design of robust control system for a parametric uncertain jet engine power plant

DESIGN OF ROBUST CONTROL SYSTEM FOR A PARAMETRIC UNCERTAIN JET

ENGINE POWER PLANT

Prof. P.S.V. Nataraj

Systems & Control Engineering Indian Institute of Technology Bombay Mumbai, India

• Introduction to Jet Engine

• Various Gas Turbine Configurations

• Performance and Control of Turbojet Gas Turbine Engine

• Control Requirements of aero Gas Turbine

• Design Steps for the Controller Design

• Robust Control Design Techniques

Overview

FUNDAMENTALS

• For an aircraft turbine engine, it is necessary to achieve maximum thrust with minimum engine weight.

• Control system should ensure all components operate at mechanical or thermal limits for at least one of the engine’s critical operating conditions.

FUNDAMENTALS …Cont.

• Single input single output controls are used on commercial engines where emphasis is on economic operation.

• Multivariable controls provide enhanced performance for military engines especially those with variable flow path geometry.

• In both case the maximum use of the available performance within the structural and aero-thermal limitations is required.

Afterburner flow DistributorNozzle

Digital ElectronicControl Unit

MainFuel

Reheat

Compressor Variable Geometry ( VG) Main Burners

Reheat burners

Nozzle actuators

Fuel in

Manual Fuel Control Linkage

Engine & System Feedback back

JET ENGINE WITH INTEGRATED CONTROLS

Gearbox

Hydromec-hanical

Systems

• Afterburner accomplishes short period increase in the rated thrust of the gas turbine engine

• Objective is to improve the take off, climb and maximum speed characteristics of jet propelled aircraft

Thrust Augmentation By Afterburning

• A Single Spool Turbojet Power Plant consists of an intake, a compressor, a combustion chamber, a turbine and a propelling nozzle.

• A Twin Spool turbofan power plant consists of an intake, a low pressure compressor, high pressure compressor, a combustion chamber, a high pressure turbine, a low pressure turbine, a mixer, and a propelling nozzle.

Various Gas Turbine Configurations

• Dynamic equations are obtained using the power balance, continuity and energy equations.

• The steady state power balance equation simply implies that the power output of the turbine must equal the power absorbed by the compressor.

Dynamic Performance of Turbojet Engine

• The increased gas turbine engine complexity has resulted in a corresponding increase in the complexity of the control system.

• Control requirements applied to gas turbine engines consist of ensuring safe, stable engine operation.

• An accurate and reliable control system is required to ensure needed engine performance and operational stability throughout the flight envelope. …Cont.

Control System Design

• The control system must sense Pilot’s commands, air frame requirements as well as critical engine parameters.

• It must then compute the necessary schedules and actuate system variable for total engine control over the full range of operation.

• Full authority digital electronic control system consists of two controllers, one of which is the digital electronic computer section and the other the back-up hydro mechanical section.

Control System Design…Cont.

• Relationship between the fuel flow control variable and the engine rotational speed over the flight envelope.

• Relationship between the fuel flow control variable and the engine compression ratio over the flight envelope.

• Relationship between the Power lever angle input into the control system and the rotational speed of the engine so as to incorporate a governing action and control engine thrust. ….Cont.

Control Requirements of aero Gas Turbine Engine

• Set limits on fuel flows so as to ensure safe operation of engine without excess rotational speeds, temperatures and pressures at specific stations of the engine.

• Position control of exhaust nozzle as per nozzle control schedule depending on Power Lever Angle (PLA) position.

• Reheat fuel flow selection by Pilot through PLA.

Control Requirements of Aero Gas Turbine Engine …Cont.

Control

Structure

Military

Engine

• Selection of the basic operating requirements for each element of control.

• Evaluation of engine requirements and control variables to select the mode of control that will provide the best operation.

• Selection of types of control regulators and computing system .

• Evaluation of stability requirements and basic performance requirements of the system

Design Steps for the Controller of Jet Engine

• Establishing the ability of the control components to meet the physical requirements of endurance, environment and vibration.

• Evaluating the final system by analysis or testing to establish its ability to perform as required under actual operating conditions

Design Steps for the Controller of Jet Engine …Cont.

Digital Control of Jet Engine

• The primary control objectives of gas turbine i.e., thrust, is a complex and non-linear function of flight conditions like altitude, mach number and control variables comprising of main engine fuel flow, reheat fuel flow and nozzle area.

• These parameters are to be controlled with complex schedules to a high accuracy.

…Cont.

Digital Control of Jet Engine …Cont.

• Full authority digital engine control system executes these schedules to provide the most optimum performance.

• The control system components are the hydro mechanical fuel management units with built in pumps, metering values and position feedback transducers and nozzle control system and their corresponding digital control units.

MAJOR CONTROL LOOPS OF JET ENGINE POWER PLANT

• HIGH PRESSURE COMPRESSOR ACCELERATION CONTROL LOOP

• LOW PRESSURE COMRESSOR SPEED CONTROL LOOP

• VARIABLE GEOMETRY CONTROL LOOP

• NOZZLE CONTROL LOOP

• AFTERBURNER FUEL FLOW CONTROL LOOP

MAIN FUEL FLOW CONTROLS NL COMPRESSOR VARIABLE GEOMETRY CONTROLS

NH NOZZLE AREA SCHEDULED AS FUNCTION OF

ALTITUDE AND PILOT THROTTLE POSITION AFTERBURNER FUEL FLOW SCHEDULED AS A

FUNCTION OF NOZZLE AREA AND ALTITUDE

BLOCK DIAGRAM OF HIGH PRESSURE COMPRESSOR SPEED ACCELERATION LOOP OF JET ENGINE

Fuel H/M System

Eng Model

Derivative

NH/Wf Controller

StepInput

NL: Low Pressure Compressor spool speedNH: High Pressure Compressor spool speedWf: Fuel flowH/M: HydromechanicalNHDOT : Rate of change of NH

Servo current

• * 3 Parametric uncertainties in Plant. * Uncertainty %: 5 %

Performance requirements for the inner Acceleration loop:1. Stability Margins : Gain Margin = 6 dB;

Phase Margin = 450;

2. Tracking Specifications : Rise time of closed loop transfer function between 0.4 to 0.8 sec1. No overshoot

Fuel H/M System

H/M System Eng Model

Derivative

ControllerOperating

ControllerSchedule

Throttle

Inlet temperature

Desired

LPCompressor speed

Eng Model

ServoNHWf

Current

NL: Low Pressure Compressor spool speedNH: High Pressure Compressor spool speedWf: Fuel flowH/M: HydromechanicalNHDOT : Rate of change of NH

BLOCK DIAGRAM OF THE LOW PRESSURE COMPRESSOR SPEED CONTROL OF JET ENGINE

* 5 Parametric uncertainties in Plant. * Uncertainty %: 5 %

Performance requirements for the outer Speed loop:1. Stability Margins : Gain Margin = 6 dB;

Phase Margin = 450;

BLOCK DIAGRAM OF EXHAUST NOZZLE POSITION CONTROL SYSTEM

Actual Actuator stroke

Controller Actuator

DesiredActuator strokeThrottle

Temperature

Operating

schedule

Area to stroke converter

Servo valve

Demanded Nozzle Area

Nozzle Electro Hydromechanical

System

Actuator position

LVDT : Linear variable differential transducer (to measure stroke)

Performance requirements for the VariableC Nozzle control loop:1. Stability Margins : Gain Margin = 6 dB

Phase Margin = 450

BLOCK DIAGRAM OF AFTERBURNER FUEL FLOW CONTROL LOOP

Controller

Throttle

Temperature

Operating

scheduleAfterburner H/M

System

Demanded Afterburner Fuel flow

Servo current

Actual Afterburnerfuel flow

Operating Schedule : Afterburner fuel flow as a function of nozzle area

Performance requirements for the Afterburner fuel flow control loop:1. Stability Margins : Gain Margin = 6 dB;

Phase Margin = 450;

-Compensator

PilotThrottle

InletTemperature

Operating

scheduleVG H/M System

Desired CompressorIGV position +

Actual IGV position

Input to VG H/M system : Servo valve currentOutput of VG H/M system : IGV Actuator Position

VG : Variable geometryIGV : Inlet guide vaneH/M : Hydromechanical

BLOCK DIAGRAM OF THE VARIABLE GEOMETRY CONTROL

SYSTEM OF JET ENGINE POWER PLANT

Performance requirements for the variable Geometry control loop:1. Stability Margins : Gain Margin = 6 dB;

Phase Margin = 450;

An Interval Analysis Approach for Design of Robust First Order

Compensator for Jet Engine

• Consider a strictly proper interval plant family P comprising of

plants of the form.

where interval bounds are a priori given for each uncertain coefficient qi and ri . Let C(s) be a compensator in a feedback

structure for this interval plant. If C(s) is such that it stabilizes the entire P, then C(s) is said to robustly stabilize P.

• If the compensator for an interval plant is first order, the stability of only sixteen extreme plants is necessary and sufficient to stabilize the entire interval family.

Basic Concepts

10 l;kks...........srr

q...........sqq

(s,r)pD

(s,q)pNP(s,q,r)

Block diagram of Compressor Speed control Loop of Jet Engine

Operating Schedule

Compressor Engine

Compressor Speed

DesiredCompressor SpeedThrottle

InletTemperature

Block diagram of Ndot controller of Jet Engine

InletTemperature

DesiredNdot

Ndot Schedule Compressor Engine

Compressor SpeedDifferentiator

Let 4 denote the set [1,2,3,4]. Let Ni1 (s) , i1 4 , denote the

Kharitonov polynomials associated with NP(s,q).

Algorithm for Compensator Synthesis

....................

sqsqsqsqsqqsN

….Cont.

Consider that a robustly stabilizing PI controller

is to be synthesized for an interval plant family P with Kharitonov’s polynomials N1 (s), N2 (s), N3 (s) and N4 (s) and

D1 (s), D2 (s), D3 (s) and D4 (s) for the numerator and

denominator respectively.

KKsC 2

Algorithm for Compensator Synthesis ..Cont.

The sixteen extreme plants are defined by

with i1, i2 {1,2,3,4}.

12,1 sD

Algorithm: Synthesis of first order compensator (due to Barmish et. al.)

Begin Algorithm

(i) Set up sixteen Routh tables for closed loop polynomials associated with each extreme plant.

(ii) Enforce positivity for each of the first column entries which are functions of K1 and K2. This leads to set of inequalities

involving K1 and K2.

(iii) To obtain the final controller, (K1, K2) should stabilize all

sixteen extreme plants simultaneously. Let

Ki1,i2, i1,i2 {1,2,3,4}

The compensator synthesis Algorithm given by Barmish et al.

• Denote the set of stabilizing gains corresponding to the i1th and i2th Kharitonov polynomial for the numerator and denominator, respectively.

•Solve the inequalities for each of the sixteen extreme plants independently by gridding method and obtain the set of stabilizing gains for each of them.

(iv) Obtain the desired set of stabilising gains for the interval system as intersection of these results evaluated for the sixteen extreme plants.

END Algorithm.

Remarks

• Here, the bounds for the stabilising gains are obtained in the selected range of K1 and K2. It is not possible to obtain all the

solutions. Further, the results are guaranteed only at the gridding points. It may happen that a point between the selected grid points may not be a feasible solution.

• The proposed algorithm using interval analysis evaluates the bounds for the stabilising gains, in which all feasible solutions are obtained as a set of interval boxes of specified accuracy. Any point within these interval box is a guaranteed stabilising gain for the interval system.

Begin Algorithm

i) Obtained by replacing the gridding technique in step (iii) of Barmish et al’s algorithm given above with the sub-definite computations technique (to solve the set of inequalities involving the sixteen extreme plants).

ii) Using sub-definite computation technique, obtain initial bounds on K1 and K2 in which all feasible solutions must lie.

iii) Find all feasible solutions as interval boxes of specified accuracy in the initial bounds obtained at (ii) above.

End Algorithm

Algorithm: Synthesis of first order Compensator using interval analysis

Remark 1 The solution set obtained as a set of interval boxes contains all the feasible solutions. Any point within this box is a guaranteed stabilizing gain for the interval system.

Remark 2 A necessary and sufficient condition for the existence of a robust stabilizing controller is non-emptiness of the set of gains in (iii) above.

Remark 3 The interval plant P can be stabilized by selecting any (K1,K2) K.

• A computer program has been developed for the above controller synthesis technique.

• Consider the SISO Jet engine interval plant with input as fuel flow and output as acceleration of compressor speed, Ndot.

P(s,) =

The uncertainty bounds are

qo [940, 980], r1 [97, 107], r2 [215,230]

Let us synthesise a compensator of the form

to stabilize the interval plant P(s,)

The Kharitonov polynomials for the numerator and denominator of the effective plant of acceleration loop are:

N1(s) = 940;N2(s) = 980;

D1(s) = s2 + 97s + 215; D2(s) = s2 + 107s + 215;

D3(s) = s2+107s+230; D4(s) = s2 + 97s + 230;

c 21)(

Thus, there are 8 different extreme plants. Using these extreme plants, and PI compensator C(s), the associated closed loop polynomials are derived as follows.

p1,1(s) = s3 + 97s2 + (215+940K1)s + 940K2

p1,2(s) = s3 + 107s2 + (215+940K1)s + 940K2

p1,3(s) = s3 + 107s2 + (230 + 940K1)s + 940K2

p1,4,(s) = s3 + 97s2 + (230 + 940K1)s + 940K2

p3,1(s) = s3 + 97s2 + (215 + 980K1)s + 980K2

p3,2(s) = s3 + 107s2 + (215 + 980K1)s + 980K2

p3,3(s) = s3 + 107s2 + (230 + 940K1)s + 980K2

p3,4(s) = s3 + 97s2 + (230+980K1)s + 980K2

•Routh table is set up for all the closed loop polynomials.

• Inequalities associated with each closed loop polynomial are obtained.

• These inequalities are solved using proposed interval analysis algorithm to obtain bounds on stabilizable controller parameters.

• Any value within these bounds will stabilize the uncertain plant

Results obtained• 3 Inequalities associated with each closed loop polynomial

• Thus 24 inequalities are solved using proposed interval analysis algorithm

• Initial bounds on K1 and K2 are obtained as

[0, 2.19131e6]

• Thus the given plant is stabilizable even with a very large value of K1 & K2

• However, large values of gain are not practical.

• Practical gains can be obtained by implementing additional performance constraints.

0 2 4 6 8 10 12 14 160

16Stabilizable controller gains obtained by Interval analysis

Time (sec.)

0 0.5 1 1.5 2 2.5 30

1Closed loop system response to a unit step command in NDot

Time (sec.)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

25Closed loop system response to a unit impulse command in Ndot

CONCLUSION• We have developed an interval analysis based algorithm

for evaluation of set of robustly stabilizing first order compensator for an interval plant of jet engine.

• The algorithm guarantees that all feasible robust stabilizers lie within the bounds of computed interval boxes.

• The technique guarantees stability for the entire interval plant set.

• It also finds if a robust first order compensator for an individual loop is feasible or not. If feasible, the algorithm gives the entire solution set.

• Although synthesis procedure is for robustly stabilizing compensator, appropriate controller meeting desired performance specification can also be obtained using constraints on compensator coefficients and verifying the performance through simulation in simulink.

DESIGN OF A ROBUST 2 DOF CONTROLLER FOR VARIABLE EXHAUST NOZZLE CONTROL OF

MILITARY AERO GAS TURBINE

ACTUAL ACTUATOR STROKE

RETRACT

COOLING

EXTEND

ENGINEGEARBOX

CONTROLLER OUTPUT

THROTTLE

ENGINE PARAMETERS

COOLER

HPCLPC

JET ENGINE

SV FUELOUT IN

COMBUSTOR

HYDRAULIC LINESELECTRICAL LINES

VARIABLE EXHAUST NOZZLE

HYDRAULIC ACTUATORHPTLPT

THROAT AREA EXIT AREA

Turbojet Gas Turbine Engine With Variable Exhaust Nozzle

Military aero gas turbines are fitted with variable exhaust nozzle system as it ensures that engine operating line is not affected by the increased volume of gas stream due to reheat combustion

We present a methodology, based on Quantitative Feedback Theory (QFT), for design of a robust 2 Degree of Freedom (DOF) controller for the control of variable exhaust nozzle area.

The advantage of this methodology is that a single robust controller can be used for the full flight envelope control of the variable exhaust nozzle, which is contrary to the conventional controller incorporating gain scheduling for different locations in the flight envelope

The design is done in frequency domain and the evolved robust controller stabilizes the system that does not have a distinct set of poles and zeros but a range over which each of the poles and zeros might lie.

The methodology is successfully applied to design a robust 2 DOF compensator for variable exhaust nozzle system of an experimental aero gas turbine engine.

Actual Actuator stroke

Conventional controller

Actuator

DesiredActuator Stroke

Throttle

Temperature

Operating

schedule

Area to stroke converter

Servo valve

Demanded Nozzle Area

Nozzle Electro Hydromechanical

System

Block diagram of exhaust nozzle position control system

Define frequencies Create frequency response dataDefine nominal plantDefine Controller typeDefine phase array for computing bounds

Compute bounds

Loop Shaping

Compare design to specifications

PLOT TEMPLATES

Define problemData

Design

GROUP BOUNDS

INTERSECTION OF BOUNDS

PLOT OF BOUNDS

FILTER DESIGN

Analysis

Flow chart showing basic steps in a QFT design

G (s) P (s)

Block diagram of the two degrees of freedom control system

The nozzle control system is basically a position control system functioning as closed servo loop.

The desired nozzle position is a function of altitude and throttle position.

The hydro mechanical system of a variable exhaust nozzle being a closed system is only a function of the engine speed.

The nonlinear plant dynamics are approximated using piecewise linearisation between idle and maximum operating speed of the gas turbine engine.

The model of this system at various operating speeds is integrated together to form a parametric uncertain system

Model uncertainties are also taken into account.

A robust 2 DOF controller is then designed which stabilizes the system that does not have a distinct set of poles and zeros but a range over which each of the poles and zeros might lie.

The designed controller must fulfill two objectives: Nozzle position keeping and nozzle position changing .

In the first case, the control objective is to maintain the nozzle actuator position. In the second case, the aim is to implement the change of position without oscillations and in the shortest time possible

In both situations, the operability of the system must be independent of the uncertainties in the model of the nozzle electro hydro mechanical system .

The mathematical model of the nozzle electro hydro mechanical system between the hydraulic actuator stroke Y(s) and the drive current to the servo valve I (s) at various operating speeds between idle and maximum engine speed is approximated by the transfer function:

where ‘k’ and ‘a’ are different constant parameters at various engine speeds.

Considering the model uncertainties at various speeds and integrating it with variation of ‘k’ and ‘a’ parameters owing to non-linearity, interval bounds on these parameters are specified.

Thus despite the fact that the model is non-linear, the QFT model for linear SISO systems with parametric uncertainty is used, incorporating the two degrees of freedom control system

Y(s) k = I (s) s (as + 1)

The design of the controller includes a cascade compensator, G (s) and a prefilter, F (s) (both LTI ) in order to reduce the variations in the output of the system caused by the uncertainties in plant parameters.

The system must fulfill robust stability and robust tracking specifications. For the robust stability margins, the phase margin angle should be at least 450 and the

gain margin 6 dB. Thus the robust stability specification is defined by:

P (j) G (j) = 2.3 1 + P (j) G (j)

The robust tracking, must be defined within an acceptable range of variation. This is generally defined in the time domain but is normally transferred to the frequency domain, being expressed by

TRL (j) TR (j) TRU (j)

where TR(s) represents the closed loop transfer function and

TRL(s) & TRU(s) the equivalent transfer functions of the lower

and upper tracking bounds

The 2 DOF controller design has been carried out for following uncertainty in the ‘k’ and ‘a’ parameters which are pertaining to a variable exhaust nozzle system of an aero gas turbine under development.

k [0.4, 0.9] ;

a [0.2, 0.4] ;

Acceptable range of variation of rise time for change in nozzle position is specified as 0.12 to 0.18 sec. Following set of frequencies for the design has been used.

= [0.1, 1, 4, 7, 9, 10, 12, 13, 20, 30, 50, 100]

Design Illustration

On the basis of the performance specifications and the plant templates, the robust stability and robust tracking bounds are computed

For the design of the G(s) controller, the Nichols chart is used, adjusting the nominal open loop transfer function Lo = PoG (Po

is the nominal plant) in such a way that no bounds are violated

The controller obtained is

G (s) = 0.965 ( s + 87.21) / ( 4.34 x 10-6 s2 + 2.08 x 10-3 s + 1)

With this controller, the robust stability specification is fulfilled but not the robust tracking specification. This is obtained by design of the prefilter

F(s) = 16.66 x 10-3 ( s + 60) / ( 1.54 x 10-3 s2 + 9.2 x 10-2 s + 1)

-350 -300 -250 -200 -150 -100 -50 0

Phase (degrees)

Magnitude (

Plant templates

-350 -300 -250 -200 -150 -100 -50 0

11111111

Phase (degrees)

-350 -300 -250 -200 -150 -100 -50 0

Phase (degrees)

Robust stability bounds.Robust tracking bounds.

-350 -300 -250 -200 -150 -100 -50 0

-400 -350 -300 -250 -200 -150 -100 -50 0-200

Phase ( degree)

Intersection of bounds. Shaping of Lo (jw) on the Nichols chart for the nominal plant

frequency ( rad/ sec)

10-1 100 101 102-250

frequency (rad/sec)

Frequency response of nozzle control system

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

Time ( sec)

Nozzle

Closed loop response of the nozzle control system for unit step command of nozzle actuator position

ROBUST QFT CONTROLLER DESIGN FOR TWIN SPOOL GAS TURBINE SPPED CONTROL

Acceleration of high-pressure compressor spool forms the inner loop. Direct control of acceleration, rather than speed, allows tighter control of engine acceleration thereby improving transient response and reducing mechanical stress.

The controller must fulfill two objectives: Speed keeping and speed changing. In the first case, the control objective is to maintain the gas turbine low pressure compressor spool speed following the desired speed as per control schedule

In the second case, the aim is to implement the change of speed without oscillations and in the shortest time possible.

k1, k2, k3 ,a ,b ,c : 5 Parametric uncertainties.

k1= [285 , 330] ; k2 = [7.5 , 8.0] ; k3 = [10, 15] ;a a = [0.01, 0.03] ; b= [0.4 , 0.5] ; c= [0.27 , 0.33] • Performance requirements for the inner loop:1. Stability Margins : Gain Margin = 6 dB; Phase Margin = 450;

2. Tracking Specifications : Rise time of closed loop transfer function between 0.4 to 0.8 sec1. No overshoot Performance requirements for the outer loop:1. Stability Margins : Gain Margin = 6 dB; Phase Margin = 450;

2. Tracking Specifications : Rise time of closed loop tr. fun. between 0.7 to 0.95 sec3. No overshoot

-350 -300 -250 -200 -150 -100 -50 0

150.11

Phase (degrees)

Figure 3: Plant templates for inner acceleration control loop

-350 -300 -250 -200 -150 -100 -50 0

80Figure 4 : Intersection of bounds of inner acceleration loop

w =0.1

12 13 20

Open-Loop Phase (deg)

Figure 5 : Shaping of Lo( jw ) on the Nichol's chart for the nominal plant of inner acceleration control loop

-450 -400 -350 -300 -250 -200 -150 -100 -50 0-350

Time (sec.)

0 0.5 1 1.50

Figure 6 : Close loop Inner acceleration loop response to unit step input of acceleration for the entire uncertain plant set

An Algorithm for Controller Synthesis of Jet Engine Power Plant with Parametric Uncertainties

A key concept for our work is the KTR• Given a GKS with vertex polynomial P1 and P2

• Construct two polynomials PA and PB as follows

PA = P1odd + P2

even PB = P1

even + P2odd

• The polytope defined by the polynomials P1PAP2PB

is defined as the KTR

• KTR is a two dimensional polytope in the coefficient space, with four exposed edges

Proposed Synthesis Procedure

• Consider any GKS associated with the C(s) and given P.

• Construct a KTR for each GKS.• Set up Routh tables for all the vertex polynomials

of all KTR.• Finally, solve the total set of inequalities using an

appropriate nonlinear solver to get a robustly stabilizing compensator C(s) for P.

Proposed Synthesis Procedure …Cont.

Jet Engine Application

• Consider the following single input single output jet engine interval plant with manipulated variable as fuel flow and controlled variable as compressor speed.

P(s,) =

• The uncertainty bounds are

r3 [0.004, 0.005], r2 [0.4, 0.5]

ssrsrsD

Jet Engine Application …Cont.

• Let us synthesize a lead lag compensator of the form

C(s) = Nc(s)/Dc(s) = a(1+bs)/(1+cs)

• Constraint on ‘c’ is selected in order to have a dominating effect of zero as compared to pole of compensator.

• From the GKT, the robust stability of the closed loop system is equivalent to that of the set of GKS. There are four GKS in our example:

• 1. 4.3a(1+bs) + (1+cs) [(0.001 + 0.004) s3 + 0.5s2 +s]

• 2. 4.3a(1+bs) + (1+cs) [(0.001 + 0.004) s3 + (0.1 + 0.4) s2 +s]

• 3. 4.3a(1+bs) + (1+cs) [(-0.001 + 0.005) s3 + (0.1 + 0.4) s2 +s]

• 4. 4.3a(1+bs) + (1+cs) [(-0.001 + 0.005) s3 + 0.4s2 +s]

• For each of these 4 GKS, KTR is constructed as per definition.

• 4 vertex polynomials of KTR corresponding to the each GKS are obtained.

• Routh tables constructed for all the 16 vertex polynomials.

• Enforcement of the positivity requirement for the first column of the Routh tables leads to a set comprising of 80 inequality constraints.

• This set of inequalities is solved using the UniCalc solver based on interval analysis computation technique.

• One of the feasible coefficients of the compensator in the solution set are a = 0.7025,

b = 0.5074 and c = 0.1062. • All the roots of the 16 vertex polynomials were

found to lie in the left half of the s plane thereby confirming stability.

CONCLUSION

• In this work, we have developed a new interval analysis based methodology for synthesis of a robust general order compensator for a parametric uncertain jet engine power plant.

• The technique guarantees stability for the entire interval plant set.

• It also finds if a compensator of any desired structure is feasible or not. If feasible, the algorithm gives the entire solution set.

CONCLUSION …Cont.

• Although synthesis procedure is for robustly stabilizing compensator, appropriate controller meeting desired performance specification could also be obtained using appropriate constraints on compensator coefficients and verifying the performance through closed loop simulation.

• Alternately, desired performance constraints can be derived as additional constraints and this synthesis technique can be extended to design jet engine robust compensators meeting both stability and performance requirements.

DESIGN OF ROBUSTLY STABILIZING CONTROLLERS FOR JET ENGINE USING

SET INVERSION

Introduction

• The performance requirements of modern, high technology aircraft has placed severe demands to engine control capability.

• All existing systems are subject to various disturbances and uncertainties. Mathematically, we can only approximate an existing system with a transfer function depending upon the information available about a system and the observations over a certain period of time.

• In this work, an algorithm using interval analysis approach is proposed for the synthesis of a robustly stabilizing first order compensator of a jet engine interval plant having parametric uncertainties.

Introduction (cont’d)

• Our aim is to develop algorithmic results which will enable us to determine if a robust first order stabilizer exists for a given interval plant.

• If it exists, then we obtain the set of stabilizing compensator parameters as a union of interval vectors (or boxes).

• Any value within this union represents a guaranteed robustly stabilizing compensator for the given interval plant.

Set Inversion Problem

• Let be a nonlinear function from Then the problem of set inversion can be posed as a problem of characterization of

• In this work we address the problem of characterizing S defined by a set of nonlinear inequalities

),...,( 32,1 ffff

ml RR to

)()(: 1 YfYxfRxS l

,00)(: 1fxfRxS l

Compensator Synthesis Algorithm

• Consider a interval plant family P comprising of plants of the form

• Where interval bounds are priory given for each uncertain coefficient

• A compensator C(S), if exists, which stabilizes the entire plant family P is then said to robustly stabilize P.

• If the compensator for an interval plant is of first order, the stability of only sixteen extreme plants is necessary and sufficient to stabilize the entire interval family (Barmish-1994).

lksrsrr

qsNrqsp

),(),,(

ii rq and

Synthesis of first order compensator by set inversion

• A compensator of first order of structure is designed as follows.

• Set up sixteen Routh tables (Barmish etal, 1992) for closed loop polynomials associated with each extreme plant.

• Enforce positivity for each of the first column entries which are functions of K1 and K2. This leads to set of inequalities involving K1 and K2.

• Solve the inequalities for the sixteen extreme plants by the proposed set inversion algorithm and obtain the set of stabilizing gains in the given initial bounds.

KKsC 2

Set Inversion Algorithm

• Let be the initial box . The set inversion algorithm encloses the portion S contained in X0, between two partitions Kin and Kout in the sense that

• where and Kε is the list of indeterminate boxes.

• The monotonicity test form is given as

where is the set of integers i such that properly contains zero

outin KSXK 0

inout KKK

))(()](),([)( iiii

mXXFDvfufXFMT

])(,)([)( XFDXFDXFD iii

Set Inversion Algorithm

BEGIN ALGORITHM

1. Initialize X=X0, Kin={}, Kout={}, L={X}

2. Remove all boxes from list L and evaluate FMT over all the boxes.

3. Deposit all boxes for which F(X)>0 for all i=1,…,m in the lists Kin and Kout.

4. Discard all those remaining boxes for which sup Fi (X)<=0 for any i=1,…,m.

5. Deposit all those remaining boxes for which w(X)< in list Kout

Set Inversion Algorithm (Cont’d)

6. Find all those boxes for which Fi is monotonically increasing or decreasing in every direction for, i=1,…. ,m. Apply Algorithm TPB to discard infeasible parts of these boxes.

7. Bisect all remaining boxes in the maximum width coordinate direction k, getting subboxes V1, V2 such that Deposit all these subboxes in L

8. If the list L is empty, EXIT algorithm. Else, go to step 2.

END ALGORITHM

21 VVX

Throwing part boxes Algo. (TPB)

• If f is monotonically increasing/decreasing in every direction on X, then algorithm TPB locates the subbox on which the inequality f>0 is certainly feasible, and outputs a list LX of boxes whose union is the complement of C in X

• Algorithm TPB parameterizes the line joining points

in terms of single parameter λ.

• Use Newton-Raphson to find λ*, such that f(λ*)=0

ll XXX ,...,X and ,..., 11

TPB (Cont’d)

• Construct a subbox on which f>0 is certainly infeasible.

• Using the box complementation algorithm in (Kearfott 1996), find the complement of box C in X to get a list LX such that

CXWXLw \

TPB (Cont’d)

Jet Engine Problem

• A SISO Jet engine interval plant with input as fuel flow and output as acceleration of compressor speed, Ndot

Desired NdotNdot

ScheduleComp-ensator

Engine

Differentiator

CompressorSpeed

Inlet temparature

Jet Engine Problem (Cont’d)

• The transfer function is

• Where

• Compensator of the form is required to be designed.

rsrsrs

]1.0,1.0[],230,215[],107,97[],980,940[ 0210 rrrq

KKsC 2

Values of K1 and K2

0 2 4 6 8 10 12 14 16 18 200

Conclusions

• The set inversion algorithm gives values of K1 and K2 for the system to be robustly stable.

• The algorithm guarantees that all feasible robust stabilizers lie within the bounds of computed interval boxes.

Design of Robust 2 x 2 MIMO Control System for Gas Turbine with Variable Geometry

Design Method

• The MIMO design is carried out using “Equivalent Disturbance Attenuation” method based on QFT.

• Plant uncertainties are transferred into its equivalent disturbance sets.

• Thus, the design problem becomes the disturbance attenuation problem.

• The fundamental point is that if G is chosen so that the output T satisfies the system tolerances over the entire VT set, then the original MIMO specifications are satisfied

Basic Concepts

F G PCR

R = PG ( FR – C)

T = (I + L0)-1 VT ; V = I - P0 P

A 2 x 2 MIMO Design Problem

f11 g11 p11

f22 g22 p22

Controller Structure for 2 x 2

MIMO System

Controller Synthesis Procedure

• Translation of time domain specifications into frequency specifications.

• Translation of tracking specifications into its equivalent disturbance specifications.

• Bound generation for stability and disturbance.

• Loopshaping and filter design.

Gas Turbine Application

k1 P11 = -------------------------- as2 (0.0169s+1) + s k2 (-s + 0.2125) P12 = -------------------------------------- bs2 (0.0029.s2 + 0.1622s+1) + s 0.848 k3 P22 = -------------------------- cs2 (0.0191s+1) + s k4 P21 = -------------------------- ds2 (0.0169s+1) + swhere, k1 [ 8.70,9.70] ; k2 [ 5.70,6.30]

k3 [ 1.15,1.25] ; k4 [6.20,6.50]

a [ 0.45,0.55] ; b [ 0.40,0.46]

c [ 0.63,0.64] ; d [ 0.45,0.55]

Inputs:

Fuel flow & VG

Outputs:

LP & HP Compressor Speed

Performance Specifications: • The rise time of closed loop transfer function for

speed control loop should lie between 0.3 to 0.5 sec.

• The rise time of closed loop transfer function for variable geometry loop should lie between 0.26 to 0.4 sec.

• The stability margins for both the loops are 6 dB gain margin and 45 degrees phase margin.

• The cross coupling effects are | t12/t22 |= 0.05,

| t21/t11|= 0.1

Tolerances on |tij|

Bounds on l110 and Loopshaping on l110

Bounds on l220 and Loopshaping on l220

Compensator and the filter obtained are as follows.

3.836 x105

g11= ------------------------------------ ( s+7.902 )(s2+400s+160000) 54 f11 = ---------------------- ( s+5.4 )(s+10) 1.514 x 105 (s +3.5) g22 = ------------------------------------ ( s+27.75 )(s2+80s+6400) 1.78 ( s+ 45) f22 = ---------------------- ( s+8 )(s+10)

Closed Loop Step Responses

Conclusion

• No plant templates need to be calculated and the design is obtained much faster.

• It is a very viable technique to design robust control system of modern gas turbine with variable geometry which are widely used for power generation, naval and air propulsion.

Thank You…

design of robust control system for a parametric uncertain jet engine power plant

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