 Open Access
 Total Downloads : 162
 Authors : Smrithi U S, Dr. Brinda V
 Paper ID : IJERTV5IS040652
 Volume & Issue : Volume 05, Issue 04 (April 2016)
 DOI : http://dx.doi.org/10.17577/IJERTV5IS040652
 Published (First Online): 20042016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Augmentation of Classical and Adaptive Control for Second Generation Launch Vehicles
Smrithi U S
Asst. Professor, EEE Department Ace College of Engineering Thiruvananthapuram, India
Dr. Brinda V
Head, Control Design Division Vikram Sarabhai Space Centre Thiruvananthapuram, India
Abstract Classical control technique is used for attitude control of launch vehicles worldwide, because of its established history of success. Usual method is to model launch vehicle dynamics by linear techniques to achieve adequate stability and tracking performance. Common type of feedback control system for launch vehicles is the ProportionalIntegral (PI) controller with appropriate filters to stabilize the lateral bending modes and slosh modes and also ensure sufficient robustness margins for rigid body. This paper presents a Classical Adaptive Augmentation Control (AAC) Algorithm for forward loop gain augmentation in real time, to cater to large dispersion in vehicle parameters beyond the capability of classical control system. The idea is to provide augmentation to a classical control designed autopilot when performance enhancement is required to tackle offnominal conditions arising out of modeling errors and large dispersion in estimated vehicle parameters (thrust, inertia, slosh, aerodynamics, lateral bending modes). There is high chance that such large dispersion can arise during initial design of a new generation launch vehicle before actual flight. Finally, simulation results for several credible launch vehicle failure scenarios show that the adaptive controller consistently and predictably improves performance and robustness, and achieves stability during extreme offnominal situations.
KeywordsAdaptive Controlt; Augmentation; Parameter Uncertainty; Gain Schedulling; Robustness

INTRODUCTION
As Launch Vehicles pace up along with the development of new technologies, the core targets of new designs are to enhance payload capability (performance), reliability, and safety at decreasing cost. As computational capability has become advanced, control algorithms play a major role towards achieving this aim. Global control algorithm used in current generation launch vehicles is the Proportional Integral (PI) control with gain scheduling. Although much advancement have taken place over the last few years, PI with gain scheduling control remains dominant due to its strong heritage. Generally, attitude control problem is considerate of the short period dynamics of the vehicle, where the basic aim is to achieve adequate stability and reasonably rapid response to input guidance commands, with average passivity to external disturbances. Launch vehicles are often aerodynamically highly unstable. Despite that, launch vehicle dynamics are readily modeled in literature using linear techniques to arrive at an autopilot configuration that meets the design requirements. For the flight control systems design, the consolidations of Blakelock[1], Greensite[2], and Garner[5] are quite allinclusive.
Major design problems exist because a launch vehicle is aerodynamically unstable, highly flexible and additional problems due to sloshing of liquid propellants and the inertia effects of engines. These problems are seriously aggravated for certain advanced launch vehicle configurations. Under such a scenario, classical control methods may not be fully effective in meeting the robustness margins for very large dispersions in vehicle parameters mainly because they are not known with sufficient precision before flight. Because a failure of any one of components could mean loss of the vehicle, an extensive ground testing and evaluation program is necessary to provide the maximum confidence for successful flight. This has led to the focus in adaptive control technique.
Greatest benefit of adaptive control that can be exploited is the fact that they dont require a deductive knowledge of the launch vehicle parameters with great accuracy. A survey of Adaptive Control Systems by Astrom, et al., in [6] clarifies the immense potential of the approach. In order to fully extract the benefits of adaptive control for a particular application, the adaptive control system must be designed with knowledge of the complete system to which it is to be applied [7], [8], [9]. This includes general features of the aerospace vehicle, such as controlstructure interaction, sloshing of propellants, performance of sensors, and actuator dynamics. Out of the many adaptive control schemes, the direct Model Reference Adaptive Control (MRAC) shows robustness to uncertainties with sometimes improved and more predictable performance, [10], [11], [12].
An algorithm that relies on model referencedriven gain adaptation supplemented by spectral damping is demonstrated by Jeb S Orr, et al., in [13]. The focus is on adaptive control developments that are specifically tuned for application to launch vehicles which maintains consistency with classical control system design.
This paper presents implementation and validation of a classical Adaptive Augmentation Control (AAC) algorithm for a typical 2nd generation launch vehicle, for performance enhancement in the event of large deviations in estimated vehicle parameters. AAC provides minimum augmentation when vehicle is under the control of classical controller within an expected range of parameter variations and environment uncertainty. However, if classical controller is unable to render sufficient performance due to parameter variation, external disturbances etc., the adaptive controller modifies the output of the classical feedback control law so as to maintain stability and minimize performance loss to the maximum possible extent. The analysis reveals that AAC can be implemented on board safely without affecting the
nominal and off nominal performances within bounds and at
dispersions thus avoiding vehicle failure possibilities. By
Force Equation
Fz Tc TT q lT U0
2
2
A[Cnl ] mpiU0 pi
(3)
the same time ensure best results for unpredicted severe
incorporating the AAC technique, the practice of assessing
i i 1 2
i
flight control stability using classical gain and phase margins
m [l l l w U ] i l
l i l m q i .
is not affected under justifiable assumptions.
r r c r
0 i T
r T r
The paper is organized in 6 sections. After a brief introduction of the topic in section I, section II describes the
Moment Equation
mathematical model of launch vehicle in pitch plane. Classical M y l [Tc T qi il ] T
T
T
1
qi il U 2 A [C l ]
qi il U 2 A [C l ]
T n
controller design is explained in section III. Section IV deals c T
with the classical adaptive augmentation (AAC) scheme and m pil U0 pi [I
T 2 0
i i
i i
m lrl ] m l U [I m lc2 ] m [l
i
i
l ]w
(4)
its validation is presented in section V. Finally, the paper is
i pi
r r c
r r 0 r r
r r c
concluded in section VI.
m l U m l U
m (l
l ) i (I
l
l
m l l ) i q
l
l

LAUNCH VEHICLE MODELING
r c 0
r c 0
lT q
i r r c
T r r r c T

Equations f short period dynamics
The primary objective of Launch Vehicle Attitude Control System is to orient the vehicle along the required trajectory in the presence of external disturbances. First step towards this is to model the vehicle attitude dynamics taking into account aerodynamics, control actuator dynamics, vehicle bending, propellant sloshing, variation in center of gravity (cg) and moment of inertia etc. as the time progresses. This leads to a time variant system. Using time slice approach a short
2) Slosh mode
Fuelslosh can be a severe issue in space vehicle stability and control. Dynamic effects of a sloshing liquid can be nearly approximated by replacing the liquid mass with a rigid mass and a harmonic oscillator like pendulum. Using the pendulum parameters like mass, length, hinge point location etc., the equation of motion of the ith pendulum can be represented as
period model is evolved [2] which can be assumed time invariant for a small duration, so that linear time invariant control system principles can be used. Further it is assumed to
w 2
pi pi pi
1
L pi
[U w (l0 pi
i i

Lpi ) q l pi ]
be decoupled in pitch/yaw/roll and planar analysis is carried out. Referring to [2],[3], the equations of short period dynamics can be represented as

Rigid body
Consider the geometry of vehicle in pitch plane

Bending mode
The elastic deflection at any point along the vehicle
(l, t) q
(l, t) q

is given by

2 (i)

(t ) (l )
i
(5)
(6)
represented in Fig 1.
z Fz
(1)
where (i) (l) denotes the normalized mode shape of the ith
(i)
U0 m0U0
mode in the pitch plane. q (t) is the generalized coordinate
I yy M y
(2)
due to elasticity for the ith mode in the pitch plane. It satisfies the equation
Considering the forces and moments acting on the launch vehicle due to engine inertia, aerodynamics, elasticity, slosh and actuator effects, effective force and moment equations
q (t) 2 iwiq
(i)

wi
2q(i)
Q(i)
M (i)
(7)
may be represented as
where
Q(i) and
M (i) are the generalized force and mass
respectively and are given by
Qi L f
l, t il dl
(8)
p
0
i L
i2


Actuator
M m l dl
0
(9)
The second order actuator dynamics may be represented as
A wA2 A wA2C 2 AwAA
(10)
Fig. 1. Geometry of vehicle in pitch plane

Nozzle
The second order nozzle dynamics may be represented as
N
[I m l l ]r r r c
I
r
m l
r r z wN
I
r
2 (
A


N
) 2 N wN N
(11)
frequency, damping factor, steady state errors gain margins, phase margins, etc. The method of pole placement is used for control system design of a conventional launch vehicle. Here the design/response specifications can be transformed into


State Space Representation
The complete plant dynamics represented by equations
(3) to (11) may be expressed in the state space form given by
desired locations of dominant closed loop poles. Using the model developed, the control system gains are selected so as to place the closed loop poles in the above locations. The
Kx Ax Bu
x K 1Ax K 1Bu
y Cx Du
(12)
(13)
gains are obtained as function of vehicle parameters and closed loop poles.
Control system design is carried out in two phases, first for a simplified model without slosh and flexibility. Gains are selected for good tracking, rapid response and good damping ensuring the system stability as the time progresses. In the
where 15 states are chosen considering rigid body mode, 3 bending modes, 1 slosh mode, actuator and nozzle dynamics given by
x[ ;;z;q(1) ;q(1) ;q(2);q(2) ;q(3);q(3) ; ;; N ;N ; A;A ](14) The control input is the deflection angle:
next step a suitable compensator is designed to stabilize the bending modes and sloshing modes as well as improving the rigid body margins in presence of higher order dynamics.

Simplified Autopilot Architecture

Block diagram of simplified Autopilot
u [c ]
(15)
The block diagram of launch vehicle rigid body model
The output:
y
T
(16)
with a simplified autopilot is shown in fig.2. forward loop gain and KR , the rate gyro gain.
KA is the
ag rg
where is the pitch angle, and represent the pitch
ag rg
rate sensed by angle and rate gyros respectively.
The launch vehicle model used for analysis consists of two large solid boosters strapped into a liquid core. Initial analysis is done at the atmospheric flight stage at a particular instant where aerodynamic forces are significant. Using the vehicle parameters and by considering the effect of two nozzles at the solid boosting phase, the governing equations are modified and rearranged to get required state space matrices.
C. Simplified System Modeling
For the purpose of investigating the general features of a
Fig. 2. Block diagram of Simplified Autopilot
The closed loop transfer function of the simplified rigid body model is given by
highly simplified version of the control system, equations (3)
(s)
Gs K
(20)

may be reduced to any simplified degree. Ignoring the
A c
effects of bending, sloshing and effects due to actuator dynamics, the plant dynamics may be expressed in the
c (s)
1GsH s
s 2 K
K
A c R
s (K
A c
)
transfer function form as


Design of simplified Autopilot
In the transfer function given by equation (20), the
(s)
c
(17)
values of KA , the forward loop gain and KR , the rate gyro
p (s)
s2
gain at a particular time instant have to be determined. The characteristic equation of the simple rigid body model with
where the control moment coefficient, c Tclc
(18)
simplified autopilot is given by
I yy
s2 K
Ac
KRs (K
Ac
) 0
(21)
the aerodynamic moment coefficient,
L l I yy
(19)
Equation (21) can be compared with the characteristic equation of a typical second order system



CLASSICAL CONTROL DESIGN METHOD
s2 2
s 2 0
n
n
n
(22)
Autopilot design for a launch vehicle is carried out in a conventional way using the classical control techniques [4]. This is well established method when performance criteria for control system are expressed in terms of undamped natural
where n and represent the undamped natural frequency and damping factor respectively
Comparing (21) & (22), we get
n2

sin d 2Ki cos
KA
c
(23)
K p
Ad sin D
(28)
K 2 n
(24)
where Ki is determined such that specified error constant is
R n2
met. For designing PI controller, the value of Kd is assumed to be zero in (27). Hence the design equations are
From (18) and (19), we have

D sin d sin d 2Ki cos
c =5.35 and =1.8 (25)
For a desirable rigid body natural freqency n =3 rad/sec
Ki
Ad sin
; K p
Ad sin
(29)
D
and damping factor, = 0.75, and substituting the same in
The dominant pole for =0.75 and n =3 rad/s,
(23) & (24) gives
K A = 2 and KR = 0.44 (26)
sd D 2.99138.6
(30)
An increase in value of KA provides best performance
Referring to Fig. 2.,
2
2
KAc (1 KRs)
and decrease the steady state error associated, but it affects the stability features. Hence Proportional Integral Control
G(s)H (s)
s
(31)
strategy may be introduced to maintain required performance
With respect to the dominant pole location,
capability without losing stability.

PI Controller Architecture
G(s)H Sd Ad d 1.043184.03
(32)
Integral constant, Ki
and proportional constant
K p are
obtained from equation (29) as
Ki 0.3
K p 1
(33)
The controller transfer function becomes
K
K
Gc (s) K p i
s
(1 0.3)
s
(34)
Fig.3. Classical Controller Architecture
The block diagram/architecture of PI controlled system is shown in fig. 3. Design of classical controllers like P/PI/PID controllers can be done using the classical root locus technique based on time domain approach where a controller can be designed in cascade with the system to have a pair of dominant closed loop poles which satisfy specified time
domain specifications n and . Here, without changing the position of already placed dominant poles significantly, damping factor is slightly changed so as to eliminate steady state error.
The designed values are applied on simplified system model as well as the complete plant model with a suitable compensator designed (a lag filter) for phase stabilization and tuned until a stable system with satisfactory performance is achieved.


CLASSICAL ADAPTIVE AUGMENTATION CONTROL ALGORITHM DESIGN

Control Architecture
If Gsd D Ad d is the open loop transfer
function of the system and Gc sd D is the transfer
function of PID controller with respect to the dominant pair
pole ( sd ) location, satisfying the magnitude condition for dominant pole pair to be on the root locus, the proportional, integral and derivative gains of a PID controller in general can be derived as
Fig.4. Classical Adaptive Augmentation Control
Fig. 4 represent the block diagram of augmented Adaptive Control. By working as an augmenting controller rather than the primary method of accommodating the changing flight
K sin d Ki
(27)
scenarios, the design preserves the strength of classical
d DAd sin D2
controller during nominal situations

Zones Of Adaptation
The three main zones where adaptation is expected to work are the following

Respond to tracking error

Respond to undesirable controlstructure interaction

Return to baseline control design during nominal cases


Adaptation Law [14]
A multiplicative rstorder adaptation law is used,

High pass filter
A high pass Chebyschev filter is used with a cut off frequency approximately twice of that of the rigid body frequency.

Low pass filter
A maximally flat Butterworth filter of cut off frequency approximately nearing the rigid body control frequency is chosen here. The frequency responses of High pass and Low
k k 2
(35)
pass filter chosen are shown in fig 5.
ka max a aer bka ysd ckFG 1
kmax
where er
is the model error,
ysd
is the damper signal and a,
b, c represents the error gain, damper gain and nominal gain respectively. The error term responds to tracking error, the damper term during controlstructure interaction and the nominal term for automatic reconvergence to PI controller, when adaptation is not needed. An upper and lower bound to adaptation is provided given by kmax and k 0 . For at least 6 dB robustness gain margin which corresponds to a magnitude of 2, kmax is chosen as 1.5 and k 0 is chosen as 0.5.
The output of the adaptive law is the adaptive gain ka , which is used to adjust the output of the PI controller. The adaptive gain ka is used to calculate a total forward loop gain
given by
Fig.5. Frequency response of High Pass & Low Pass Filter

VALIDATION OF ADAPTIVE AUGMENTATION CONTROL DESIGN
In order to validate the proposed control scheme, simulations of attitude control of launch vehicle are presented. The controller is tested on the typical discretized launch vehicle model to check whether the design criteria are met.
kFG k0 ka

Computation of error term, er
(36)
A. Adaptive Controller Parameters
The tuned adaptation gains are given in table 1.
Table 1. Adaptive Control Parameters
Gains
Symbol
Value
Error gain
a
0.001
Damper gain
b
100
Nominal gain
c
0.2
Error mixing constant
d
25
Maximum adaptation gain
kmax
1.5
Minimum adaptation gain
k0
0.5
Gains
Symbol
Value
Error gain
a
0.001
Damper gain
b
100
Nominal gain
c
0.2
Error mixing constant
d
25
Maximum adaptation gain
kmax
1.5
Minimum adaptation gain
k0
0.5
The error term is the part which increases the adaptive gain, when needed. A reference model is designed to attain the desired closed loop performance, similar to second order rigid body system that tracks the guidance commands.
0
1
2
r 2
r r
(37)
r
r
2rr r
0
c
er d r
where d is the error mixing constant
2) Computation of damper signal,
ysd
(38)

Test Cases
The four credible test failure scenarios selected are

Minimal adaptation during nominal plant situation

Low thrust/high inertia (dispersed by 20%)
Damper signal is a rectified signal detecting and passing undesirable highfrequency dynamics in the loop. Decay of adaptive gain is proportional to magnitude of signal.

First bending mode frequency decreased by 30%

High thrust/Low Inertia

C. Short Period Simulations
2
2
yHP H HP (s)UG
(39)
The above mentioned test cases are tested on the modeled
ysd H LP
(s)yHP
(40)
linear time invariant plant on short period basis initially, subjected to a steering step command, c
yHP and ysd are outputs of a High pass filter and a Low pass
filter being used, with transfer functions
H LP respectively.
HHP and
Fig.6. Nyquists & step response for case 1
Fig.7. Nyquists & step response for case 2
Fig.8. Nyquists & step response for case 3
Fig.9. Nyquists & step response for case 4
Fig.10: short periodVariation in adaptive gain for all cases
D.Planar Simulations
A lng period planar simulation is carried out using vehicle data from 40 to 90 seconds with proper gain scheduling and performance of adaptation is evaluated. Here, the steering command generated by guidance is applied to the
plant with time varying parameters. The plant parameters, the steering command signal and the reference model vary at each instant. Fig.11 shows the variation in scheduled gains KA & KR
Fig. 11: Variation of scheduled gains, KA & KR
Fig. 12: Case 1: Attitude Plot
Fig. 13: Case 3: Attitude Plot
Fig. 14: Case 4: Attitude Plot
Fig.15: long periodVariation in adaptive gain for all cases
E. Analysis of Simulation Results
The simulation results for different test cases selected show that PI controller with suitable compensation works satisfactorily for nominal cases as well as for dispersions upto 20% in plant parameters. For extreme offnominal situations tested, AAC takes the control.
In the first test case, the nominal plant is simulated. Here, the baseline controller renders reasonable performance. Thus, the contribution from AAC is minimal, as expected. From the nyquist plot shown in fig. 6, the rigid body margins obtained from short period simulation are Gain margin = 7 dB; Phase margin= 48 deg; Aero margin = 12.3 dB. For slosh mode, Phase margin= 30 deg. For first bending mode, phase margin= 47.8 deg. Step response shows an acceptable overshoot of less than 30 percent and tracking error less than 1 deg. In the planar simulation also (fig. 12), commanded attitude profile is followed satisfactorily. In Fig 10 and fig 15,
ka is maintained close to 0.5, so that total adaptive loop gain,



CONCLUSIONS
An adaptive augmenting strategy has been incorporated into the designed pitch axis dynamics of a typical launch vehicle to back support the classical controller (PI) designed, so as to improve performance and handle extreme off nominal situations. Several credible test cases were selected for validating the performance supremacy of proposed controller. Simulation results show that adaptive augmentation provided sufficient performance improvement, and avoided loss of vehicle for extreme offnominal cases. By acting as augmenting controller, classical control is maintained as the primary controller thereby preserving the strength and legacy of classical control for nominal as well as bounded dispersion cases. The method is proved to be suitable for ensuring safety of new generation launch vehicles designed for advanced missions during their initial flight testing phase. As forward work, a strict assessment of proposed scheme in a full sixdegree of freedom nonlinear environment may be done to validate its use in a relevant flight environment.
REFERENCES

J. Blakelock, Stability and Control of Aircraft and Missiles, 2nd Edition WileyInterscience, 1991

A. Greensite, Analysis and Design of Space Vehicle Flight Control Systems, Volume I – Short Period Dynamics, NASA CR820,1967

A. Greensite, Analysis and Design of Space Vehicle Flight Control Systems, Volume XV Elastic body equations, NASA CR834,1970

A. Greensite, Analysis and Design of Space Vehicle Flight Control Systems, Volume VII Attitude Control during Launch, NASA CR 826,1967

Garner D, Control theory Handbook, NASA TMX53036, 1964

Astrom K. J., Theory and Applications of Adaptive Control – A Survey, Automatica 19(5), pages 471486, 1983

Boskovich B., and R. E. Kaufmann, Evolution of the Honeywell First Generation Adaptive Autopilot and Its Applications to F94, F101, X 15, and X20 Vehicles, AIAA Journal of Aircraft, Volume 3, No. 4, pages 296304, 1966
kFG
is equal to 1.
In the second case, a model error is created by decreasing

Sta of the Flight Research Center, Experience with the X15 adaptive ight control system, technical report, NASA Flight Research Center, TN D6208, 1971
torque and increasing inertia by 30%. Here, PI controller
deteriorates in its performance. AAC increases the adaptive gain owing to model error and improves performance and stability margins. The increase in adaptation gain is visible in both fig. 10 and fig. 15.
An unstable situation is created in third test case by introducing an unexpected extreme dispersion (30% decrease) in first bending mode frequency and mass. The PI controller degrades here to retain stability. There is a chance that the command signal gets in phase with the excited plant, increasing its amplitude further till stability is lost. AAC decreases the adaptive gain and maintains stability. Planar simulation (fig. 13) supports the short period analysis. The decrease in adaptation gain is shown in both fig. 10 and fig. 15.
In the last case, a high thrust/low inertia situation is experimented, which leads to complete loss of stability of vehicle as the PI controller gain becomes excessive. AAC decreases gain and stability is gained back. Same result is established during planar simulation as well (fig. 14).

M. Thompson and J. Welsh, Flight Test Experience With Adaptive Control Systems, technical report, NASA Flight Research Center, 1970.

Landau. I. D., A Survey of Model Reference Adaptive Techniques:
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Kreisemayer. G., and B. Anderson, Robust Model Reference Adaptive Control, IEEE transactions on Automatic Control AC31(2), pages 127133, 1986

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July 2000

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APPENDIX
Notations used
A = reference area; D = drag
Cn =normal force coefficient
Fz = total force acting parallel to vehicle body axis, Z
Ts = ungimballed thrust
I yy
= moment of inertia of reduced about pitch axis
TT =total thrust
I 0 = moment of inertia of rocket engine about its c.g.
I r = moment of inertia of rocket engine about swivel point
KA = servo amplifier gain
KI = integrator gain
KR = rate gyro gain;
Vw = wind velocity parallel to Z ' axis
lc = distance from origin of body axis to engine swivel point
lr = distance from c.g. of rocket to engine swivel point
l =distance from c.p. in pitch plane to origin of body axis
l pi =distance from hinge point of ith pendulum to axis origin
L =length of vehicle
Lpi =length of ith pendulum
L =aerodynamic toad per unit angle of attack
ml = reduced mass/length along vehicle longitudinal axis
m0 = reduced mass of vehicle
V U0 = forward velocity of vehicle
U 0 =acceleration
z = perturbation velocity of vehicle parallel to Z axis
= perturbation angle of attack
= flight path angle
= pendulum angle
pi
= rocket engine deflection angle
c = command signal to rocket engine
N = nozzle deflection angle
A = actuator deflection angle
a , N = relative damping factor for actuator, nozzle
i = relative damping ratio for ith bending mode
= perturbation attitude angle
c= attitude command signal
(l, t) =bending deflection
i =slope of ith bending mode
i
mpi
= mass of ith pendulum
= normalized mode function for the ith bending mode
mT = total mass of vehicle
,
N
N
a
= udamped natural frequency for actuator, nozzle
mR = mass of rocket engine
i
M = generalized mass of ith bending mode
M = total perturbation moment about pitch axis
X
q(i) = generalized coordinate of ith bending mode
Q(i) = generalized force (moment) of ith bending mode
Tc =control thrust
, =frequency of the ith bending mode, pendulum
i
i
pi