Experimental Application of Extremum Seeking on an Axial-Flow ...
Experimental Application of Extremum Seeking on an Axial-Flow Compressor Hsin-Hsiung Wang
Dept. of Mech. Engineering University of Maryland
Simon Yeung
Division of Eng. and Applied Science California Institute of Technology
Abstract
We show an application of the method of extremum seeking to the problem of maximizing the pressure rise in an axial ow compressor. First we apply extremum seeking to the Moore-Greitzer model and design a feedback scheme actuated through a bleed valve which simultaneously stabilizes rotating stall and surge and steers the system towards the equilibrium with maximal pressure. Then we implement the scheme on a compressor rig in Richard Murray's laboratory at the California Institute of Technology. We perform stabilization of rotating stall via air injection and implement extremum seeking through a slow bleed valve. The experiment demonstrates that extremum seeking ensures the maximization of the pressure rise starting on either side of the stall inception point. The experiment also resolves a concern that extremum seeking requires the use of periodic probing|the amplitude of probing needed to achieve convergence is far below the noise level of the compressor system (even outside rotating stall).
1 Introduction
Even though the methods of \extremum seeking" [1, 3, 4, 5, 7, 8, 12, 14, 15, 16, 17] have been in existence since the 1950's, much before the theoretical breakthroughs in adaptive linear control of the 1980's, a rigorous proof of stability did not exist before the recent paper [11]. However, the idea to use extremum control for maximizing the pressure rise in an aeroengine compressor is not new. As far back as in 1957, George Vasu of the NACA (now NASA) Lewis Laboratory published his experiments in which he varied the fuel ow to achieve maximum pressure [18]. Our main objective in the present paper is to apply the extremum seeking scheme to an axial- ow compressor to maximize its pressure rise. We brie y summarize the extremum seeking scheme in Section 2. In Section 3 we rst present an application of extremum seeking to a compressor model with bleed valve actuation. We achieve maximization of the compressor pressure rise in the presence of an uncertain \compressor characteristic" whose argument of the maximum is unknown. The main results of the paper are experimental and The work of the rst and the third authors was supported in part by the Air Force Oce of Scienti c Research under Grant F49620-961-0223 and in part by the National Science Foundation under Grant ECS-9624386. The work of the second and the rst author was supported in part by the Air Force Oce of Scienti c Research under Grant F49620-95-1-0409. y Corresponding author.
Miroslav Krsticy
Dept. of AMES UC San Diego
they are shown in Section 4 where we implement extremum seeking on an axial- ow compressor in Richard Murray's laboratory at the California Institute of Technology. The extremum seeking is implemented via a slow bleed valve, while rotating stall stabilization is performed with air injection, as in [2]. The results demonstrate the e ectiveness of extremum seeking in maintaining maximal pressure rise while preventing rotating stall (of either large or small amplitude).
2 Extremum Seeking Scheme
In [11] we described a control scheme which can enforce convergence to the peak of an output equilibrium map. Consider a general SISO nonlinear model x_ = f (x; u) (2.1) y = h(x) ; (2.2) n where x 2 IR is the state, u 2 IR is the input, y 2 IR is the output, and f : IRn IR ! IRn and h : IRn ! IR are smooth. Suppose that we know a smooth control law u = (x; ) parameterized by a scalar parameter . Assumption 2.1 There exists a smooth function l : IR ! IRn such that f (x; (x; )) = 0 if and only if x = l() : (2.3) Assumption 2.2 For each 2 IR, the equilibrium x = l() of the closed-loop system is locally exponentially stable. Assumption 2.3 There exists 2 IR such that (h l)0 ( ) = 0 (2.4) 00 (h l) ( ) < 0 : (2.5) Thus, we assume that the output equilibrium map y = h (l()) has a maximum at = . Our objective is to develop a feedback mechanism which maximizes the steady state value of y but without requiring the knowledge of either or the functions h and l. Our feedback scheme is shown in Figure 1. It is an extension of a simple method for seeking extrema of static nonlinear maps [3]. The parameters in Figure 1 are selected as !h = !!H = !!H0 = O(!) (2.6) 0 !l = !!L = !!L = O(!) (2.7) 0 k = !K = !K = O(!) ; (2.8) where ! and are small positive constants and !H0 , !L0 , and K 0 are O(1) positive constants. a also needs to be small.
+l
^
6
k s
- x_ = f (x ; (x ; )) y = h(x)
!l s + !l
-
y
ly ? s +s!h
6
a sin !t
Figure 1: A peak seeking feedback scheme. Theorem 2.1 ([11]) Consider the feedback system in Figure 1 under Assumptions 2.1{2.3. There exists a ^ ball of initial conditions around the point x ; ; ; = (l( ) ; ; 0 ; h l( )) and constants ! ; ; and a such that for all ) ; 2 (0 ; ), and a 2 (0 ; a), the ! 2 (0 ; ! solution x(t) ; ^(t) ; (t) ; (t) exponentially converges to an O(! + + a)-neighborhood of that point. Furthermore, y(t) converges to an O(! + + a)-neighborhood of h l( ).
3 Application to a Compressor Model
We now apply the peak seeking scheme of Section 2 to the Moore-Greitzer compressor model [13]: Z 2 p p R C + 2 R sin sin d (3.1) R_ = 3 0 Z 2 p C + 2 R sin d (3.2) _ = ? + 21 0 1 _ = 2 ( ? T ) ; (3.3) where is the mass ow coecient, is the pressure rise coecient, R is a quadratic function of the amplitude of the rotating stall, and the other quantities are de ned in [19, Table 1]. The throttle ow T is related to the pressure rise through the throttle characteristic (3.4) = 12 (1 + C0 + T )2 ; where is the throttle opening.
3.1 Pressure peak seeking for the surge model
For clarity of presentation, we rst consider the model (3.1){(3.3) restricted to the invariant manifold R = 0: _ = ? + C () (3.5) 1 (3.6) _ = 2 ( + T ) : This model is, in fact, the surge model introduced by Greitzer [6], which describes limit cycle dynamics in centrifugal compressors. The function C () represents the \compressor characteristic," which is a typical S-shape function. Our objective is to converge to the peak of this characteristic and operate the compressor with maximum pressure. Thus, we denote x = (; ) and y = . In [9] we showed that a control law of the form ? + 2 (cp ? c ) (3.7)
=
stabilizes equilibria parameterized by ?. If the design parameters are chosen to satisfy 2 > , c > 0, c = c + 12 > 0, and c d () > max C (3.8) c d (which is nite), then the control law (3.7) achieves global exponential stability of equilibria parameterized by ?. In order to apply the peak seeking scheme, we rst check that all three assumptions from Section 2 are satis ed: 1. From (3.4){(3.7), ? is given by ? = ? (0 ) = 1+C0 + 2 (c 0 ? c C (0 )) ; (3.9) where 0 is the equilibrium value of . In view of (3.8), it is clear that the function ? () is invertible. Thus, for each value of ?, there?is only one ? equilibrium ( ; ) = ?? 1 (?) ; C ?? 1 (?) ; which means that Assumption 2.1 is satis ed. 2. As we indicated above, it was proved in [9] that (3.8) guarantees that the equilibrium is exponentially stable not only locally but also globally, hence Assumption 2.2 is satis ed. 3. Following ?the notation in Section 2, y = h 4 ? 1 l(?) = C ? (?) . The Moore-Greitzer model (3.1){(3.3) is scaled so that C () always has a maximum ? at = 1. Since ? () is bijective, C ?? 1 (?) has a maximum at ? = 1 + C0 + 2 (c ? c C (1)) ; (3.10) that is, ? C ?? 1 0 (? ) = 0 (3.11) ? 00 ? 1 C ? (? ) < 0 : (3.12) Hence, Assumption 2.3 is satis ed. Since Assumptions 2.1{2.3 are satis ed, we can apply the peak seeking scheme given in Figure 2 with ? = ?^ + a sin !t : (3.13) By Theorem 2.1, for suciently small !, , and a, (t) converges to an O(! + + a)-neighborhood of 1 and (t) converges to an O(! + + a)-neighborhood of its maximum value C (1). The application of the peak seeking scheme to the surge model (3.5), (3.6) will make clearer the application to the full Moore-Greitzer model (3.1){(3.3).
3.2 Simulations for the full MG model
In Figure 2, we replace the surge model with the full model and extend the stabilizing controller by R ; _ terms with coecients cR ; d , respectively (the stabilizing control laws are derived in [10]). The bifurcation diagrams are shown in Figure 3, in which the solid lines represent stable equilibria and the dash lines represent unstable equilibria. The output function, the solid curve mapping in the vs. ? ? ? in Figure 3, is not
-? +l 6 +l ?^
6
p
-
Stabilizing feedback 2 (c ? c )
k s
!l s + !l
2.7
2.65
I
2.6
l ? s +s!h
2.55
6
2.5 0.98
Peak seeking feedback Figure 2: Peak seeking scheme for the surge model (3.5), (3.6). a sin !t
3
2.75
-
Surge model (3.5), (3.6)
Figure 4: A trajectory under the peak seeking feedback for ! = 0:03 and k = 0:4. 1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
2.75
2 2.7
2.5 1
2
R
1.5 1
−100
4
??
0
?
100
−2 −200
3
2
2
1
1
0 −200
0
??
0 −2
?
100 2.5 0.98
Figure 5: A trajectory under the peak seeking feedback for ! = 0:03 and k = 1:5.
? ? ? Figure 3: Bifurcation diagrams for the case = 0:9, = 1:42, with the full-state controller. The control gains are cR = 30, c = 7 and c = 20. −100
0
100
I
2.55
−100
4
3
2.65
2.6
−1
0.5 0 −200
0
−1
0
1
2
continuously di erentiable but we can show that the average equilibrium is O(a)-close to the point (R ; ; ) = (0 ; 1 ; C(1)), that it is to the right of the peak, and that it is exponentially stable. We now present simulations of the peak seeking scheme for a compressor with C0 = 0 ; C0 = 0:72 ; = 1:42 ; = 0:9 ; = 4, and with a stabilizing controller whose parameters are = 1:42 ; c = 2 ; c = 4 ; cR = 7 ; d = 0. Our rst simulation employs a peak seeking scheme with a = 0:05 ; ! = 0:03 ; !h = 0:03 ; !l = 0:01 ; and k = 0:4. The trajectory is shown in Figure 4, where the darker curve represents the trajectory and the lighter lines represent the axisymmetric and stall characteristics. The trajectory starts from an equilibrium on the stall characteristic and converges to a small periodic orbit near the peak of the compressor characteristic. If the peak seeking parameters are selected di erently, for example, if k increased to k = 1:5, the new shape of the trajectory is shown in Figure 5. In this case the convergence is smoother and faster but the periodic orbit is farther from the peak. (Note, however, that, given more time, the periodic orbit would slowly approach the peak.) Even though faster adaptation throws the periodic orbit further to the right of the peak, the periodic orbit remains in the \ at" region of the compressor
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
characteristic where variations in mass ow result in only minor variations of the pressure rise . As explained in the previous section of the paper, the convergence of the trajectory to a close neighborhood of the peak is the result of regulating ?^ (t) to a neighborhood of ? . Figure 6 shows the time traces of ?^ (t) ? ?^ (0) for the trajectories in Figures 4 and 5. In this case ?^ (0) = ?1:2 and ? = ?0:90, which means that ? ? ?^ (0) = 0:30. This explains why the trajectory in Figure 4 converges closer to the peak than that in Figure 5. Since a permanent presence of the periodic perturbation is undesirable, we now show that it can be disconnected after a short peak seeking period. Figure 7 shows the pressure transient and steady state up to t = 2550, at which time, k and a are set to zero. The disconnection of adaptation makes the trajectory transition from the periodic orbit to an equilibrium on the compressor characteristic, just to the right of the peak. To make the transient more visible, we have set R(2550 + 0) = 0:06 because R(2550 ? 0) is practically zero. Figure 7 also shows that the pressure variations in steady state under the peak seeking feedback are hardly noticeable, especially if compared to a large gain in the DC value of the pressure.
4 Experiment Results on Caltech Rig
A prerequisite for experimental validation of the scheme from Section 3 is the availability of a high-bandwidth bleed valve for stabilization of rotating stall. However, as shown in [2], rotating stall can also be stabilized by air injection. In this section we combine the air injection rotating stall controller from [2] with the extremum
0.375
0.8
Fig. 5
0.7
0.37
0.365
0.6
?^
0.5
0.4
Fig. 4
0.3
0.2
0.355
0.35
0.345
0.34
0.1
0 0
0.36
500
1000
1500
t
2000
2500
0.335 110
3000
2.75
Fig. 5
b
2.7
Fig. 4
2.65
+l
6
2.6
2.55
2.5 0
500
1000
1500
t
2000
2500
3000
Figure 7: Time response of the pressure rise. At t = 2550 we disconnect the peak seeking feedback.
seeking scheme from Section 2 to achieve maximization of pressure rise. Peak seeking is implemented via a bleed valve whose bandwidth is 51 Hz.
4.1 Stall stabilization
Stall stabilization is performed by air injection in a onedimensional on-o fashion. When the measured amplitude of the rst mode of the stall cell is above a certain threshold, all three injectors are fully open. Otherwise they are closed. The set point of the compressor is varied by a bleed valve. The characteristic of the pressure rise with respect to the bleed angle b is shown in the Figure 8. Note that higher bleed angle means lower overall throttle opening. There is a clear peak in the characteristic curve. The points to the left of the peak are axisymmetric equilibria. The points to the right of the peak are stabilized low amplitude stall equilibria.
4.2 Filter design
We implement the extremum seeking scheme in a con guration shown in Figure 9. In the theoretical analysis in [11] noise was not considered and a high-pass lter s s+!h was employed. Because of noise in the experiment we use a band pass lter. From the power spectrum analysis, we learn that the noise is above 150 Hz. Also we know that the stall frequency is about 65 Hz. Since the lter should cut out both the high frequency noise and the stall oscillations, as well as DC, we choose the pass band to be 4 to 6 Hz, and implement it as 3rd order Butterworth lter. We do not use a low pass lter !l s+!l .
4.3 Experimental Results 4.3.1 Initial point at axisymmetric characteristic
We select the integrator gain as 600 and set the frequency of the perturbation to 5 Hz. From Figure 8 we
115
120
125
130
b
135
140
145
150
155
Figure 8: vs. bleed angle b.
Figure 6: Time response of ?^ = ?^ (t) ? ?^ (0). 4
Rig - withCaltech Air Injection
k s a sin !t
-
l ? Band-pass lter 6
Peak seeking feedback
Figure 9: Peak seeking scheme for the Caltech rig. know that the peak is around 134{139. We set the initial bleed angle at 110, the farthest point to the left in Figure 8, and set the perturbation to 3 . The perturbed bleed angle is shown in Figure 10 and the pressure rise response is shown in Figure 11. Comparing the peak pressure rise 0:372 of Figure 8 to that of Figure 11, we see that they are the same.
4.3.2 Initial point at nonaxisymmetric characteristic
For the compressor control, the most important issue is to control the system to avoid the stall that causes the deep pressure drop. Since the air injection can control stall for a reasonable interval, we can set the initial point at the stall characteristic. We select the initial point as 150 bleed valve angle because from Figure 8 we can see that 150 is the largest angle we can achieve without a deep drop of the pressure rise. In this case we choose the gain of the integrator as 400. The perturbation signal is set to 3 as in the axisymmetric case. The perturbed bleed angle is shown in Figure 12 and the pressure rise response is shown in Figure 13. The peak pressure in Figure 13 coincides with that in Figure 8. A closer look at Figures 12 and 13 shows that the rotating stall amplitude reduces as the bleed angle reduces. The e ect of peak seeking on a system starting in rotating stall is to bring it out of stall without pushing the operating point away from the peak and reducing the pressure rise. In both Figures 10 and 12 one can observe uctuations of the mean of the bleed angle at the peak. Comparing with Figures 11 and 13, we see that these uctuations occur at the same time when pressure drops resembling stall inception occur. Peak seeking reacts to
140
155
135 150 130
b
b
125
120
145
140
115 135 110
105 0
5
10
15
20
t
25
30
35
40
45
130 0
50
Figure 10: The time response of the bleed angle initiating from the axisymmetric characteristic.
5
10
t
15
t
15
20
25
Figure 12: The time response of the bleed angle initiating from the stall characteristic. 0.42
0.4 0.4 0.38
0.38
0.36
0.34
0.36 0.34 0.32 0.3
0.32 0.28 0.3 0
0.26 5
10
15
20
t
25
30
35
40
45
50
Figure 11: The time response of the pressure rise initiating from the axisymmetric characteristic.
this by pushing the operating point further to the right on the axisymmetric characteristic and then slowly returning it to the peak.
References
[1] K. J. Astrom and B. Wittenmark, Adaptive Control, 2nd edition, Reading, MA: Addison-Wesley, 1995. [2] R. D'Andrea, R. L. Behnken, and R. M. Murray, \Active control of rotating stall using pulsed air injection: A parametric study on a low-speed, sxial slow compressor," In Proceedings of SPIE, volume 2494, pages 152{165, Orlando Florida, 1995. [3] P. F. Blackman, \Extremum-Seeking Regulators," in J. H. Westcott, Ed., An Exposition of Adaptive Control, New York, NY: The Macmillan Company, 1962. [4] C. S. Drapper and Y. T. Li, \Principles of optimalizing control systems and an application to the internal combustion engine," ASME, vol. 160, pp. 1{16, 1951, also in R. Oldenburger, Ed., Optimal and Self-Optimizing Control, Boston, MA: The M.I.T. Press, 1966. [5] A. L. Frey, W. B. Deem, and R. J. Altpeter, \Stability and optimal gain in extremum-seeking adaptive control of a gas furnace," Proceedings of the Third IFAC World Congress, London, 48A, 1966. [6] E. M. Greitzer, \Surge and rotating stall in axial
ow compressors|Part I: Theoretical compression system model," Journal of Engineering for Power, pp. 190{198, 1976. [7] O. L. R. Jacobs and G. C. Shering, \Design of a single-input sinusoidal-perturbation extremum-control system," Proceedings IEE, vol. 115, pp. 212-217, 1968. [8] V. V. Kazakevich, \Extremum control of objects with inertia and of unstable objects," Soviet Physics, Dokl. 5, pp. 658661, 1960. [9] M. Krstic, D. Fontaine, P. V. Kokotovic, and J. D. Paduano, \Useful nonlinearities and global bifurcation control of jet engine surge and stall," submitted to IEEE Transactions on Automatic Control, 1996.
0
5
10
20
25
Figure 13: The time response of the pressure rise initiating from the stall characteristic. [10] M. Krstic and H. H. Wang, \Control of Deep-Hysteresis Aeroengine Compressors|Part II: Design of Control Laws," Proceedings of the 1997 American Control Conference, Albuquerque, NM. [11] M. Krstic and H. H. Wang, \Design and stability analysis of extremum seeking feedback for general nonlinear systems," Proceedings of the 1997 Conference on Decision and Control, San Diego, CA, TA02-3. [12] M. Leblanc, \Sur l'electri cation des chemins de fer au moyen de courants alternatifs de frequence elevee," Revue Generale de l'Electricite, 1922. [13] F. K. Moore and E. M. Greitzer, \A theory of post-stall transients in axial compression systems|Part I: Development of equations," Journal of Engineering for Gas Turbines and Power, vol. 108, pp. 68{76, 1986. [14] I. S. Morosanov, \Method of extremum control," Automatic & Remote Control, vol. 18, pp. 1077-1092, 1957. [15] I. I. Ostrovskii, \Extremum regulation," Automatic & Remote Control, vol. 18, pp. 900-907, 1957. [16] A. A. Pervozvanskii, \Continuous extremum control system in the presence of random noise," Automatic & Remote Control, vol. 21, pp. 673-677, 1960. [17] J. Sternby, \Extremum control systems: An area for adaptive control?" Preprints of the Joint American Control Conference, San Francisco, CA, 1980, WA2-A. [18] G. Vasu, \Experiments with optimizing controls applied to rapid control of engine pressures with high amplitude noise signals," Transactions of the ASME, pp. 481{488, 1957. [19] H. H. Wang, M. Krstic, and M. Larsen, \Control of deephysteresis aeroengine compressors|Part I: a Moore-Greitzer type model," Proceedings of the 1997 American Control Conference, Albuquerque, NM.
Dept. of Mech. Engineering University of Maryland
Simon Yeung
Division of Eng. and Applied Science California Institute of Technology
Abstract
We show an application of the method of extremum seeking to the problem of maximizing the pressure rise in an axial ow compressor. First we apply extremum seeking to the Moore-Greitzer model and design a feedback scheme actuated through a bleed valve which simultaneously stabilizes rotating stall and surge and steers the system towards the equilibrium with maximal pressure. Then we implement the scheme on a compressor rig in Richard Murray's laboratory at the California Institute of Technology. We perform stabilization of rotating stall via air injection and implement extremum seeking through a slow bleed valve. The experiment demonstrates that extremum seeking ensures the maximization of the pressure rise starting on either side of the stall inception point. The experiment also resolves a concern that extremum seeking requires the use of periodic probing|the amplitude of probing needed to achieve convergence is far below the noise level of the compressor system (even outside rotating stall).
1 Introduction
Even though the methods of \extremum seeking" [1, 3, 4, 5, 7, 8, 12, 14, 15, 16, 17] have been in existence since the 1950's, much before the theoretical breakthroughs in adaptive linear control of the 1980's, a rigorous proof of stability did not exist before the recent paper [11]. However, the idea to use extremum control for maximizing the pressure rise in an aeroengine compressor is not new. As far back as in 1957, George Vasu of the NACA (now NASA) Lewis Laboratory published his experiments in which he varied the fuel ow to achieve maximum pressure [18]. Our main objective in the present paper is to apply the extremum seeking scheme to an axial- ow compressor to maximize its pressure rise. We brie y summarize the extremum seeking scheme in Section 2. In Section 3 we rst present an application of extremum seeking to a compressor model with bleed valve actuation. We achieve maximization of the compressor pressure rise in the presence of an uncertain \compressor characteristic" whose argument of the maximum is unknown. The main results of the paper are experimental and The work of the rst and the third authors was supported in part by the Air Force Oce of Scienti c Research under Grant F49620-961-0223 and in part by the National Science Foundation under Grant ECS-9624386. The work of the second and the rst author was supported in part by the Air Force Oce of Scienti c Research under Grant F49620-95-1-0409. y Corresponding author.
Miroslav Krsticy
Dept. of AMES UC San Diego
they are shown in Section 4 where we implement extremum seeking on an axial- ow compressor in Richard Murray's laboratory at the California Institute of Technology. The extremum seeking is implemented via a slow bleed valve, while rotating stall stabilization is performed with air injection, as in [2]. The results demonstrate the e ectiveness of extremum seeking in maintaining maximal pressure rise while preventing rotating stall (of either large or small amplitude).
2 Extremum Seeking Scheme
In [11] we described a control scheme which can enforce convergence to the peak of an output equilibrium map. Consider a general SISO nonlinear model x_ = f (x; u) (2.1) y = h(x) ; (2.2) n where x 2 IR is the state, u 2 IR is the input, y 2 IR is the output, and f : IRn IR ! IRn and h : IRn ! IR are smooth. Suppose that we know a smooth control law u = (x; ) parameterized by a scalar parameter . Assumption 2.1 There exists a smooth function l : IR ! IRn such that f (x; (x; )) = 0 if and only if x = l() : (2.3) Assumption 2.2 For each 2 IR, the equilibrium x = l() of the closed-loop system is locally exponentially stable. Assumption 2.3 There exists 2 IR such that (h l)0 ( ) = 0 (2.4) 00 (h l) ( ) < 0 : (2.5) Thus, we assume that the output equilibrium map y = h (l()) has a maximum at = . Our objective is to develop a feedback mechanism which maximizes the steady state value of y but without requiring the knowledge of either or the functions h and l. Our feedback scheme is shown in Figure 1. It is an extension of a simple method for seeking extrema of static nonlinear maps [3]. The parameters in Figure 1 are selected as !h = !!H = !!H0 = O(!) (2.6) 0 !l = !!L = !!L = O(!) (2.7) 0 k = !K = !K = O(!) ; (2.8) where ! and are small positive constants and !H0 , !L0 , and K 0 are O(1) positive constants. a also needs to be small.
+l
^
6
k s
- x_ = f (x ; (x ; )) y = h(x)
!l s + !l
-
y
ly ? s +s!h
6
a sin !t
Figure 1: A peak seeking feedback scheme. Theorem 2.1 ([11]) Consider the feedback system in Figure 1 under Assumptions 2.1{2.3. There exists a ^ ball of initial conditions around the point x ; ; ; = (l( ) ; ; 0 ; h l( )) and constants ! ; ; and a such that for all ) ; 2 (0 ; ), and a 2 (0 ; a), the ! 2 (0 ; ! solution x(t) ; ^(t) ; (t) ; (t) exponentially converges to an O(! + + a)-neighborhood of that point. Furthermore, y(t) converges to an O(! + + a)-neighborhood of h l( ).
3 Application to a Compressor Model
We now apply the peak seeking scheme of Section 2 to the Moore-Greitzer compressor model [13]: Z 2 p p R C + 2 R sin sin d (3.1) R_ = 3 0 Z 2 p C + 2 R sin d (3.2) _ = ? + 21 0 1 _ = 2 ( ? T ) ; (3.3) where is the mass ow coecient, is the pressure rise coecient, R is a quadratic function of the amplitude of the rotating stall, and the other quantities are de ned in [19, Table 1]. The throttle ow T is related to the pressure rise through the throttle characteristic (3.4) = 12 (1 + C0 + T )2 ; where is the throttle opening.
3.1 Pressure peak seeking for the surge model
For clarity of presentation, we rst consider the model (3.1){(3.3) restricted to the invariant manifold R = 0: _ = ? + C () (3.5) 1 (3.6) _ = 2 ( + T ) : This model is, in fact, the surge model introduced by Greitzer [6], which describes limit cycle dynamics in centrifugal compressors. The function C () represents the \compressor characteristic," which is a typical S-shape function. Our objective is to converge to the peak of this characteristic and operate the compressor with maximum pressure. Thus, we denote x = (; ) and y = . In [9] we showed that a control law of the form ? + 2 (cp ? c ) (3.7)
=
stabilizes equilibria parameterized by ?. If the design parameters are chosen to satisfy 2 > , c > 0, c = c + 12 > 0, and c d () > max C (3.8) c d (which is nite), then the control law (3.7) achieves global exponential stability of equilibria parameterized by ?. In order to apply the peak seeking scheme, we rst check that all three assumptions from Section 2 are satis ed: 1. From (3.4){(3.7), ? is given by ? = ? (0 ) = 1+C0 + 2 (c 0 ? c C (0 )) ; (3.9) where 0 is the equilibrium value of . In view of (3.8), it is clear that the function ? () is invertible. Thus, for each value of ?, there?is only one ? equilibrium ( ; ) = ?? 1 (?) ; C ?? 1 (?) ; which means that Assumption 2.1 is satis ed. 2. As we indicated above, it was proved in [9] that (3.8) guarantees that the equilibrium is exponentially stable not only locally but also globally, hence Assumption 2.2 is satis ed. 3. Following ?the notation in Section 2, y = h 4 ? 1 l(?) = C ? (?) . The Moore-Greitzer model (3.1){(3.3) is scaled so that C () always has a maximum ? at = 1. Since ? () is bijective, C ?? 1 (?) has a maximum at ? = 1 + C0 + 2 (c ? c C (1)) ; (3.10) that is, ? C ?? 1 0 (? ) = 0 (3.11) ? 00 ? 1 C ? (? ) < 0 : (3.12) Hence, Assumption 2.3 is satis ed. Since Assumptions 2.1{2.3 are satis ed, we can apply the peak seeking scheme given in Figure 2 with ? = ?^ + a sin !t : (3.13) By Theorem 2.1, for suciently small !, , and a, (t) converges to an O(! + + a)-neighborhood of 1 and (t) converges to an O(! + + a)-neighborhood of its maximum value C (1). The application of the peak seeking scheme to the surge model (3.5), (3.6) will make clearer the application to the full Moore-Greitzer model (3.1){(3.3).
3.2 Simulations for the full MG model
In Figure 2, we replace the surge model with the full model and extend the stabilizing controller by R ; _ terms with coecients cR ; d , respectively (the stabilizing control laws are derived in [10]). The bifurcation diagrams are shown in Figure 3, in which the solid lines represent stable equilibria and the dash lines represent unstable equilibria. The output function, the solid curve mapping in the vs. ? ? ? in Figure 3, is not
-? +l 6 +l ?^
6
p
-
Stabilizing feedback 2 (c ? c )
k s
!l s + !l
2.7
2.65
I
2.6
l ? s +s!h
2.55
6
2.5 0.98
Peak seeking feedback Figure 2: Peak seeking scheme for the surge model (3.5), (3.6). a sin !t
3
2.75
-
Surge model (3.5), (3.6)
Figure 4: A trajectory under the peak seeking feedback for ! = 0:03 and k = 0:4. 1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
2.75
2 2.7
2.5 1
2
R
1.5 1
−100
4
??
0
?
100
−2 −200
3
2
2
1
1
0 −200
0
??
0 −2
?
100 2.5 0.98
Figure 5: A trajectory under the peak seeking feedback for ! = 0:03 and k = 1:5.
? ? ? Figure 3: Bifurcation diagrams for the case = 0:9, = 1:42, with the full-state controller. The control gains are cR = 30, c = 7 and c = 20. −100
0
100
I
2.55
−100
4
3
2.65
2.6
−1
0.5 0 −200
0
−1
0
1
2
continuously di erentiable but we can show that the average equilibrium is O(a)-close to the point (R ; ; ) = (0 ; 1 ; C(1)), that it is to the right of the peak, and that it is exponentially stable. We now present simulations of the peak seeking scheme for a compressor with C0 = 0 ; C0 = 0:72 ; = 1:42 ; = 0:9 ; = 4, and with a stabilizing controller whose parameters are = 1:42 ; c = 2 ; c = 4 ; cR = 7 ; d = 0. Our rst simulation employs a peak seeking scheme with a = 0:05 ; ! = 0:03 ; !h = 0:03 ; !l = 0:01 ; and k = 0:4. The trajectory is shown in Figure 4, where the darker curve represents the trajectory and the lighter lines represent the axisymmetric and stall characteristics. The trajectory starts from an equilibrium on the stall characteristic and converges to a small periodic orbit near the peak of the compressor characteristic. If the peak seeking parameters are selected di erently, for example, if k increased to k = 1:5, the new shape of the trajectory is shown in Figure 5. In this case the convergence is smoother and faster but the periodic orbit is farther from the peak. (Note, however, that, given more time, the periodic orbit would slowly approach the peak.) Even though faster adaptation throws the periodic orbit further to the right of the peak, the periodic orbit remains in the \ at" region of the compressor
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
characteristic where variations in mass ow result in only minor variations of the pressure rise . As explained in the previous section of the paper, the convergence of the trajectory to a close neighborhood of the peak is the result of regulating ?^ (t) to a neighborhood of ? . Figure 6 shows the time traces of ?^ (t) ? ?^ (0) for the trajectories in Figures 4 and 5. In this case ?^ (0) = ?1:2 and ? = ?0:90, which means that ? ? ?^ (0) = 0:30. This explains why the trajectory in Figure 4 converges closer to the peak than that in Figure 5. Since a permanent presence of the periodic perturbation is undesirable, we now show that it can be disconnected after a short peak seeking period. Figure 7 shows the pressure transient and steady state up to t = 2550, at which time, k and a are set to zero. The disconnection of adaptation makes the trajectory transition from the periodic orbit to an equilibrium on the compressor characteristic, just to the right of the peak. To make the transient more visible, we have set R(2550 + 0) = 0:06 because R(2550 ? 0) is practically zero. Figure 7 also shows that the pressure variations in steady state under the peak seeking feedback are hardly noticeable, especially if compared to a large gain in the DC value of the pressure.
4 Experiment Results on Caltech Rig
A prerequisite for experimental validation of the scheme from Section 3 is the availability of a high-bandwidth bleed valve for stabilization of rotating stall. However, as shown in [2], rotating stall can also be stabilized by air injection. In this section we combine the air injection rotating stall controller from [2] with the extremum
0.375
0.8
Fig. 5
0.7
0.37
0.365
0.6
?^
0.5
0.4
Fig. 4
0.3
0.2
0.355
0.35
0.345
0.34
0.1
0 0
0.36
500
1000
1500
t
2000
2500
0.335 110
3000
2.75
Fig. 5
b
2.7
Fig. 4
2.65
+l
6
2.6
2.55
2.5 0
500
1000
1500
t
2000
2500
3000
Figure 7: Time response of the pressure rise. At t = 2550 we disconnect the peak seeking feedback.
seeking scheme from Section 2 to achieve maximization of pressure rise. Peak seeking is implemented via a bleed valve whose bandwidth is 51 Hz.
4.1 Stall stabilization
Stall stabilization is performed by air injection in a onedimensional on-o fashion. When the measured amplitude of the rst mode of the stall cell is above a certain threshold, all three injectors are fully open. Otherwise they are closed. The set point of the compressor is varied by a bleed valve. The characteristic of the pressure rise with respect to the bleed angle b is shown in the Figure 8. Note that higher bleed angle means lower overall throttle opening. There is a clear peak in the characteristic curve. The points to the left of the peak are axisymmetric equilibria. The points to the right of the peak are stabilized low amplitude stall equilibria.
4.2 Filter design
We implement the extremum seeking scheme in a con guration shown in Figure 9. In the theoretical analysis in [11] noise was not considered and a high-pass lter s s+!h was employed. Because of noise in the experiment we use a band pass lter. From the power spectrum analysis, we learn that the noise is above 150 Hz. Also we know that the stall frequency is about 65 Hz. Since the lter should cut out both the high frequency noise and the stall oscillations, as well as DC, we choose the pass band to be 4 to 6 Hz, and implement it as 3rd order Butterworth lter. We do not use a low pass lter !l s+!l .
4.3 Experimental Results 4.3.1 Initial point at axisymmetric characteristic
We select the integrator gain as 600 and set the frequency of the perturbation to 5 Hz. From Figure 8 we
115
120
125
130
b
135
140
145
150
155
Figure 8: vs. bleed angle b.
Figure 6: Time response of ?^ = ?^ (t) ? ?^ (0). 4
Rig - withCaltech Air Injection
k s a sin !t
-
l ? Band-pass lter 6
Peak seeking feedback
Figure 9: Peak seeking scheme for the Caltech rig. know that the peak is around 134{139. We set the initial bleed angle at 110, the farthest point to the left in Figure 8, and set the perturbation to 3 . The perturbed bleed angle is shown in Figure 10 and the pressure rise response is shown in Figure 11. Comparing the peak pressure rise 0:372 of Figure 8 to that of Figure 11, we see that they are the same.
4.3.2 Initial point at nonaxisymmetric characteristic
For the compressor control, the most important issue is to control the system to avoid the stall that causes the deep pressure drop. Since the air injection can control stall for a reasonable interval, we can set the initial point at the stall characteristic. We select the initial point as 150 bleed valve angle because from Figure 8 we can see that 150 is the largest angle we can achieve without a deep drop of the pressure rise. In this case we choose the gain of the integrator as 400. The perturbation signal is set to 3 as in the axisymmetric case. The perturbed bleed angle is shown in Figure 12 and the pressure rise response is shown in Figure 13. The peak pressure in Figure 13 coincides with that in Figure 8. A closer look at Figures 12 and 13 shows that the rotating stall amplitude reduces as the bleed angle reduces. The e ect of peak seeking on a system starting in rotating stall is to bring it out of stall without pushing the operating point away from the peak and reducing the pressure rise. In both Figures 10 and 12 one can observe uctuations of the mean of the bleed angle at the peak. Comparing with Figures 11 and 13, we see that these uctuations occur at the same time when pressure drops resembling stall inception occur. Peak seeking reacts to
140
155
135 150 130
b
b
125
120
145
140
115 135 110
105 0
5
10
15
20
t
25
30
35
40
45
130 0
50
Figure 10: The time response of the bleed angle initiating from the axisymmetric characteristic.
5
10
t
15
t
15
20
25
Figure 12: The time response of the bleed angle initiating from the stall characteristic. 0.42
0.4 0.4 0.38
0.38
0.36
0.34
0.36 0.34 0.32 0.3
0.32 0.28 0.3 0
0.26 5
10
15
20
t
25
30
35
40
45
50
Figure 11: The time response of the pressure rise initiating from the axisymmetric characteristic.
this by pushing the operating point further to the right on the axisymmetric characteristic and then slowly returning it to the peak.
References
[1] K. J. Astrom and B. Wittenmark, Adaptive Control, 2nd edition, Reading, MA: Addison-Wesley, 1995. [2] R. D'Andrea, R. L. Behnken, and R. M. Murray, \Active control of rotating stall using pulsed air injection: A parametric study on a low-speed, sxial slow compressor," In Proceedings of SPIE, volume 2494, pages 152{165, Orlando Florida, 1995. [3] P. F. Blackman, \Extremum-Seeking Regulators," in J. H. Westcott, Ed., An Exposition of Adaptive Control, New York, NY: The Macmillan Company, 1962. [4] C. S. Drapper and Y. T. Li, \Principles of optimalizing control systems and an application to the internal combustion engine," ASME, vol. 160, pp. 1{16, 1951, also in R. Oldenburger, Ed., Optimal and Self-Optimizing Control, Boston, MA: The M.I.T. Press, 1966. [5] A. L. Frey, W. B. Deem, and R. J. Altpeter, \Stability and optimal gain in extremum-seeking adaptive control of a gas furnace," Proceedings of the Third IFAC World Congress, London, 48A, 1966. [6] E. M. Greitzer, \Surge and rotating stall in axial
ow compressors|Part I: Theoretical compression system model," Journal of Engineering for Power, pp. 190{198, 1976. [7] O. L. R. Jacobs and G. C. Shering, \Design of a single-input sinusoidal-perturbation extremum-control system," Proceedings IEE, vol. 115, pp. 212-217, 1968. [8] V. V. Kazakevich, \Extremum control of objects with inertia and of unstable objects," Soviet Physics, Dokl. 5, pp. 658661, 1960. [9] M. Krstic, D. Fontaine, P. V. Kokotovic, and J. D. Paduano, \Useful nonlinearities and global bifurcation control of jet engine surge and stall," submitted to IEEE Transactions on Automatic Control, 1996.
0
5
10
20
25
Figure 13: The time response of the pressure rise initiating from the stall characteristic. [10] M. Krstic and H. H. Wang, \Control of Deep-Hysteresis Aeroengine Compressors|Part II: Design of Control Laws," Proceedings of the 1997 American Control Conference, Albuquerque, NM. [11] M. Krstic and H. H. Wang, \Design and stability analysis of extremum seeking feedback for general nonlinear systems," Proceedings of the 1997 Conference on Decision and Control, San Diego, CA, TA02-3. [12] M. Leblanc, \Sur l'electri cation des chemins de fer au moyen de courants alternatifs de frequence elevee," Revue Generale de l'Electricite, 1922. [13] F. K. Moore and E. M. Greitzer, \A theory of post-stall transients in axial compression systems|Part I: Development of equations," Journal of Engineering for Gas Turbines and Power, vol. 108, pp. 68{76, 1986. [14] I. S. Morosanov, \Method of extremum control," Automatic & Remote Control, vol. 18, pp. 1077-1092, 1957. [15] I. I. Ostrovskii, \Extremum regulation," Automatic & Remote Control, vol. 18, pp. 900-907, 1957. [16] A. A. Pervozvanskii, \Continuous extremum control system in the presence of random noise," Automatic & Remote Control, vol. 21, pp. 673-677, 1960. [17] J. Sternby, \Extremum control systems: An area for adaptive control?" Preprints of the Joint American Control Conference, San Francisco, CA, 1980, WA2-A. [18] G. Vasu, \Experiments with optimizing controls applied to rapid control of engine pressures with high amplitude noise signals," Transactions of the ASME, pp. 481{488, 1957. [19] H. H. Wang, M. Krstic, and M. Larsen, \Control of deephysteresis aeroengine compressors|Part I: a Moore-Greitzer type model," Proceedings of the 1997 American Control Conference, Albuquerque, NM.
