subject to the same stability property for the zero solution, is equivalent to the existence of K
? 0; and
Proof: The theorem can be easily proven by setting K 1 noting that
with T the similarity transform, in the proof of Theorem 3.3.
For pitchfork bifurcation, i.e., ~
the situation is again different. We adopt an approach as in
[3] by setting the linear term of the controller to zero. In fact, by the proof of Theorems 3.1 and 3.3,
the nonzero gain K 1 will result in ~ -
thereby changing the pitchfork bifurcation into transcritical
bifurcation for which only one branch of the bifurcated solution can be stable. Hence this is not a
desirable situation unless for some exceptional situations. It is noted that with K the eigenvectors
of L
0 corresponding to the critical eigenvalue at Hence both row
and column eigenvectors of the critical eigenvalue are invariant under feedback control. Since L
there holds '   L
(0). Thus the zero solution of the feedback system changes its stability
at well. The stabilizability of the bifurcated solution is given by the following result.
Theorem 3.6 Consider the nonlinear control system in (9) with output feedback control law in (10)
under the same hypothesis as in Theorem 3.3. Then for the case there exists a feedback control
law subject to K that ensures ~
changes the pitchfork bifurcation
from subcritical into supercritical, if and only if
Note that the expression of ae is the same as in (16).
Proof: By the proof of Theorem 3.2 and 'B there holds
~
~
~
Setting yields that ~ -
~
e2 can be solved from
~
Combined with 'x
x
Substituting the above into the expression of ~
~
~
~
~
~
there exists K 2 6= 0 such that ~
Conversely, stability of the
bifurcated solution implies that
~
~
that in turn implies that ae 6= 0, and ~
by the hypothesis that ~
It is noted that terms of higher order than two do not have effect on stability. Thus the stabilizing
controllers can be taken as quadratic ones. Moreover if the output measurements consist of linear
combination of state-variables, then the observability condition implies that ~
In this case,
stabilizability for pitchfork bifurcation is equivalent to ae 6= 0. For bifurcation stabilization using state
feedback in the case of pitchfork bifurcation, the stabilizability condition is again equivalent to ae 6= 0,
because state feedback is a special case of output feedback satisfying observability.
4 Output Feedback Stabilization for Hopf Bifurcation
It is assumed that the linearized system matrix L(fl) as in (2) has a pair of complex (critical) eigen-values
other eigenvalues are stable.
As explained in the previous section, this assumption has no loss of generality. The problem to be
investigated in this section is stabilization of Hopf bifurcation with output feedback control, if the Hopf
bifurcation born at for the nonlinear system in (1) is unstable. We consider first the case when
the pair of critical modes corresponding to the pair of complex eigenvalues -(fl) are neither controllable,
nor observable. According to PBH test [12], both left and right eigenvectors corresponding to the pair
of critical eigenvalues satisfy
It follows that when the feedback controller (10) is employed, L
retains the pair
of the critical eigenvalues \Sigmaj ! c at  as the left and right eigenvectors of L  corresponding to the pair of critical eigenvalues, respectively. Then there hold '
thus Hopf bifurcation is again born at which the zero solution changes its stability as fl crosses
[11]. The next result gives the condition on stabilizability of Hopf bifurcation.
Theorem 4.1 Consider the nonlinear control system in (9) with output feedback control law in (10).
Suppose that ~ - 2 ? 0 with ~ -(") as in (6), and the critical modes of L 0 is neither controllable, nor
observable in the sense that 'B simplicity, assume in addition that ~
0 be as in the proof of Theorem 3.1. Let - and - be as in Section 2.2,
and define -   and -   by
\GammaL
where L
. Then there exists a feedback control law that stabilizes Hopf bifur-
cation, if and only if there exists a linear feedback control law stabilizes Hopf bifurcation.
Moreover the existence of stabilizing feedback control law is equivalent to that
and the none critical eigenvalues of L
0 lie on open left half plane for some K 1
Proof: Denote ~ -   (") as the function in (6) for controlled system, and ~ -
2 as the first coefficient of
~
-   ("). From the proof of Theorem 3.1,
by using 'B
there hold
The above equalities imply that
Although Q
due to the feedback term. In fact,
The coefficient ~ -
2 can now be computed as
~
jo
By -
~
where ' and OE are as in (30). For any square nonsingular matrices M , and M +NF ,
whenever FM \Gamma1 N is a scalar. The expression for ~ -
2 as in (29) can now be verified. Hence stabilization
of Hopf bifurcation with nonlinear controllers in (10) is equivalent to the existence of K 1 6= 0 such that
~
and the rest of eigenvalues of L
remain in open left half plane.
The result in Theorem 4.1 is similar to stabilizability of pitchfork bifurcation in Theorem 3.2 where
nonlinear controllers do not offer any advantage over linear ones in terms of bifurcation stabilization.
Hence if the pair of critical modes of the linearized system are uncontrollable and unobservable, linear
controllers are adequate for bifurcation stabilization. Moreover a necessary condition for stabilizability
of Hopf bifurcation as in Theorem 4.1 is that
denotes the Euclidean norm in R p . If the above holds, all K 1 such that ~ -
can be
easily parameterized, especially for scalar K 1 that is a collection of limited number of finite intervals,
or semi-infinite intervals, while stability of the non-critical eigenvalues of L 0 can be tested using root
locus of (22) as argued in the previous section.
In the rest of the section, we study the case when stabilizability condition in Theorem 4.1 is not
additional sensors have to deployed such that H 1 r 6= 0 is valid in order for Hopf
bifurcation to be stabilizable. Although linear controllers can be investigated, it is much easier to
consider the class of nonlinear controllers in (10) with K discussed in [2]. The next result
generalizes the result in [2] to output feedback stabilization.
Theorem 4.2 Consider the nonlinear control system in (9) with output feedback control law in (10)
subject to K as in (6), - as in Subsection 2.2, and the critical
modes of L 0 is observable through linearized output measurement y x. Then there exists a feedback
control law that stabilizes the Hopf bifurcation, if and only if
Re
~
jo
0: (32)
If the above condition holds, stabilizing controllers can be taken as quadratic ones.
Proof: With K critical eigenvalues and left/right eigenvectors are invariant under feed-
back. Moreover
~
~
by It follows that
~
~
~
By the property of the quadratic term,
~
~
After lengthy calculation, we finally obtain
~
jo
Because ~
bifurcation of uncontrolled system is unstable. Hence stabilization requires ~ -
to hold that implies the condition in (32). Conversely, if (32) holds, then there exists K 2 such that
~
ensures stability of Hopf bifurcation for the feedback system. Since only quadratic term
of the nonlinear controller is involved in determination of ~ -
2 , the stabilizing controller can be taken as
quadratic one.
It is worth to point out that the condition (32) in Theorem 4.2 reduces to that of [2] if ~
~
r] is a nonzero real number, that was derived for state feedback control laws.
5 Applications to Rotating Stall Control
Rotating stall is primarily flow instability in axial flow compressors of gas turbines, induced by nonlinear
bifurcations. It can not only lead to large penalties in performance, but also cause catastrophe. Hence
there is a growing interest in suppression of rotating stall using feedback control in order to extend the
stable operating range, and to improve turbine-based aeroengines for axial flow compressors. A number
of control laws are proposed in [8, 14, 16, 20] for rotating stall control, and shown to be effective based
on low order Moore-Greitzer model [18]. In this section established results on stationary bifurcation
in Section 3 will be applied to Moore-Greitzer model that yield identical results as in [8, 14, 16, 20],
thereby validating our results on bifurcation stabilization. Clearly our results are more general, and
apply to more broad bifurcation instability problems, other than rotating stall.
The post-stall model developed by Moore and Greitzer is of the form
where \Phi is the average flow rate, \Psi the pressure rise, R the amplitude square of the disturbance flow
(R and u the actuating signal implemented with throttle, that are all non-dimensionalized. An
obvious equilibrium
It can be easily shown that there exists such that the above equilibrium is stable for
unstable for
with
Thus the equilibrium as in (36) is the local zero solution for both \Psi e and \Phi e are functions
of fl. Moreover the linearized system at the origin possesses exactly one zero eigenvalue at that
implies that fl c is the critical value. The equilibrium in (36) at the critical value of fl is determined as
See [16, 14, 20]. Clearly the nonlinear system (37) can be expanded in the same form as (9) with
c
~
c
. Thus the critical mode of the linearized system is uncontrollable. By the assumption
in [16, 14, 20], oe ? 0, and fi ? 0. For the uncontrolled system,
0:
Therefore it is transcritical bifurcation, and bifurcated solution for R e ? 0 is unstable. Corollary 3.5 is
now applied to compute linear state feedback gain K 1 that stabilizes the bifurcated solution at
Recall that R is the amplitude square of the disturbance flow. Thus R e ! 0 has no physical meaning.
Let T be the required similarity transform as in (23). Then
Straightforward computation gives
23that yield a = \Gamma2
of Corollary 3.5 are equivalent to
Note that (iii) implies that (ii) is true, and (i) is reduced
used. After simplification, the stabilizing state feedback gain satisfies
c
that are exactly the same as in [14]. Taking
that is the same condition obtained in [8]. Taking k
that implies that 0:5 as in [16, 20] is stabilizing too.
It should be pointed out that the results of [16, 20] can also be obtained from Theorem 3.6 directly
that is skipped. A more interesting case is that condition (44) can be obtained using Theorem 3.2 as
well. Indeed, set with A the amplitude of disturbance flow. Then (35) can be written as
Together with (33), and (34), Moore-Greitzer model has a new coordinates (\Psi; \Phi; A), and rotating stall
corresponds to subcritical pitchfork bifurcation that is unstable due to ~
and the output measurement be given by
Then it is easy to see that the critical mode of the linearized system at neither controllable,
nor observable in light of
that are of the same form as (15) with I. By Theorem 3.2, linear controllers suffice for stabilization
that has the form
Direct computation gives
"\Gamma2
6oe
oe
Hence the stabilizing conditions in Theorem 3.2 are equivalent to
c
that are exactly the same as in (44) as
6 Conclusion
This paper investigated bifurcation stabilization using smooth local output feedback controllers for
parameterized nonlinear systems where the critical mode of the linearized system is uncontrollable.
Stabilizability conditions were established for both the case where the critical mode is linearly unob-
servable, and observable through output measurement. The later case includes state feedback as a
special case. It was shown that nonlinear controllers do not offer any advantage over the linear ones for
bifurcation stabilization if the critical mode of the linearized system is unobservable. For the case that
the critical mode of the linearized system is observable, it was shown that linear controllers are adequate
for stabilization of transcritical bifurcation, and quadratic controllers are adequate for stabilization of
pitchfork and Hopf bifurcations, respectively. Stabilization conditions for nonlinear bifurcations with
single critical parameter were characterized in explicit form that can be used to synthesize stabilizing
controllers, if they exist. The applicability of the bifurcation stabilization results was demonstrated for
rotating stall control of axial flow compressors.



--R

"Bifurcation analysis of nonuniform flow patterns in axial-flow gas compressors,"
"Local feedback stabilization and bifurcation control, I. Hopf bifurcation,"
"Local feedback stabilization and bifurcation control, II. stationary bifurcation,"
"Stabilization of a class of nonlinear systems by a smooth feedback control,"
"A simplified approach for control of rotating stall - Part 1/2,"
"Nonlinear control design for systems with bifurcations with applications to stabilization and control of compressors,"
"Control of rotating stall in a low-speed axial flow compressor using pulsed air injection: Modeling, simulations, and experimental validation,"
"Rotating stall control via bifurcation stabilization,"
"Asymptotic stability and feedback stabilization,"
"Nonlinear bifurcations,"
Elementary Stability and Bifurcation Theory
Linear Systems
"Bifurcation control via state feedback for single input nonlinear systems: Part I and Part II"
"The feedbacks which soften the primary bifurcation of MG3"
"Backstepping designs for jet engine stall and surge control,"
"Active control of compressor stall inception: A bifurcation-theoretical approach,"
"Bifurcation analysis of axial flow compressor stability,"
"A theory of post-stall transients in axial compressors: Part I - development of the equations,"
"Subanalytic sets and feedback control,"
"Nonlinear analysis and control of rotating stall in axial flow compressors,"
Introduction to Applied Nonlinear Dynamical Systems and Chaos
--TR
bifurcations;linear controllability observability;bifurcation stabilization;projection method;nonlinear systems