In this paper, the polar controller is applied to the three-state, one-mode Moore-Greitzer Compressor model. A benchmark is first established with a backstepping controller. The polar control method is then explained, and compared to the backstepping controller. The polar controller is used successfully to control the surge and stall problem in the presence of both disturbances and uncertainties

Abdallah, Chaouki T.

Horspool, K. R.

English

University of New Mexico

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Publications Engineering Publications
6-1-1999
On the rotating surge and stall and the polar control
method
Chaouki T. Abdallah
K. R. Horspool
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Abdallah, Chaouki T. and K. R. Horspool. "On the rotating surge and stall and the polar control method." Proceedings of the 1999
American Control Conference (1999): 2668-2672. https://digitalrepository.unm.edu/ece_fsp/185
Proceedings of the American Control Conference 
San Diego, California June I999 
On the Rotating Surge and Stall and the Polar Control Method 
K. R. Horspool and C. T. Abdallah 
The University of New Mexico, EECE Dept. 
Albuquerque, NM 87131, USA 
{horspool, chaouki}@eece .urn. edu 
Abstract 
In this paper, the polar controller is applied to 
the three-state, one-mode Moore-Greitzer Compres- 
sor model. A benchmark is first established with a 
backstepping controller. The polar control method 
is then explained, and compared to the backstepping 
controller. The polar controller is used successfully 
to control the surge and stall problem in the presence 
of both disturbances and uncertainties. 
1. Introduction 
The rotating stall and surge control problem in 
axial flow compressors has been studied in several 
recent papers [1]-[3], and others. The control mo- 
tivation being that at the maximum efficiency op- 
erating condition, the compressor is at an unstable 
equilibrium, and therefore without sufficient control, 
the compressor will leave the axisymmetric design 
flow, and enter either a rotating stall, deep surge, 
or classic surge. Rotating stall is an inherently two- 
dimensional local compression system oscillation. I t  
manifests itself as a region of severely reduced flow 
that rotates at a fraction of the rotor speed. Surge is 
a one-dimensional axisymmetric global compression 
system oscillation that can cause flameout and en- 
gine damage. Traditionally, designers have avoided 
the problems of rotating stall and surge by accepting 
a lower efficiency of the compression system. 
One model to describe the instabilities of the com- 
pressor is the three-state, one-mode Moore-Greitzer 
model found in [4] and [5]. Several variations exist on 
the basic model, and in this paper one such variation 
is used to evaluate the control law required for stabi- 
lization. To achieve stabilization, several techniques 
have been applied by various authors: backstepping 
[5], nonlinear robust control [3], and nonlinear distur- 
bance rejection [2]. These three control techniques 
will be discussed, and the recently introduced po- 
lar control method [6] will be applied, and compared 
to previous works. The recently published method 
of stabilizing nonlinear systems via polar control [6 ] ,  
provides an efficient, multistage logic controller with 
adjustable parameters. The design of the controller 
is based on a Lyapunov approach using backstep- 
ping, where the designer has the freedom to cancel 
unwanted nonlinearities during backstepping. How- 
ever, in the standard backstepping approach, one 
cannot expect to cancel the nonlinearities exactly. 
Therefore in the polar control design, an error term 
is defined, and the system is then modified to re- 
flect the error and its derivative, making the polar 
controller robust in the presence of uncertainties. 
Section 2. 
presents an overview of a benchmark backstepping 
design for a surge and stall model. Section 3. presents 
the polar controller design. Section 4. deals with con- 
troller design for disturbance rejection while section 
5. deals with the robust control problem. Finally, 
our conclusions are given in section 6. 
This paper is organized as follows. 
2. Backstepping Design Overview 
The backstepping technique developed in [5] is 
first reviewed to establish a benchmark for analyz- 
ing the control law. To ensure that the controllers 
are evaluated evenly, the initial conditions used in [3] 
are used throughout this paper. 
The first model utilized is found in equations (2.7), 
(2.8), and (2.9) of [5], and is duplicated in equation 
(1) below. This model has already been translated 
to the origin such that the equilibrium state is at 
= Re = 0. The Control law has also been 
defined to be U = g(q5 + 1 - %). 
= 
4 = -II, - - 143 - 3R4 - 3R 2 2  
R = u R ( - ~ + - + ~  -R) (1) 
$ ) = U  
Where R ,  the normalized stall cell squared ampli- 
tude, is restricted to being a nonnegative number. 
Note that + is the mass flow translated to the origin, 
@JT is the mass flow through the throttle, and II, is 
the pressure rise translated to the origin. ?I, is al- 
lowed to be negative, but has a lower bound defined 
by II, = Q - QCo - 2, where Q is the untranslated 
pressure rise, and is restricted to being a positive 
number due to physical limitations. QCo, p, and o 
0-7803-4990-6199 $10.00 0 I999 AACC 2668 
are constant parameters and ,8 and cr are required to 
be positive. Note that R = 0 is an invariant set, and 
defines a plane. Therefore if the system moves to a 
state where R = 0, it will stay on that plane, and 
the system could be represented as a two state sys- 
tem. This is what KrstiC and KokotoviC have noted 
and used to start their design process in the section 
"Design for No-Stall Model" in [5 ] .  They have also 
noted that an upper bound exists for R as defined 
by equation (2.11) in [5], and have exploited these 
two properties to simplify their design. By careful 
examination of ( l ) ,  it can be seen that once q5 is 
forced to  the zero state, R will follow, and therefore 
it is not necessary to control R directly. This pro- 
cess was started by KrstiC and KokotoviC and will be 
used throughout this paper. Following the procedure 
in [5] a control law was found to be 
Where CO = c1 + p, c1 2 0, and c2 > 0 make the 
system globally asymptotically stable. Note that 
when R = 0, the control law reduces to a linear con- 
troller. The closed-loop system defined by (1) and (2) 
was simulated using initial conditions $0 = -0.25, 
RO = 0.5, and $0 = 0, and system parameters 
(T = 3.6 and c1 = c2 = 1. The results of the sim- 
ulation are shown in Figure 1. 
0 0.5 1 1.5 I 2.5 3 5.5 4 4.5 5 
-151 ' * " " " "
Figure 1: KrstiC and KokotoviC's Backstepping Con- 
troller, and the control effort U. 
3. Polar Control 
In order to implement the polar controller, three 
steps must be considered on the m-dimensional 
space R". The first step to consider is the simplex 
partition of the unit sphere in Rm into m + 1 
partition vectors [SI. Next, the polar partition is 
established to ensure that an overlapping partition 
exists on R" for 0 5 w 5 A; where w is a sector 
width parameter that defines the amount of overlap 
between partitions. Finally, the polar controller is 
established. In this paper, we will only consider a 
polar controller for a system with a strict feedback 
structure; which is in contrast to the embedded 
polar controller for strict feedback systems. The 
difference between the two structures is that the 
former has only one design step, and relies on the 
strict feedback nature of the system. The latter 
utilizes strict feedback as well, but develops a 
controller to directly influence each state of the 
plant. Effectively the system with one the one-step 
controller design is used to drive the plant to a 
stability manifold and once on the manifold the 
system will stabilize. The embedded polar controller 
works to directly stabilize each state of the system 
until the entire plant reaches the desired operating 
point. 
The simplex partition is defined on the m- 
dimensional space, and is a collection of m + 1 unit 
length vectors (II1, I I 2 ,  ..., IIm+1} satisfying the fol- 
lowing conditions: 
rITrIi = l , i = 1 , 2  ,..., m + l  
rITllj = q,i # j  
II, + II2 + + IIm+1 = 0 (3) 
IIrIi-IIjll = d , i # j  
where 11 
is defined by the open sets 
is the 2-norm. The polar partition of Rm 
Ci(w) = (5' E Rm : KT (&) > w , }  (4) 
where w > 0, 
control is defined as 
= -IIi, i = 1,2,  ..., m + 1. The polar 
where m is the dimension of the control input, and 
where 
hi = max(7i, 0) 
= -I&; i = 1,2,  ..., m + 1 (8) 
2669 
The function K(llSll) is a monotonically increasing 
scalar function of IlSll, and the condition on w is to 
ensure that an overlapping partition exits on W. 
While the backstepping controller made use of the 
knowledge of the nonlinearities of the system, the po- 
lar controller does not. The polar controller merely 
dominates the nonlinearities, and thus provides a de- 
gree of robustness. Since the backstepping controller 
can utilize the nonlinearities to the designer’s advan- 
tage, the controller can be made to use less control 
effort than the polar controller, but overall the polar 
controller can be made more robust. 
Using the model of (1) while following the ap- 
proach used in [5], avoiding cancellations, using the 
fact that R has an upper bound, and that R is neg- 
ative semidefinite, a polar controller was developed. 
Letting $c = -$q52 - $$3, and noting that $c can be 
rewritten as -3 (4 + $) 4 + Eq5, the first equation 
of model (1) can be transformed into 
2 
NOW letting$des(4,R) = ( C I + ! ) ~ - ~ R = Q ~ - ~ R ,  
and defining S = 1c, - $des, we obtain 
6 = -s-c1q5- - 1 (4+ ; )24 -3R4  
2 
s = Q S+c1++-  4 + -  4+3R4 [ x ;I2 ] 
+3aR (-24 - q52 - R) + U (9) 
R = U R ( - ~ ~ - ~ ~ - R R )  
Now applying the rules for polar control, and noting 
that for this particular case m = 1, was chosen 
such that Vi = 1 and V2 = -1. The term K (IlSll) 
was chosen to be K * IlSll, where K is a positive real 
constant that can be varied to  increase the gain of the 
controller. By increasing the gain of the controller, 
the nonlinearities and any perturbations or distur- 
bances can be dominated by the controller U, and 
thus global asymptotic stability is achieved. Using 
the same values as before for a, Q, and c1, and choos- 
ing values of K = 1, and w = 0.3, the closed loop 
system was simulated. The initial conditions are the 
same as before, except that the initial condition for S 
was determined by evaluating S = $-$des at $0 and 
$des,O, and was found to  be -1.3. The results of the 
simulation are shown in Figure 2. There are several 
comments worth noting about the polar controller. 
First, it can be shown using Lyapunov techniques 
that the system is asymptotically stable. Second, 
notice that the time scale in Figure 2 is twice that in 
I , , ,  
‘0 1 2  3 4 5 6 7 8 0 D 
Figure 2: Polar Controller, and the control effort U 
Figure 1. This was only done to  show the richness 
of the transient behavior with the polar controller; 
however, both systems show approximately the same 
levels of decay after 5 seconds of simulation. A third 
and most important result is that the initial magni- 
tude of the control effort is about 4 times less with 
the polar controller than with the backstepping con- 
troller of [5]. This result is extremely important if 
one is to implement this controller in hardware since 
the magnitude of the control effort can determine the 
type of hardware used. Another important result to  
notice is that while 1c, has a definite lower bound, S 
does not. 
4. Disturbance Rejection 
In [2], Haddad et al. have recognized that one of 
the main contributors for the system to enter either 
a rotating stall or surge, is the presence of distur- 
bances. The model becomes in this case, 
3 1  6, = 2 2  
A = --(TA (-24 - 42 - A2) + Plwi 
-1c, - -q52 - -43 - 3A24 - 3A2 + P2w2 
1 
2 (10) 
$ = U  
for the system of [2], and converting to polar form 
yields 
2670 
CA (-24 - 42 - A2) + BIWI 1 A = -  
2 
where PI, /32 are positive constants and w1, w2 are 
L2 external disturbances as defined in [2]. To an- 
alyze the system, the same disturbances used in 
[2] were applied i.e, w1 = 0.1e-0.5t sint and wp = 
0.5e-0.05tsin0.7t. Using the model of ( l ) ,  the con- 
trol law defined in [5], and adding the disturbance, 
the system was not able to yield valid results from 
the ODE integrator. By making the substitution 
R = A2,  and forming the system (lo), the ODE 
integrator gave valid results, and after substituting 
R = A2 into (2), the system of (10) was simulated as 
shown in Figure 3. Note that in [2], the control uses 
5 1 
Figure 3: Backstepping controller with L2 distur- 
bance, and the control effort U. 
a negative U, whereas in this paper the control uses a 
positive U. The polar controller of (11) was also able 
to stabilize the system as shown in Figure 4. Notice 
that with the disturbance added, the controller ef- 
fort is larger. The control effort can be reduced and 
smoothed out by simply increasing the gain of the 
controller. The gain of the polar controller was in- 
creased from K = 1 to K = 2, as shown in Figure 5. 
Specific attention is drawn to this result since this is 
one of the main advantages of the polar controller. 
By simply increasing the gain of the control law, a 
level of robustness is achieved, and the disturbance is 
rejected much easier. While the difference is subtle 
for this example, it will be shown later that the in- 
crease in gain can yield significant reductions. One 
very important observation is that for the controller 
in [2], the compressor flow $J is highly oscillatory, 
and takes a long time to die out. With the polar 
controller, the compressor flow $J, has a smaller os- 
cillation frequency and dies out quicker. This is true 
Polar Conlmller with Dislutbanca 
5 I 
, (  
0 5 10 15 20 25 30 35 40 45 50 
-5 
Conwol ElfM U 
0 5 10 15 20 25 30 35 40 45 50 
-51 " " " " ' J 
Figure 4: Polar Controller with L2 Disturbance,and 
the control effort U .  Gain K = 1. 
for both values K = 1, and K = 2. It can be shown 
that with higher gains, the compressor flow 4 will 
die out quicker with virtually no oscillation. The 
initial control effort does tend to increase, thus cre- 
ating a practical upper bound on the size of the gain 
K.  Nevertheless, the polar controller is able to reject 
the disturbances more efficiently than the controller 
of [2]. 
5. Robust Control 
In this section, the system of (1) is again mod- 
ified to study robustness, and the polar controller 
obtained earlier is applied. To obtain the uncertain 
model, we add the term A@ to  the first modified 
equation where, 
A@ = klII(q5)sinw4 
n(4) = tan-'[p(4 - U)] - tan-' [p(4 - b)]  
so that, 
3 1  4 = -$J - ~4~ - -41~ 2 - 3A24 - 3A2 + A@ (12) 
and after transforming to polar form yields the fol- 
lowing equations: 
1 
+3uA2 [-24 - 42 - A2] + U 
1 
2 
A = - 0 A ( - 2 4 - 4 ~ - A ' )  
267 1 
P&r Convollsr wlh Di.isrbams 
0 5 10 I5 20 25 30 35 40 45 50 
Cmtml EHat U 
4 
Polar Cqnlrmler 
0 2 b 6 8 10 12 14 16 18 XI 
;h , , , , Conlml,EIlonU , , , , ~ 
J 
0 5 10 15 20 25 30 35 40 45 50 
41 
Figure 5: Polar Controller with Lz Disturbance, and 
control effort U. Gain K = 2. 
Using the values kl = 0.3751r, w = 2.0, a = 0, 
b = 1, and p = 1, the system was simulated, and the 
results are shown in Figure 6 where K = 2. Haddad 
et al. [3] have stabilized the system with virtually no 
oscillation and a little overshoot; whereas the polar 
controller has a large overshoot. The advantage of 
the polar controller is that  one simple controller has 
been developed, and stabilization was achieved by 
increasing the gain. The controller by Haddad et 
al. is a much more complicated controller, and their 
robust controller is significantly different then the 
controller found for disturbance rejection; whereas 
the polar controller used here is the same controller 
used throughout. 
6. Conclusion 
The polar controller has been shown to be an effec- 
tive control method for the rotating stall and surge 
problem. While the controller tends to  oscillate, it is 
a simple controller that has a lower peak amplitude 
for the control effort than the backstepping controller 
of [5]. The polar controller was also shown to sup- 
press disturbances, and by simply increasing the gain 
of the controller the overall control effort could be re- 
duced. The polar controller was also shown to have 
a degree of robustness by increasing the gain of the 
controller. The ability to  increase the gain allows for 
one controller to  be used for both the disturbance 
rejection, and robust control models. This is in con- 
trast to  the controllers developed in [3] and [2], where 
different and complicated controllers are developed 
for each case. In future research, the polar controller 
will be modified to  control both rotating stall and 
surge. In future research, more work will be done to  
Figure 6: Polar Controller with Perturbation for Ro- 
bust Control. Gain K = 2. 
study the effects of different forms of K(llSll), and 
varying other parameters such as w and the degree 
of the polynomial in IlSll. 
References 
[l] A. Banaszuk and A. Krener. Design of controllers 
for mg3 compressor models with general charac- 
teristics using graph backstepping. In Proceedings 
of the American Control Conference, volume 2, 
pages 977-981, Albuquerque, NM, 1997. 
[2] W. Haddad, V. Chellaboina, and J .  Fausz. Op- 
timal nonlinear disturbance rejection control for 
nonlinear cascade systems. In Proceedings of the 
American Control Conference, volume 1, pages 
448-452, Albuquerque, NM, 1997. 
[3] W. Haddad, V. Chellaboina, J. Fausz, and 
A. Leonessa. Optimal nonlinear robust control for 
nonlinear uncertain cascade systems. In Proceed- 
ings of the American Control Conference, vol- 
ume 1, pages 403-407, Albuquerque, NM, 1997. 
[4] M. Krstit, I. Kanellakpoulos, and P. KokotoviC. 
Nonlinear and Adaptive Control Design. John 
Wiley & Sons, Inc., New York, NY, 1st edition, 
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[5] M. KrstiC and P. KokotoviC. Lean backstepping 
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[6] J .  Medanic and M. Yuan. Logic controller de- 
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2672 


On the rotating surge and stall and the polar control method

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On the rotating surge and stall and the polar control method

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