pitch control of flex launchvehicles subirpatra
TRANSCRIPT
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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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Date:
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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
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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.5.12 Commanded pitch rate (c )and actual pitch rate () 43
Fig.5.13 Enlarged view of the steady-state tracking of commanded pitch rate (c ) 44
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
the Launch vehicle for designed autopilots
Fig.5.18 Variation in the Rise time during atmospheric flight of 47
the Launch vehicle for designed autopilots
Fig.5.19 Variation in the Phase margin during atmospheric flight of 47
the Launch vehicle for designed autopilots
Fig.5.20 Variation in the Gain margin during atmospheric flight of 48
the Launch vehicle for designed autopilots
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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.6.25 Commanded pitch rate (c )and actual pitch rate () 71
Fig.6.26 Enlarged view of the steady-state tracking of commanded pitch rate (c ) 71
Fig.6.27 Variation in the overshoot during atmospheric flight of 72
the Launch vehicle for designed autopilots
Fig.6.28 Variation in the Rise Time during atmospheric flight of 72
the Launch vehicle for designed autopilots
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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
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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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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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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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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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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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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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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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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
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.
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.
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.
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
for designed autopilots
6.16 Stability margins
6.16.1Gain Margin
Fig.6.30 Variation in the Gain Margin during atmospheric flight of the Launch vehicle
for designed autopilots
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
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[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),