robust stabilization of jet engine compressor in …14]. krstic et al [17] in their work, on jet...

Abstract— Compressors for jet engines in operation experience disturbances such as variations in the states of the system, mass flow, and pressure. These disturbances sometimes result in surge and stall instability problems, which adversely affects its performance. In this work, first we modify the Moore and Grietzer three-state model for compressors to include disturbance (noise signals) and then use the method of integrator backstepping, coupled with saturation functions to develop robust controllers for the stabilization of the compressor. Also, we develop robust observers for estimating the states of the system in situation where there exist some states that cannot be measured. Implementing the developed controllers on the system, simulation results showed that stability was achieved. Also, the observer designed for the jet compressor was able to provide accurate state estimates.

Index Terms— Integrator backstepping, observer, robust control, stall and surge, unmeasured states

I. INTRODUCTION

N the operation of jet engine compressor, there is the need to control surge and stall, so as to ensure stability; and in turn reduce machine

damage that may arise from excessive vibrations and high thermal loading. A number of research works with regard to stability analysis of jet compressor engines have evolved over the years. Some of these works looked at modeling the system while others considered the stability dynamics. Moore and Greitzer [12] proposed a three-state nonlinear model that characterized the dynamics of the behavior of a compressor. The control of surge and rotating stall in compressors has been investigated by a number of researchers see [13,

John A. Akpobi (corresponding author) is a Professor in the

Department of Production Engineering, University of Benin, Benin City, Nigeria (corresponding author’s phone number: +2348055040348; e-mail: [email protected]).

Aloagbaye I. Momodu is with the Department of Production Engineering University of Benin, Benin City, Nigeria. e-mail: [email protected]).

14]. Krstic et al [17] in their work, on jet engine compressor, used integrator backstepping to avoid cancellation of useful nonlinearities in the stabilization analysis. Jan and Olgav [9] in their work used the backstepping method to design a closed couple valve for controlling surge and stall in compressors. Feng and Shih-Chiang [3], developed an adaptive controller regulating for rotating stall and surge in Jet engines using a function approximation approach. The concept of integrator backstepping in stability analysis is well expounded in literature [4, 6, 17, 18, 19]. In actual operation of Jet compressors, the controllers developed in these models do not stabilize the system when it is subjected to uncertainties such as system modeling errors, in-service changes amongst others, and compressor disturbances (noise) such as speed fluctuations, combustion noise etc., which significantly reduces the efficiency of the compressor [10, 15, 16]. Consequently, in this work, the aim is to resolve these instability problems associated with compressors subjected to disturbances (noise). In addressing these problems, we develop robust controllers for the stabilization of the compressors in the presence of disturbances (noise signals) using the integrator backstepping method, coupled with saturator. Also, we develop controllers for observer design for the compressor, in the situation where there is no stall, and the pressure rise is an unmeasured state (unmeasured state).

II. PROBLEM FORMULATION We begin with the basic three state Moore and

Grietzer model [12] representing the compressor

dynamics for a Jet Engine. This is given as:

2 2(2 )R R R (1)

Robust Stabilization of Jet Engine Compressor in the Presence of Noise and Unmeasured States

John A. Akpobi, Member, IAENG and Aloagbaye I. Momodu

I

Proceedings of the World Congress on Engineering 2011 Vol III WCE 2011, July 6 - 8, 2011, London, U.K.

ISBN: 978-988-19251-5-2 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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2 33 13 3

2 2R R (2)

u (3)

where R is the normalized stall cell squared

amplitude.

1

Where is the mass flow

2co

is the pressure rise and

co is a constant.

u is the input or control

The system represented by equations 1-3 does not

have noise signals. Using Integrator backstepping,

the stabilized system is given as:

2 2(2 )R R R (4)

33 1

13

2Z C R (5)

.

3 2 3Z C Z (6)

A. Introduction of Disturbances (Noise signals)

In this section we modify the basic Moore and

Greitzer model to include noise, and then develop

stabilizing controller for it. Introducing noise as:

1 , 2 , 3 to equations (1-3), the system is

modified as:

2 21(2 )R R R (7)

2 32

3 13 3

2 2R R (8)

3u (9)

B. Definitions The following are some definitions needed for the development of the results: Lyapunov Function [1, 5, 7, 8, 18, 19] A Lyapunov function is defined as follows:

Let: : nV R R R be a 1C function defined

in a domain nD R that includes the origin. Then the Lyapunov function, ( , )V t x , which must

satisfy the following conditions:

1. V is proper at the equilibrium state ex :

( )nx R V x (10)

that is, V is a compact subset of some

neighbourhood O of ex for each >0

small enough. 2. V is positive definite on O:

( ) 0eV x and

( ) 0V x , ex O x x (11)

For ex x in O there is some time

1 1, 0t T t and some control

1(0, )tu U admissible for x such that the

trajectory ( , )x u resulting from the

control and this initial state,

( ( )) ( )V t V x ( ( )) ( )V t V x and

( ( )) ( )V t V x

3. ( , )V x t as x (12)

This third property is referred to as radially unbounded or uniformly unbounded or weakly coercive. Saturation Function

We define saturation function ( , )i as follows:

( , ) sgn( ).min{ , }i i i i i i (13)

Where i is the saturation level

sgn(.) is the signum function defined as:



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1 0

sgn( ) 0 0

1 0

if

if

if

(14)

III. THEOREM 1 Given a feedback control system with disturbance signal at each subsystem of the form:

( , , )x f x u (15)

, ,n m nx R u R R

Then, the existence of a controller of the form:

( , ( , ))u k x , where (.) is a saturation

function such that 0dV

dt or

2dVx

dt is a

necessary and sufficient condition for the resulting closed loop control system to be robustly globally(locally) asymptotically stable. Proof The proof of the theorem requires both necessity and sufficiency conditions to be satisfied. Details of the proof of the theorem can be found in [2].

A. Methodology for stabilization of the Compressor

Using equation (14), we introduce saturators in to

the system as follows:

2 21 1 1(2 ) ( , )R R R (16)

2 3

2 2 2

3 1

2 2 3 3 ( , )R R

(17)

3 3 3( , )u (18)

IV. THEOREM 2 For the Compressor system bombarded with noise signals represented by equations (7-9), asymptotic stabilization of the system is obtained with the choice of control law:

2 3 0 1

2

1

2 2

2 22 1 2 2

2

2

u c z k

R k R Rk

z R k Rz

k c z R k R k

(19)

where

1

21 1 1( , )

kR

(20)

1 1

3 3 2 21 1

01 1

1

21

2

1

1 1

2 221 1

2

1

3 1.5

0.751.5

0.75 0.75

R R

k

zRR

zR R

R

(21)

1

21 1

1 2

12 3 3 3 2

1 1 1 22

1 1

2 21 1 1 1

2 2

12 2

1 2 12

33 2

2

3 1.5

0.75 0.75

0.75

Rk z R

R R z

R

R R R R

z

R R

(22)

1

21

2 1 2

2 12

3 3

1.5 1.5 1.5

k R c zR

zR

(23)

2 1z k (24)



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2 21 2 2

3 2 22 2 2

332

.

2 2 2

2 1.5

0.5 1.5 1.5

0.5 1.5 3 3

1 ( , )

c z R k R z

z kz k zz

k k Rz kR

k

(25)

and 1 2,c c are constants with 1 2, 0c c

A. Proof of theorem 2

Let

1 1 1 1

1

21 1 1

, ( , )

( , ) 1 1

R

kR

(26)

Where

1

21 1 1( , )

kR

Therefore, substituting into equation (19) 2

21 1 1

2 ( 1)

( 2 1) ( , )

R R R k

R k k

2 2

1 1 1

2 2

2 ( , )

R k R R k R

k R R

2 21 1 1( , )R R k R

(27)

2 1 1 1( , )k

R

;

21 1 1( , )k R

(28)

, Hence

21 1 1 1 1 1( , ) ( , )R R R

2R R R

1R R (29)

Using the Lyapunov function,

2

( ) 2

RV R (30)

We have: ( )

( ) .V R

V R RR

. 1R R R

= 2 1R R (31)

This is negative definite 1R and 0 . Hence there would be Local asymptotic stability.

But is not the control so we introduce 2z to

track error.

Set 2 1 1 1 1, ( , )z R (32)

2 1 1 1 1, ( , )z R (33)

Substituting into equation (16), 2

2

22 1 1 1

2 ( 1)

( 1) ( , )

R R R z k

R z k

2 2 22 2 2 2( 1) 2 2 1z k z kz z k k

22

22 2 2

1 1 1

(2 2 2

2 2 1)

( , )

R R R z k

z kz z k k

2

2 2 2

R R k R R

z R k Rz

(34)

From 2 1 1 1 1, ( , )z R

2 1z k

Which gives:

2z k (35)

From 2 3

2 2 2

3 1

2 2 3 3 ( , )R R

2 32

.

2 2 2

3 13

2 2

3 ( , )

z R

R k

(36)

Substituting for , we have:

3 22 2 2 2

3 2 2 22 2 2 2 2 2

2 3 22 2

2 2 2 2

32 2 1

2

6 3 3 3 51

2 3 3 3 3 1

3 1 3 ( , )

z z kz z k k

z kz z k z kz kz

z z k k k

R z k R k

3

2 2 2

2 2 32 2

2

.

2 2 2

1.5 0.5

1.5 1.5 0.5

1.5 3 3 1

( , )

z z z

kz k z k

k Rz kR

k

(37)



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Define 22

22 2( , )

2 2

zRV R z

2 2 2V RR Z Z

22 2

3 22 2 2

2 32 2 2

.

2 2 2

2

1.5 0.5 1.5

1.5 0.5 1.5 3

3 1 ( , )

R R k R R z R k Rz

z z kz

z k z k k Rz

kR k

2

2 22

3 2 22 2 2

322

.

2 2 2

2 1.5

0.5 1.5 1.5

0.5 1.5 3 3

1 ( , )

R R k R R

R k R z

z kz k zz

k k Rz kR

k

(38)

Selecting the control law using

1( )V

g R keR

(39)

Where ( )g R = all terms in R, that are multiplied

by 2Z

1e = 2Z and k is a constant, 0k

2 21 2 2

3 2 22 2 2

32

.

2 2 2

2 1.5

0.5 1.5 1.5

0.5 1.5 3 3

1 ( , )

c z R k R z

z kz k z

k k Rz kR

k

(40)

Substituting into equation (38), 3 2 2 2

2 1 2V R k R R c z

2 21 21R R k c z

2V is negative definite for

11, 0, 0R k c , hence there is local

asymptotic stability. Substituting for into equation (37)

2 22 1 2 2Z c z R k R (41)

Next we define

3 2 2 1 1 1 2 2 2, , ( , ), ( , )z R z ,

with

2 2 1 1 1 2 2 2

2 21 2 2

3 2 22 2 2

32

.

2 2 2

, , ( , ), ( , )

2 1.5

0.5 1.5 1.5

0.5 1.5 3 3

1 ( , )

R z

c z R k R z

z kz k z

k k Rz kR

k

(42)

2 21 2

3 22 2 2

2 33 2

2

.

2 2 2

2

1.5 0.5 1.5

1.5 0.5 1.5

3 3 1

( , )

c z R k R

z z kz

z k z k k

Rz kR

k

(43)

13 2

2 11 2

1

23 21

2 2 2

3

21 1

3 2

1 1

2 21 1

2

.

2 2 2

2

1.5 0.5 1.5

1.5 0.5

1.5 3 3

1 ( , )

Rc z R

z z zR

z zR R

RRz

R

k

(44)

3 3 33 2

2

3 31 2

1 2

z z zz R z

R z

z zk

(45)

3 0 1 1 2 2 2z k k R k z k (46)

3 0 1 1 2 2 2z u k k R k z k (47)

where,



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1 13 3 2 21 1

01 1

1

21

2

1

1 1

2 221 1

2

1

3 1.5

0.751.5

0.75 0.75

R R

k

zRR

zR R

R

(48)

1

21 1

1 2

12 3 3 3 2

1 1 1 22

1 1

2 21 1 1 1

2 2

12 2

1 2 12

33 2

2

3 1.5

0.75 0.75

0.75

Rk z R

R R z

R

R R R R

z

R R

(49)

and

1

21

2 1 2

2 12

3 3

1.5 1.5 1.5

k R c zR

zR

(50)

Define 222

323 2 3( , , )

2 2 2

zzRV R z z (51)

3 2 2 3 3V RR z z z z (52)

2

2 2

2 2

2 1 2

0 1 1

3

2 2 2

3

2

2

R k R RR

z R k Rz

z cz R k R

u k kRz

k z k

V

(53)

Using the control law in equation (39),

2 3 0 1 1 2 2 2u c z k k R k z k (54)

Hence, .

3

2 3 2 2 22 3 1 2V c z R k R R c z

.

3

2 22 3 1 2

2 1

V c z c z

R R k

(55)

3V is negative definite for

1 21, 0, , 0R k c c ,

hence there is local asymptotic stability. Substituting for u into equation (47) produces

.

3 2 3z c z (56)

The resulting feedback system is: .

2

2 2 2

R R k R R

z R k Rz

(57)

.2 2

2 1 2 2z c z R k R (58)

.

3 2 3z c z (59)



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V. SIMULATION RESULTS I

Simulation results of the unstable system without disturbances are shown in Figs. 1-4

Fig. 1. Trajectory for R (without noise)

Fig. 2. Unstable trajectory for (without noise)

Fig. 3. Unstable trajectory for (without noise)

Fig. 4. Phase portrait of against R (without noise)

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Time(s)

R

0 5 10 15 20 25 30-2

-1.5

-1

-0.5

0

0.5

1

1.5

Time(s)

0 5 10 15 20 25 30-4

-3

-2

-1

0

1

Time(s)

0 0.2 0.4 0.6 0.8 1-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

R



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Simulation results of the system when subjected to disturbances are shown in Figs. 5-7.

Fig. 5. Plot of R in the presence of noise (disturbances).

Fig. 6. Trajectory of in the presence of noise (disturbances).

Fig. 7. Trajectory of in the presence of noise (disturbances).

Simulation results of the stabilized system (without noise) are shown in Figs. 8-10.

Fig. 8. Stabilized trajectory of R without noise (disturbance)

0 5 10 15 20 25 30-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time(s)

R

0 5 10 15 20 25 30-3

-2

-1

0

1

2

Time(s)

0 5 10 15 20 25 30-5

-4

-3

-2

-1

0

1

2

Time(s)

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Time(s)

R



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Fig. 9. Stabilized trajectory of without noise (disturbance)

Fig.10. Stabilized trajectory of without noise (disturbance)

Simulation results of the stabilized system which was subjected to disturbances are shown in Figs. 11-13.

Fig.11. Stable trajectory of R with noise (disturbance) using the robust controller

Fig.12. Stable trajectory of with noise (disturbance) using the robust controller

0 5 10 15 20 25 30-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time(s)

0 5 10 15 20 25 30-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time(s)

0 5 10 15 20 25 30-0.5

0

0.5

1

1.5

2

Time(s)

R

0 5 10 15 20 25 30-0.5

0

0.5

1

1.5

2

Time(s)



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Fig.13. Stable trajectory of with noise (disturbance) using the robust controller

VI. OBSERVER DESIGN

(INTEGRATOR BACKSTEPPING WITH AN

UNMEASURED STATE)

In observer design the following question is addressed: Is it possible to estimate, on the basis of external information provided by passed input and output signals, the magnitude of an internal state at time, t?

A. Problem formulation for the observer Recall the compressor equations (1-3):

2 2(2 )R R R

2 33 13 3

2 2R R

u

With regard to the Jet compressor model, let us treat the pressure rise as state not measurable. This state

could be estimated as and the state estimation

error which converges exponentially to zero is

estimated from: ˆ

ˆ (60)

With the assumption of no stall (R= 0), the model is

rewritten as: .

2 33 1

2 2

(61)

.

u (62)

Hence with (65), we have:

2 33 1ˆ

2 2 (63)

ˆ u (64)

(65)

B. Observer design In designing the observer to estimate the

state ,

is added as an observer.

Let the error variable, 1ˆ ( )z

(66)

21

3( )

2 which is chosen to avoid cancellation

of the useful nonlinearity, 31

2

Due to the presence of , we introduce a nonlinear

damping term s

Select 21s d (67)

2 31 1

3( )

2d (68)

2 31

3ˆ

2z d

(69)

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

Time(s)



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or

2 31

3ˆ

2z d

(70)

Substituting into (68),

2 2 3 31

3 3 1

2 2 2z d

3 31

1

2z d

(71)

Select Lyapunov function

21

2V (72)

3 31

1

2V z d

4 41

1

2z d

(73)

Completing the square,

4

2 2

21

1 1

1

2

2 2

V z

dd d

(74)

2

4

1

1

2 2V z

d

(75)

Since 2 is the error of an exponentially converging

observer, we augment the function V with a

quadratic term in

1 21

V Vd

2

1 21

V Vd

(77)

2 24

1 2 21 1

1

2 4V z

d d

(78)

2

42

1

1 3

2 4z

d

(79)

Hence, there would be global asymptotic stability

for z = 0

Recall, 1ˆ ( )z (80)

1ˆ ( )z

1( ).u

2 31

1

( ) 3 1ˆ.

2 2

( ) .

u

(81)

Let 2

22 1

2

1, , ,

2 2V z V z

d

2. .

12

,V z zd

(82)

2

1 21

,2

V Vd



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24

2 21

2 312

21

1 3

2 4

( ) 3 1ˆ

2 2

( ).

V zd

u

zd

24

21

2 31

21

2

1 3

2 4

( ) 3 1ˆ.

2 2

( ).

d

u

z

zd

(83)

The choice of control,

2 31

2

12

( ) 3 1ˆ.

2 2

( )

u cz

d z

(84)

yields

2.4

2 21

2 31

2

2 31 12

21

2

1 3

2 4

( ) 3 1ˆ

2 2

( ) ( ) 3 1ˆ

2 2

( ) . (85)

Vd

cz

z

d z

zd

Completing the square yields: 2

4 22

1

2 22 1 1

22

1 3

2 4

( ) ( )

czd

d z zd

24 2

21

2 2 21

22 2 2

1 3

2 4

( )

2 4

czd

d zd d d

24 2

21

2 21

22 1

1 3

2 4

( ) 3

2 4

czd

d zd d

(86)

4 2 22 2

1 2

1 3 1 1

2 4V cz

d d

(87)

This also shows a global asymptotic stability with 2Vnegative definite Substituting (84) into (81) yields:

2

12

1

( )

( ) .

z cz d z

(88)

Hence the resulting closed loop system is

3 31

1

2z d

(89)

2

12

1

( )

( ) . (90)

z cz d z

(91)

where 21

1

( )3 3d

Hence we have the resulting feedback system as: 3 3

1

1

2z d

(92)

222 1

21

3 3

3 3 .

z cz d z d

d

(93)

(94)



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VII. SIMULATION RESULTS II

Using the MATLAB Simulink® software to simulate the developed Observer system, the following results shown in Figs. 14-17, were obtained:

Fig. 14.Time history of (stabilized)

Fig. 15.Time history of the estimate state

Fig. 16. Time history of the tracking error

Fig. 17. Comparing and

0 5 10 15 20 25 30-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time(s)

0 5 10 15 20 25 30-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time(s)

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

Time(s)

0 5 10 15 20 25 30-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time(s)

ˆ,

Pressure Rise,

Pressure Rise Estimate,



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VIII. DISCUSSION OF RESULTS All the controllers developed for the jet engine compressor, were simulated using the Matlab Simulink® software. In developing the solutions, we

used 1 2 1 2 1c c d d . Figs. 1-3 show the

trajectories for the unstable signals produced by the Moore-Grietzer three state model.

Fig. 4 shows the phase portrait of R and in the Moore-Grietzer model without noise. The phase portrait in Fig. 4, again shows that the Moore-Grietzer model for compressor is unstable. In Figs. 5-7, it is seen that the disturbance introduced to the system by the white noise caused a significant increase (intensity) in the instability of the system. Figs. 8-10 represent stable trajectory obtained without noise in the system. Implementing the robust controller developed on the unstable system subjected to noise signals, the results show clearly that the resulting system is stabilized as shown in Figs. 11-13. The results for the observer developed for the system, are shown in Figs.14-17. From the simulation studies, it is seen that using the developed observer on the compressor, results in stability; for the situation where a state that cannot be measured exist, in addition to the absence stall during operation. In

this case, is considered as the state that cannot be measured. The tracking error shown in Fig. 16 shows the error in the state estimation dropped from 0.3 to 0 in 5s, and remained at zero as time progressed from

5s. From Fig. 17, it is seen that is sufficiently

estimated by within the first 2 s then peaks low to a value, thereafter there is accurate state estimation from 8 s onwards. Thus, the observer provides accurate estimates of the unmeasured state .

IX. CONCLUSION We have developed in this work, a robust controller to stabilize the jet engine compressor system in the presence of noise (disturbances). Simulation studies showed that the developed controller was robust to handle the stabilization of compressor system in the presence of noise (disturbances) using integrator backstepping coupled with saturators. The saturators introduced, significantly reduced the effect of noise. Also, the observer developed, accurately estimated the unmeasured state of the compressor.

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[3] T. Feng and L. Shih-Chiang, (2005), “An Adaptive Control for Rotating Stall and Surge of Jet Engines – A Function Approximation Approach”, in Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, pp.

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[5] D. E. Koditschek, “Adaptive Techniques for Mechanical Systems”, Proceedings of the 5th Workshop on Adaptive Systems, New Haven, CT, 1987, pp. 259-265.

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