Using LQG/LTR Optimal Control Method to Improve Stability and

International Scholarly Research NetworkISRN ElectronicsVolume 2012, Article ID 134580, 8 pagesdoi:10.5402/2012/134580

Research Article

Using LQG/LTR Optimal Control Method to Improve Stabilityand Performance of Industrial Gas Turbine System

Fereidoon Shabaninia and Kazem Jafari

School of Electrical Engineering, Shiraz University, Shiraz, Fars, Iran

Correspondence should be addressed to Fereidoon Shabaninia, [email protected]

Received 27 April 2012; Accepted 3 July 2012

Academic Editors: C. W. Chiou and M. Hopkinson

Copyright © 2012 F. Shabaninia and K. Jafari. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

The gas turbine is a power plant, which produces a great amount of energy for its size and weight. Its compactness, low weigh, andmultiple fuels make it a natural power plant for various industries such as power generation or oil and gas process plants. In any ofthese applications, the performance and stability of the gas turbines are the end products that strongly influence the profitability ofthe business that employs them. Control and analyses of gas turbines for achieving stability and good performance are importantso that they have to operate for prolong period. Effective control system design usually benefits from an accurate dynamic model ofthe plant. Characteristic component parts of the system are considered in this model. Gas turbine system is described by specifiedthermodynamic equations that can be used for defining its model. This paper introduces an optimal LQG/LTR control methodfor a gas turbine. Analysing the gas turbine dynamic in time and frequency domain by using proposed control compared to PIDcontroller is followed. Applying this optimal control method can provide good performance and stability for the component partsof system.

1. Introduction

The object of a control system is to make the outputs behavein a desired way by manipulating the plant inputs [1]. So thecontrol systems are being required to deliver more accurateand better overall performance in the face of difficult andchanging operating conditions [2]. Many complex engineer-ing systems are equipped with several actuators that mayinfluence their static and dynamic behavior. Systems withmore than one actuating control input and more than onesensor output may be considered as multivariable systems ormulti-input multioutput (MIMO). The control objective formultivariable systems is to obtain a desirable behavior ofseveral output variables by simultaneously manipulatingseveral input channels [3]. The feedback control systems arewidely used in design controller. Optimal control design isone choice for many MIMO systems. Linear Quadratic Gaus-sian and Loop Transfer Recovery (LQG/LTR) is an optimalcontrol problem whose name is derived from the fact thatit assumes a linear system, quadratic cost function, and Gaus-sian noise.

In some practical circumstances, the dynamics of con-trolled plant may not be exactly modeled, and there may besystem disturbances and measurement noises in the plant.The LQG/LTR controller can provide good performance andguaranteed stability in the face of such noises [3, 4]. Moredetails about this method can be consulted in [1, 3, 5–7].Effective control system design usually benefits from an accu-rate model of the plants.

The system to be investigated is a gas turbine. Variousapproaches such as Fuzzy [8], MPC [9], and Neural Network[10] are used for control of different types of this system. Theaim of the present paper is to design an optimal LQG/LTRcontroller for this system. Gas turbines are widely used assource of power generation. Some features such as ease ofinstallation and maintenance, high reliability, and quickresponse have made it an attractive means of producingmechanical energy [11]. There are literature on the modelingof gas turbine that described the system by mathematicaldynamic equations [12, 13] or transfer function block dia-gram [14, 15]. Gas turbine model dynamic equations in [9]are used for control object. In this model, all states, inputs,

2 ISRN Electronics

and outputs are related by nonlinear mathematical equa-tions.

Next sections are concerned with a general description ofthe gas turbine system, definition of its model, LQG/LTRmethod, and introducing state space model and simulationresults. Matlab mfile commands are used for this idea. PID(Proportional Integrated Derivative) control [1] is applied tothis system to compare results.

2. System Description

Gas turbines are designed for many different purposes. Inthe petroleum industry, they are commonly used to drivecompressors for transporting gas through pipelines orgenerators that produce electrical power. The gas turbine is aconstant flow cycle with a constant addition of heat energy. Itis commonly referred to as the Brayton cycle [16, 17]. Gasturbine consists basically of a compressor, combustionchamber, and turbine. The compressor draws air from theambient and increases the pressure of the incoming air. Thepressurized and hot air then used as combustion air incombustion chamber, in which a fuel is injected and com-busted continuously. The hot pressurized gases are thenexpanded slightly above atmospheric in turbine and cause toproduce rotational speed at output. When compressing theair, power is required and when expanding the hot gasthrough the turbine, power is generated [11]. The gas turbineis the best suited prime mover when the needs at hand such ascapital cost, time from planning to completion, maintenancecosts and fuel cost, are considered [18].

The design of any gas turbine must meet essential criteriabased on operational considerations such as high efficiency,high reliability and thus high availability, ease of service, easeof installation and commission, flexibility to meet variousservice and fuel needs, conformance with environmentalstandards, and auxiliary and control systems. The two fac-tors, which most effect high turbine efficiencies, are pressureratios and temperature. High-pressure ratios and turbineinlet temperatures improve efficiencies on the simple cyclegas turbine. It should also be noted that the very high-pres-sure ratios tend to reduce the operating range of the turbinecompressor [18].

Auxiliary systems and control systems must be designedcarefully, since they are often responsible for the downtime inmany units. Control systems provide acceleration time andtemperature time controls for startups as well as controlvarious antisurge valves. At operating speeds they must reg-ulate fuel supply and monitor vibrations, temperatures, andpressures throughout the entire range [18].

3. Gas Turbine Model

Gas turbine system is described by nonlinear mathematicalequations that used thermodynamic principle rules such asconservation of total mass, energy, and ideal gas equation[2, 12, 17, 19]. The investigation of steady state behavior ofgas turbine in terms of static turbine characteristic is atraditional area in engineering [12].

Nonlinear state equations of gas turbine system in [9, 20]describe the gas turbine model and are used for controlobjects in this paper. This model is based upon the character-istics of the component parts of system such as pressure andtemperature in compressor, plenum, turbine, and rotationalspeed.

The compressor pressure (pcomp) is relative to air massflow rates in compressor (mcomp), plenum (mbl) blow offvalve (mbl) inlet pressure ambient temperature, and com-pressor temperature according to nonlinear dynamic equa-tion as follows [9, 20]:

dpcomp

dt= γR

Vcomp

[mcompTcpout −

(mplmbl

)Tcomp

], (1)

where

Tcpout = Tin

(pcomp

pin

)(γ−1)/γηcomp

,

mpl =

√√√√√2pcompA2comp

(pcomp − ppl

)

ξRTcomp.

(2)

Nonlinear dynamic equation of turbine temperature is

dTtbin

dt

= RTTtbin

ptbinVcc

[γT

(mthrTccin

cpcomp

cpT− mtbTtbin +

powercpT

)

−Ttbin(mthr − mtb)

],

(3)

where

Tccin = Tpl

(Ptbin

ppl

)(γ−1)/γ

. (4)

Rotational speed described by

dN

dt

=[mtbcpT(Ttbin−Ttbout)−mcompcpcomp

(Tcpout−Tin

)] 1NI

,

(5)

where γ = cp/cv is specific heat capacity ratio, R is gasconstant (Nm/kg K), I is inertia (Kg m2), ξ is friction factor,and η is efficiency.

Other parameters state by specific nonlinear dynamicequations same as these at [9, 20].

The state space model of gas turbine is defined with 5inputs as

U =[mcomp mbl mthr mtb power

]. (6)

And 7 states as

X =[pcomp Tcomp ppl Tpl ptbin Ttbin N

]. (7)

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uB

+ +

+ +

A

Kf

C

y

−1s I

x

x

Figure 1: The Kalman filter.

To CRT display

To CRT display

Temperature

Speed

Accelerationrate

Start upShutdown

Manual

Minimumvalueselectlogic

FSR

To turbine

Fuelsystem

Fuel

To CRTdisplay

Figure 2: Simplified control schematic.

Parameters of this model are defined as bellows:

mcomp, mbl, mthr, , mtb: Mass flows through compres-sor, blow off, throttle, and turbine, respectively(kg/s).

pcomp, ppl, ptbin: Pressure at compressor, plenum,and turbine, respectively (pa).

Tcomp, Tpl, Ttbin: Temperature at compressor, ple-num, and turbine, respectively (k).

The system dynamic equations can be expressed asfollows

·X= f (X ,U). (8)

4. LQG/LTR Method

Linear quadratic Gaussian or LQG problem is a methodbased on optimal control theory. The main results of thistheory are stated in this section. But fuller details can bereferred to [1, 3, 5–7]. This method is one which allows thedesigner to shape the principle gains of the return ratio ateither the input or the output of the plant, to achieve requiredperformance or robustness specifications. Stability is ob-tained automatically [3].

Suppose the plant is generally described by the dynamicequations in the form of state-space representation asfollows:

x(t) = Ax(t) + Bu(t) + Γw(t),

y(t) = Cx(t) + v(t),(9)

where x(t) ∈ Rn ,u(t) ∈ Rm, and y(t) ∈ Rq are the states,inputs, and outputs vectors, respectively, A ∈ Rn×n, B ∈Rn×m, Γ ∈ Rn×p, C ∈ Rq×n are the system states, inputs ofplant, inputs of disturbance, and outputs matrices, respec-tively. The system disturbance w(t) and the measurementnoise v(t) are p- and q-dimensional uncorrelated Gaussianwhite noise processes with zero-mean, and the associatedcovariance matrices are defined as

E{wwT

}=W ≥ 0, E

{vvT

}= V � 0,

E{wvT

}= 0,

(10)

where E{·} is an expectation function operator, and W andV are the system disturbance and measurement noise covari-ance matrices, respectively. The problem is then to devise afeedback control low which minimizes the cost:

J = limT→∞

E

{∫ T

0

(zTQz + uTRu

)dt

}, (11)

where Z = Mx is some linear combination of the states, andR = RT � 0, Q = QT ≥ 0 are weighting matrices.

The solution to the LQG problem is prescribed by theseperation principle, which states that the optimal result isachieved by adopting the following procedure. First obtainan optimal estimate x of the state x, optimal in the sense thatE{(x−x)T(x−x)} is minimized, and then use this estimate asif it was an exact measurement of the state to solve the deter-ministic linear quadratic control problem. The point of thisprocedure is that it reduces the problem to two sub problems,the solutions to which are known.

The solution to the first subproblem that of estimatingthe state is given by Kalman filter theory. Figure 1 shows theblock diagram of a Kalman filter, which is seen to have thestructure of a state observer; it is distinguished from otherobservers by the choice of gain matrix k f . Note that theinputs to the Kalman filter are the plant input and outputvectors, u and y, and that its output is the state estimatevector x.

The second subproblem is to find the control signalwhich will minimize the cost:

J =∫ T

0

(zTQz + uTRu

)dt. (12)

On the assumption that

·x= Ax + Bu (13)

4 ISRN Electronics

0

Uncontrolled systemLQR controlled system

LQG controlled systemLQG/LTR controlled system

0 5 10 15

Time (s)

3.5

3

2.5

2

1.5

1

0.5

×105

Com

pres

sore

pre

ssu

re (

Pa)

(a)

8

7

6

5

4

3

2

1

0

−1

Turb

ine

tem

pera

ture

(K

)



0 5 10 15

Time (s)

(b)



0 5 10 15

Time (s)

0

−0.01

−0.02

−0.03

−0.04

−0.05

−0.06

−0.07

−0.08

−0.09

Rot

atio

nal

spe

ed (

rev/

s)

(c)

Figure 3: Step response of the compressor pressure (a), turbine temperature (b), and rotational speed (c) respect to compressor mass flowrate.

the solution to this is to let the control signal u be a linearfunction of the state:

u = −Kcx, (14)

where Kc is the state feedback matrix gain.Kc and k f are obtained from solution of algebraic Riccati

equations [3, 5].By substitute (14) in to (13):

·x= (A− BKc)x, (15)

as the equation of the close loop system.From Figure 1, The state equation of the Kalman filter is

to be

d

dtx =

(A− Kf C

)x + Bu + Kf y. (16)

It is shown that the kalman filter and optimal state feedbackare asymptotically stable [3].

The equation of the combined Kalman filter-optimalstate feedback scheme is

d

dt

[xx

]=[

A −BKc

K f C A− Kf C − BKc

][xx

]+

[ΓwKf v

]. (17)

If ε define as state estimation error, ε = x − x, then aboveequations can be rewritten as

d

dt

[xε

]=[A− BKc BKc

0 A− Kf C

][xε

]+

[Γw

Γw − Kf v

], (18)

which shows that the close loop eigenvalues of the LQGcompensated plant are just the union of eigenvalues of the

ISRN Electronics 5

PID controlLQG/LTR control

12

10

8

6

4

2

0

−20 5 10 15 20 25 30 35 40 45 50

×104C

ompr

esso

r pr

essu

re (

Pa)

Time (s)

(a)

7

6

5

4

3

2

1

0

−10 20 40 60 80 100 120 140

Time (s)

Turb

ine

tem

pera

ture

(K

)


(b)

Time (s)

0

−0.01

−0.02

−0.03

−0.04

−0.05

0.06−−0.07

−0.08

−0.09

−0.1

Rot

atio

nal

spe

ed (

rev/

s)

0 5 10 15 20 25


(c)

Figure 4: Step response of the compressor pressure (a), turbine temperature (b), and rotational speed (c) respect to compressor mass flowrate for LQG/LTR and PID control.

optimal state feedback scheme with those of the Kalmanfilter. The overall scheme is therefore internally stable underthe stated assumption [3].

5. Loop Transfer Recovery (LTR)

Since both the optimal state feedback regulator and theKalman filter have such good properties, it might be expectedthat LQG compensator would generally yield good robust-ness and performance. Unfortunately, this is not the caseand the LQG designs can exhibit arbitrarily poor stabilitymargins [3].

The LQG loop transfer function can be made to approachfilter transfer function with its guaranteed stability margins ifKc, state feedback gain, is designed to be large by specificprocedure [1, 3, 5, 6]. It is necessary to assume that the plant

model is minimum phase and that it has at least as manyinputs as outputs. Alternatively, the LQG loop transfer func-tion can be made to approach state feedback transfer func-tion by designing k f in the Kalman filter to be large using spe-cific procedure. Again, it is necessary to assume that the plantmodel is minimum phase and that it has at least as manyoutputs as inputs [1, 3, 5, 6].

6. State Space Description

The state space detailed model of gas turbine at a given oper-ation point helps to understand the dynamic behavior of sys-tem. A nonlinear model can capture the dynamic behavior ofgas turbine system [12]. By using nonlinear equations de-scribed at [9, 20], a linear time invariant, LTI, model of thegas turbine is developed for control purposes. LTI state space

6 ISRN Electronics

16

14

12

10

8

6

4

2

0

−20 0.5 1 1.5 2 2.5 3

Time (s)

Rot

atio

nal

spe

ed (

rev/

s)×10−8

LQR controllerLQG/LTR controller

(a)

0 0.5 1 1.5 2 2.5 3

Time (s)


×10−5

7

6

5

4

3

2

1

0

−1

Turb

ine

tem

pera

ture

(K

)

(b)

Figure 5: Impulse response of the rotational speed (a) and turbine temperature (b) respect to power.

0 1 2 3 4 5 6 7 8 9 10

Time (s)

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

−0.2

−0.4

Turb

ine

tem

pera

ture

(K

)


Figure 6: Impulse response of the turbine temperature respect tomass flow rate of compressor.

model derived by using Jacobian method [1] at operationcondition for MS5002D General Electric gas turbine system.Matlab and Control toolbox used for creating this model.Validation of this model is done by measurement data withMatlab mfile commands [21] and so at [9].

The MS5002D mechanical drive gas turbine is used todrive a centrifugal load compressor to compress treated gasand deliver export gas at 90 bar pressure for export viapipeline. The gas turbine is that part of the mechanical drivegas turbine, exclusive of control and protection devices, inwhich fuel and air are processed to produce shaft horsepower.Gas turbine axial air compressor has 17 stages with 10.75 : 1pressure ratio. The output shaft speed is 4670 rpm with

32.5 MW output power. Combustion section has 12 multiplecombustors with reverse flow type.

Control of the gas turbine in providing the shaft horsepower required by the operation or process is accomplishedusing parameters such as fuel flow, compressor inlet pressure,compressor discharge pressure, shaft speed, compressor inlettemperature, and turbine inlet or exhaust temperature [19].

The SPPEDTRONIC MARK VI turbine control is thecurrent state of the art control for General Electric turbine.It contains a number of control protection and sequencingsystems designed for reliable and safe operation of the gasturbine [22].

Control of the turbine is done mainly by startup, speed,acceleration, synchronization, and temperature control. Fig-ure 2 illustrates the three control modes and the means offuel control in relation to the fuel command signal. Sensorsmonitor the turbine speed, temperature, and compressordischarge pressure to determine the operating conditions ofthe unit. When it is necessary for the turbine control to alterthe turbine operating conditions because of changes in loador ambient conditions, it is accomplished by modulatingthe flow of fuel to the turbine. For example, if the exhausttemperature starts to exceed its permitted value for a givenoperating condition, the temperature control circuit willcause a reduction in the fuel supplied to the turbine andthereby limit the exhaust temperature.

Operating conditions of the turbine are sensed andutilized as feedback signals to the SPEEDTRONIC controlsystem. There are three major control loops—startup, speed,and temperature, which may be in control during turbineoperation. These loops command fuel stroke reference(FSR), the command signal for fuel. The outputs of thesecontrol loops are connected to a minimum value select algo-rithm as shown in Figure 2. The minimum value select algo-rithm selects the lowest FSR called for by the control loopsand passes the result to the FSR controller. The action of

ISRN Electronics 7

150

100

50

0

−50

−100

−150

−200

−250

−300

Sin

gula

r va

lues

(dB

)

−350

Frequency (rad/s)

10−2 100 102 104 106 108 1010

SV diagram of main system

(a)

SV diagram of return ratio at input for LQG/LTR controlled system100

0

−100

−200

−300

−400

−500

−600

−700

−800

Sin

gula

r va

lues

(dB

)

Frequency (rad/s)

10−2 100 102 104 106 108 1010

(b)

Figure 7: Singular value diagram of main system (a) and return ratio at input for LQG/LTR controlled system.

this circuit is similar to a low-voltage selector. The lowestvoltage output of the control loops is allowed to pass thegate to the fuel control system as the controlling FSR voltage.Using this method of FSR selection, switching between thecontrol modes of speed, temperature, and startup controltakes place without any discontinuity. The controlling FSRwill establish the fuel input to the turbine at the valuerequired by the system which is in control. Displays on theturbine control panel CRT indicate which of the controlsystems is controlling FSR [22].

7. Simulation Results

Matlab mfile commands used for simulating gas turbinesystem [21]. The value of inlet temperature is equal to 321 kthat can be changed. The state weighting matrix Q and thecontrol weighting matrix R are chosen by use Bryson’s rule[5]. Often, this choice is just the starting point for a trial anderror iterative design procedure aimed at obtaining desirableproperties for the close loop system [5].

The gain matrix of state feedback controller Kc isobtained as:

Kc =

⎡⎢⎢⎢⎢⎢⎣

1.1481 −2055 1.3219 −3967.8 0.0016442 −1.7117 −1.3466e-007−1.1648 2057.9 −1.3435 4025.5 −0.0020121 2.0259 1.578e-007−1.1653 2076.7 −1.3454 4041.4 0.074836 −45.208 −1.4127e-007

0.00087393 −19.741 0.0011058 −21.969 −0.12122 60.776 −2.0258e-007A B C D E F G

⎤⎥⎥⎥⎥⎥⎦∗1e−6, (19)

where A: 1.463e − 010, B: 7.8539e − 007, C: −8.3254e −010, D: 8.4805e− 006, E: 4.6762e− 008, F: −1.3999e− 00 =and G: 4.3014e − 013.

Analysis the gas turbine dynamic in time and frequencydomain by using LQG/LTR control method and with com-pare to PID controller is flowed. Step response, impulseresponse and principal gain consider for simulation. Figure 3shows the unit step response of the compressor pressure, tur-bine temperature and rotational speed respect to compressormass flow rate for LQR, LQG and LQG/LTR controller designof proposed gas turbine. So that to compare the results, unitstep response of these parameters for LQG/LTR and PIDcontrol is shown in Figure 4.

Simulation results for rotational speed and turbine tem-perature by using impulse function as input to power depictin Figure 5 for LQR and LQG/LTR controller design. Also

Figure 6 shows impulse response of the turbine temperaturerespect to mass flow rate of compressor.

The structure singular value can be used to analyze theability of a given design on achieves robust performance[1, 3]. Singular value or principal gain plot for main systemand controlled system by LQG/LTR are shown in Figure 7.

8. Conclusion

This paper introduces the gas turbine systems. Control andsatisfactory performances are important concept for thissystem. By using thermodynamic equations, a nonlinear dy-namic model is specified to study control objects. A gasturbine type is considered and its control system is described.Analyses the system is followed by making LTI state spacemodel obtained of linearization nonlinear dynamic model

8 ISRN Electronics

and apply suggested LQG/LTR optimal control method. ByLQG/LTR controller, and set parameters of this control byspecified procedure, simulation results in time and frequencydomain are achieved. To compare results, we use PID controlmethod for described system model.

System analysis shows response of uncontrolled dynam-ics and effect of LQR, LQG, and LQG/LTR controller de-signed and PID control. LQG/LTR controller improvedprincipal gain diagram to attain good performance. WithLTR method a satisfactory return ratio is recovered at theplant input.

Settling times of system characteristic component partsby using LQG/LTR controller are shorter, and stability ofsystem makes it better by applying this optimal controlmethod.

References

[1] S. Skogestad and I. Postlethwaite, Multivariable FeedbackControl Analysis and Design, Wiley, New York, NY, USA, 2000.

[2] A. Maddi, A. Guessoum, and D. Berkani, “Using linearquadratic gaussian optimal control for lateral motion ofaircraft,” Proceedings of World Academy of Science, Engineeringand Technology, vol. 37, pp. 285–289, 2009.

[3] J. M. Maciejowski, Multivariable Feedback Design, Addison-Wesley, Boston, Mass, USA, 1989.

[4] H. L. Tsai, “Generalized linear quadratic gaussian and looptransfer recovery design of F-16 aircraft lateral control system,”Engineering Letters, vol. 14, p. 1, 2007.

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[6] G. Stein and M. Athans, “The LQG/LTR procedure formultivariable feedback control design,” IEEE Transactions onAutomatic Control, vol. 32, no. 2, pp. 105–114, 1987.

[7] M. Athans, “A tutorial on the LQG/LTR method,” in Proced-dings of the American Control Conference, 1986.

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[9] B. Veroemen, “Model predictive control of gas turbine instal-lation,” WFW Report, 1997.

[10] N. Chiras, C. Evans, and D. Rees, “Nonlinear gas turbinemodeling using feedforward neural networks,” in Proceedingsof the ASME TURBO Expo 2002:Controls, Diagnostics, andInstrumentation, Cycle Innovations, Marine, Oil and GasApplications, pp. 145–152, nld, June 2002.

[11] O. Bolland, Thermal Power Generation, Compendium, 2008.[12] P. Ailer, I. Santa, G. Szederkenyi, and K. M. Hangos, “Non-

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[13] R. T. Jagaduri and G. Radman, “Modeling and control ofdistributed generation systems including PEM fuel cell and gasturbine,” Electric Power Systems Research, vol. 77, no. 1, pp. 83–92, 2007.

[14] N. Kakimoto and K. Baba, “Performance of gas turbine-basedplants during frequency drops,” IEEE Transactions on PowerSystems, vol. 18, no. 3, pp. 1110–1115, 2003.

[15] J. Mantzaris and C. Vournas, “Modelling and stability ofa single-shaft combined cycle power plant,” InternationalJournal of Thermodynamics, vol. 10, no. 2, pp. 71–78, 2007.

[16] H. Cohen, G. F. C. Rogers, and H. I. H. Saravanamuttoo, GasTurbine Theory, Longman, Scientific and Technical, 1996.

[17] A. M. Y. Razak, Industrial Gas Turbine: Performance andOperability, Woodhead Publishing limited and CRC pressLLC, 2007.

[18] M. P. Boyce, Gas Turbine Engineering Handbook, Gulf Profes-sional Publishing, 3rd edition, 2006.

[19] T. Giampaolo, Gas Turbine Handbook: Principles and Practices,The Fairmont Press, 3rd edition, 2006.

[20] C. H. J. Meuleman, Measurment and Unsteady Flow Modelingof Centrifugal Compressor Surge, University of Technology,Eindhoven, Netherlands, 2002.

[21] Matlab Refrence Guide, The Mathworks, 2010.[22] GE Power System, “SPEEDTRONIC MARK VI Turbine

Control System,” Phases 9 & 10 Refinery, 2007.

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Using LQG/LTR Optimal Control Method to Improve Stability and