Pitch Control of Flex LaunchVehicles SubirPatra

7/21/2019 Pitch Control of Flex LaunchVehicles SubirPatra

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Pitch Control of Flexible Launch Vehicle

A dissertation submitted in partial fulfillments of

the requirement for the degree of

Master of Technology

By

Subir Patra

Roll No.09301025

Under the guidance of

Prof. Hari B.Hablani

Department of Aerospace Engineering

INDIAN INSTITUTE OF TECHNOLOGY BOMBAY

June, 2011


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Dissertation Approval for M. Tech

The dissertationtitled Pitdl Control of lesible Launch Vehicle by 8umr Patra

(09301025)s approvedfordegreeofMasterofTecbnology.

Examiner

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Guide

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ecl r tion

I declare that this written submission represents my ideas in my own words and where others ideas or

words have been included, I have adequately cited and referenced the original sources. I also declare that

I have adhered to all principles of academic honesty and integrity and have not misrepresented,

fabricated or falsified any idea/data/fact/source in my submission. I understand that any violation of the

above will be a cause for disciplinary action by the Institute and can also evoke penal action ITomthe

sources which have thus not been properly cited or ITomwhom proper permission has not been taken

when needed.

cfMo \ .

SubirPatra

09301025)

,

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Certificate

This is certified that this M.Tech Project Report titled Pitch Control of Flexible Launch Vehicle by

Subir Patra is approved by me for submission. Certified further that , to the best of my knowledge the

report represents work carried out by the student.

Prof.Hari B.Hablani

Guide

iv


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Abstract

Due to the use of lightweight composites, launch Vehicles of recent times are more flexible, with their

modal frequencies lower and hence closer to the control bandwidth than earlier. This causes a

destabilizing control-structure interaction in the launch vehicle control loops. The structural modes,

therefore, as in the past, need to be considered in the design of control systems for launch vehicles. The

scope of this project is limited to the pitch control of a flexible launch vehicle in its first stage, tracking

an optimum trajectory to a desired point in space with desired velocity. First, an optimum pitch profile

of a launch vehicle is presented. Second, a pitch controller for a rigid launch vehicle is designed using

the classical control theory. Actuators, deflecting the engine nozzle, modelled as first- and second-order

dynamics are considered. A launch vehicle is modelled as a slender beam, and its modal frequencies and

shapes are determined using Ansys. The first bending mode in the pitch plane is considered in the design

of the pitch controller, and a detailed study of its interaction with the controller is undertaken. In order to

gain stabilize the mode, an unsymmetrical notch filter, tuned with the first bending mode of the launch

vehicle in the control loop, is used. Stability analysis is carried out by means of the root locus, Bode, and

Nyquist plots. Stability margins are determined over entire flight duration at an interval of 20s. Based on

the specifications of gain margin, phase margin, and stability margin, a zone of exclusion that satisfies

these specifications is drawn in a Nyquist plot to show clearly the stability of the designed controller.

Step responses are examined at each 20s interval of the flight time to verify that the time-domain

specifications (percentage overshoot, rise time, settling time) are met.

Keywords: Flexible launch vehicle, notch filter, gains stabilization.

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Table of contents

List of figures ix

List of tables xiii

Nomenclature xiv

1. Introduction 1

1.1 Motivation 1

1.2 Project objective and scope of the work 3

1.3 Organization of report 4

2. Gravity Turn: Modelling and simulation 5

2.1 Dynamics of gravity Turn 5

2.1.1 Gravity turn Trajectories: Force and Acceleration-Normal frame 6

2.1.2 Analytical solution of tangential velocity vs flight path angle 10

3. Optimal Pitch Profile 13

3.1 Launch vehicle trajectory optimization 13

3.2 Optimal ascent trajectory 14

4. Pitch Control of Launch Vehicle Rigid Body Dynamics with FirstOrder Actuator 15

4.1 Rigid body model of launch vehicle 15

4.2 Details of parameter variation staring from launch to completion of first stage 18

4.3 Determination of and in the first stage of a flight 20

4.4 Time Slice Approach 21

4.5 Gain design 22

4.6 Design requirement 23

4.6.1 Stability margins 23

4.7 Ramp response 27

4.7.1 Ramp response (at flight time , t = 20sec and

=0.2.) 27

4.7.1.1 Tracking error rate from Simulink model 28

4.7.1.2 Tracking error from analytical calculation 29

4.7.2 Ramp response (

= 0 and flight time = 20sec) 30

4.7.3 Ramp response (at flight time t=100sec) 31

4.7.3.1 Tracking error rate from Simulink model 32

4.7.3.2 Tracking error rate from analytical calculation 33

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4.8 Nyquist plot with zone of exclusion (t=20sec) 33

5. Pitch Control of Launch Vehicle with Second-Order Actuator dynamics 34

5.1 Launch vehicle rigid body dynamics 34

5.2 Gain design 36

5.2.1 Gain Schedule 36

5.3 Loop transfer function 38

5.4 Input parameters (at time 20sec) 38

5.5 Close loop poles 42

5.6 Ramp response (at flight time t=20sec) 42

5.6.1 Tracking error rate from Simulink model 43

5.7 Stability margin (by zone of exclusion) 44

5.8 The updated feedback gains 45

5.9 Time response Analysis 45

5.9.1 Design specifications 45

5.10 Frequency domain analysis 47

5.10.1 Variation in the Phase margin 47

5.10.2 Variation in the Gain margin 48

6. Pitch Control of Flexible Launch Vehicle 49

6.1 Flexible body dynamics 496.2 Controller Design Using Gain Stabilization 51

6.2.1 Controller design to track commended pitch rate 52

6.3 Bending Frequency determination 52

6.3.1 Mode Shapes 52

6.4 Generalized mass 53

6.5 Slope 54

6.6 Design specification 55

6.7 Loop transfer function 56

6.7.1 Input to launch vehicle autopilot 56

6.8 Stability analysis : Nyquist Plot 60

6.9 Stability analysis : Bode Plot 60

6.10 Loop transfer function with notch filter 63

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6.11 Notch filter location 63

6.12 Nyquist Plot with Notch filter 66

6.13 Closed-loop step response 68

6.14 Closed-loop Ramp response 70

6.14.1 Tracking error rate (deg/sec) 71

6.15 Time response Analysis 72

6.16 Stability margins 73

6.16.1 Gain Margin 73

6.16.2 Phase margin 74

6.17 Variable frequency notch filter 75

6.18 Simulation results general time varying pitch command 75

6.18.1 Pitch Command 75

6.18.2 Commanded input (thetac) and output 76

6.18.3 Tracking error 76

6.18.4 Actuator deflection 77

6.19 Simulation result with a stair-like pitch rate command 77

6.19.1 Commanded pitch rate and actual pitch rate 78

6.19.2 Commanded pitch and actual pitch 796.19.3 Tracking error 79

6.19.4 Actuator deflection 80

7. Conclusion and Future Work 81

7.1 Conclusion 81

7.2 Future work 81

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List of figures

Figure No Title

Page

No

Fig1.1 Flexibility of launch vehicle and sensor location 2

Fig.1.2 Rigid body response and flex response 2

Fig.2.1 Force acting on satellite booster (T=Thrust, D =Drag) 5

Fig.2.2 Launch vehicle boost trajectory,is the flight path angle 6

Fig.2.3(a-b-c-d) Gravity turn trajectory 8

Fig.2.3(e) Pitch profile 9

Fig.2.3 (f) Gravity turn trajectory 9

Fig.2.4 The gravity turn 11

Fig.3.1 Mission Pitch Profile 14

Fig.4.1 Force acting in Pitch Plane 15

Fig.4.2 Launch vehicle autopilot for a simplified Rigid Body 17

Fig.4.3 Variation of Thrust profile (Tc) in First Phase of Flight 18

Fig4.4 Mass centre of launch vehicle 18

Fig.4.5 The aerodynamic load per unit of angle of attack 19

Fig.4.6 Moment of Inertia (Iyy ) vs Flight time 20

Fig.4.7 Control moment coefficient 20

Fig. 4.8 Aerodynamic moment coefficient 21

Fig.4.9 Gain Schedules for forward gain 22

Fig.4.10 Gain Schedules for feedback gain 22

Fig. 4.11 Zone of exclusion 24

Fig. 4.12 Root Locus for Simplified autopilot (at time t=20 sec) 25

Fig. 4.13 Bode plot of the open loop launch vehicle 25

Fig. 4.14 Step response of the system 26

Fig.4.15 Enlarged view of the step response in steady-state 26

Fig.4.16 Close-loop ramp response( = 0.2) 27

Fig.4.17 Commanded pitch rate (c )and actual pitch rate () 28

Fig.4.18 Enlarged view of the steady-state tracking of commanded pitch rate (c ) 29

Fig.4.19 Ramp response ( = 0) 30

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Fig.4.20 Ramp response (t=100 sec) 31

Fig.4.21 Commanded pitch rate (c )and actual pitch rate ()

(at flight time t=100sec) 32

Fig.4.22 Enlarged view of the steady-state tracking of commanded pitch rate (c )

Fig.4.23 Nyquist plot with zone of exclusion 33

Fig. 5.1 Launch vehicle autopilot with 2nd order actuator 34

Fig.5.2 Gain Schedules for forward gain 36

Fig.5.3 Gain Schedules for feedback gain 37

Fig.5.4 Launch vehicle autopilot with 2nd order actuator and integrator 37

Fig. 5.5 Step response of rigid body system to a pitch step command (at t=20sec) 39

Fig.5.6 Bode plot of rigid body system (at t=20 sec) 39

Fig. 5.7 Step response of rigid body system to a pitch step command 40

for updated gain(at t=20sec)

Fig.5.8 Bode plot of rigid body system for updated gain (at t=20 sec) 40

Fig.5.9 Root locus plot of rigid Launch vehicle 41

Fig.5.10 Enlarge portion of root locus showing close loop poles 41

Fig.5.11 Close-loop ramp response 42



Fig.5.14 Nyquist diagram with zone of exclusion 44Fig5.15 Rate feedback gain 45

Fig.5.16 Variation in the overshoot during atmospheric flight of 46

the Launch vehicle for designed autopilots

Fig.5.17 Variation in the Settling time during atmospheric flight of 46


Fig.5.18 Variation in the Rise time during atmospheric flight of 47


Fig.5.19 Variation in the Phase margin during atmospheric flight of 47


Fig.5.20 Variation in the Gain margin during atmospheric flight of 48


x


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Fig.6.1 Launch Vehicle autopilot 51

Fig.6.2 Bending mode (1st and 2nd bending mode) 52

Fig.6.3 Bending frequency variation with time 53

Fig.6.4 Generalised mass of launch vehicle in 1st phase of flight 54

Fig.6.5 Slope changes in first phase of flight in sensor location 55

Fig.6.6 Launch vehicle auto pilot with notch filter location-1 57

Fig.6.7 Launch vehicle auto pilot with notch filter location-2 57

Fig .6.8 Root locus of flexible launch vehicle (uncompensated) 58

Fig.6.9 Flexible response & rigid body response (at t=20 sec) 59

Fig.6.10 Step response with flexibility (at t=20sec) 59

Fig.6.11 Nyquist plot of the controller without the notch filter, and zone of exclusion 60

Fig.6.12 Bode plot of flexible launch vehicle (uncompensated) 61

Fig. 6.13 Notch filter poleZero pattern 62

Fig.6.14 Bode plot of notch filter 63

Fig.6.15 Step response for different Place of Notch filter in the control loop 64

Fig.6.15 Frequency Bode magnitude plot of the gain stabilised system 65

Fig.6.16 Bode plot of flexible launch vehicle (uncompensated) 65

Fig.6.17 Frequency Bode magnitude plot of the gain stabilised system 66

Fig.6.18 Nyquist plot of the controller with the notch filter, and zone of exclusion 66

Fig.6.19 Root locus plot of flexible launch vehicle 67Fig6.20 Root locus plot of the flexible launch vehicle 68

Fig6.21 Compensated flexible mode response 68

Fig6.22 Compensated rigid body response & flexible mode response 69

Fig.6.23 Step response of gain stabilized system 69

Fig.6.24 Ramp response of a gain stabilized system 70



Fig.6.27 Variation in the overshoot during atmospheric flight of 72


Fig.6.28 Variation in the Rise Time during atmospheric flight of 72


xi


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List of TablesTable

No Title

Page

Number

Table1 Final conditions 13

Table2 Stage data for a launch vehicle 13

Table3 Initial conditions (Kourou Launch site for European space Agency) 13

Table4 Tracking error rate (from Matlab) 28

Table5 Tracking error rate (from Matlab) (for = 0) 30

Table6 Close loop poles 42

Table7 Tracking error rate (from Matlab) (at flight time t=20s) 43

Table8 Input for modal analysis 52

Table.9 Tracking error rate (from Matlab) (at flight time t=20s and = 0) 70

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Nomenclature

Perturbation angle in Pitch

L Aerodynamic force

Control location

Actuator deflection

Angle of attack

Control thrust

Mass of engine

Moment of inertia-pitch

Le Distance between engine CG and gimbal point

Actuator damping Actuator natural frequency

Amplifier gain

Engine servo gain

Rate gyro gain

Integrator gain

Distance of origin of body axis system to engine swivel point

Distance from centre of pressure in pitch plane to origin of body axis system.

Total mass of launch vehicle

Generalized coordinate of ith

bending modes

Generalized force of ith

bending mode in (pitch) plane

Generalized coordinate of ith

bending modes

Laplace operator

Time

Command signal to rocket engine.

1 Damping ratio of 1stbending mode

Damping ratio of actuator

Attitude commanded angle

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Control moment coefficient

Aerodynamic moment coefficient

NLg

1 Negative slope of 1stbending mode in pitch plane

1 undamped natural frequency of 1stbending mode.

Undamped natural frequency of actuator

xv


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1

Chapter1

Introduction

1.1Motivation

Launch vehicles have very complicated dynamic characteristics due to structural vibrations of

their slender body, fuel sloshing, aerodynamic effects and engine gimbal dynamics. However,

launch vehicles follow simple trajectories determined in advance. The guidance and control

systems of a launch vehicle act together for the vehicle to fly a path, taking it to some desired

terminal conditions. The vehicle is designed to maximize the payload for a given takeoff

weight, and the inert weight of the structure is reduced to a minimum. The trajectory is

designed to cause minimum aerodynamic load in atmospheric phase. The trajectories have

dispersion due to imperfections such as variation of thrust-time curve, aerodynamic

coefficients, autopilot errors, and so forth. These all need not be to be corrected by the

guidance system in the atmospheric phase, because once the vehicle is outside the atmosphere

there is sufficient time for correcting the trajectory. Hence the speed of response is not high

in the atmospheric phase of the flight.

Flexibility must be considered by the control designer if the lowest frequency of the launch

vehicle vibration is less than about six times the desired control bandwidth. Otherwise there

is a possibility that this mode will be destabilized by the control system (the control effort

spills over outside the control bandwidth and destabilizes the vibration mode). Launch

vehicles have bending modes that can be excited by control motion. In one experiment, this

interaction manifested itself as a servo-elastic instability on a test bench when a feedback

loop was closed from a gyro in the nose to the control surface at the tail. The inertial force

associated with the controls excited the first bending mode of the launch vehicle, and a sensor

feedback designed for the rigid launch vehicle had a wrong sign. This is a classic case of thesensor separated from the actuator by compliance.

The flexibility effect is incorporated in the dynamic model for the control system design due

to following reasons

i) In a flexible launch vehicle, inertial navigation system measures the attitude and angular

velocities of the deflection as well as the rigid body motion, and feed these signals back to the

control loop. If the bending vibration frequencies are near the control system frequency,


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2

feeding back of the deflection degrades the control system stability and, in the worst case,

makes it unstable. Moreover, the interaction of control forces with elastic deformations could

cause undesired excitations, leading to resonance.

Fig1.1 Flexibility of launch vehicle and sensor location [19]

The main objective of the pitch controller is to control rigid body pitch angle (

R) shown in

Fig.1.1. For the case of flexible launch vehicle the rigid body pitch angle alone cannot be

sensed. The sensed angle (S) by the sensor, located in the front portion of the launch vehicle,as shown in Fig.1.1,is equal to the sum of the rigid body angle (R ) and the localdeformation angle(F).The anglesR ,F and S versus time are shown in Fig.1.2, and = +

Fig.1.2 Rigid body response and flex response [19]


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The feedback signal will thus be based not only on the rigid vehicle state but also on

contribution from the vehicle flexibility. Thus structural interaction is caused by the control

loop through the actuator and the sensors. This control-structure interaction could cause

divergent oscillations leading to a structural failure of the launch vehicle. So the control

system design need to ensure that control- structure interaction is stable and does not lead to

diverging oscillations.

ii) When the launch vehicle deforms, the local angle of attack along the length of the vehicle

changes, which introduces additional aerodynamic load on the launch vehicle, increasing the

bending moment along the launch vehicle. Thus aeroelasticity is introduced in the control

system.

1.2 Project objective and scope of the work

The principal objective of this project is to design a pitch control system of a flexible launch

vehicle. A prerequisite for designing a launch vehicle autopilot is its mission profile,

reference trajectory, and overall vehicle configuration. Here the launch vehicle is assumed to

be a continuous beam steered along a prescribed trajectory. Aerodynamic, propulsive and

gravity forces are considered to determine the reference trajectory.

The vehicle dynamics changes continuously during the flight. A launch vehicle is a time-

varying parameter system. To use the classical frequency domain techniques the system is

converted to a fixed parameter system and it is assumed that the vehicle parameter remain

constant over a short period of time. This is called the time slice approach. This work focuses

on the first stage of flight. In the flight stages other then the first, the launch vehicle length

becomes smaller so the flexible frequency is greater than the six times of the control

bandwidth; hence it does not affect the control system so appreciably.

As stated above, we have used the time slice approach to design the autopilot for pitch

control. Using this approach we design an autopilot that will:

i) Stabilize the vehicle

ii) Provide desired speed of response to guidance command, and

iii) Provide adequate stability margin which shows a good performance during the flight.

The efforts made here to design a launch vehicle divide into three phases:


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4

i) An optimized pitch profile of a representative launch vehicle is computed.

ii) A rigid body analysis is carried out to determine the performance characteristics of the

control system.

iii) The flexible body analysis is carried out to determine performance characteristics by

incorporating the first bending mode in the dynamics model for the first stage of flight.

1.3 Organization of Report

In the beginning of this project work, literature survey and preliminary planer dynamic model

and simulation were done to understand the gravity turn trajectories. This is summarized in

Chapter2. In Chapter3, an optimal pitch profile of a launch vehicle is presented. Controller

design of a rigid launch vehicle in frequency domain is presented in Chapter4. In this chapter

a nozzle dynamics is modelled as first order. The gain schedule for the first stage of flight is

obtained by taking appropriate values of time-varying parameters of the launch vehicle at

various flight instants. In chapter5, the controller design of a rigid launch vehicle is carried

out by modelling actuator dynamics as second-order actuator including engine-moment-of-

inertia. In the Chapter6, we consider the flexibility of the launch vehicle in the control system

design. We have considered the first bending mode in the autopilot design. In order to gain

stabilize the mode, an unsymmetrical notch filter tuned with the first bending mode of the

launch vehicle in the control loop is used. Stability analysis is carried out by means of the

Root locus, Bode Plots, and Nyquist plots.


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5

Chapter 2

Gravity Turn: Modelling and Simulation

2.1 Dynamics of gravity turn [17]

Since the performance of a launch vehicle depends on the amount of fuel it carries, the only

mass that can be reduced is its structural mass. Further, because the launch vehicle must be

able to withstand high launch loads, which are mostly axial as shown in Fig.2.1, its transverse

strength is sometimes sacrificed to gain longitudinal strength and to decrease structural mass.

Velocity

v

Drag, D

Thrust, T mg

Fig.2.1 Force acting on satellite booster (T=Thrust, D =Drag) [Wiesel, W .E, McGraw-

Hill]

Launch vehicles are thus weak in the transverse direction; hence their trajectory is designed

to pass through the atmosphere at zero angle of attack. It follows that thrust vector must be

aligned with the velocity vector of the vehicle at all times during the flight, as shown in

Fig.2.1. Also, during the flight to a designated position vector in space orbit, the vehicle

must obviously be rotated from its vertical position of launch to horizontal position at burn

out, with a desired final velocity. This transpires through dynamics automatically by what is

known as gravity turn. To analyze this motion, we need equations of motion in tangent-

normal coordinates.


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2.1.1Gravity turn Trajectories: Force and Acceleration-Normal frame [17]

Velocity, vut= unit vector tangent to the trajectoryun= unit vector normal to the trajectory

Centre of mass Local horizonDrag, D

Thrust, T mg

h

Y C

(Trajectorys centre of curvature)

X

Fig.2.2 Launch vehicle boost trajectory,is the flight path angle [Curtis, H., OrbitalMechanics for Engineering Student, Elsevier, 2007, p.552]

Satellite launch vehicle forces during powered ascent is illustrated in Fig.2.2, where

T= thrust produced by the nozzle at the base acting along the vehicles longitudinal axis

aligned with or vD=aerodynamic drag force, opposite to v, D=

1

2 a v2ACD , (2.1)


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where,a = atmoshphere density , A = frontal area of vehicle, CD = coeff. of drag,mg = gravitational force along the vertical.

Force along u

t: T

D

mgsin

Force along un : mgcoswhereis the flight path angle, the angle of v relative to the local horizonAcceleration

Tangential acceleration: at=dv

dt (2.2)

Normal acceleration:For flat earth an = v (2.3)For spherical earth an = v + v2RE +h cos (2.4)

RE = spherical earth radius.

Newtons law F=maexpressed in the osculating plane, along utand un

Tangential:

dv

dt =

T

m D

m gsin (2.5)Normal acceleration: = (2.6)where , is given above.Hence, the flight path angle ()is governed by

v = g v2RE + h

cos (2.7)

Down range distance:x = RERE +h

vcos ( 2.8)h = vsin (2.8)

Variation of g with altitude

=

1 + 2 (2.9)


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Numerical methods are used to solve Eqs. (2.5-9).While doing so, one must account for the

variation of thrust, mass, atmospheric pressure, gravity and. A numerical example of thisgravity turn is shown below. However, in this example, all variations just stated are not

considered, nor is the atmospheric drag considered. In this simulation, the initial flight path

angle at t=0 is taken to be 850 and the desired velocity is 7.71 km/sec corresponding to a

satellite in a circular orbit at an altitude of ~327km, explained more fully later. From

Fig.2.3b and Fig.2.3f, however, we see that the launch vehicle has achieved a higher

altitude~500km. This is because here we consider initial perturbation angle as = 850

Fig.2.3 Gravity turn trajectory (a-b-c-d)


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Fig.2.3 Pitch profile (e)

Fig.2.3 (f) Gravity turn trajectory [17]


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arbitrarily to initiate a gravity turn trajectory. To get a desired altitude with a desired velocity

we have to choose0in such a way that the desired altitude is achieved. This is in fact atwo -point boundary value problem.

As stated earlier, to keep the transverse lifting load close to zero, the angle of attack is

controlled to be close to zero by keeping the launch vehicle longitudinal axis aligned with the

instantaneous velocity vector. At lift off, the launch vehicle is vertical and = 900. Afterclearing the tower and gaining speed, the vernier thrusters or gimballing of the main engines

produce a small, programmed pitchover, establishing an initial flight path angle 0slightly 1)

For flat earth, the ,Eq. (2.7), is rewritten as = g

vcos (2.11)

We eliminate time t by dividing v equation (2.10) with equation (2.11); the integration ofthe subsequent equation of

yields

logv = logcos 01+t a n

2

1 tan2 + constant (2.12)


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11

Because1+sin1sin = 1+tan

2

1tan2

2 (2.13)

Eq. (2.12) solution can also be written as

vcos 1 + s i n1 sin

o2

= constant (2.14)

Note that at t = tf(the final time), = 0because the velocity is horizontal then; therefore,Eq.(2.14) yields: constant = v(tf)= vf = The desired velocity.

The normalized velocity (v

vf)is plotted in Fig.2.4 for o=2, starting with the initial value at

t = 0 = 85 degrees chosene earlier for Figs.2.3a-f. Indeed, Fig.2.4 is the normalizedversion of Fig.2.3a.

Fig.2.4 The gravity turn [17]

The change in the flight path angle from 900to 850is caused by inducing a small pitch rate

of the launch vehicle. We see in Fig.2.3f that, with a small initial perturbation of

from 900after a short vertical ascent, the path angle bends over naturally under the


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influence of gravity as the launch vehicle is accelerated. Clearly, the launch vehicle must also

be pitched at the same rate as [Fig.2.3d], Eq. (2.11), and the resulting velocity profile mustyield the desired altitude and the desired horizontal velocity at timet = tf. For a circular orbit

of a satellite injected by a launch vehicle, the final altitude and the final velocity are related

as:

vf = RE +hf (2.15)where = 398600.4415km 3

s2= earths gravitational constant. For example, for a satellite

orbit at a low altitude ofhf = 327.3298 km, vf = 7.71km

s. The corresponding velocity,

altitude, downrange, flight path angle and the rate profiles are illustrated in Fig.2.3,obtained

by integration of motion equations (2.7)-(2.8). We observe in Fig.2.3a that the desired

velocity is achieved; however, as noted earlier, the final altitude, 500 km, in fig.2.3b and

Fig.2.3f is higher then the desired altitude 327.3298 km. This is a consequence of selecting

the initial = 850arbitrarily.To determine the pitch profile that achieves desired velocity atdesired altitude is a two-point boundary value problem, related to maximizing the payload

(satellite) mass for a specified horizontal velocity at fixed terminal time subjected to the

terminal constraint of a desired altitude. One simple version of an optimal pitch program

under some simplifying condition is given by [7]

= arctantano ct (2.16)where, the constants 0 and c are chosen to satisfy the terminal conditions. This optimalsolution is referred to as a linear pitch steering law. For details see Ref.7.


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13

Chapter -3

Optimal Pitch Profile

3.1 Launch vehicle trajectory optimization

The main objective of the launch vehicle is to place a satellite in an orbit satisfing the

requirements of a particular mission, a geosynchronous transfer orbit (GTO) for instance. For

one particular GTO, a set of, final conditions are:

Final conditions

Apogee 42161 km radius

Perigee 300 km altitude

Latitude 0

Path angle from North 90 deg

Table1.Final conditions [7]

The rocket Nozzle area for the three stages estimated as 2.96, 0.78 and 0.06 sq.m respectively

[7].

Stage Structure(tonnes) Propellant(tonnes) VaccumThrust(kN) Burn time(sec)

1 17.5 157 4*748 138

2 4.325 34 760 130

3 1.2 10.7 62 735

Table2. Stage data for a launch vehicle [7]

Numerical method based on optimal control theory and constraint optimization algorithms is

applied to compute an ascent trajectory [7].

Initial Condition

Longitude 307.23deg.

Latitude 5.43 deg.

Radius 6378.14 km.

Air velocity 0.0 km.

Flight path azimuth from north 90 deg.

Table3.Initial conditions (Kourou Launch site for European space Agency)


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3.2 Optimal ascent trajectory [14]

The optimal pitch profile for the launch vehicle (Ariane-40) with above parameters [table 1-

3] is shown below in Fig.3.1. This plot is taken from Ref.14. This pitch profile is used in

Chapter -6 as an input to a Simulink model.

Fig.3.1 Mission Pitch Profile


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Chapter 4

4. Pitch Control of Launch Vehicle Rigid Body Dynamics with FirstOrder

Actuator

4.1 Rigid body Model of Launch Vehicle

Here only the pitch-plane dynamics is studied. Several assumptions are made that allow for

simplification of the equations. The equations of motion of a launch vehicle are complicated

by the fact that the vehicle has time-varying mass and inertia. There can also be relative

motion between various masses within the vehicle, such as fuel sloshing, engine gimbal

rotation, and vehicle flexibility. The derivation of equations of motion neglects nozzle inertiaand sloshing effect.

Fig.4.1 Force acting in Pitch Plane [4]

Useful relationships

Flight-Path Angle: =-Dynamic Pressure (

): =

1

2

2

Aerodynamic Forces: =, =


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The system dynamics, neglecting the nozzle inertia and sloshing effect, can be written in the

form [2]

q

= l

Iyy +

Tc lc

Iyy +

Cmqd

2uq d

Iyy

(4.1)

In terms of notations used in Ref.2, Eq. (4.1) can be written more compactly as

(using = WV

)

q = mw w + m + mqq (4.2)where mq =

Cmqd

2uq d

Iyy , mw =

lIyy

, m =Tc lc

Iyy

= WV

and for = and q , the perturbation model, Eq.4.2, can be written as

= mw V + m + mq = + mq + c (4.3)The aerodynamic damping provided by the mq term is usually very small and can be

neglected in the first cut design [2]. The vehicle transfer function is then given by

= P =

cs2 (4.4)

where c = Tc lcIyy = control moment coefficient. = LlIyy =aerodynamic moment coefficient.

Here we consider a case where the control is provided by secondary injection thrust vector

control where effect of inertia of control effectors can be neglected and the actuator is

considered as a first order actuator, i.e.,

+ Kc = Kcc (4.5)Taking Laplace transform with zero initial states, we obtain the transfer function relating and as followsc = G =

Kc

s + Kc (4.6)


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Fig.4.2 Launch vehicle autopilot for a simplified Rigid Body

The close-loop transfer function of the signal flow diagram shown in Fig.4.2 is given by

C =KsGP

1+KsGP(1+Kr s) (4.7)

=cKs Kc

s3 + Kc s2 +

cKs KrKc

s + Kc (

cKs

)

(4.8)

=K

s + P(s2 + 2scc + c2 ) (4.9)

Equating coefficient of like powers ofs, we get

cKs =

c2 (Kc 2cc)

Kc+

(4.10)

Kr =

c2 + 2cc

Kc

2cc

+

cKsKc (4.11)

The gain schedule for the entire flight duration is obtained by taking appropriate values of

andc.


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4.2 Details of parameter variations starting from launch to the

completion of the first stage

The variations of various parameters of the model are shown below

Fig.4.3 Variation of Thrust profile ()in First Phase of Flight [11]The mass of a launch vehicle changes due to burning of propellant during the flight of launchvehicle, mass centre of a launch vehicle changes its location with flight time [Fig.4.4]. The

change of mass centre of a launch vehicle for the first two stages is shown below. Here centre

of gravity is determined from the nose of the launch vehicle.

Fig4.4 Mass centre of launch vehicle

1350000

1400000

1450000

1500000

1550000

1600000

0 20 40 60 80 100 120 140 160

Thrust(N)

Flight Time(sec)

0

5

10

15

20

25

30

35

0 50 100 150 200 250 300

Xcg(m)

Flight Time(sec)


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The aerodynamic load per unit of angle of attack for a launch vehicle

Fig.4.5 The aerodynamic load per unit of angle of attack [11]

Aerodynamic load data at three instants of time (0.5 sec, 72 sec and 152 sec) are taken from

Ref.11. At any other time in this time interval the load is determined by interpolation

[Fig.4.5]. Total mass of launch vehicle and its cg position changes during the flight time. This

changes moment of inertia of the launch vehicle during the flight time. Here we have

determined the varying moment of inertia during the first stage of a launch vehicle [Fig.4.6].

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

1000000

0 20 40 60 80 100 120 140 160

Lalpha(N/rad)

Flight Time(sec)

Aerodynamic load per unit angle of attack


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Fig.4.6 Moment of Inertia ()vs Flight time

4.3 Determination of and in the first stage of a flightc = Tc lcIyy = control moment coefficient. = LlIyy =aerodynamic moment coefficientSee their time variation in Figs.4.7-4.8

Fig.4.7 Control moment coefficient

0

200000

400000600000

800000

1000000

1200000

0 20 40 60 80 100 120 140 160

MomentofIn

ertia(Kgm

2)

Flight Time(sec)

0

1

2

3

4

5

6

7

8

9

10

0 20 40 60 80 100 120 140 160

c(1/sec

2)

Flight Time(sec)


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Fig. 4.8 Aerodynamic moment coefficient [11]

4.4 Time Slice Approach

The simplification used to design of autopilot is called time slice approach .the various

parameters of the vehicle and trajectory continuously changes during flight .the vehicle

related parameter are mass ,moment of inertia and centre of gravity and the trajectory related

parameter are vehicle velocity, attitude and flight path angle .

Thus vehicle dynamics continuously changes during the flight and makes it a time varying

parameter system. To enable use of classical frequency domain techniques, this system is

converted to fixed parameter system by assuming the system parameter remain constant over

a short period of time .thus parameter of vehicle dynamics are worked out for short segment

of trajectory to cover the entire trajectory. A small perturbation motion is then considered

about the nominal position and control system parameter or gains are designed to have

satisfactory and well damped response for the perturbed motion .this study is repeated at

various segment of trajectory and suitable gains at various segment are obtained. This gives a

scheduled of control gain to be used during flight. This approach used to design flight control

system and give quite satisfactory results.

Here efforts to be made to use this approach for first stage of flight for a typical launch

vehicle.

0

0.5

1

1.5

2

2.5

3

3.5

0 20 40 60 80 100 120 140 160

alpha(1/sec

2)

Flight Time(sec)

Angular accleration Per unit angle of attack


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4.5 Gain design

The gain schedule for the entire flight duration can be obtained by taking the appropriate

values of

and

at various flights instant and solving for Ks andKr .

i) Design bandwidth ()of 6 r/s and damping coefficient (c) of 0.65 has been chosen.ii) The First order Actuator =30 rad/sec [2].

Figs. (4.9-10) shows the variation of Gains (Forward gain & Feedback gain) of a typicallaunch vehicle.

Fig.4.9 Gain Schedules for forward gain

Fig.4.10 Gain Schedules for feedback gain


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4.6. Design requirement

4.6.1 Stability margins

Specifications on the stability margins need to take care of following [2]

i) Approximations in the analytical model used for vehicle and other control elements inside

the loop such as actuator, sensors etc.

ii) Uncertainties of parameter values of the model.

iii) The autopilot performances such as damping ratio overshoot etc.

The following stability margins for nominal case have good performances during flight trials

[2].

1. Gain margin >6dB

2. Negative gain margin 300

4. Stability margin R=

1 + G H

min > 0.5

5. Attenuation for higher modes>10db

Based on these specifications, Ref.2 shows an elliptical zone of exclusion in the GH plane.

The GH plot of the controller should not enter this zone to ensure that stability margin

specifications are not violated.

The method of obtaining the equation for zone of exclusionboundary is illustrated bellow

for the following specifications:

Positive gain margin =6dB

Negative gain Margin = -6dB

Phase margin =300

The boundary of zone of exclusion for the above specification is given by [Ref.2]:

x2

0.752+ y

2

0.5852= 1


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Fig. 4.11 Zone of exclusion

Time-domain characteristics of rise time less than 1 second, settling time of less than 3

seconds, percentage overshoot less than 20%, and with a steady state error of less than 2%,

for controlling the pitch angle.

Autopilot gains and launch vehicle parameters are initially set for the flight time t=20sec

from Fig. (7-8-9-10).

6.5 0.265 -0.2 4.3

The close-loop poles of the system with gain Ks = 6.5 are shown with cross marks on the

root locus. We see the close-loop poles are in the left half plane of the s-plane.


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Fig. 4.12 Root Locus for Simplified autopilot (at time t=20 sec)

The gain and phase magnitude (Bode) plot of a rigid launch vehicle and the step response to a

pitch command are shown in Fig.4.13 and Fig.4.14 respectively. While phase margin of the

controller is 50 degrees, the gain margin is infinite.

Fig. 4.13 Bode plot of the open loop launch vehicle


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Fig. 4.14 Step response of the system

We can see from the Bode plot and step response plot that the design meets the specifications

[2]. The enlarged view of the steady-state portion of the step response is shown in Fig 4.15

below.

Fig.4.15 Enlarged view of the step response in steady-state


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From Fig.4.15, the steady state error can be found as 7E-3. Now we will determine the

steady- state error analytically for comparison.

Considering the autopilot schematic of Fig.4.2 the steady sate response of the system to a

step commandc is given by [11]

= 11

(4.17)

For = 0.2 sec2, c = 4.3sec2 & Ks = 6.5we get ss = 0.99289degreeThen, the steady state error ss error = (1 ss ) = 7.104E 3 degree.The higher the value of Ks, the smaller the steady state error. There is limit to the permissible

value of . So we need to null the error by other means; this is done by an integral control inthe forward loop as shown in the next chapter.

In the next chapter we will replace the first-order actuator with a second-order actuator. The

inertial force produced by engine gimballing will then be in the rigid body equation. We use

an integral gain in the forward loop to make steady-state error zero.

4.7 Ramp response

4.7.1 Ramp response (, = = .)

Fig.4.16 Close-loop ramp response ( = .)


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The ramp response to a pitch ramp command is shown in Fig.4.16. Here the tracking error

grows continuously. From the ramp response plot, we determined the following tracking

error rate:

Tracking error(deg)

at t=3ses 0.264

at t=8sec 0.30

Tracking error rate(o/sec) (0.3-0.264)/5=7.2E-3 deg/sec.

Table.4 Tracking error rate (from Matlab)

4.7.1.1 Tracking error rate from Simulink model

Fig.4.17 Commanded pitch rate ( ) and actual pitch rate ( )The enlarged view the of steady-state portion of the pitch rate is shown in Fig 4.18 below,


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Fig.4.18 Enlarged view of the steady-state tracking of commanded pitch rate ( )4.7.1.2 Tracking error rate from analytical calculation

From Ref.2 we can write tracking error rate ()=

For c = 10s , = 0.2 sec2, c = 4.3sec2and Ks = 6.5 we get tracking error rate( ) =7.1047957E 3(o/sec) which agrees with the simulation results above. Because thisrate error is small, the grown in the tracking error is significant. However, as flight

continues,

becomes larger and the error growth also become significant, as will be shown

in Sec.4.7.3.


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4.7.2 Ramp response ( = = )

Fig.4.19 Ramp response ( = )

We determine the tracking error rate from Matlab plot as below,

Tracking error(deg)

At time t=3sec. (4-3.75)=0.25

At time t=8 sec. (8-7.75)=0.25Tracking error rate( /Sec) 0

Table.5 Tracking error rate (from Matlab) ( = )Tracking error analytically [2]:

ytrackng ,error = Kr = 0.25 1 = 0.25 deg.From Ref.2 we can write tracking error rate (

)=


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As = 0,tracking error rate ( ) = 0, as found above via simulation.

4.7.3 Ramp response (at flight time t=100sec)

Autopilot gains and launch vehicle parameters are now set for the flight time t=100sec from

Fig. (7-8-9-10).

4.5 0.28 3.2 sec2 6.1sec2

Fig.4.20 Ramp response (t=100 sec)

From the Fig.4.20 we can see that tracking error grows significantly with time. From the

Simulink model commanded pitch rate ( c) and actual pitch rate ( ) are plotted together.


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4.7.3.1Tracking error rate from Simulink model

Fig.4.21 Commanded pitch rate ( ) and actual pitch rate ( ) (at flight time t=100sec)

Fig.4.22 Enlarged view of the steady-state tracking of commanded pitch rate ( )


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Chapter-5

5. Pitch Control of Launch Vehicle with Second-Order Actuator dynamics

5.1 Launch vehicle rigid body dynamics

In launch vehicle the control torque is generated by gimballing the engine or by deflecting

the flexible nozzle, so we need to consider actuator as a higher order transfer function

(second and third order) transfer function. In the previous case we have considered first order

actuator.

The dynamic equation considering the influence of engine inertia is written by including the

as follows [2],Iyy = Tc lc + L l + me Lelc (5.1)

In transfer function form it can be written as

= G = c

meLe

Tc

(s2 +Tc

meLe)

(s2 ) (5.2)

write, Kd = c me LeTc

G=c +Kd s2

s2 (5.3)

The actuator transfer function written as [2],

c = P =

a2s2 + 2aas + a2 (5.4)

Fig. 5.1 Launch vehicle autopilot with 2ndorder actuator


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By using the procedure in [chapter-3], we get a characteristic equation for this system as

follows [2]

s4 + 2aa + KsKr Kda2s3 + a2 + KsKda2s2 + KsKra2c 2aas+ cKs a2 = 0 [Ref. 2] (5.6)

We can consider this characteristic equation as a product of two quadratic equations as

s2 + 2ccs +c2s2 + 2s + 2 = 0 (5.7)Here, c and c define the desired pole location of the system and and define theremaining two poles.

Equating coefficients of like powers of s, we get [Ref.2]

2aa + KsKr Kda2 = 2ac + 2 (5.8)a2 + Ks Kda2 = 2 + c2 + 2cc . 2 (5.9)KsKra2c 2aa = c2. 2 + 2. 2cc (5.10)c Ks a2 = 2c2 (5.11)Using the equation (5.8) to (5.11) we get [Ref.2],

Ks =1

c L1Kd c214c2P[2cc .c2Kd ]1Kd c21Kd c214c2+2cc Kd (2cc .c2Kd ) (5.12)

Kr =2ccKd L + 1 Kd c2P

L1 Kd c21 4c2 P[2ccc2Kd ] (5.13)

Where,

Kd =

Kdc (5.14)

M = 2aa 2cc (5.15)N = a2 c2 2cc2aa 2cc (5.16)

L = + c2

a2

N (5.17)


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P =2aaa2 +

c2a2 M +2cca2 N (5.18)

5.2 Gain design

Design value:

c =6 rad/sec and c = 0.65a = 50 rad/ sec and a = 0.7[Ref. 2]

using equation (5.12) to (5.18).The unknowns (forward and feedback gains,Ks and Kr ) are

found out for the different values of and c throughout the complete flight path.Here variables parameters are ,c and Tc[chapter-3]5.2.1 Gain Schedule

Forward gain ()

Fig.5.2 Gain Schedules for forward gain


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Feedback gain()

Fig.5.3 Gain Schedules for feedback gain

Fig.5.4 Launch vehicle autopilot with 2nd

order actuator and integrator [Ref.11]


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38

5.3 Loop transfer function

GH= Ks(s+K i

s) wa

2

s2+2a w a s+wa2 c me LeTc (s2

+

Tc

m e Le )

(s2 ) (1+Krs) ...(5.20)= Ks Kra2c me LeTc s+Kis

s2+ Tcm e Le

s2

1

s2+2awa s+w a2 (s +1

Kr) ...(5.21)

5.4 Input parameters (at time 20sec) [Ref.11]

Control thrust Tc = 1454568N

Control moment coefficient (c

) = 4.3sec-2

Aerodynamic moment coefficient () = -0.2 sec-2

Mass of engine [me]= 437.36 kg.

Distance between engine CG and gimbal point (Le )= 0.7680 m.

From the gain, at flight time t=20sec we set the autopilot

gains Ks = 3.1 . Kr = 0.2531 and Ki = 0.4 . We chose Ki by trial and error method. Ki is

chosen such that control system gives satisfactory performance. For the range Ki = 0.4 to 0.9

we get good performance of the control system so we chose Ki = 0.4. By using the gain

schedules from Fig.(5.2-3) the settling time, rise time gain and phase margin specifications

are met [Figs(5.5-6)], but the overshoot specification is not met. So we need to increase

feedback gain values to introduce enough damping in the system.

The step response to pitch command and Bode magnitude and phase plots are shown in

Fig5.5 and Fig 5.6 respectively.


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Fig. 5.5 Step response of rigid body system to a pitch step command (at t=20sec)

Fig.5.6 Bode plot of rigid body system (at t=20 sec)

From the Fig.5.5, we can see that settling time and rise time meet there specifications but

overshoot is too large, so we needs to increase the feedback gain. Now we set the rate

feedback (Kr) value to 0.3. The modified Step response and Bode plot are shown in Fig5.7

and Fig5.8 respectively.


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Fig. 5.7 Step response of rigid body system to a pitch step command for updated gain

(at t=20sec)

Fig.5.8 Bode plot of rigid body system for updated gain (at t=20 sec)


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41

We can see with these gains values all the specification is met [2]

The root locus of the rigid body system is shown in Fig.5.9. The closed loop pole of the

system with gain Ks = 3.1 is shown with cross mark on the root locus.

Fig.5.9 Root locus plot of rigid Launch vehicle

Fig.5.10 Enlarge portion of root locus showing close loop poles


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5.5 Close Loop poles

Close loop poles

Actuator Pole -33.5653 +29.6407i

-33.5653 -29.6407i

Aerodynamic pole -3.3505 + 4.9097i

-3.3505 - 4.9097i

Rate gain zero -0.3945

Table6.Close Loop Poles

It is seen from the Fig. (5.9-10), the close loop poles are located on the left half of the s-

plane. So the system is stable.

5.6 Ramp response (at flight time t=20sec)

Autopilot gains and launch vehicle parameters are now set for the flight time t=100sec from

Fig. (7-8-9-10).

3.1 0.3 0.2 sec2 4.3sec2

Fig.5.11 Close-loop ramp response


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From the Fig.5.11 we can see that tracking error grows with time. From the ramp response

plot, we have determined the tracking error rate.

Tracking error(deg)

At time t=3sec. 0.322

At time t=8 sec. 0.3249

Tracking error rate( /Sec) 5.8E-3 deg/sec

Table.7 Tracking error rate (from Matlab) ( = )5.6.1 Tracking error rate from Simulink model

From the Simulink model commanded pitch rate (

c) and actual pitch rate (

) are plotted

together.

Fig.5.12 Commanded pitch rate ( ) and actual pitch rate ( )


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Fig.5.13 Enlarged view of the steady-state tracking of commanded pitch rate ( )

5.7 Stability margin (by zone of exclusion)

Fig.5.14 Nyquist diagram with zone of exclusion


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From the Nyquist plot we can see that GH plot does not enter the zone of exclusion. So we

conclude that the stability margin specifications are met [Ref.2].

5.8 The updated feedback gains

Fig5.15 Rate feedback gain

5.9 Time response Analysis

Step response analysis to check the time domain specifications (like percentage overshoot,

rise time, settling time) are met.

5.9.1 Design specifications




We determined the percentage overshoot, rise time and settling time over the entire flight

zone at an interval of 20 sec. Plots are made by simply adding the values by straight line.


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Fig.5.16 Variation in the overshoot during atmospheric flight of the Launch vehicle for

designed autopilots

Fig.5.17 Variation in the Settling time during atmospheric flight of the Launch vehicle

for designed autopilots

0

2

4

6

8

10

12

14

16

18

20

0 20 40 60 80 100 120 140 160

Overshoot(%)

Flight Time(sec)

0

0.5

1

1.5

2

2.5

3

3.5

0 20 40 60 80 100 120 140 160

Settlingtime(sec)

Flight Time(sec)


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Fig.5.18 Variation in the Rise time during atmospheric flight of the Launch vehicle for

designed autopilots

5.10 Frequency domain analysis

Here phase margin and gain margins are determined at 20 sec interval of flight time to cover

the first phase of flight .Gain and Phase margins are determined by selecting gains from the

gain schedules [Fig.5.2 and Fig.5.15]. Plots are made by simply adding the values of phase

margin and gain margin at 20 sec interval of time by straight line.

5.10.1 Variation in the Phase margin

Fig.5.19 Variation in the Phase margin during atmospheric flight of the Launch vehiclefor designed autopilots

0

0.1

0.2

0.3

0.4

0.5

0.6

0 20 40 60 80 100 120 140 160

RiseTime(sec)

Flight Time(sec)

0

5

10

1520

25

30

35

40

45

0 20 40 60 80 100 120 140 160

PhaseMargine(deg)

Flight time(sec)


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5.10.2 Variation in the Gain margin

Fig.5.20 Variation in the Gain margin during atmospheric flight of the Launch vehiclefor designed autopilots

Stability margin for a case have good performance during flight trials [2].


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Chapter-6

Pitch Control of Flexible Launch Vehicle

6.1 Flexible body dynamics [Ref.11]

Flexibility of a launch vehicle is of primary concern to the control engineer because the

sensing instrument (rate gyro) picks up not only rigid body motion but also local elastic

deflection. This introduces an unstable control-structure interaction. This unstable control-

structure interaction could cause divergent oscillations, ending up with structural failure of a

launch vehicle.

Vehicle flexibility influence on the autopilot design is demonstrated by assuming the first

bending mode. Here we assumed that the bending modes are primarily excited by launch

vehicle engine deflection (First-order effect) [11]. The actuator input to the launch vehicle is

only the rigid vehicle state but also contribution due to vehicle flexibility.

Bending equation [Ref.11]

Bending deflection Wx, t = qi(t)ini (x) ...(6.1)q i + 2iiq i + i2qi = FiM i ...(6.2)Generalizes force for ith bending mode

Fi = f(x, t)iL0 dx ...(6.3)Generalizes mass for ith bending mode

Mi = m(x)i(x)2dxl

0 ...(6.4)

Considering only the first-order effect, the bending modes is excited by rocket engine

deflection. Considering only the first bending mode, the corresponding generalized force for

a concentrated force is determined as follows. Recall that the Dirac delta function (xa) is

defined by[13],


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So we can write [Ref.11]

F1=

[T

c + m

eL

e

]1

x

(x

L

T)

x

=[Tc + meLe] 1x(x LT)xL0 = [Tc + me Le]1LT ...(6.5)

The point of application of thrust force is at engine swivel point, x=LT . We normalized

bending modes at engine swivel point by taking1LT = 1, therefore Eq(6.5) becomes[11]

s2 + 2aw1s + w12q1 = 1

M1 (meLe s2 + Tc) ...(6.6)

1 =

m e LeM 1

(2+ m e Le

)

(s2+2a w1s+w12)

...(6.7)

From ref. 2 we can write f + q1NLG1 (6.8)WhereLG denotes the point on the vehicle where a gyro is located and NLG

1 = LRigid body equation [2]

The rigid body pitching motion is coupled to other dynamic modes of the system by

considering the first order effect the rigid body equation becomes

I r = Tc lc + Llr + meLelc ...(6.9)

Hence, = me LeTc

(s2+Tc

m e Le)

(s2) ...(6.10)

Actuator equation [2]

=-2 2 + 2 ...(6.11)Actuator transfer function:

= =

2

2+2 +2 ...(6.12)


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Gyro output [11]

Rate integrating gyroPG = r + NLG1 q1 ...(6.13)

LG denotes the point on the vehicle where the gyro is located

Rate gyro RG = Krs(r + NLG1 q1) ...(6.15)The feedback signal is

Fed = PG + RG

Fig.6.1 Launch Vehicle autopilot [11]

Fig.6.1 shows the pitch controller of the launch vehicle based on the preceding equations. Its

performance will be discussed subsequently.

6.2 Controller Design Using Gain Stabilization

Conventional control design approaches utilised decoupled single input and single outputmodels of the launch vehicle dynamics and classical control techniques to specify feedback

control loop structure and set the gains of the controller to obtain a controller that satisfy

design requirement. To compensate the flexible modes of the launch vehicle, A notch filter is

used which is designed to attenuate frequencies associated with the flexible modes. This

ensures that the signal produced by the sensors at these frequencies will be sufficiently

attenuated so as to cause no instability problems. This is called gain stabilisation.


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6.2.1 Controller design to track the commanded pitch rate

Our objective is to design a pitch plane autopilot to track the commanded pitch from the

guidance system. The autopilot control loop is shown in Fig.6.1. The design process is to

design the pitch plane autopilot for the rigid body first by setting the autopilot gains. After

adding the flexible modes autopilot gains are readjusted, so that performance specifications

are met. However if all specifications cannot be meet by just adjusting the gain values, a

notch filter is added in the forward loop which is tuned for the first bending mode. The

transfer function of the notch filter is given by [2],

Fnotch =s2 + 211s + 12s2 + 2

2

2s +

22

22

12 (6.16)

Where 1is equal to the frequency of the mode that interacts with the controller.6.3 Bending Frequency determination

Here we have determined mode shape of a representative launch vehicle. The launch vehicle

is assumed to be a free-free beam with circular cross-section. We have used Ansys to

determine the mode shapes and modal frequencies. Carbon-fibre-reinforced polymer is

assumed as material of launch vehicle. The parameters of the simulated launch vehicle are

shown below:

Input for Modal analysis

Length of the beam 52m

Second moment of cross sectional area(I) 1.13m4

Cross-sectional area of the beam 3.79 m2

Modulus of Elasticity 150 GPa

Poisson ratio 0.33

Table8.Input for modal analysis

6.3.1 Mode Shapes

Fig.6.2 bending mode (1stand 2

ndbending mode)


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The modal frequencies depend upon the mass and length of the launch vehicle. So during the

atmospheric flight the launch vehicle modal frequencies change with time. Here we have

determined the frequencies at three different points of flight time. At any other time in this

time interval these are determined by interpolation. Fig.6.3 shows the lowest frequency of the

launch vehicle. The operation points are 0 sec, 72 sec and 138 sec. [Ref.11]. Fig6.3 shows the

lowest frequency of the launch vehicle.

Fig.6.3 Bending frequency variation with time[11]

6.4. Generalized mass [12]

During the atmospheric flight aeroelastic forces act on the launch vehicle, causing high

frequency vibrations. In general, elastic body dynamics is expressed in terms of natural

vibration modes of the vehicle, in following manner [12]

q

i +2

i

iq

i +

i2qi =

Fi

Mi

(6.17)


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W(x,t)= qini tix (6.18)Generalized force for ith bending mode

Fi= f(x, t)iL

0 dx (6.19)

Generalized mass for ith

bending mode

Mi = m(l)i(x)2dxl0 (6.20)The variation of the generalised mass of the lowest frequency mode and the total mass of a

launch vehicle are shown in Fig.6.4 below. At each time instant the generalized mass is

determined by interpolation from the known values at 0, 72sec, and 138 sec.

Fig.6.4 Generalised mass of launch vehicle in 1stphase of flight [11]

6.5 Slope ( )We determined the slope of the lowest frequency of the mode at the point of gyro location.

The gyro is located at 15 m from the nose of the launch vehicle. We measure the slope at

three different time instant (0 sec, 72 sec, and 138 sec) at gyro location. At any other time the

slope is determined by interpolation [Fig.6.5].


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.

Fig.6.5 Slope changes in first phase of flight in sensor location[11]

6.6 Design specification

The control system is specified to have a settling time less then 3 seconds and a peak

overshoot less then 20%. For stability and robustness, it is desirable to have a minimum of

6dB gain margin and 30 degree phase margin. For robustness with regards to noise and

saturation problem, it is desirable to have minimum 10 dB attenuation of the flexible mode

peaks [2].


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6.7 Loop transfer function

GH= Ks(s+K i

s)

wc2

s2+2a wa s+w a2 c me LeTc(s2+

Tcm e Le

)

(s2 ) m e Le

M 1(s2+

Tcm e Le

)

(s2+21w1 s+w12) NLg1 (1+Krs) ... (6.17)

Denoting A=c me LeTc and B=me LeM1 NLg1

GH =KsKra2(A B) s+K i (s2+

Tcm e Le

)s+ 1K r(s2+Cs+D)

ss2+2a wa s+w a2s2 (s2+21w1 s+w12) ... (6.18)

Where C=21a A

AB and D=A12B

AB

6.7.1 Input to launch vehicle autopilot at the flight time t=20sec [11]

The actuator frequency () =50 rad/s, damping coefficient () =0.7. [14]Damping coefficient for 1stflexible mode = 0.5% [9]

Mass of engine [me] = 437.36 kg [11]

Distance between engine CG and gimbal point (Le ) = 0.7680 m. [11]

Control thrust (Tc)=1454568 N

Generalized mass (1) = 33600 KgControl moment coefficient () = 4.3sec-2Aerodynamic moment coefficient ( ) = -0.2 sec-2Slope per unit length at gyro location (NLg

1 ) = 0.0896 rad/m.

First modal frequency (at t=20 sec) =17.1 rad/sec.

Fig.6.6 and fig.6.7 show the pitch controller of the flexible launch vehicle with notch filter. In

Fig.6.6, the notch filter filters the control torque command before the actuator, whereas in

Fig.6.7 the notch filter filters the sensor output. These are called location-1 and location-2.

Intuitively, the notch filter after the sensor causes lag in the measurement, whereas the notch


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filter before the actuator filters out the modal frequency component from the control torque

command. So, intuitively the location-1 is superior to the location-2. So will be seen

subsequently.

Fig.6.6 Launch vehicle autopilot with notch filter location-1

Fig.6.7 Launch vehicle autopilot with notch filter location-2


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The root locus of pitch controller, shown in Fig.6.1 without notch filter, is shown in Fig.6.8.

The close-loop poles of the system for gain = 3.1 are shown with cross-marks on the rootlocus. We see bending poles on the right side of the s-plane implying unstable control-

structure interaction. This instability arises because of noncolocation of the sensor and the

actuator shown in Fig.6.2.

Fig .6.8 Root locus of flexible launch without Notch-filter


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The step response to a pitch command is shown in Fig.6.9. Flexible and rigid body responses

are shown separately. The flexible response shows diverging oscillations.

Fig.6.9 Flexible response and rigid body response (at t=20 sec)

The total response to a pitch command is shown below, compared with the rigid body

response.

Fig.6.10 Step response with flexibility (at t=20sec)


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6.8 Stability analysis: Nyquist Plot

Fig.6.11 Nyquist plot of the controller without notch filter, and zone of exclusion (at

t=20sec)

From the Nyquist plot we see that GH plot enters the zone of exclusion. So we conclude that

the stability margin specifications are not met [2].

6.9 Stability analysis: Bode Plot

The loop gain (GH) plotted against frequency (jw) is shown below Fig 6.12. As the structural

damping is very low (0.5%), there is sharp peak occurring in GH plot at the frequency of


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17.1 rad/sec and the amplitudes of the peak is well above the zero dB line.

Fig.6.12 Bode plot of flexible launch vehicle (uncompensated)

So, clearly, we need to provide attenuation of the flexible mode. The effect of the nozzle

inertia effects (Tail-wags-dog effect) is included in the model. This exhibits a pronounced

hump in the GH plot beyond the TWD frequency. There is adequate attenuation above the

TWD frequency; the gain hump does not introduce TWD oscillations.

The method of stabilizing the mode by providing attenuation is called gain stabilization

of the modes. The mode peak in the gain plot depends upon the structural damping ratio

assumed for the modes. We assume a structural damping ratio of 0.5%. There may be

uncertainties or error in prediction of frequencies and mode shape data. We have assumed 5%

error in the prediction of modal frequency data. Hence caution exercised when notch filter is

used for attenuation of a particular mode.

We provided an unsymmetrical notch filter which notches the modal frequency with a

required attenuation, and there is about 1.5 dB attenuation beyond the notch frequency. The

corresponding Bode plot is shown in Fig.6.14. The transfer function of the notch [2],


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Fnotch =s2 + 211s + 1

2

s2 + 222s + 22

22

12

2 = 0.4,1 = 0.004,1 = 17.95 and 2 = 17.1,

The notch filter transfer function, numerically, is

292.4s2 + 40.94s + 94200

322.2s2 + 4511s + 94200

The Notch filter pole-zero patterns is shown in Fig.6.13 and the frequency response is shown

in Fig. 6.14.

Fig. 6.13 Notch filter polezero pattern


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Fig.6.14 Bode plot of notch filter

6.10 Loop transfer function with notch filter

GH= Ks(s+K i

s)

wc2

s2+2a wa s+w a2 c me LeTc(s2+

Tcm e Le

)

(s2 ) m e Le

M 1(s2+

Tcm e Le

)

(s2+21w1 s+w12) NLg1 (1+Krs) s2+211s+12s2+222s+22

22

12

(6.19)

Taking A=c me LeTc and B=me LeM1 NLg1

GH =KsKra2(A B) s+K i (s2+

Tcm e Le

)s+ 1K r(s2+Cs+D)ss2+2a wa s+w a2s2 (s2+21w1 s+w12)

s2+211s+12

s2+222s+22

22

12 (6.20)

where C=21a A

AB and D=A12B

AB

6.11 Notch filter location

To determine a suitable place for the notch filter in the control loop, we have considered

location-1, and location-2 in Fig 6.6 and Fig. 6.7 respectively. In location-1, the filter is

before the actuator and it filters the actuator command, so as to filter out the frequency


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component that would otherwise cause excitation of the elastic mode. In location-2, the filter

filters attitude and rate sensor measurements.

Fig.6.15 Step response for different Place of Notch filter in the control loop

In the case of location-2, the step response in steady-state is oscillatory about one, whereas in

location-1 the step response settles to unity without oscillations. This is because, as

experienced earlier, in the case of location-1, the filter filters out the modal frequency

component of the control torque command. So we use the notch filer locaton-1 in the

subsequent autopilot design.

Here we assume that there is 5% error in computation generalized mass and modal frequency

determination. At flight time t=20 sec the computed first bending frequency is 17.1 rad/sec.

With 5% error, modal frequency varies from 17.1 rad/sec to 17.95 rad/sec. Now we willcheck whether this Notch filter [2] is suitable for this variation of frequency. The frequency

response of notch filter is shown in Fig.6.14.

The Bode plot of the gain stabilized system with notch filter for bending frequency 17.1

rad/sec is shown in Fig.6.16.


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Fig.6.16 Frequency Bode magnitude plot of the gain stabilised system

From Fig.6.16 we can see that stability margin specifications are met. This design approach

seems promising.

The Bode plot of the controller without notch filter when the bending frequency is 17.9

rad/sec (compared to the earlier 17.1 rad/sec) compared to the earlier17.2 rad/sec is shown in

Fig.6.16. As before, the controller is unstable.

Fig.6.16 Bode plot of flexible launch vehicle (uncompensated)


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Now we use the same Notch filter. Bode plot with notch filter is shown below in Fig.6.17.

We see from the controller is now stable and that it satisfies the performance specifications

[2].

Fig.6.17 Frequency Bode magnitude plot of the gain stabilised system

6.12 Nyquist Plot with Notch filter

Fig.6.18 Nyquist plot of the controller with notch filter, and Zone of exclusion (at t=20

sec)


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From the Nyquist Plot [Fig.6.18] we see that GH plot does not enter in the zone of

exclusion. So we conclude that control system performance satisfies the stability margin

specifications [Chapter4].

The root locus of the controller with bending filter is shown in Fig.6.19. The close-loop poles

of the system with the gain Ks = 3.1 are shown with the cross-marks on the root-locus. We

see that the flexible poles move to the left side of the imaginary axis with a significant

damping coefficient. The change of damping ratio of the flexible poles is from -0.0564 to

0.0113.

Fig.6.19 Root locus plot of flexible launch vehicle

The enlarged view of the root locus is shown below in Fig.6.20. The locations of the close-

loop flexible poles are shown by the cross-mark.


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Fig6.20 Root locus plot of the flexible launch vehicle

As all the flexible poles are on left side of the s-plane and far from imaginary axis, the system

is stable. The compensated flexible response is shown in Fig.6.23.

6.13 Closed-loop step response

Fig6.21 Compensated flexible mode response


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Fig6.22 Compensated rigid body response and flexible mode response

Fig.6.23 Step response of gain stabilized system


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6.14 Closed-loop Ramp response

The close-loop ramp response of the controller with the notch filter is shown in Fig.6.24.Though not clearly visible, here the tracking error grows with time. We determine the

tracking error rate from Fig.6.24 and these are stated in the table below.

Fig.6.24 Ramp response of a gain stabilized system

Tracking error(deg)

At time t=3sec. 0.3247

At time t=8 sec. 0.3624

Tracking error rate (deg/Sec.) 7.54E-3 deg/sec.

Table.9 Tracking error rate (from Matlab) ( = = .)


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6.14.1 Tracking error rate (deg/sec.)

We can also determine the tracking error rate from the Simulink model [Fig.6.25-6.26].

Fig.6.25 Commanded pitch rate ( ) and actual pitch rate ( )

Fig.6.26 Enlarged view of the steady-state tracking of commanded pitch rate ( )The steady-state tracking error rate from Fig.6.26 is 7E-3 deg/sec. which agrees with the

results from Fig.6.24.


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6.15 Time response Analysis

Step response analysis to check the time domain specifications (like percentage overshoot,

rise time, settling time) are met.




Fig.6.27 Variation in the overshoot during atmospheric flight of the Launch vehicle for

designed autopilots

Fig.6.28 Variation in the Rise Time during atmospheric flight of the Launch vehicle for

designed autopilots

0

2

4

6

8

10

12

14

16

18

20

0 20 40 60 80 100 120 140 160

Overshoot(%)

Flight Time(sec)

0

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80 100 120 140 160

RiseTime(Sec)

Flight Time(sec)


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Fig.6.29 Variation in the Settling time during atmospheric flight of the Launch vehicle


6.16 Stability margins

6.16.1Gain Margin

Fig.6.30 Variation in the Gain Margin during atmospheric flight of the Launch vehicle


0

0.5

1

1.5

2

2.5

3

3.5

0 20 40 60 80 100 120 140 160

SettlingTime(Sec)

Flight Time(Sec)

0

2

4

6

8

10

12

14

16

0 20 40 60 80 100 120 140 160

GainMargine(dB)

Flight time(sec)


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6.16.2 Phase margin

Fig.6.31 Variation in the Phase Margin during atmospheric flight of the Launch vehiclefor designed autopilots

6.17 Variable frequency notch filter

The notch frequency variation in the first phase of flight is shown below, which satisfy all

design requirements [2].

Fig.6.32 variation of notch filter frequency during atmospheric flight of the Launch

vehicle for designed autopilots

0

5

10

15

20

25

30

35

40

0 20 40 60 80 100 120 140 160

PhaseMargine

(dB)

Flight time(sec)


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6.18 Simulation results general time varying pitch command

Fig.6.33 shows the time-varying pitch command for 0 t 140 extracted from the optimalpitch command profile, Fig.3.1. This time-varying command is inputted to the controller

shown in Fig.6.6 (with the notch filter before the actuator). Here we have used the time slice

approach. By keeping parameters frozen for a short interval of time, the complete simulation

is done for the first-stage of flight. The simulation results are shown in figs.6.34-6.36 and the

performance of the controller is satisfactory.

6.18.1 Pitch Command

Fig.6.33 Commanded pitch (thetac)


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6.18.2 Commanded input (thetac) and output

Fig.6.34 Commanded pitch (thetac) and output pitch (theta)

6.18.3 Tracking error

Fig6.35 Pitch tracking error


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6.18.4 Actuator deflection

Fig6.36 Nozzle deflection

6.19 Simulation result with a stair-like pitch rate command

Fig.6.37 shows a stair-like, discrete, pitch rate command profile .The simulation results from

the Simulink model are given in Figs. (6.38-6.41), and, again the pitch tracking performance

is satisfactory.


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Fig.6.37 Commanded pitch rate profile

6.19.1 Commanded pitch rate and actual pitch rate

Fig.6.38 Commanded pitch rate angle and output pitch rate


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6.19.2 Commanded pitch and actual pitch

Fig.6.39 Commanded pitch angle and output pitch.

6.19.3 Tracking error

Fig.6.40 Tracking error


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6.19.4 Actuator deflection

Fig6.41 Nozzle deflection


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Chapter7

Conclusion and Future Work

7.1 Conclusion

In this project, we present a pitch control design methodology for a flexible launch vehicle

using classical control theory. We begin with the control system design of a simplified rigid

launch vehicle. We use a first-order actuator. A rigorous study is carried out to determine the

gain schedules for the first stage of the launch vehicle. The transfer function of this simplified

model is developed.

The autopilot design for controlling the pitch incorporating the nozzle inertia is presented

next. Detailed gain scheduling is carried out for the first stage of the launch vehicle. Pitch

step response is examined to check if the time domain specifications (like percentage

overshoot, rise time, settling time) are met. This is done at a 20 second interval of first-stage

flight duration. Frequency-domain analysis is conducted to determine the gain margin and the

phase margin of the controller for the entire first stage of the flight.

We then present the design of an autopilot for controlling the pitch of a flexible launch

vehicle. Here we examine the gain stabilization method. An unsymmetrical notch filter is

added in the loop before actuator. The filter is centred at the first flexible mode. 5% error in

determination of the modal frequency shape and generalized mass is considered. The notch

filter is designed to meet the performance requirement of the control system for this

uncertainty. The gains are selected to meet the rigid body control requirements. If any of the

specifications are not met, the gains are adjusted to meet the specifications. We also

determine the step response at 20 second interval of the flight time of the first stage of the

launch vehicle. Bode plots, Nyquist plots and zone of exclusion are drawn to check if the gain

margin and the phase margin requirements are met for the entire first stage of the flight.

7.2 Future work

i) The pitch controller design and analysis of the launch vehicle indicates that the classical

control theory is sufficient to meet the stability and performance requirements. However

adaptive control concepts may be used in conjunction with the classical approach to improve

the performance and increase robustness of the controllers.


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ii) In this project we have included only first bending mode in the control system design. We

suggest to incorporate at least first three modes in the control system design.

iii) Here we have carried out what is known as a short period analysis we need to study the

vehicle performance using a detailed simulation model where all nonlinearities and

disturbances including atmospheric disturbance (wind, gust), longitudinal and lateral

acceleration and yaw control should be carried out. Such a study would be a combination of

six-degree-of freedom simulation and long period analysisand it would includes the effect

of autopilot performance parameters on the overall trajectory.


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References

[1]Jenkiss, K. and Roy, J., Pitch Control of a Flexible Launch Vehicle,IEEE Transactions

on Automatic Control,ISSN: 0018-9286

[2] Kadam,N.V., Flight control system for Launch vehicle and Missiles,Allied Publishers

private Limited

[3] A.E Bryson, Jr., Control of Spacecraft and Aircraft, Princeton university Press,

Princeton, New Jersey

[4] Plaisted, C.E., and Leonessa, A., Expendable Launch Vehicle Adaptive Autopilot

Design,AIAA Guidance, Navigation and Control Conference and Exhibit18 - 21 August

2008, Honolulu, Hawaii

[5] Matlab control system ToolBox Users guide,The Math works Inc, 1996

[6]Geissler,E.D., Wind Effect on Launch Vehicle, Technivision Services

Slough,England,1970

[7] Noton,M. Spacecraft Navigation and guidance,Springer-verlag London Limited, 1998.

[8]Franklin,G.F, Powell,J.D,and Emami-Naeini,A., Feedback Control of Dynamic

Systems, 5th Edition, Pearson Education, Delhi,2006

[9] Jiann-Woei, J., Abram, A., Robert, H., Nazareth, B., Charles, H., Stephen, R., and Mark,

J., Ares I Flight Control System Design

[10] Kisabo, A.B., Agboola, F., Adebimpe, O.A. and Lanre, A.M., Autopilot Design for a

Generic based Expendable Launch Vehicle Using Linear Quadratic Gaussian (LQG) Control

Approach,European Journal of Scientific Research,Vol.50, No.4 (2011),

Pitch Control of Flex LaunchVehicles SubirPatra

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