1

AIAA-2002-4969

MODELING AND SIMULATION OF LAUNCH VEHICLE DIGITAL AUTO PILOT

Ashok Joshi, Professor, Department of Aerospace Engineering, Non-member, IIT Bombay, Mumbai, IndiaP G Pramodh, Scientist/Engineer PSLV Project, Non-member, VSSC, Thiruvananthapuram, India

AbstractThis study is concerned with the effect of non-linearities in the configuration design of DigitalAuto Pilot (DAP) in launch vehicles. An electro-hydraulic actuator model of a launch vehiclecontrol system is considered for analysis of non-linearities. Various non-linear effects likesaturation (in current and stroke limit), dead zoneand coulomb friction are taken into account. DAP,which is an interface between the guidance systemand control system, is designed to cater to themodel (linear/ non-linear) adopted for the actuator.In the actuator alone case, without considering thetotal flight regime and vehicle model, theperformance is found to be satisfactory for linear aswell as non-linear actuator models. In the actuator–vehicle combination, when the simulation is carriedout for the total flight regime considering thevehicle model, the performance of the linear / non-linear actuator model is dependant on DAPconfiguration. This study brings out the fact thatthe DAP configuration is specific to the actuatormodel, so that satisfactory performance of launchvehicle control system can be ensured only bychoosing proper configuration for DAP, based onconsideration of non-linearities in actuator model.

NomenclatureD Drag Force (N)FA Actuator Force (N)FN Force on the Engine/ Nozzle (N)IR Inertia of Vehicle about Swivel Point (kg.cm2)IXY Inertia of Vehicle about Pitch Axis (kg.cm2)ka Actuator Servo Amplifier Gain (mA/V)kC Flow Pressure Coeff. of Servo valve (cc/s/ksc)kL Equivalent Stiffness of Mounting (kgf/cm)kP LVDT Sensitivity of Servo valve (V/rad)kq Linear Equivalent Gain of Servo valve (cm2/s)kv Spool Valve Current by Displacement (cm/mA)L Actuator Lever Arm (cm)Lp1 Length of Slosh Pendulum (m)lc Distance of Center of Force-Engine Swivel (m)lp1 Distance of Pendulum Hinge -Vehicle CG (m)lr Distance of Rocket CG-Engine Swivel point (m)

lα Distance of Vehicle Nose-Body Axis Origin(m)m0 Reduced mass of Vehicle (Kg)mp1 Mass of Slosh Pendulum (Kg)mr Mass of Rocket engine (Kg)Pc No load Pump Delivery Pressure (ksc)PL Load Pressure in Actuator (ksc)Ps Fluid supply Pressure in Actuator (ksc)Qm Maximum Pump Delivery rate (cc/s)Qp Pump Flow (cc/s)Qv Control flow from Servo valve (cc/s).TC Control Thrust (N)TF Forward Thrust (Un-gimballed thrust) (N)Tv Actuator Spool valve time constant (s)Xa Actuator Piston movement (cm)Xv Servo valve Orifice opening (cm2)Yv Spool Displacement (cm)δN Engine/ Nozzle deflection angle (rad)θ Pitch Angle (deg)Γp1 Pendulum Angle for first Slosh Pendulum(rad)µC Control Moment coefficient (sec-1)µ p1 Slosh Force coefficient ( Oxidizer tank) (sec-2)µe Rocket Engine Inertia-Vehicle Inertia ratioµα Aerodynamic Force coefficient (sec-1)ωp1 Slosh Frequency ( Oxidizer tank) (rad/s)ξp1 Slosh Damping Ratio ( Oxidizer tank)

IntroductionLaunch vehicles require a control system tostabilize them as well as to give trajectory relatedcommands. In modern launch vehicles, this task isperformed by a Digital Auto Pilot or a DAP. In thedesign of DAP, it is necessary to represent all thephysical effects as accurately as possible.However, in many cases, linearized models ofvarious subsystems are used for the design process.Such an approach can work if there are largecontrol margins available, but a more efficient andoptimal control design needs the mathematicalmodels which have greater fidelity. In general,such models are non-linear in nature andsometimes it is necessary to determine if a DAPdesigned for a linearized system will work withsame fidelity, if the models are nonlinear. Thepresent study explores this aspect of the DAPdesign in launch vehicles.

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A literature survey has been carried out, regardingdifferent types of controllers and controller designmethods, currently in practice[8]. Variousalgorithms for the selection / design of controller /auto-pilot are also discussed. In addition, the typesof non-linearities present in the launch vehiclemodels, particularly in actuator systems, arediscussed in detail. Objective of the present studyis to carry out modeling and simulations of electrohydraulic actuator used for nozzle angularpositioning and assess its performance without andwith non-linearities taken into account.

Linear Modeling & Simulation of Actuator[3,8,9]

The critical elements of the linear actuator modelare servo valve, hydraulic pump, actuator chamberand piston and engine gimbal dynamics. Nozzledynamics is also considered in the modeling. Inthis case, various non-linearities associated withthe electro hydraulic actuator like saturation, deadzone, coulomb friction, preload, on-off hysterisis,backlash etc. are not considered. The followingequations describe the mathematical model, whichis shown graphically in fig. 1.

Servo Valve Dynamics:ka. kq kv / (1+ Tv s) = 1741/(1+0.002s) (1)

1741.69

0.002s+1

spool valve (ka kv kq) / (1+Tv s)

9205

kl

0.11025

kc

0.25s +6.28s +369.3875s+888.12502

s +101.1s +2589.8s+1579.42

forward filterGc(s)

142122.3

s +533.06s+142122.32

filter1

command dc

0.00491

Vo / Be

394780s

s +879.64s+3947802

deltaA

deltaN

Scope2

Scope1

S3S2S1

73.32

15910s +66667s2

L/(Jes*s+Be s)

73.32

L

s

1

Integrator

Fd

18.6

Ba

0.0526

1/Ap

0.0526

1/ Ap

143.2

kp

0.013638

1/L

Fig. 1 Linear SIMULINK Model of the Actuator

Forward Filter Transfer Function:0.25s3 +6.28 s2 +369.3785s +888.1250s3 + 101.1 s2 + 2589.8 s +1579.4 (2)

Engine Gimbal Transfer Function: 73.32 (15910 s2+ 66667 s) (3)

Feedback Filter Transfer Function: 142122.3 (s2 + 53 3.06s+142122.3) (4)

The numerical simulation is carried out withMATLAB-SIMULINK using fourth orderRunge-Kutta algorithm with step size of 1 ms.Simulation results are analyzed for a duration of1.5 s, with and without considering thenozzle/engine gimbal dynamics in the forwardloop. Step response parameters like rise time, peakovershoot and settling time, are computed for boththe cases and are found to match approximatelywith the physical case. (Figs. 2&3 and Table 1)

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Table.1 Comparison of Simulation Results and Design Specifications for Linear Actuator

Parameters Nozzle deflection (δN) Actuator deflection (δA) Specification (δA)

Rise time (ms) 55 80 80 ± 20

Overshoot (%) 33 19.5 20 ± 2

Settling time (ms) 680 260 250 ± 30

Fig 2. Linear Actuator Step Response

Fig 3. Linear Actuator Step response

Modeling of DAP for Linear Actuator. [10,11,12]

Digital Auto Pilot (DAP) for Linear Actuatorconsists of Attitude Filter, Forward Path Gain,Vehicle Transfer Function, Feedback FilterTransfer Function and Sensor Transfer Function. Inthe present case, an existing design of such a DAPis used for assessing the adequacy of the actuatormodelling. The various transfer functions are asgiven below.

Attitude Filter Transfer Function:

0.75 + 0.75 z-1 [11] (5)

1 + 0.5 z-1

Forward Path Gain, Ka = 0.85 (6)

Forward Filter Transfer Function:

0.241702 +

1.3374555 z -1 + 1.4128671 z -2 +

1 + 0.1517240 z -1 - 0.0586190 z -2

-1.314151 z -1 + 0.465605 z -2 +

1- 0.7496074 z -1 +0.2760324 z -2

-0.1122301 z -1 + 0.1120510 z –2

1 - 1.839706 z-1 + 0.840520 z –2 (7)

Vehicle Transfer Function:

µc µc = 5.656, µ∝ = 0.0214 (8)

s2 - µ∝

Feedback Filter Transfer Function:

0.75+0.75z-1

1+ 0.5 z-1 (9)

Sensor Transfer Function:- Position Gyro [11]

o 6168.5 / (s2 + 109.955s + 6168.5)- Rate Gyro [11]

o 12090.265/(s2+153.94 s +12090.265)

(10)

Figure 4 shows the schematic of the total system,which is simulated.

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Fig 4. SIMULINK Model of Linear Actuator Coupled with DAP

theta c

In1 Out1

linear actuator

limit

1

0.002

ki

-0.11223z +0.112051z -1 -2

1-1.839706z +0.84052z -1 -2

-1.314151z +0.465605z -1 -2

1-0.7496074z +0.2760324z -1 -2

1.3374555z +1.4128671z -1 -2

1+0.1517240z -0.0586190z -1 -2

Zero-Order

Hold2

Zero-Order

Hold1

5.656s

s +0.02142

Transfer Fcn1

1

s

thetadot

To Workspace1

theta

To Workspace

Scope1

Scope

0.75+0.75z -1

1+0.5z -1

Rate path filter

12090.265

s +153.93s+12090.2652

Rate gyro dynamics

6168.5

s +109.955s+6168.52

Position gyro dynamics

0.81

Kr

0.85

Ka

0.241702

1

z

0.75+0.75z -1

1+0.5z -1

Attitude filter

Linear Actuator Coupled with DAP.

Numerical simulation of linear actuator with DAPis carried out using fourth order Runge-Kuttaalgorithm with step size of 1 ms. The followingassumptions are made for the simulation.

1. Error in yaw and roll is neglected.2. Rate control logic is not applied.3. Gain scheduling is not considered.

Step response characteristics (Fig. 5) show asimilar result, in comparison to that of actuatoralone case. It is observed that the system is stable;whereas the settling time is observed to be higher.Bode diagrams (fig. 6) also show a stable system.

Fig 5. Linear Actuator Coupled with DAP

Fig 6. Linear actuator coupled with

Fig. 6 Linear Actuator Coupled with DAP

Non Linear Modeling/Simulation of Actuator[7,8,9]

The electro hydraulic actuator is modeled here asan actuator-nozzle combination. In the presentmodeling of the electro hydraulic actuator, fewnon-linear effects are considered, as in the actualsystem these effects also contribute to the systemperformance. In some cases, the accuracy ofrepresentation of physical dynamics is poor forlinear models. Therefore, the consideration to thevarious non-linearities is expected to make themodel response characteristics closer to the actualsystem specifications.

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Servo Valve Non-linearity[9]

Spool valve displacement YV directly controls theflow into the actuator chamber by varying theorifice opening, XV. Here the variation of XV withrespect YV is non linear. When the spooldisplacement is between a specified limit of ∆1 and∆2 in forward or reverse direction, the variation islinear with one slope. When YV is between ∆2 andYVmax, slope in forward or reverse direction isdifferent. Therefore when the total range of spooldisplacement is considered, the variation is non-linear with two slopes. The non-linearity can beincorporated in the model in the form of a look uptable with seven data points as given below. It maybe noted here that in the design of DAP, portopening Xv is considered as a state variable, andcomputed from this table.

Table 2 Servo Valve Characteristics

Xv Yv

-0.0403 -0.0380

-0.0046 -0.0023

-0.0008 -0.0008

0.0000 0.0000

0.0008 0.0008

0.0046 0.0023

0.0403 0.0380

Hydraulic Flow Non-linearity[9]

The control flow from the servo valve is given by,

Qv = KN.xv. PS -γPL . sgn (PS-γPL)Where γ = sgn Xv (11)

Here Qv is dependant on PS and PL. Supplypressure PS is not constant and it varies dependingon the pump delivery rate Qp. If the internalleakage of the pump is neglected, it can deliver amaximum flow rate of Qm for a wide range ofoutput pressures up to Pc.

When the flow demand is less than Qm, the pumpoutput pressure increases proportionally up to P0.The pump supply pressure PS for a delivery rate,Qp< Qm can be obtained as,

Ps = P0 – [ (P0 – Pc) / Qm]*Qp Where Pc = 205,P0 = 212, Qm = 360 & Qp = PL *K (12)

In addition to the above, the following non-linearities are also considered.

Saturation of servo valve current: ±10 mASaturation of actuator stroke: ± 10.7 cmDead zone: ± 0.000035 VCoulomb friction at actuator side: 0.256 Kgf

Modeling of Non-linear Actuator[7,8,9]

Modeling is based on servo valve equations,hydraulic pump model, accumulator dynamics andforce balance equations, which are as given below.

Servo Valve Equations:Yv(s) / i v(s) = kv / (1+ Tvs) [9] (13)

Forward Loop Filter Transfer Function [9] :

(s+2.5) (s2+ 22.62s + 1421)0.25 (s+ 0.625) (s2 + 100.5s +2527) (14)

Force Balance Equations:

Engine Gimbal Transfer Function [13] :

73.32

15910 s2 + 66667 s (15)

Feedback Filter Transfer Function: 142122.3 s2 + 533.06s+142122.3 (16)

Fig. 7 shows the schematic SIMULINK diagram ofthe non-linear actuator.

6

1

O u t 1

9 2 0 5

k l

0 . 1 1 0 2

k c

1 4 2 1 2 2 . 3

s + 5 3 3 . 0 6 s + 1 4 2 1 2 2 . 32

f i l t e r 1

0 . 0 0 4 9V o / B e

d e l t a A

d e l t a N

0 . 0 0 3 8

0 . 0 0 2 s + 1

S p o o l d y n a m i c s S e r v o v a l v e

n o n l i n e a r i t y

1 3 . 8

S e r v o a m p l r

g a i n ( k a )

S c o p e 2

S c o p e 1

S a t u r a t i o n

( s t r o k e l i m i t )

S a t u r a t i o n

( c u r r e n t )

S 1

1 4 3 .

L V D T

s e n : t y ( k p )

7 3 . 3 2

1 5 9 1 0 s + 6 6 6 6 7 s2

L / ( J e s * s + B e s )

7 3 . 3 2

L

s

1

I n t e g r a t o r

0 . 2 5 s + 6 . 2 8 s + 3 6 9 . 3 8 7 5 s + 8 8 8 . 1 2 5 03 2

s + 1 0 1 . 1 s + 2 5 8 9 . 8 s + 1 5 7 9 . 43 2

F o r w a r d f i l t e r

G c ( s )

F d

1603*sgn (u ) *sq r t ( sq r t ( ( u ) ^2 ) )

sg n ( u )

2 1 2 - 0 . 0 1 9 4 4 * u

d u / d t

D e a d Z o n e

C o u l o m b

F r i c t i o n

2 5 6

C o m m a n d

d e l t a C

0 . 0

1 / A p

0 .

1 / A p

0 .

1 / L _ 1

0 . 0

1 / L

- K - B a

1

I n

Fig 7 SIMULINK Model of Non-linear Electro-hydraulic Actuator (with nozzle dynamics)

Numerical simulations are carried out to verify thestep response & frequency response characteristics.Actuator commands of 10%, 50% and 100% areused as inputs to the control power plant and theoutputs are actuator deflection (δA) and nozzledeflection (δN)

Fig 8 - Step Response for 10% Command

The results of simulation, in terms of the step inputresponses, are presented in figures 8 to 13.

Fig 9. - Step Response for 50% Command

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Fig 10 - Step Response for 100% Command

Fig 11 – Step Response for 10% Command

Table 3 Time Response of Non-linear Actuator

Fig 12 - Step Response for 50% Command

Fig 13 - Step Response for 100% Command

Command10% 50% 100%Parameters

WithNozzle

Dynamics

WithoutNozzle

Dynamics

WithNozzle

Dynamics

WithoutNozzle

Dynamics

WithNozzle

Dynamics

WithoutNozzle

DynamicsRise time (ms) 115 115 120 130 210 220

% Overshoot 13.72 35.1 34.3 45 25.1 31.3Settling time (ms) 830 950 220 410 300 450

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Non-linear Actuator Coupled with DAPHere, the non-linear actuator model is coupled withpresent DAP designed using classical approach fora linear actuator, and the same is modeled inSIMULINK. (See Fig. 14)

1

Out1

In1 Out1

nonlinear actuatorlimit

1

0.002

ki

-0.11223z +0.112051z -1 -2

1-1.839706z +0.84052z -1 -2

-1.314151z +0.465605z -1 -2

1-0.7496074z +0.2760324z -1 -2

1.3374555z +1.4128671z -1 -2

1+0.1517240z -0.0586190z -1 -2

Zero-Order

Hold2

Zero-Order

Hold1

5.656s

s +0.02142

Transfer Fcn1

1

s

thetadot

theta

Scope1

Scope

0.75+0.75z -1

1+0.5z -1

Rate path filter

12090.265

s +153.93s+12090.2652

Rate gyro dynamics

6168.5

s +109.955s+6168.52

Position gyro dynamics

0.81

Kr

0.8

Ka

0.2417

1

z

Command

deltaC 0.75+0.75z -1

1+0.5z -1

Attitude filter

1

In

Fig 14 SIMULINK Model of Non linear actuatorcoupled with DAP Designed for Linear Actuator

An autopilot designed for a linear actuator willgive satisfactory performance with a linear actuatoronly. When this is coupled with a non-linearactuator, the step response characteristic showsunstable behavior. Here the DAP used is beingdesigned for a linear actuator, with out consideringthe non-linearities. (See Fig. 15)

Fig 15 - Step Response for 50% command

DAP Design Using LQR Approach [1,6,12]

Here the design of the DAP for the pitch loop of atypical launch vehicle is carried out using LinearQuadratic Regulator approach. The non-linearitiesare treated as operating points, for the purpose ofdesign. Slosh effects in both the tanks (fuel andoxidizer) are considered while, vehicle flexibilityand engine inertia effects are neglected. Dead zone,saturation and servo valve non-linearity areincorporated in the model in a look up table format.

Mathematical modeling [6]

Mathematical model of the launch vehicle isformulated from the force equation, momentequation and slosh equation.

Force equation:Fz = mp ax Γp + mr ax δ + Tc δ - (TF – D ) θ - Nα (17)Moment equation:IXYθ = Tc lc δ + Nα lα α - mp lp ax Γp + ( IR + mr lr lc ) δ (18)

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Slosh Equation:

Γp1 + 2 ξp1 ωp1 Γp1 + ωpk2 Γp1 =

1 ∑ F1 + (lp1 - Lp1) θ +

Lp1 m0

1 mp1 ax Γp1 - mp2 ax Γp2 - mr lr ωd

2 δc + Lp1 m0

2 mr lr ξd ωd δN - Tc + mr lr ωd

2 δN +

( (TF – D ) + Nα) θ +

(lp1 - Lp1) [ (µc - µc ωd2) δN + µα θ - µ p1 Γp1 –

µ p2 Γp2 + µe ωd2 δc - 2ξd ωd µe δN ] (19)

Modeling of the hydraulic actuator is based onthe hydraulic flow equations and the forcebalance equations, which are,Xv = -512 XA - 500 Xv + 3755 δc (20)

δN = - 4.19 δN – 3106.67 δN + 42.42 XA (21)

XA = 0.0053 XA + 2.63 XA – 192.83 δN (22)

The above mathematical model of the launchvehicle, along with electro-hydraulic actuator, isconverted to the standard state space format, usingthe standard procedures.

dx/dt= Ax + Bu y= Cx (23)

The state vector is defined as,

{x}={θ,θ,Γp1,Γp1,Γp2,Γp2,δN,δN,XA,XA,Xv}T (24)

and δc is taken as the control input vector. Further,both θ and dθ/dt are considered as the outputs fromthe system. For the present design, the system isfound to be nonlinear as well as time varying, dueto propellant consumption and therefore, fivedifferent time instants, 0 s, 40 s, 80 s, 120 s, 150 sare considered for obtaining the vehicle parametersfrom the data documents. Since, nonlinear effectslike saturation, dead zone and servo valve non-linearity are also considered, state variables XA andXV show nonlinear behaviour.

For each of these time instant, the coefficients ofmatrix A and row vector B, are computed for 10%,50% and 100% command input.

Optimal Gain Matrix Computation [6]

Optimal gain matrix for each set of A and Bmatrices is worked out using the LQR technique. Inthe present case Q and R are selected as a diagonalmatrix having the elements as reciprocal of thesquare of the maximum specified values for thestate vector variables and control variablerespectively. If any of the state or control variablevalue is obtained as more than the allowablespecification, all the elements are normalized withrespect to this large value, in order to satisfy theconstrains. This is repeated until the specificationsare satisfied and the final Q and R matrices areobtained, from which the gain vector K can becomputed from the MATLAB control tool boxfunction, ‘LQRD’. The above steps are repeated forall the five different time instants and the optimalgain vectors are computed for them

Minimal State Observer Design [6]

In the present case all the state variables are notavailable for feedback. Only the state variables θand dθ/dt are sensed in the present system usingposition and rate gyros and thus we need toestimate the rest of the state variables. This is doneusing the concept of minimum state observer. Thestate observer estimates the state variables based onthe measurements of the sensed output and controlvariables. The estimator design is done inMATLAB using Ackerman’s formulae. The statevector is partitioned into two parts xa and xb. Thestate variable xa is equal to the output y, which canbe directly measured and xb is the measurableportion of the state vector, as shown below.

xa Aaa Aab xa Ba

--- = -------------- --- + --- {u} xb Aba Abb xb Bb

xa

y = [ 1 0 ] --xb (25)

The eigen-values used in the Ackerman’s formulaeare selected as –1 for all the nine states that need tobe estimated. Gain matrix is of order 9×2, and forboth the columns same gain vectors are assumed.

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thetadot

theta

K

[0/0/ I 10]

Sum4

Sum3

Sum1

Sum

StepSaturation

Multiport

Switch

K

Ke

s

1

Integrator

K

I / Ke

gain1.mat

gain3.mat

gain2.mat

Dot Product1

Demux

Demux

Dead Zone

K

Bb - Ke.Ba

x' = Ax+Bu

y = Cx+Du

Actuator +Vehicle

K

Abb - Ke.Aab

K

Aba - Ke.Aaa

-K-

Forward

(servo amplr)

Gain

Fig 16 SIMULINK Model of DAP Using LQR

A new DAP is designed (see figure 16), using thegain matrices generated from the above analysisand it can be seen that all non-linearities areincluded in the model.

New DAP Coupled with Non-linear Actuator

Non-linear model of the electro hydraulic actuatorconsidered earlier is coupled with the DAPdesigned using the LQR approach. Simulation ofthe total system is carried out in MATLAB-SIMULINK. Numerical simulation is carried outusing fourth order Runge-Kutta algorithm with stepsize of 1 ms. Step response and frequency responsecharacteristics show a result similar to that ofactuator alone case. In the step response it isobserved that the system is stable and thecharacteristics are close to the specifications. Stepresponse characteristics like rise time, percentageover shoot and settling time are observed asincreasing with command percentage as expected.Settling time obtained for 100% command isslightly above the specification. Frequencyresponse characteristics of the system also showthat the performance is satisfactory and response isvery close to the specifications. (See Figs. 17-20).

Step response 10%command

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time(sec)

Th

eta

Fig. 17 Step Response for 10% Command

Step response- 100% command

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time(sec)

Th

eta

Fig. 18 Step Response for 100% Command

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Fig. 19 Frequency Response for 10% Command

Fig. 20 Frequency Response for 100% Command

ConclusionsThis paper brings out the dependence of the designof a typical launch vehicle Digital Auto Pilot(DAP) on the type of modeling adopted for thecontrol actuator. Depending on linear / nonlinearmodel adopted for the actuator, DAP design needsto be different. In the actuator alone case bothlinear as well as non-linear actuators givesatisfactory results. When coupled with DAPdesigned for linear actuator the simulation resultsfor linear actuator are found to be stable. When thenon-linear model of the actuator is coupled withDAP designed for linear actuator, the simulationresults show unacceptable response. The newdesign of DAP is carried out using LQR techniquein MATLAB. The performance of the nonlinearactuator, when coupled with the new design ofDAP, is verified in the simulation and responseparameters are found to match the specifications. Itis concluded that, the DAP designed for linearactuator becomes inadequate for non-linearactuator model and satisfactory performance ispossible only by a new design of DAP.

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control design of a missile autopilot, AIAA 98-4321, (1998), pp. 1209-1214.

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6. Ogata.K , Modern Control engineering ,Prentice -Hall, New Delhi,1992.

7. Sam. K. Zachariah, L40 EGC systemmathematical model in Actuator–Nozzleconfiguration based on Q-series hardware testresults., Internal technical report,VSSC/CASD/l40/14/99.

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9. Sam K Zachariah & BB Das, Modeling andanalysis of an electro hydraulic servosystem for

launch vehicle application, First NationalWorkshop in Aerospace servo system, 1998.

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AcknowledgementsThis work was carried out as part of an M. Tech.Dissertation at the Department of AerospaceEngineering, Indian Institute of Technology (IIT)Bombay, under the sponsorship from VSSCThiruvananthapuram. Authors wish to thank theVSSC and Centre for Aerospace Systems Designand Engineering, IIT Bombay, for the supportextended to this work.