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CLARKSON UNIVERSITY
HYBRID ADAPTIVE ASCENT FLIGHT CONTROL FOR A FLEXIBLE LAUNCH VEHICLE
A Dissertation
by
BRIAN D. LEFEVRE
DEPARTMENT OF MECHANICAL AND AERONAUTICAL ENGINEERING
Submitted in Partial Fulfillment of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
in Mechanical Engineering
August 30, 2010
Accepted by the Graduate School
_______________ ____________________________ Date Dean
Copyright © 2010 by Brian D. LeFevre
All Rights Reserved
ii
The undersisned have examined the dissertation entitled
Hybrid Adaptive Ascent Flight Control for a Flexible Launch Vehicle
presented by Brian D. LeFevre, a candidate for the degree of Doctor of Philosophy, and hereby
certifu that it is worthy of acceptance.
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Professor Ratneshwar JhaMechanical and Aeronautical Engineering Dept.
&o, A,,,a 4l n^.J,'Professor Goodarz Ahmadi
Mechanical and Aeronautical Engineering Dept.
Mechanical and Aeronautical Engineering Dept.
tll
Professor Erik BolltMathematics and Computer Science Dept.
PiergiovanniMarzocca
Professor Robert SchillinsElectrical and Computer Engineeri
“It is not the critic who counts, nor the man who points out where the strong man stumbled, or where a doer of deeds could have done them better. The credit belongs to the man in the arena whose face is marred by dust and sweat and blood, who strives valiantly, who errs, and who comes up short again and again, who knows the great enthusiasms, the great devotions, and spends himself in a worthy cause. The man who at best knows the triumph of high achievement and who at worst, if he fails, fails while daring greatly, so that his place will never be with those cold timid souls who know not victory nor defeat.”
- Teddy Roosevelt
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ABSTRACT
For the purpose of maintaining dynamic stability and improving guidance command tracking
performance under off-nominal flight conditions, a hybrid adaptive control scheme is selected
and modified for use as a launch vehicle flight controller. This architecture merges a model
reference adaptive approach, which utilizes both direct and indirect adaptive elements, with a
classical dynamic inversion controller. This structure is chosen for a number of reasons; the
properties of the reference model can be easily adjusted to tune the desired handling qualities of
the spacecraft, the indirect adaptive element (which consists of an online parameter identification
algorithm) continually refines the estimates of the evolving characteristic parameters utilized in
the dynamic inversion, and the direct adaptive element (which consists of a neural network)
augments the linear feedback signal to compensate for any nonlinearities in the vehicle
dynamics. The combination of these elements enables the control system to retain the nonlinear
capabilities of an adaptive network while relying heavily on the linear portion of the feedback
signal to dictate the dynamic response under most operating conditions.
To begin the analysis, the ascent dynamics of a launch vehicle with a single 1st stage rocket
motor (typical of the Ares I spacecraft) are characterized. The dynamics are then linearized with
assumptions that are appropriate for a launch vehicle, so that the resulting equations may be
inverted by the flight controller in order to compute the control signals necessary to generate the
desired response from the vehicle. Next, the development of the hybrid adaptive launch vehicle
ascent flight control architecture is discussed in detail. Alterations of the generic hybrid adaptive
control architecture include the incorporation of a command conversion operation which
transforms guidance input from quaternion form (as provided by NASA) to the body-fixed
angular rate commands needed by the hybrid adaptive flight controller, development of a
Newton’s method based online parameter update that is modified to include a step size which
regulates the rate of change in the parameter estimates, comparison of the modified Newton’s
method and recursive least squares online parameter update algorithms, modification of the
neural network’s input structure to accommodate for the nature of the nonlinearities present in a
launch vehicle’s ascent flight, examination of both tracking error based and modeling error based
neural network weight update laws, and integration of feedback filters for the purpose of
preventing harmful interaction between the flight control system and flexible structural modes.
To validate the hybrid adaptive controller, a high-fidelity Ares I ascent flight simulator and a
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classical gain-scheduled proportional-integral-derivative (PID) ascent flight controller were
obtained from the NASA Marshall Space Flight Center. The classical PID flight controller is
used as a benchmark when analyzing the performance of the hybrid adaptive flight controller.
Simulations are conducted which model both nominal and off-nominal flight conditions with
structural flexibility of the vehicle either enabled or disabled.
First, rigid body ascent simulations are performed with the hybrid adaptive controller under
nominal flight conditions for the purpose of selecting the update laws which drive the indirect
and direct adaptive components. With the neural network disabled, the results revealed that the
recursive least squares online parameter update caused high frequency oscillations to appear in
the engine gimbal commands. This is highly undesirable for long and slender launch vehicles,
such as the Ares I, because such oscillation of the rocket nozzle could excite unstable structural
flex modes. In contrast, the modified Newton’s method online parameter update produced
smooth control signals and was thus selected for use in the hybrid adaptive launch vehicle flight
controller. In the simulations where the online parameter identification algorithm was disabled,
the tracking error based neural network weight update law forced the network’s output to diverge
despite repeated reductions of the adaptive learning rate. As a result, the modeling error based
neural network weight update law (which generated bounded signals) is utilized by the hybrid
adaptive controller in all subsequent simulations.
Comparing the PID and hybrid adaptive flight controllers under nominal flight conditions in
rigid body ascent simulations showed that their tracking error magnitudes are similar for a period
of time during the middle of the ascent phase. Though the PID controller performs better for a
short interval around the 20 second mark, the hybrid adaptive controller performs far better from
roughly 70 to 120 seconds. Elevating the aerodynamic loads by increasing the force and moment
coefficients produced results very similar to the nominal case. However, applying a 5% or 10%
thrust reduction to the first stage rocket motor causes the tracking error magnitude observed by
the PID controller to be significantly elevated and diverge rapidly as the simulation concludes. In
contrast, the hybrid adaptive controller steadily maintains smaller errors (often less than 50% of
the corresponding PID value). Under the same sets of flight conditions with flexibility enabled,
the results exhibit similar trends with the hybrid adaptive controller performing even better in
each case. Again, the reduction of the first stage rocket motor’s thrust clearly illustrated the
superior robustness of the hybrid adaptive flight controller.
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ACKNOWLEDGMENT
This research was primarily supported by the NASA Graduate Student Researchers Program
(Grant No. NNM 04-AA03H) through the Marshall Space Flight Center in Huntsville, Alabama.
I would like to gratefully acknowledge the collaborative efforts of Dr. Mark Whorton (former
Branch Chief of Guidance, Navigation, and Mission Analysis at NASA Marshall), and of Nhan
Nguyen and Kalmanje Krishnakumar from the Intelligent Systems Division at the NASA Ames
Research Center. I would also like to thank my faculty advisor here at Clarkson University,
Professor Ratan Jha, for all of the guidance and support that he has offered me throughout my
academic career. In addition, I extend my appreciation to my examining committee of Professors
Goodarz Ahmadi, Erik Bollt, Pier Marzocca, and Bob Schilling for donating their valuable input
and time to my pursuit of a doctorate degree.
I find it hard to express how much my family and friends have meant to me throughout all
phases of my life. I search for the perfect words because they are an invaluable resource. It is
their support that fortifies my strength, and their compassion that enriches my spirit. First, to all
the friends I have made during my time at Clarkson: I couldn’t have done this without you. To
my mom and dad, Nancy and David: thank you for recognizing when I needed a little nudge to
get motivated, and for instilling in me an unwavering faith in my abilities. You have given me
more intangible gifts than I can ever hope to repay. To my brother Jason and sister-in-law Eileen:
thank you for always welcoming me into your home, and for being a couple that I can turn to for
help under any circumstances. I wish you both the very best as you welcome your beautiful
daughter Bethany into this world, and as your lives continue to grow as a family. To my fiancée,
Adrienne: thank you for being my closest source of counsel, and thank you for all of the
encouragement and love that you continue to give me. I am also grateful that you have chosen to
stay the course with me, despite my degree progress at Clarkson evolving from a sprint, to a
marathon, to an expedition of sorts. Finally, I would like to dedicate this work to my grandpa and
grandma, Arthur and Mary Helen Machell. Though they have both passed on while I have been
away in graduate school, I can not say that I have lost them because their love lives on in my
memory and in my heart. I think of them often and trust that we will meet again someday. In
closing, a sincere thanks goes to all of my friends and family; may you savor such blessings as I
have been given.
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NOMENCLATURE
A, B = matrices characterizing the error dynamics
iC = aerodynamic force/moment coefficient
jiC , = aerodynamic force/moment coefficient derivative
cref = aerodynamic reference length
Di = neural network input vectors
e = error dynamics state vector
[ ]Teeee ψθφ≡E = Euler angle tracking error
)(ˆ ke = prediction error used in online parameter update
FARP = aerodynamic force at designated reference point
Frkt = thrusting force generated by rocket motor
Fthrust = rocket thrust magnitude
F1, F2, G = dynamic inversion matrices
H = angular momentum
Ht = angular momentum transferred from the spacecraft
I ≡ Iij = time-varying inertia tensor
KI = integral feedback gain matrix
KP = proportional feedback gain matrix
l = lever arm for jet damping force
m& = mass flowrate out of rocket nozzle
M ≡ [L M N]T = external moment acting on the spacecraft
Maero ≡ [Laero Maero Naero]T = aerodynamic moment
MARP = aerodynamic moment at designated reference point
MRCS ≡ [LRCS 0 0]T = roll control moment
Mrkt ≡ [Lrkt Mrkt Nrkt]T = moment created by thrust vectoring
P = solution to Lyapunov equation concerning error dynamics
q ≡ [qo qx qy qz]T = spacecraft attitude quaternion
qc = commanded quaternion
qd = dynamic pressure
R = learning rate matrix for modeling error based update law
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rARP ≡ [xARP yARP zARP]T = position vector from cm to aerodynamic reference point
rgim ≡ [xeg yeg zeg]T = position vector from cm to origin of gimbal axes
S = aerodynamic reference area
Tjet = jet damping torque
uad = adaptive portion of feedback signal
ue = linear portion of feedback signal
V = spacecraft velocity magnitude
W = neural network weight matrix
α = angle of attack
β = sideslip angle
β = set of basis functions applied to neural network inputs
Γ = learning rate for tracking error based update law
δ ≡ [LRCS θp θy]T = control signal vector
∆t = sample time of flight control system
ε = angular acceleration modeling error
ζp, ζq, ζr = damping ratios for reference model
)(kθ = state vector used in online parameter update
θp = pitch gimbal angle
θy = yaw gimbal angle
)(kΘ = matrix of parameter estimates at kth time step
µ = step size for Newton’s method-based parameter update
ρ = atmospheric density
σ ≡ [1 α β]T = state vector used in dynamic inversion
ω ≡ [p q r]T = angular velocity
ωc = angular velocity command
dω& = desired angular acceleration
ωe = angular velocity tracking error
ωm = angular velocity output of reference model
ωn ≡ diag(ωp ωq ωr) = matrix of natural frequencies for reference model
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TABLE OF CONTENTS
Chapter 1 Introduction ................................................................................................. 1
1.1 Launch Vehicle Flight Control: Past & Present............................................ 2
1.2 Research Objectives...................................................................................... 5
Chapter 2 Rigid Launch Vehicle Ascent Dynamics ................................................... 8
2.1 Flow of Angular Momentum ........................................................................ 9
2.2 Aerodynamic Loads .................................................................................... 12
2.3 Thrust Vectoring & Roll Control................................................................ 13
2.4 Equations of Rotational Motion.................................................................. 17
Chapter 3 Hybrid Adaptive Ascent Flight Control.................................................. 20
3.1 Control Architecture ................................................................................... 21
3.2 Online Parameter Identification.................................................................. 25
3.3 Neural Network Design .............................................................................. 29
3.4 Feedback ‘Flex Filtering’............................................................................ 31
3.5 Simulink Diagram Representation.............................................................. 34
Chapter 4 Ascent Flight Simulation .......................................................................... 41
4.1 SAVANT Ascent Simulator & Existing PID Flight Controller.................. 42
4.2 Selection of Adaptive Laws ........................................................................ 46
4.3 Rigid Body Performance............................................................................. 48
4.4 Flexible Body Performance ........................................................................ 69
4.5 Rigid vs. Flexible Performance Comparison .............................................. 89
Chapter 5 Concluding Remarks................................................................................. 92
5.1 Methods, Results, & Conclusions............................................................... 92
5.2 Contributions of this Work ......................................................................... 97
5.3 Future Work ................................................................................................ 99
References....................................................................................................................... 101
Appendix A – Matlab Code & Simulink Diagrams.................................................... 104
x
LIST OF FIGURES
Figure 1. Progression of launch vehicle design. ............................................................................ 2
Figure 2. Ares I body-fixed coordinate system.............................................................................. 9
Figure 3. Rocket nozzle gimbal axes. .......................................................................................... 14
Figure 4. Thrust vector rotation. .................................................................................................. 15
Figure 5. Hybrid adaptive launch vehicle ascent flight control architecture. .............................. 22
Figure 6. Single-hidden-layer sigma-pi neural network. ............................................................. 29
Figure 7. Feedback ‘flex filter’ Bode diagram............................................................................. 33
Figure 8. Simulink model of the hybrid adaptive ascent flight controller. .................................. 34
Figure 9. Hybrid adaptive controller – Reference model subsystem........................................... 35
Figure 10. Hybrid adaptive controller – PI element subsystem................................................... 35
Figure 11. Hybrid adaptive controller – Dynamic inversion subsystem. .................................... 36
Figure 12. Hybrid adaptive controller – Signal routing subsystem. ............................................ 36
Figure 13. Hybrid adaptive controller – Neural network subsystem. .......................................... 37
Figure 14. Neural network – Tracking error based weight update subsystem............................. 38
Figure 15. Neural network – Modeling error based weight update subsystem. .......................... 38
Figure 16. Hybrid adaptive controller – Online parameter identification subsystem.................. 39
Figure 17. Hybrid adaptive controller – Flex filter subsystem. ................................................... 40
Figure 18. Baseline PID ascent flight control architecture. ......................................................... 43
Figure 19. Data embedded in SAVANT...................................................................................... 45
Figure 20. Nominal rigid Ares I – Parameter identification algorithm comparison.................... 46
Figure 21. Nominal rigid Ares I – gimbal angle commands........................................................ 54
Figure 22. Nominal rigid Ares I – roll torque command. ............................................................ 55
Figure 23. Nominal rigid Ares I – angular velocities. ................................................................. 56
Figure 24. Nominal rigid Ares I – tracking errors. ...................................................................... 57
Figure 25. 95% thrust, rigid Ares I – gimbal angle commands. .................................................. 58
Figure 26. 95% thrust, rigid Ares I – tracking errors................................................................... 59
Figure 27. 90% thrust, rigid Ares I – gimbal angle commands. .................................................. 60
Figure 28. 90% thrust, rigid Ares I – tracking errors................................................................... 61
Figure 29. 110% Aerodynamic load, rigid Ares I – gimbal angle commands............................. 62
Figure 30. 110% Aerodynamic load, rigid Ares I – tracking errors. ........................................... 63
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Figure 31. 120% Aerodynamic load, rigid Ares I – gimbal angle commands............................. 64
Figure 32. 120% Aerodynamic load, rigid Ares I – tracking errors. ........................................... 65
Figure 33. Nominal rigid Ares I – Tracking error norm. ............................................................. 66
Figure 34. 95% thrust, rigid Ares I – Tracking error norm.......................................................... 66
Figure 35. 90% thrust, rigid Ares I – Tracking error norm.......................................................... 67
Figure 36. 110% Aerodynamic load, rigid Ares I – Tracking error norm. .................................. 67
Figure 37. 120% Aerodynamic load, rigid Ares I – Tracking error norm. .................................. 68
Figure 38. Rigid Ares I – Influence of aerodynamic load on tracking error norm. ..................... 68
Figure 39. Nominal flexible Ares I – gimbal angle commands. .................................................. 74
Figure 40. Nominal flexible Ares I – roll torque command. ....................................................... 75
Figure 41. Nominal flexible Ares I – angular velocities.............................................................. 76
Figure 42. Nominal flexible Ares I – tracking errors................................................................... 77
Figure 43. 95% thrust, flexible Ares I – gimbal angle commands. ............................................. 78
Figure 44. 95% thrust, flexible Ares I – tracking errors. ............................................................. 79
Figure 45. 90% thrust, flexible Ares I – gimbal angle commands. ............................................. 80
Figure 46. 90% thrust, flexible Ares I – tracking errors. ............................................................. 81
Figure 47. 110% Aerodynamic load, flexible Ares I – gimbal angle commands........................ 82
Figure 48. 110% Aerodynamic load, flexible Ares I – tracking errors........................................ 83
Figure 49. 120% Aerodynamic load, flexible Ares I – gimbal angle commands........................ 84
Figure 50. 120% Aerodynamic load, flexible Ares I – tracking errors........................................ 85
Figure 51. Nominal flexible Ares I – Tracking error norm. ........................................................ 86
Figure 52. 95% thrust, flexible Ares I – Tracking error norm. .................................................... 86
Figure 53. 90% thrust, flexible Ares I – Tracking error norm. .................................................... 87
Figure 54. 110% Aerodynamic load, flexible Ares I – Tracking error norm. ............................. 87
Figure 55. 120% Aerodynamic load, flexible Ares I – Tracking error norm. ............................. 88
Figure 56. Flexible Ares I – Influence of aerodynamic load on tracking error norm.................. 88
Figure 57. Nominal Ares I – Influence of flexibility on tracking error norm.............................. 90
Figure 58. 90% thrust – Influence of flexibility on tracking error norm. .................................... 90
Figure 59. 120% Aerodynamic load – Influence of flexibility on tracking error norm............... 91
xii
PUBLICATIONS
Journal Papers:
• LeFevre, B., and Jha, R., “Hybrid Adaptive Ascent Flight Control for a Flexible
Launch Vehicle,” Proceedings of the Institution of Mechanical Engineers, Part G:
Journal of Aerospace Engineering (submitted).
• LeFevre, B., and Jha, R., “Hybrid Adaptive Launch Vehicle Ascent Flight Control,”
American Institute of Aeronautics and Astronautics: Journal of Guidance, Control,
and Dynamics (submitted).
• LeFevre, B., and Jha, R., “Attitude Dynamics of a Square Solar Sailcraft During Spin-
Deployment,” Proceedings of the Institution of Mechanical Engineers, Part G:
Journal of Aerospace Engineering (accepted Aug. 16, 2010).
Conference Proceedings:
• LeFevre, B., and Jha, R., “Hybrid Adaptive Launch Vehicle Ascent Flight Control,”
AIAA Guidance, Navigation and Control Conference and Exhibit, AIAA Paper 2009-
5958, Chicago, IL, Aug. 10-13, 2009.
• LeFevre, B., and Jha, R., “Launch Vehicle Ascent Flight Control Augmentation via a
Hybrid Adaptive Controller,” AIAA Guidance, Navigation and Control Conference
and Exhibit, AIAA Paper 2008-7130, Honolulu, HI, Aug. 18-21, 2008.
xiii
1
Chapter 1
Introduction
Chapter 1 Introduction
Cutting edge aerospace research plays a key role in the never-ending quest to explore our
solar system. Outlined in the Vision for Space Exploration policy in 2004 (and later finalized by
the NASA Authorization act of 2005), the space program was given the goal of developing
vehicles and technology that can carry the United States back to the Moon, and enable
exploration of what lies beyond. The Space Shuttle is unable to meet this objective because it
was only designed for trips to Earth orbit, and thus can not withstand the rigors of the 25,000
mph exit and re-entry into Earth’s atmosphere that is required by a mission to another celestial
body. As a result, the Constellation program and the Ares series of launch vehicles were born.
The Ares class of spacecraft (shown in Fig. 1) has evolved into a family of long and slender
rockets, not unlike the Saturn class of rockets which flew under the direction of the Apollo
program from the 1960’s to the early 70’s. Though this enables NASA to draw on years of
research and experience gained from the Apollo missions, the integration of technological
advancements which have taken place since man last set foot on the Moon is crucial in
broadening mission capabilities for Constellation. Ultimately, every facet of launch vehicle
1
technology must undergo a thorough re-examination if manned space exploration is to continue
its growth in a safe, reliable, and efficient manner.
1
Saturn V Ares V
Ares I
Space Shuttle
Figure 1. Progression of launch vehicle design.
1.1 Launch Vehicle Flight Control: Past & Present The flight control system’s ability to accurately guide a launch vehicle along a desired
trajectory is vital to the successful execution of mission plans. The overwhelming majority of
aerospace guidance systems that are currently certified for implementation in hardware revolve
around a gain-scheduled classical linear feedback controller. When designing a classical flight
controller for any spacecraft, the vehicle dynamics must be modeled with great accuracy in order
for the control system to function as intended when the vehicle is in operation. This requirement
can make the design process very time consuming and expensive. In addition, if the vehicle
experiences flight conditions outside of those considered by the numerical model used in the
2
design process, the nominal performance and stability specifications associated with the classical
controller are no longer necessarily satisfied. Adaptive control systems have the ability to adjust
to changing system dynamics and reshape their control output accordingly, while preserving
stability and performance of the system. This added flexibility also decreases developmental
costs by lessening the need for an extremely accurate plant model. If utilized aboard a launch
vehicle, the enhanced protection from un-modeled effects or unanticipated disturbances afforded
by the adaptive controller would greatly increase the likelihood of mission success.
A number of studies have been conducted in recent years which focus on extending the
benefits of an adaptive control system to launch vehicle flight control, and several different
fundamental approaches have been utilized. A few of the resulting control architectures include
one which uses two neural networks that work in tandem to compensate for lumped uncertainty
and reconstruction error,1 a controller that forms its output from sensory and reward signals that
are interpreted by a learning algorithm modeled after the human brain,2 a linear-adaptive
technique which uses disturbance observers to compensate for dynamic inversion error and
subspace stabilization to guide the behavior of the error state,3 a L1 based adaptive method which
uses the output of a low pass filter to guide the parameters of the control law and guarantee
certain measures of performance such as the desired transient tracking behavior,4 and a model
reference direct adaptive controller that calls upon Lyapunov stability theory to control the
motion of the launch vehicle in the presence of environmental and dynamic uncertainty.5 To
further develop the feasibility of adaptive space vehicle flight control, NASA commissioned the
Advanced Guidance and Control program at the Marshall Space Flight Center (MSFC). In
conjunction with several universities and contractors, systems were developed that included a
dynamic inversion-based controller whose adaptive elements monitor and compensate for
3
variations in control effectiveness while online,6 an adaptive backstepping controller which
utilizes process identification to offset actuator failures and online trajectory command reshaping
to generate an attainable flight path for the degraded vehicle,7,8 a sliding mode controller which
incorporates a sliding mode disturbance observer and gain adaptation to counteract the effects of
bounded uncertainties while maintaining low feedback gains,9 and a suboptimal ‘θ–D’ control
technique which reduces online computational expenses typically associated with optimal control
of a nonlinear system by perturbing the cost function.10
While the benefits of a strictly adaptive framework (such as those mentioned above) cannot
be ignored, methods of quantifying the stability of adaptive systems have yet to be accepted by
the aerospace community on par with traditional stability margin metrics for linear systems.
Favoring an incremental approach, the next step towards obtaining a truly robust flight controller
involves blending the heritage and flight-proven abilities of classical linear feedback control with
the flexibility of an adaptive system. Thus, a combination of these two architectures has been
proposed for implementation aboard the next generation of launch vehicles.11 However, as
compared to the number of studies that concentrate on purely adaptive control schemes, limited
research has been dedicated to the integration of classical and adaptive control. A number of
these works focus on the augmentation of a model reference linear feedback controller with a
neural network whose purpose is to accommodate for uncertainty or unmodeled dynamics by
contributing directly to the control signal.12-14 A similar model reference direct adaptive control
scheme, which includes a control hedging method that prevents the adaptive element from
adapting to selected input characteristics (e.g., actuator position and rate constraints), has been
extended to launch vehicles.15,16 Yet another direct adaptive control approach was developed to
augment the existing linear feedback flight control architecture and preserve stability in the
4
presence of reduced control effectiveness which could be attributed to interaction with structural
bending modes.17 Muse and Calise also introduced the concept of ‘virtual hedging’ to enable
stable adaptation of the controller’s states even when contributions from the adaptive elements
are deemed unnecessary and ignored.18
1.2 Research Objectives The current study revolves around the ascent flight control system of Ares I, since it is
scheduled to be the first launch vehicle to fly under the Constellation program. In addition, the
Ares I is aerodynamically unstable since the center of mass lies aft of the center of pressure (i.e.,
the point through which all aerodynamic loads could be resolved into a single force). This further
predicates the need for a robust guidance control system. Consequently, a hybrid adaptive
control scheme (originally developed at the NASA Ames Research Center for stability recovery
of damaged aircraft)19-22 was selected for development as an Ares I flight controller. This hybrid
adaptive control approach has not been previously considered for application to launch vehicles.
The hybrid adaptive controller (so named because it utilizes both direct and indirect adaptive
networks) contains a reference model that describes the desired handling qualities of the vehicle,
a classical linear feedback element which operates on the tracking error, a direct adaptive control
element which augments the linear feedback signal in an attempt to cancel out the modeling error
arising from nonlinearities in the vehicle dynamics, a dynamic inversion operation that generates
control signals from desired angular rates, and an indirect adaptive element which tracks the
evolution of characteristic parameters contained in the dynamic inversion. The combination of
these classical and adaptive elements enables the flight controller to retain the flexibility of an
adaptive system while relying heavily on the linear feedback element to dictate the control signal
under most operating conditions. In turn, this produces small tracking errors while allowing the
5
adaptive learning rates to be kept at a minimum. Maintaining a low adaptive learning rate
minimizes the possibility of developing high-gain control and its associated adverse effects (i.e.,
high frequency oscillation in the control signal).
The main objectives of this research are to modify the hybrid adaptive control architecture for
use as an Ares I launch vehicle ascent flight controller, investigate its performance in numerical
simulations that encompass both nominal and off-nominal flight conditions, and compare this
performance with that of a classical gain-scheduled linear feedback flight controller. To achieve
these objectives, this research makes the following major contributions:
• The ascent dynamics of the Ares I launch vehicle are characterized in Chapter 2.
Equations of motion are developed and subsequently linearized so that the flight
controller’s dynamic inversion can generate control signals via matrix algebra.
• Modifications of the hybrid adaptive flight control architecture that are necessary for
integration with Ares I are described in Chapter 3. These alterations include:
o Conversion of guidance command input from quaternion form (as provided by
NASA) to the body-fixed angular rate commands required by the flight controller.
o Examination of a recursive least squares and a modified multidimensional
Newton’s method based online parameter identification algorithm which serves as
the indirect adaptive element. These two algorithms are selected for their
distinctly different convergence speed, and the effects of rapid changes in the
parameter estimates are discussed in the results.
o Modification of the input structure of the neural network (i.e., the direct adaptive
component) so that it can recreate the nonlinear terms contributing to the dynamic
response of the launch vehicle.
6
o Examination of tracking error based versus modeling error based neural network
weight update laws. Differences in the derivation of each update law yield
drastically different behavior of the network output.
o Integration of signal filters into the feedback loop which serve to prevent harmful
interaction between the flight control system and structural bending modes.
• To validate the performance of the hybrid adaptive launch vehicle flight control system, a
high-fidelity Ares I ascent simulator23 is obtained from MSFC which contains a classical
gain-scheduled proportional-integral-derivative (PID) ascent flight controller for
comparative purposes. Chapter 4 discusses the properties of this simulator (referred to as
SAVANT) and the baseline linear flight controller, in addition to presenting results of
ascent simulations which compare the tracking performance of the hybrid adaptive and
baseline PID flight controllers under the following circumstances:
o Rigid and flexible body vehicle dynamics are considered with structural
flexibility effects appropriately disabled or enabled within SAVANT.
o Nominal and off-nominal (i.e., reductions of the 1st stage rocket motor’s thrust
and increases in the aerodynamic loads) flight conditions are considered in both
rigid and flexible cases.
A summary of results obtained and conclusions reached, as well as a discussion of potential
future enhancements, is given with the closing remarks in Chapter 5.
7
2
Chapter 2
Rigid Launch Vehicle Ascent Dynamics
Chapter 2 Rigid Launch Vehicle Ascent Dynamics
Given that the flight controller developed herein utilizes dynamic inversion, equations of
motion are sought which characterize the rigid body dynamics of the 1st stage of ascent for a
launch vehicle (i.e., from liftoff to separation of the 1st and 2nd stages). It is a requirement of such
equations that they capture the dependence of the spacecraft’s response on the control input and
other important state variables. Following their development, the equations of motion are
linearized so that the desired control signal may be obtained by inverting the nominal dynamics
with simple matrix algebra. The choice of linearizing assumptions is guided by the desire to
characterize the dominant behavior of the launch vehicle during ascent.
This study focuses on a flight controller which generates control signals based on angular rate
commands and appropriate feedback. Thus, the analysis begins by considering the flow of
angular momentum which governs the rotational rigid body dynamics of a launch vehicle during
its 1st stage ascent phase of flight. The development of these equations accounts for the fact that
vehicle mass varies greatly throughout ascent, since a significant percentage of the overall
vehicle weight at launch is attributed to the propellant contained in the rocket. Other related
8
time-dependent quantities include the location of the center of mass, the components of the
inertia tensor, the rate of mass flow out of the rocket nozzle, and the propulsive force generated
by the rocket motor.
2.1 Flow of Angular Momentum First, consider a rigid Ares I launch vehicle with a body-fixed coordinate system whose origin
lies at the center of mass (cm) as shown below.
2
Top View
cm
The x, y, and z body
describe the rotation
be written which g
surrounding environ
1
where the vector
spacecraft, I is the in
M
subsequent notation
produces the followi
y
Figur
-fixed ax
al motion
overns t
ment24
ertia tens
[ ML=
abides by
ng relatio
z
x
e 2. Ares
es are ref
of the ve
he transf
or, and
TN ]
ω
the impl
nship
y
I body-f
erred to
hicle in
er of an
HM ≡= &
contains
[ qp=
icit defin
z
ixed coordinate system.
as the roll, pitch, and yaw axes, respectively. To
this reference frame, the following equation can
gular momentum between the rocket and its
( )Iωdtd (1)
the resultant external moments acting on the
]Tr is the angular velocity vector. Note that all
ition that I ≡ I(t). Expanding the time derivative
9
2 (2) tHHωωIM && +×+=
The term represents the rate of angular momentum transfer from the variable mass system.tH& 24
It includes any angular momentum flux created by the reduction in mass due to the burning of
propellant (i.e., effects that would be contained in the term ) and the flow of the resulting
exhaust gases out of the rocket nozzle.
ωI&
The 1st stage solid rocket motor that powers Ares I during ascent is descendant from the twin
booster rockets that propel the Space Shuttle into orbit. These solid-fuel rockets are made up of a
cylindrical metal outer casing that is lined with propellant via a casting process. At launch, an
incendiary device at the nose of the rocket causes the inner surface of the propellant ‘tube’ to
ignite. Given that all particles involved in this combustion interact in a manner which is
consistent with Newton’s third law of equal and opposite reactions, the following fundamental
assumption can be made: “The angular momentum flux is fully conserved during the
transformation from solid propellant to gases that takes place at the burning surface. This angular
momentum is eventually carried out of the system by the gases through the exit nozzle.”25 This
identity enables the term to be reduced to the angular momentum flux contributions from jet-
damping.
tH&
25 Jet-damping refers to the damping of the motion of the rocket’s longitudinal (i.e.,
roll) axis which is caused by the transfer of angular momentum about the transverse axes. When
the rocket rotates about a transverse axis, the exhaust gases still contained within the rocket
acquire angular momentum as well. This interaction places a reactionary torque on the body of
the rocket which damps the motion about the transverse axes and enhances the gyroscopic
stability of the roll axis. Taking this effect into consideration, Eq. (2) now becomes
3 jetTHωωIM −×+= & (3)
10
where Tjet signifies the jet-damping torque. In component form about the body-fixed axes, it is
given by25
4 [ ]Tjet rqm 02 ⋅= l&T (4)
where is the mass flow rate out of the rocket exhaust nozzle, and is the effective lever arm
(see Fig. 2) between the center of mass of the vehicle and the mass flow center of the exhaust
nozzle. The mass flow center is located in the exit plane of the rocket exhaust nozzle and is
nominally aligned with the geometric center of a circular nozzle. The absence of any jet-damping
torque about the roll axis has been proven analytically in Ref. 25. In scalar form, Eq. (3) can now
be written as
m& l
(5a) prIpqIrqIqrIIrIqIpIL xyxzyzyyzzxzxyxx −+−+−+++= )()( 22&&&
5 (5b) qmpqIqrIprIprIIrIqIpIM yzxyxzzzxxyzyyxy222 )()( l&&&& −−+−+−+++=
(5c) rmqrIprIqpIpqIIrIqIpIN xzyzxyxxyyzzyzxz222 )()( l&&&& −−+−+−+++=
Observe that the equations of motion above do not account for any variation in the jet-
damping torque due to rotation of the rocket nozzle about its pitch or yaw gimbals. Within the
scope of this study, it is assumed that the engine gimbal angles change slowly with respect to
time (i.e., their angular accelerations are sufficiently small) so that the resulting “tail-wags-dog”
effect is negligible.
The external disturbances which act on a launch vehicle during ascent consist of contributions
from a number of sources. Most notably, these include the aerodynamic loading, rotation of the
thrust vector about the engine gimbals, and the moment created by the roll control system (RCS).
Taking these three main sources into account, the resultant moment can be expressed as
6 RCSrktaero MMMM ++= (6)
11
where Maero contains the aerodynamic moments, Mrkt describes the torque on the launch vehicle
due to rotation of the rocket’s thrust vector, and MRCS is the moment created by the RCS. Each of
these quantities is evaluated in detail in the following sections.
2.2 Aerodynamic Loads When computing the aerodynamic moments, it is important to note that the rocket’s center of
mass continually shifts as propellant is consumed. To accommodate for this shift, the
aerodynamic moments are expressed as [ ]Taeroaeroaeroaero NML=M
7 ARPARPARPaero FrMM ×+= (7)
where MARP contains the aerodynamic moments about a fixed aerodynamic reference point
(ARP) on the vehicle, FARP contains the aerodynamic forces about this same reference point, and
is the position vector of the ARP with respect to the center of mass.
SAVANT dictates that the aerodynamic moments about the ARP are given by
[ TARPARPARPARP zyx=r ]
23
8 [ ]TnmlrefdARP CCCScq βα βα ,,⋅=M (8)
Similarly, the aerodynamic forces at the ARP are specified as23
9 [ ]TzyxdARP CCCSq αβ αβ ,, −−⋅=F (9)
where qd represents the dynamic pressure, S is the aerodynamic reference area, cref is the
aerodynamic reference length, α is the angle of attack (or pitch angle), β is the sideslip (or yaw)
angle, the vector [ Tnml CCC βα βα ,, ] contains the moment coefficients about the ARP, and the
vector [ Tzyx CCC αβ αβ ,, −− ] contains the coefficients of force acting through the ARP.
Observe that SAVANT’s linearized aerodynamic model takes into account the fact that launch
vehicles are designed to operate with minimal angular velocity ω in a small flight envelope
around α = 0 and β = 0. This allows the model of the rolling moment and drag force coefficients
12
to be independent of the aerodynamic angles, and eliminates coupling in the remaining
aerodynamic coefficients. Also, considering the vehicle’s geometry and lack of airfoils,
SAVANT dictates the use of a single aerodynamic reference area S and aerodynamic reference
length cref. The dynamic pressure, as a function of the varying atmospheric density ρ and
increasing launch vehicle velocity magnitude V, is defined by
10 2
21 Vqd ρ= (10)
Observe that the negative signs in Eq. (9) account for the orientation of the body-fixed
coordinate axes (i.e., drag acts in the negative x-direction and lift acts in the negative z-
direction). Also note that the aerodynamic force and moment coefficients, while modeled as
explicit linear functions of α and β as per Eqs. (8) and (9), are implicitly dependent on Mach
number. That is, )(MaCC ≡ . This is consistent with modeling methods implemented in the
SAVANT ascent flight simulator.23
By evaluating Eq. (7), the aerodynamic moments in scalar component form are
[ ]βα βα ,, yARPzARPlrefdaero CzCyCcSqL −−⋅= (11a)
11 ( )[ ]xARPzARPmrefdaero CzCxCcSqM −+⋅= ααα ,, (11b)
( )[ ]xARPyARPnrefdaero CyCxCcSqN ++⋅= βββ ,, (11c)
2.3 Thrust Vectoring & Roll Control The purpose of the roll control system is to provide sufficient control authority over the
rolling motion of the spacecraft which cannot be supplied by rotation of the rocket nozzle alone.
To accomplish this task, small transverse thrusters are positioned near the nose of the vehicle to
generate a control torque about the roll axis. The resulting moment contribution from the RCS is
12 [ ]TRCSRCS L 00=M (12)
13
where LRCS is the magnitude of the control moment about the spacecraft’s longitudinal roll axis.
As indicated in Eq. (12), the thrusters of the RCS are aligned such that they generate no moment
about either of the body-fixed transverse axes.
The vast majority of the control torque required during ascent is generated by a rotation of the
rocket nozzle. Within the scope of this study, the flow of exhaust gases is assumed to be
axisymmetric about a vector which is normal to the circular exit plane of the rocket nozzle and
aligned with its geometric center (i.e., a vector which passes through the nominal mass flow
center). Thus, the thrust generated by the rocket can be modeled as a resultant force which acts
through both the mass flow center and the point of intersection of the engine gimbal axes. The
body-fixed gimbal axes of the rocket nozzle are introduced below.
3
zgimy
cmx
xgim
ygim
z
Figure 3. Rocke
In Fig. 3 above, ygim represents the pitch gim
The orientation of these axes is chosen such t
will generate a positive pitching moment in th
for the yaw gimbal. Given consecutive positiv
gimbals ( ),yp θθ → the vector describing the
as shown in Fig. 4. ( 210 uuu →→ )
t
b
h
e
e
nozzle gimbal axes.
al axis and zgim represents the yaw gimbal axis.
at a positive rotation of the pitch engine gimbal
reference frame affixed to the cm, and likewise
angular displacements about each of the engine
line of action of the resultant thrust is altered
4
14
a) Pitch gimbal displacement. b) Yaw gimbal displacement.
Figure 4. Thrust vector rotation.
After displacement about the pitch gimbal by an angle θp, the thrust vector is characterized by
13 01 uRu ⋅= p (13)
where is the thrust vector after this first rotation, is the initial thrust vector, and R1u 0u p is a
rotation matrix defining the angular displacement about the pitch gimbal axis. The well-known
form of such a rotation matrix is given by
14 (14) ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
)cos(0)sin(010
)sin(0)cos(
pp
pp
p
θθ
θθR
To arrive at the final thrust vector orientation, the intermediate vector is rotated by an angle
θ
1u
y about the yaw gimbal axis. The realigned thrusting force is then
15 12 uRu ⋅= y (15)
where is the thrust vector after this second rotation and R2u y is the corresponding rotation
matrix. The components of the matrix describing a rotation about the yaw gimbal are as follows.
16 (16) ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −=
1000)cos()sin(0)sin()cos(
yy
yy
y θθθθ
R
15
Combining Eqs. (13) and (15) yields
17 02 uRRu ⋅⋅= py (17)
which, after evaluation, results in the following equation describing the transformation of the
thrust vector in the reference frame defined by the engine gimbal axes
18 (18) 02
)cos(0)sin()sin()sin()cos()sin()cos()cos()sin()sin()cos()cos(
uu ⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−
−=
pp
ypyyp
ypyyp
θθθθθθθθθθθθ
Given that the resultant thrust is initially aligned with the longitudinal axis of the spacecraft (i.e.,
) the propulsive force generated by the rocket is found to be [ TthrustF 0010 ⋅=u ] rktF
19 (19) ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
)sin()sin()cos()cos()cos(
p
yp
yp
thrustrkt Fθ
θθθθ
F
where has been replaced by to clarify the notation, and F2u rktF thrust is the magnitude of the
thrusting force created by the rocket engine. To transform the thrust vector back to the body-
fixed coordinate system whose origin is at the center of mass, the signs of the y-axis and z-axis
components must be reversed (see Fig. 3). This produces
20 (20) ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=)sin(
)sin()cos()cos()cos(
p
yp
yp
thrustrkt Fθ
θθθθ
F
The moment created by a rotation of the rocket nozzle [ ]Trktrktrktrkt NML=M is then
21 rktgimrkt FrM ×= (21)
and is the location of the point of intersection of the engine gimbal axes
O' with respect to the center of mass in the body-fixed coordinate system. Evaluation of Eq. (21)
yields the following scalar components
[ Tegegeggim zyx=r ]
16
( )ypegpegthrustrkt zyFL θθθ sincossin += (22a)
22 ( )ypegpegthrustrkt zxFM θθθ coscossin +−= (22b)
( )ypegypegthrustrkt yxFN θθθθ coscossincos −−= (22c)
2.4 Equations of Rotational Motion Combining Eqs. (11), (12), and (22) as dictated by Eq. (6) produces the following expressions
for the resultant external moments acting on the launch vehicle during its ascent
[ ]
( RCSypegpegthrust
yARPzARPlrefd
LzyF
CzCyCcSqL
+++
−−⋅=
θθθ
βα βα
sincossin ,,
) (23a)
23 ( )[ ]
( )ypegpegthrust
xARPzARPmrefd
zxF
CzCxCcSqM
θθθ
ααα
coscossin ,,
+−+
−+⋅= (23b)
( )[ ]
( )ypegypegthrust
xARPyARPnrefd
yxF
CyCxCcSqN
θθθθ
βββ
coscossincos ,,
−−+
++⋅= (23c)
After reassembly from Eqs. (5) and (23), the equations of rotational motion for ascent of a rigid
launch vehicle with a single gimbaled exhaust nozzle can be written as
[ ] ( )
prIpqIrqIqrIIrIqIpI
LzyFCzCyCcSq
xyxzyzyyzzxzxyxx
RCSypegpegthrustyARPzARPlrefd
−+−+−+++=
+++−−⋅
)()(
sincossin22
,,
&&&
θθθβα βα (24a)
24 ( )[ ] ( )
qmpqIqrIprIprIIrIqIpI
zxFCzCxCcSq
yzxyxzzzxxyzyyxy
ypegpegthrustxARPzARPmrefd
222
,,
)()(
coscossin
l&&&& −−+−+−+++=
+−+−+⋅ θθθααα (24b)
( )[ ] ( )
rmqrIprIqpIpqIIrIqIpI
yxFCyCxCcSq
xzyzxyxxyyzzyzxz
ypegypegthrustxARPyARPnrefd
222
,,
)()(
coscossincos
l&&&& −−+−+−+++=
−−+++⋅ θθθθβββ (24c)
The equations of motion given above must be linearized in terms of the control variables
before they can be implemented in the hybrid adaptive flight controller’s dynamic inversion
scheme that is described in Chapter 3. Describing the dynamics as a linear function of
measurable state variables and the control signals enables the flight controller to calculate the
17
nominal control input necessary to impart the desired guidance corrections via simple matrix
algebra. This well-known approach keeps computation costs low so as to maintain feasibility
when implemented in real time.
To linearize Eq. (24), two simplifications are made. First, it is noted that the angular velocity
of the spacecraft remains small during the ascent phase of flight. This is a valid
assumption for launch vehicles because structural loads, due in large part to aerodynamic drag,
become quite significant soon after takeoff. Consequently, the angular velocity is kept at a
minimum during ascent to limit bending stress on the frame of the spacecraft. Considering this
fact, the equations of motion become
[ Trqp=ω ]
[ ] ( )
rIqIpI
LzyFCzCyCcSq
xzxyxx
RCSypegpegthrustyARPzARPlrefd
&&& ++=
+++−−⋅
sincossin,, θθθβα βα (25a)
25 ( )[ ] ( )
qmrIqIpI
zxFCzCxCcSq
yzyyxy
ypegpegthrustxARPzARPmrefd
2
,,
coscossin
l&&&& −++=
+−+−+⋅ θθθααα (25b)
( )[ ] ( )
rmrIqIpI
yxFCyCxCcSq
zzyzxz
ypegypegthrustxARPyARPnrefd
2
,,
coscossincos
l&&&& −++=
−−+++⋅ θθθθβββ (25c)
where the second order terms in the scalar components of ω are omitted. The other simplifying
observation to be made is that the engine gimbal angles remain small throughout ascent. To
further support this assumption, it is noted that the high-fidelity SAVANT ascent flight simulator
dictates an engine gimbal saturation angle of 10 degrees. Thus, considering the small angle
identities of and produces aa ≡)sin( 1)cos( ≡a
[ ] ( )
rIqIpI
LzyFCzCyCcSq
xzxyxx
RCSyegpegthrustyARPzARPlrefd
&&& ++=
+++−−⋅
,, θθβα βα (26a)
26 ( )[ ] ( )
qmrIqIpI
zxFCzCxCcSq
yzyyxy
egpegthrustxARPzARPmrefd
2
,,
l&&&& −++=
+−+−+⋅ θααα (26b)
18
( )[ ] ( )
rmrIqIpI
yxFCyCxCcSq
zzyzxz
egyegthrustxARPyARPnrefd
2
,,
l&&&& −++=
−−+++⋅ θβββ (26c)
The linearized equations of rotational motion for ascent of a rigid launch vehicle with a single
gimbaled exhaust nozzle, given by Eq. (26), can also be expressed in matrix form as
27 δgσfωfωI ⋅+⋅+⋅=⋅ 21dtd (27)
where contains a bias term and the two aerodynamic angles, and the control
signal inputs to the plant are contained in
[ Tβα1=σ ]
[ ]TypRCSL θθ=δ . The coefficient matrices shown
in Eq. (27) are given by
28
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
++−
+−
−−
=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
ββ
αα
βα
,,
,,
,,
2
2
21
0
0
0000
1
00
00000
yARPnrefxARPd
egthrust
zARPmrefxARPd
egthrust
yARPzARPlref
d
eg
eg
egegthrust
thrust
CxCcCySq
yF
CxCcCzSqzF
CzCyCc
Sq
xx
zyF
Fm
m
f
gfl&
l&
(28)
Observe that the first column of (corresponding to the bias term in σ) captures both
aerodynamic and thrusting moment contributions. Reorganizing Eq. (27) yields
2f
29 δGσFωFω ⋅+⋅+⋅= 21& (29)
where the coefficient matrices of this state-space type representation are
30 (30) gIGfIFfIF ⋅=⋅=⋅= −−− 12
121
11
19
3
Chapter 3
Hybrid Adaptive Ascent Flight Control
Chapter 3 Hybrid Adaptive Ascent Flight Control
By combining classical and adaptive control elements, the hybrid adaptive control scheme
relies heavily on linear feedback signals to dictate the nominal response while it also enables the
flight control system to compensate for unmodeled effects or unanticipated changes in the
vehicle dynamics. In addition to increasing the overall robustness of the control system, this
approach places less emphasis on modeling all possible flight conditions to a high degree of
accuracy. Supplementing these features are feedback signal filters which are integrated within
the hybrid adaptive flight controller for the purpose of preventing potential interaction between
the flight control system and structural bending modes. The resulting hybrid adaptive ascent
flight control architecture has not been previously considered for application to launch vehicles.
The hybrid adaptive flight control system utilizes both direct and indirect adaptive elements
integrated with a classical model reference dynamic inversion controller which calculates the
control signals necessary to maintain desired angular rate commands. The classical linear portion
of the feedback signal is formed by applying predetermined gains to the components of the
tracking error. The direct adaptive element is a neural network (NN) which augments the rate
20
command input to the dynamic inversion. The indirect adaptive element is comprised of a
parameter estimation algorithm which adjusts the inversion matrices that describe the dynamic
response of the plant, thereby not affecting the control input to the dynamic inversion directly
(hence the ‘indirect’ designation). The combination of these two adaptive components allows the
dynamic inversion to track the actual behavior of the plant while the NN compensates for any
residual error between the command input to the dynamic inversion and the nonlinear response
of plant. Given that these nonlinearities remain small, distributing the adaptation in this manner
also minimizes the possibility of developing high gain feedback and its associated ill effects. In
the present study, the hybrid adaptive flight controller is used to accommodate for the evolving
dynamic behavior of a launch vehicle during ascent. This requires the interpretation of guidance
commands in quaternion form, investigation of different parameter estimation algorithms, and
alteration of the input structure and weight update law contained in the NN. Also, since the long
and slender design of Ares I causes structural flexibility to become a concern, signal filters which
inhibit interaction between the flight control system and structural bending modes are
incorporated within the feedback loop. Description of the generic hybrid adaptive approach,
implemented in a flight controller for a rigid aircraft, can be found in Refs. 19-22.
3.1 Control Architecture Guidance system input consists of a pre-determined quaternion command qc which defines
the desired trajectory. This quaternion command generates an angular velocity command ωc via
a control conversion operation which is governed by26
31 c
cxcyc
xcczc
yczcc
zcycxc
c
qqqqqq
qqqqqq
ωq ⋅
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
−−
−−−−
=
0
0
0
21
& (31)
21
where and the quaternion derivative is discretely approximated as [ Tzcycxccc qqqq0=q ]
32 t
kckckc ∆
−= − )1()(
)(
qqq& (32)
The subscripts (k) and (k-1) indicate quantities sampled at the current and previous time step,
respectively, and ∆t is the sample time of the flight control system. The hybrid adaptive ascent
flight controller is shown schematically in Fig. 5. Elements that have been added are outlined in
green and elements whose characteristics have been modified are outlined in blue.
5
Figure 5. Hybrid adaptive launch vehicle ascent flight control architecture.
The angular velocity command is processed by a first-order reference model which dictates
the desired handling characteristics of the spacecraft. This produces a reference angular velocity
ωm and a reference angular acceleration according to mω&
33 cnmnm ωωωωω =+& (33)
where ),,( rqpn diag ωωω=ω is the reference model frequency matrix. These natural frequencies
are chosen to generate a desirable transient response from the plant while satisfying all actuator
position and rate limit constraints. The reference angular velocity is then compared to angular
velocity feedback from the plant ω to form the tracking error ωe in accordance with
ωc ωm
.
ω
ω,σ,δ
qc ue δ
ωm
+.
β(Di)
∆f
uad
ωd ωe ( )σωω ,,1df &− PI Ares I
NN
Control Conv.
Flex Filter
Param. Update
Ref. Model
22
34 ωωω −= me (34)
Subsequently, the linear portion ue of the feedback control vector is generated by applying a
proportional-integral (PI) control scheme to the angular velocity tracking error as shown.
35 (35) ∫+=t
eIePe d0
τωKωKu
Proper selection of the proportional KP and integral KI gains to ensure good behavior of the
tracking error dynamics will be discussed shortly. To track the output of the reference model, the
desired angular acceleration is specified as dω&
36 ademd uuωω −+= && (36)
where the adaptive control signal uad is designed to cancel out the inversion error arising from
the difference between the linear model utilized in the dynamic inversion and the true nonlinear
behavior of the spacecraft dynamics. Given that the nonlinearities remain small, this enables the
desired angular acceleration to approach the reference angular acceleration while the tracking
error trends to zero asymptotically. The linearized dynamic model described by Eq. (29) is
ultimately inverted to obtain the control signal δ which will produce the desired angular
acceleration given the current state of the plant dynamics as described by
37 ( )σFωFωGδ 211 −−= −
d& (37)
It is important to note that the dynamic inversion matrices F1, F2, and G, whose nominal values
are given by Eq. (30), are constantly being adjusted by the parameter identification algorithm to
reflect the dynamic response of the spacecraft during ascent.
Returning to the issue of proper PI gain selection, an examination of the error dynamics
reveals constraints which aid in maximizing the stability of the system. To begin with, consider
23
the form of the modeling error ε arising between the dynamic inversion and the response of the
spacecraft. Expressed as a difference in angular accelerations, this error is
38 dωωε && −= (38)
Expanding the desired angular acceleration in terms of its components and noting the definition
of the tracking error given by Eq. (34) yields
39 adeeadem uuωuuωωε +−−=+−−= &&& (39)
Rearranging terms produces the following expression
40 ( )εuuω −+−= adee& (40)
Finally, expressed in matrix form, the error dynamics of the system are described by
41 ( )εuBAee −+= ad& (41)
where and T
Te
tTe d ⎥
⎦
⎤⎢⎣
⎡= ∫ ωωe
0
τ
42 (42) ⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡−−
=I0
BKKI0
API
The matrix A describes the effect that the PI portion of the hybrid adaptive controller has on
the tracking error dynamics. Given that the gain matrices KP and KI are diagonal (and thus
commutative), the characteristic polynomial of matrix A is given by
43 (43) 0KλKλ =++ IP2
where λ represents a vector of eigenvalues and the power function operate element-wise. This
quadratic equation has the following roots
44 21
2
42 ⎟⎟⎠
⎞⎜⎜⎝
⎛−±
−= I
PP KKKλ (44)
24
Maximizing the negative real part of these eigenvalues insures that any transients introduced to
the system will decay as quickly as possible. Doing so requires
45 4
2P
IKK ≥ (45)
Introducing the damping ratio ζ allows Eq. (45) to be rewritten as
46 ( ) PIPI KKζKKζ =⇒= 21
22 24 (46)
where . Also, observe that the homogeneous characteristic equation describing the
dynamics of a second order system in the frequency domain is given by
10 ≤< ζ
47 (47) 0ωζω =++ 22 2 nnss
Equating the natural frequencies of the reference model and the system dynamics will make
certain that the system exhibits good low-gain tracking error performance.19 Therefore, the PI
gain matrices are specified as
48 (48) )2,2,2(
),,( 222
rrqqppP
rqpI
diagdiag
ωζωζωζωωω
==
KK
Each damping ratio is chosen to be 10 ≤< ζ . This choice of gains is also affirmed by Eq. (46),
whose results are verified by equating the coefficients of Eqs. (43) and (47).
3.2 Online Parameter Identification Throughout ascent, the behavior of the launch vehicle is constantly evolving due to changes in
the amount of thrust produced by the rocket, variation in the moments of inertia as propellant
burns, variation in the dynamic pressure, changes in characteristic aerodynamic parameters such
as force and moment coefficients, and perturbation of the location of the center of mass. These
variations are reflected in the dynamic inversion matrices, whose nominal values are given by
Eq. (30). To track these changes, two different process identification approaches are
25
investigated: a recursive least squares (RLS) method, and a parameter update procedure derived
from a modified multidimensional Newton’s iteration.
Given that sensed angular acceleration feedback ω is not directly available, the rotational
motion of the launch vehicle is approximated in discrete form by
&
49 )()1()1()( ˆ kkkk t eωωω +⋅∆+= −− & (49)
where ∆t is the sample time of the flight controller, the subscript (k) denotes a quantity sampled
at the kth time step, and is prediction error in the k)(ˆ ke th sample. The angular acceleration term
can then be expanded according to the linearized equations of motion, given by Eq. (29).
Combining like terms and rearranging yields
50 (50) )1()1()()(ˆ −− ⋅−= kTkkk θΘωe
The characteristic plant parameters are contained in
51 [ ])1()1(2)1(1)1( −−−− ⋅∆⋅∆⋅∆+= kkkTk ttt GFFIΘ (51)
and is a vector of measurable state variables and control signals. [ TTk
Tk
Tkk )1()1()1()1( −−−− = δσωθ ]
The RLS method is a recursive formulation which attempts to minimize the square of the
prediction error by computing the optimal matrix of updated parameter estimates at each time
step. It also employs adaptive directional forgetting which serves to reduce the contributions
from errors observed in the remote past. Consequently, the matrix of parameter estimates Θ is
updated by the RLS recursive relation27
52 )()(
)1()1()1()( ˆ
1 kk
kkkk e
θCΘΘ ⋅
⎥⎥⎦
⎤
⎢⎢⎣
⎡
+
⋅+= −−
− ξ (52)
The supplemental identification variable ξ is defined as
53 (53) )1()1()1()( −−− ⋅⋅= kkTkk θCθξ
26
and C is a square covariance matrix (of proper dimension) which monitors the sensitivity of the
parameter adjustment to each state and control variable. If 0)( >kξ , C is updated according to
54 )(
1)(
)1()1()1()1()1()(
kk
kTkkk
kk ξε +
⋅⋅⋅−= −
−−−−−
CθθCCC (54)
where the second supplemental scalar ε is formulated as follows
55 )(
)1()1()(
1
k
kkk ξ
ϕϕε −
−
−−= (55)
The adaptive directional forgetting factor ϕ is ultimately calculated for each sampling period as
56 )(
)(
)()(
)()()(
1)( 1
11
)1()1ln()1(1
k
k
kk
kkkk ξ
ξηξ
υηξγϕ
+⎥⎥⎦
⎤
⎢⎢⎣
⎡−
++
+++⋅++=− (56)
and the final set of auxiliary variables is defined by
57 )(
)()()(
ˆˆ
k
kTk
k λη
ee ⋅= (57)
58 ( )1)1()1()( += −− kkk υϕυ (58)
59 ⎥⎥⎦
⎤
⎢⎢⎣
⎡
+
⋅+= −−
)(
)()()1()1()( 1
ˆˆ
k
kTk
kkk ξλϕλ
ee (59)
When initializing the RLS algorithm, the following values have been shown to facilitate good
performance in previous studies of self-tuning controllers:27 main diagonal elements of the
covariance matrix , directional forgetting factor 3)0( 10=iiC 1)0( =ϕ , 001.0)0( =λ , ,
and
6)0( 10−=υ
99.0=γ .
Introduced as a significantly more conservative approach to the parameter identification
process, the convergence of the following modified multidimensional Newton’s iteration is
slower. This is brought on by two distinguishing factors: the multidimensional Newton’s method
27
approach operates on the prediction error itself (not the squared prediction error),28 and an
adjustable step size is incorporated in this study to modulate the rate of change in the parameter
estimates. As such, the update for each scalar parameter estimate Θij proceeds according to
60 1ˆ
ˆ~−
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂
∂⋅⋅−=
ij
jjijij Θ
eeΘΘ µ (60)
where ijΘ~ is the previous value of the parameter estimate, µ is a step size which regulates the
speed of adaptation, the subscript ij denotes row and column indices for a matrix, and the
subscript j simply indicates the jth component when attached to a vector. Given the definition of
the prediction error in Eq. (50), evaluating the partial derivative yields
61 iij
j θΘe
−=∂
∂ˆ (61)
Substituting this result into Eq. (60) and expressing the modified Newton’s method weight
update law as a recursive relationship yields
62 )1(
)()1()(
ˆ
−− +=
ki
kjkijkij θ
eΘΘ µ (62)
Observe that indicates the j)(ˆ kje th component of the prediction error evaluated at the kth time
step, and likewise for the other quantities. To prevent excessively large changes in the parameter
estimates when the state variables or control signals become arbitrarily small, the parameter
update is only executed when the absolute value of is greater then a prescribed threshold.
The value of this modified Newton’s method update threshold and the step size µ, in addition to
the information used to initialize , is presented with the results in Chapter 4.
)1( −kiθ
)0(Θ
28
3.3 Neural Network Design The task of the neural network in the hybrid adaptive ascent flight controller is to compensate
for the difference between the truly nonlinear launch vehicle dynamics and the linearized model
implemented in the dynamic inversion. To do so, the hybrid controller utilizes a single-hidden-
layer sigma-pi neural network.19,20 The structure of the sigma-pi NN is illustrated in Fig. 6.
6
D1 Π
Π
Figure 6. Single-hidden-layer sigma-pi neural network.
A sigma-pi NN is a linearly parameterized adaptive network whose output uad is given by
63 (63) ),,,( 4321 DDDDβWu Tad =
where W is a matrix of variable weights and β contains a sufficiently rich set of basis functions
such that, when applied to the network inputs (D1, D2, D3, D4), the inversion error can be
accurately reconstructed at the network output (i.e., the basis functions contained in β, when
applied to the network inputs, must be able to recreate all of the nonlinear terms contributing to
the dynamic response of the launch vehicle). Considering the nonlinear ascent dynamics
described by Eq. (24), the inputs to the NN are specified as
64 [ ]TTT δσωD 11 = (64a)
[ ]TTT rqp ωωωD =2 (64b)
uad
D4
D3
D2
W
Π
Σ
Basis Functions
29
[ ]yypp θθθθ cossincossin3 =D (64c)
[ ]ypypypyp θθθθθθθθ coscossincoscossinsinsin4 =D (64d)
The set of basis functions which describes β can then simply be the identity matrix, since each
term eliminated by linearization of the dynamics is now a distinct input to the NN. As a result
65 [ ]T43214321 ),,,( DDDDDDDDβ = (65)
Two unique approaches to updating the weight matrix W are investigated within this study.
The first to be analyzed is a weight update law founded on work by Rysdyk and Calise in the
area of model reference direct adaptive control.13 In this form, the progression of the weight
update is regulated by the tracking error. Recall that the parameter e contains both the angular
velocity tracking error ωe and its integral. This update law also incorporates a gain Г > 0 to
control the learning rate and an e-modification term µ > 0 to enhance robustness even without
persistent excitation. As such, modification of the network weights proceeds according to
66 ( )WPBePBβeW TT µ+Γ−=& (66)
where the matrices A and B are defined in Eq. (42), P is the solution to the Lyapunov equation
ATP + PA = -I, and the vertical bars indicate the Euclidean norm. A Lyapunov based stability
proof for this weight update law can be found in Ref. 13.
The second weight update law to be evaluated is driven by the modeling error ε instead of the
tracking error. This adaptive law was designed by Nguyen and Jacklin for a direct adaptive flight
controller which employs neural networks.20 It uses a least-squares approach to minimize the
magnitude of the term in the error dynamics, which are characterized by Eq. (41). As
a result, the tracking error is driven to zero asymptotically primarily by the PI portion of the
flight controller. The modeling error based NN weight update law is
( εuB −ad )
67 ( ) ( )TT εWβRβW −+−= −11 κ& (67)
30
where monitors the level of persistent excitation in the system, and is the
modeling error calculated as the difference between the (in this case estimated) angular
acceleration feedback from the plant ω and the desired angular acceleration . The matrix R
is a positive-definite matrix of adaptive learning rates which is updated by
RββT=κ dωωε && −=
& dω&
68 ( ) RRββR T11 −+−= κ& (68)
This adaptive law has been proven to be stable via a Lyapunov analysis, while at the same time it
lowers the sensitivity to unmodeled dynamics present in the system (even when high learning
rates are present in R to force faster convergence).20 Initialization of the weight matrix W and
the learning rate matrix R is discussed in Chapter 4.
3.4 Feedback ‘Flex Filtering’ The potential for harmful interaction between the flight control system and structural bending
modes was recognized as a concern during the development of Ares I because of the launch
vehicle’s strikingly long and slender shape. Consequently, feedback signal ‘flex filters’ were
developed with the purpose of removing the influence of structural flexibility from guidance
tracking error measurements by canceling out sensed frequency content which is generated by
lateral bending of the spacecraft. A single flex filter, formed by superimposing a low pass filter
and a notch filter, is devoted to processing the feedback signals in each of the roll, pitch, and yaw
channels. The low pass filter component is dedicated to eliminating high frequency content from
the second and higher bending modes, while leaving sufficient bandwidth for the flight
controller. The task of the notch filter component is to cancel out low frequency content
attributed to the first bending mode. Though each flex filter for the Ares I is designed to
accommodate for a 10% deviation in the modal frequencies, special attention is given to the
attenuation characteristics of the notch filter component because a sufficient perturbation of the
31
first bending mode frequency (~0.96 Hz) could shift it to within close proximity of the rigid
body control bandwidth (~0.13 Hz).11 This would give the guidance system the ability to
potentially excite the first structural bending mode and destabilize the dynamics. Thus, even an
optimal flex filter design is a trade-off between allowing enough control bandwidth for the
guidance system (i.e., sensors, flight controller, and actuators) to function in an unadulterated
manner and ensuring that flexibility effects are adequately suppressed. A technique to improve
this approach of addressing structural flexibility by feedback filtering is discussed in Section 5.3.
The flex filter design utilized in conjunction with the hybrid adaptive ascent flight controller
is taken directly from the classical linear feedback flight control system which is included as part
of the SAVANT launch vehicle ascent simulator.23 In the context of this study, the flex filters are
used to process only the angular velocity feedback ω = [p q r]T and not the aerodynamic state or
control signal feedback (σ and δ, respectively). Though these two unfiltered quantities contribute
to the response of the parameter estimation algorithm and the output of the neural network,
neither the aerodynamic model nor the control actuator model included in SAVANT incorporates
structural flexibility effects. Accordingly, the feedback signals σ and δ describe purely rigid
body motion and are thus suitable for input directly to the adaptive elements. Further justifying
the lack of filtering of the control signals δ is the assumption that the structural compliance
associated with deformation of the engine gimbals or the roll control thrusters is minimal.
Observe that lateral symmetry of the launch vehicle’s structure dictates the use of identical
flex filters when processing pitch rate q and yaw rate r feedback. In addition, the torsional
flexibility of Ares I about its longitudinal axis has drawn little attention from NASA. This is due
primarily to very low torsional compliance and high corresponding modal frequencies. However,
to maintain consistency with the linear feedback flight controller included in SAVANT, a third
32
identical flex filter is integrated within the hybrid adaptive ascent flight controller for the purpose
of processing roll rate feedback p.
The transfer function of each flex filter in the discrete time domain HF (z) is characterized by
69 )()()( zHzHzH NLPF ⋅= (69)
where HLP (z) and HN (z) are the discrete transfer functions of the low pass and notch components
of the filter, respectively. The discrete transfer function of each component is given by
70
81572891.03258134.31967275.56835243.390246813.05026520.32034847.55026520.390246813.0)(
60273723.05010434.1025423465.0050846929.0025423465.0)(
234
234
234
234
+−+−+−+−
=
+−++
=
zzzzzzzzzH
zzzzzzzH
N
LP
(70)
The Bode response of a flex filter with the discrete transfer function HF (z) and sample time of
0.02 seconds is shown in Fig. 7.
7
-80
-60
-40
-20
0
20
Mag
nitu
de (d
B)
10-1 100 101 102-250
-200
-150
-100
-50
0
50
100
Frequency (rad/s)
Pha
se (d
eg)
Figure 7. Feedback ‘flex filter’ Bode diagram.
33
3.5 Simulink Diagram Representation To validate the hybrid adaptive ascent flight control system, it must interface with the
SAVANT ascent simulator in the MATLAB/Simulink environment. Screenshots of the Simulink
block diagrams created for this research are shown in the following pages.
8
Figure 8. Simulink model of the hybrid adaptive ascent flight controller.
Figure 8 shows the Simulink model of the hybrid adaptive controller, with its inputs and
outputs shaded in yellow. The input and output signals of each subsystem are labeled with the
notation that is established in this chapter (with the following slight alteration: ωc is indicated by
omega_c, etc.). The orange ‘Goto’ and ‘From’ blocks are used to route signals without cluttering
up the graphical interface, and the ‘Unit Delay’ blocks are used to delay a signals current value
for one discrete sample period. The value of this discrete sample time is contained in the constant
34
block ‘dt’ located in the lower left corner of Fig. 8. The ‘From Workspace’ block that contains
the guidance input omega_c is colored blue to indicate that its data is tabulated by Eq. (31)
offline and stored in the workspace. This is because the trajectory of the spacecraft (and thus, the
quaternion command) is established before launch. The ‘Manual Switch’ at the bottom of the
diagram is used to select angular velocity feedback ω that has been either processed or
unprocessed by the flex filters when structural flexibility is enabled or disabled, respectively.
9
Figure 9. Hybrid adaptive controller – Reference model subsystem.
Figure 9 shows the contents of the ‘Reference Model’ subsystem. Observe that discrete-time
integrators are used wherever integration of a signal is necessary, since the flight controller
operates on a fixed discrete sample time. The integration algorithm defaults to a forward (or left-
hand) Euler method.
10
Figure 10. Hybrid adaptive controller – PI element subsystem.
35
Figure 10 shows the contents of the ‘PI Controller’ subsystem. The thick vertical bar at the
bottom of the diagram is used to multiplex signals into a single vector.
11
Figure 11. Hybrid adaptive controller – Dynamic inversion subsystem.
Figure 11 shows the contents of the ‘Dynamic Inversion’ subsystem. The thick vertical bar with
the white midsection on the left hand side is a bus bar used to route multiple signals in a single
bus. Note the visual difference from the bars used to form/split a vector from/into its
components.
12
Figure 12. Hybrid adaptive controller – Signal routing subsystem.
36
Figure 12 shows the contents of the ‘Signal Routing’ subsystem that functions to shape and route
the control signals in the format required by SAVANT.
13
Figure 13. Hybrid adaptive controller – Neural network subsystem.
Figure 13 shows the contents of the ‘Neural Network’ subsystem. The subsystem connected to
the input port ‘delta’ is used to generate the NN input signals that are trigonometric functions of
the gimbal angles, given by Eqs. (64c,d), and the weight update method is selected via a manual
switch.
14
37
Figure 14. Neural network – Tracking error based weight update subsystem.
Figure 14 shows the contents of the ‘Tracking Error Based Weight Update’ subsystem. The
function block labeled ‘f(u)’ calculates the scalar magnitude of its input.
15
Figure 15. Neural network – Modeling error based weight update subsystem.
38
Figure 15 shows the contents of the ‘Modeling Error Based Weight Update’ subsystem that is
utilized by the hybrid adaptive controller to generate all results presented herein.
16
Figure 16. Hybrid adaptive controller – Online parameter identification subsystem.
Figure 16 shows the contents of the ‘Parameter Identification’ subsystem. Observe that the
parameter update method is selected via a manual switch. Diagrams of the ‘RLS Update
Algorithm’ and ‘Newton’s Method Update Algorithm’ subsystems are omitted because of their
size and complexity, however both algorithms are described fully in Section 3.2. The operations
performed downstream of the manual switch are necessary to extract the updated estimates of F1,
F2, and G from the matrix Θ.
17
39
Figure 17. Hybrid adaptive controller – Flex filter subsystem.
Finally, Fig. 17 shows the contents of the ‘Flex Filters’ subsystem. Each of the roll, pitch, and
yaw channels has its own filter, however (as supplied with SAVANT) the coefficients of their
discrete transfer functions are identical. These coefficients are stored in vectors for the numerator
and denominator of both the low pass and notch component of the filter.
40
4
Chapter 4
Ascent Flight Simulation
Chapter 4 Ascent Flight Simulation
Before an adaptive controller is ever considered for full-scale testing in hardware, the
advantages which it can provide over a purely classical controller must be clearly illustrated in
appropriate numerical simulations. This is especially true of flight control systems, be it for an
aircraft or a space vehicle, since safety of the crew and mission success is of the utmost
importance. Gain scheduled linear feedback control systems are currently entrenched as the
heavy favorite of designers in the aerospace industry, given their extensive knowledge base
resulting from decades of research and development. To prove their legitimacy, adaptive flight
control systems must not only perform adequately under nominal conditions, increased attention
must be drawn to the ability of an adaptive controller to maintain stability and improve
performance during off-nominal or unanticipated flight conditions.
To validate the performance of the hybrid adaptive flight controller, a high-fidelity Ares I
ascent flight simulator (referred to as SAVANT) is obtained from the NASA Marshall Space
Flight Center which includes both a detailed launch vehicle dynamic model and a baseline gain
scheduled PID linear feedback ascent flight controller for comparison. The properties of this
41
baseline flight controller and the launch vehicle model incorporated into SAVANT are discussed
in detail in Section 4.1. The sample time of the hybrid adaptive controller is chosen to be ∆t =
0.02 seconds so that it operates at the same frequency as the baseline PID flight control system.
All of the results presented herein are generated by prescribing the reference model natural
frequencies to be ωn = diag(1.2, 0.8, 0.8), setting the damping ratios to ζp = ζq = ζr = 1/sqrt(2),
and initializing the neural network weights W at zero. The following values (representing the
best available approximation of the nominal magnitude of that particular quantity at launch)
serve as the final startup condition which initializes the matrix of parameter estimates that
is manipulated by the parameter identification algorithm: lb/s, ft, F
)0(Θ
41017.1 xm −=& 42=l thrust =
2.85x106 lb, (xeg yeg zeg) = (-42 0 0) ft, 0=dq , and
71 (71) 2
834
383
436
sftlb 1088.21053.11039.51053.11088.21007.1
1039.51007.11026.1⋅⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−=
xxxxxx
xxxI
Selection and tuning of the online parameter identification algorithm and the neural network
weight update law is discussed in Section 4.2. Results are then presented in Sections 4.3 and 4.4
which compare the performance of the hybrid adaptive controller to the baseline PID controller
when the ascent simulation incorporates purely rigid body or full flexibility effects in both
nominal and off-nominal configurations.
4.1 SAVANT Ascent Simulator & Existing PID Flight Controller The baseline linear feedback flight controller, developed at MSFC, is a classical PID
controller which operates at 50 Hz. The feedback gains are scheduled by velocity and the input
to the controller is comprised of Euler angle and angular velocity tracking error. A schematic of
this control architecture is shown in Fig. 18.
42
18
δ
Figure 18. Baseline PID ascent flight control architecture.
Guidance input to the PID flight control system consists of the commanded quaternion qc
which depicts the desired trajectory, and an angular velocity command which is
provided to minimize the rotational motion of the launch vehicle. Since the spacecraft is
expected to follow some curvilinear ascent trajectory (and would require non-zero angular
velocity at certain points to do so), it is clear that both commands cannot always be satisfied
simultaneously. In addition, the angular velocity command provides no information about
the desired flight path. Thus, as outlined in Chapter 3, guidance input to the hybrid adaptive
flight controller consists solely of the quaternion command q
0ω =cˆ
cω
c. It is then the job of the hybrid
controller’s reference model to smooth the resulting angular velocity command ωc in a manner
that produces an acceptable dynamic response from the launch vehicle.
To generate the desired feedback for the PID ascent flight controller, the Euler angle tracking
error is formed by comparing the commanded quaternion to quaternion
feedback from the spacecraft according to
[ Teeee ψθφ=E ]
][ Tzyxo qqqq=q 23
72 ( ) ( ) ∗⋅= qqΩqqE cTce sign2 (72)
where the sign operator returns 1 when the operand is positive and -1 when the operand is
negative, indicates the quaternion conjugate, and the matrix Ω
evaluated over an arbitrary quaternion a is given by
[ Tzyxo qqqq −−−=∗q ]
q, ω
^ Ee, ωe ^ qc, ωc = 0
PID Ares I Flex Filter
43
73 (73) ( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−=
0
0
0
aaaaaaaaaaaa
xyz
xzy
yzx
aΩ
The angular velocity tracking error is calculated as the difference between the angular
velocity command and the response from the launch vehicle ω, that is
eω
0ω =cˆ
74 ωωωω −=−= ce ˆˆ (74)
After they are formed, the tracking errors Ee and are processed by identical flex filters in
each of the roll, pitch, and yaw channels for the purpose of removing the influence of structural
flexibility from the feedback measurements. The properties of these flex filters are described in
Section 3.4. Following this operation, the PID control signal δ (which consists of the roll torque
command and the two engine gimbal angle commands) is formed by a summation of scheduled
proportional, integral, and derivative gains applied to the filtered Euler angle tracking error, the
filtered and discretely integrated Euler angle tracking error, and the filtered angular velocity
tracking error, respectively. Note that the PID gains of this baseline classical ascent flight
controller are not necessarily the same as the gains used by the PI portion of the hybrid adaptive
controller.
eω
ˆ
The SAVANT launch vehicle ascent simulator was developed through a partnership between
bd Systems and NASA MSFC.23 It was created in MATLAB/Simulink and employs fully
nonlinear 6 degree-of-freedom equations of motion in conjunction with a Dormand-Prince
(ode45) numerical integration scheme to depict the translational and rotational behavior of the
Ares I during ascent. The dynamics of the launch vehicle are tracked in the inertial reference
frame via the quaternion as well. Contained within the Ares I plant model are subsystems which
track the variation of gravitational effects with altitude above an oblate Earth, evolution of mass
44
properties as propellant is consumed, performance of the first stage rocket motor, second order
dynamics of the engine gimbal actuators, aerodynamic forces and torques, tail-wags-dog or
rocket nozzle inertia effects, upper stage propellant slosh, and the influence of flexibility on the
vehicle dynamics and sensor feedback. Certain time-dependent parameters, such as vehicle
inertia, center of mass location, overall vehicle mass, rocket thrust magnitude, upper stage
propellant slosh model coefficients, and bending mode shapes and frequencies, have been
tabulated in advance and are loaded into the simulation (see Fig. 19 for examples).
19
1 2 3 4 50
2
4
6
Mach Number
Cm
, α (1
/deg
)
0 40 80 120
0
2
4x 106
Time (s)
1st S
tage
Thr
ust (
lb)
0 40 80 120
150
200
250
Time (s)
Axi
al c
m lo
catio
n (ft
)
0 40 80 1200
1
2x 106
Time (s)
I xx (
lb*f
t*s2 )
0 40 80 120
1
2
3x 108
Time (s)
I yy (
lb*f
t*s2 )
Figure 19. Data embedded in SAVANT.
SAVANT models the sloshing of liquid propellant for the upper stage rocket motor as an
attached spring-mass-damper system in both lateral (y and z) directions, with the longitudinal
location of the point mass being determined by the liquid level in the tank. Spatial dependence of
various atmospheric properties is captured through the use of the 1976 U.S. Standard
Atmosphere model. Prevailing atmospheric wind data is introduced to the simulation as a
function of altitude. Aerodynamic force and moment coefficients are stored in look-up tables as a
function of Mach number, with the resulting aerodynamic forces and torques being calculated by
linear functions (given in Section 2.2) of α and β. Bending mode frequencies (which vary
45
significantly throughout ascent) and shapes were generated by a NASTRAN finite element
model analogous to that of a free-free beam, with symmetry of the launch vehicle dictating
identical bending modes in each of the lateral directions. See Ref. 17 for plots of the mode
shapes and corresponding frequencies. For validation purposes, structural flexibility can be
enabled or disabled within SAVANT. The model simulates the first 120 seconds of ascent, or
from liftoff until shortly before the 1st stage solid rocket motor is expended and jettisoned.
4.2 Selection of Adaptive Laws Before the performance of the hybrid adaptive flight controller can be compared to that of the
baseline PID flight controller, nominal rigid body ascent simulations are conducted with the
hybrid adaptive controller in the loop for the purpose of selecting one of the parameter
identification algorithms presented in Section 3.2, and one of the neural network weight update
laws presented in Section 3.3. The results of these component selection studies are presented
below. Observe that disabling structural flexibility within SAVANT also bypasses the flex
filters, effectively removing them from the feedback loop for all rigid body ascent simulations.
20 Recursive Least Squares Multidimensional Newton’s Method
0 20 40 60 80 100 120-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
0 20 40 60 80 100 120-0.02
-0.01
0
0.01
0.02
0.03
0.04
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
Figure 20. Nominal rigid Ares I – Parameter identification algorithm comparison.
46
The first adaptive component to be evaluated is the online parameter identification algorithm.
The behavior of the RLS and modified multidimensional Newton’s method algorithms is
analyzed with the neural network temporarily disabled. Even under nominal flight conditions, the
aggressive nature of the RLS algorithm consistently produced high frequency oscillation in both
of the engine gimbal commands (as illustrated in Fig. 20). This occurred despite attempts to
correct this behavior, such as varying the initial values of the identification variables (e.g., the
covariance matrix and the directional forgetting factor) and expansion of the discrete sample
interval between parameter updates. Not only would these highly oscillatory command signals
encounter problems with actuator rate limits, they also have the potential to excite the structural
bending modes of the launch vehicle. As a result, the RLS parameter update method was
discarded since the possibility of control/structure interaction has come under much scrutiny
from NASA. Alternatively, Fig. 20 also shows that the modified multidimensional Newton’s
iteration was able to refine the parameter estimates in a manner that produced significantly
smoother gimbal commands. Consequently, all results generated from this point forward by the
hybrid adaptive ascent flight controller are done so via use of the modified multidimensional
Newton’s method online parameter identification algorithm. Tuning the modified Newton’s
method parameter update for good tracking error performance and smooth control command
output resulted in a step size of µ = 1x10-4, updating each parameter estimate only when the
corresponding state variable satisfies the threshold condition of ( ) 3105 −> xabs iθ , and only
performing the parameter update process every 10 samples.
The other adaptive component to be studied is the neural network weight update law. The
online parameter identification algorithm is disabled during this process. Comparing the behavior
of the weight update laws under nominal flight conditions revealed that the tracking error based
47
update law, given by Eq. (66), forced the adaptive signal uad to diverge regardless of repeated
reductions of the learning rate Γ. Consequently, it was unable to generate any presentable results
and was removed from consideration as part of the hybrid adaptive ascent flight control system.
One possible source of this instability is the sensitivity of the tracking error based weight update
law to dynamics that are left un-modeled by the flight controller,20 such as tail-wags-dog effects
or upper stage propellant slosh. On the other hand, the modeling error based weight update law
described by Eqs. (67) and (68) produced smooth and bounded signals with the adaptive learning
rate matrix R initialized as the identity matrix. As a result, the hybrid adaptive ascent flight
controller utilizes the modeling error based neural network weight update law to generate all of
the results presented from this point forward.
4.3 Rigid Body Performance The first round of performance studies is conducted with flex dynamics disabled within
SAVANT. Recall that the flex filters are bypassed (i.e., HF (z) = 1) in the control loop during the
execution of all rigid body ascent simulations. Observe that, for all ascent simulations conducted
herein, the pitch gimbal angle θp tends to diverge as the simulation concludes at 120 seconds.
This is because the 1st stage solid rocket motor has consumed nearly all of its propellant at that
time and, consequently, the thrust it produces is dropping off rapidly (see Fig. 19). As a result,
the flight control system is attempting to achieve the desired corrective moment by increasing the
angular deflection of the rocket nozzle. Figures 21 through 24 compare the results of nominal
rigid body ascent simulations where the control commands are issued by either the baseline PID
or hybrid adaptive ascent flight controller. Figures 21 and 22 reveal that both of the flight
controllers generate engine gimbal angle and roll torque commands of similar magnitude under
nominal conditions, and thus place comparable demands on the control actuators. It is also clear
48
that the command signals issued by the PID controller tend to oscillate much more than the
command signals generated by the hybrid adaptive controller. This is particularly apparent in the
behavior of the gimbal angle commands during the first 30 seconds of flight (see Fig. 21) and in
the nature of the roll torque command signal (see Fig. 22). The roll torque command generated
by the PID controller oscillates about zero for the duration of ascent, whereas the command
issued by the hybrid adaptive controller contains several peaks separated by periods of very little
control input. Considering the true ‘bang-zero-bang’ nature of the RCS actuator dynamics
(which is due to the momentary firing of the thrusters) the roll torque command generated by the
adaptive flight controller is an achievable alternative to the PID command. Figures 23 and 24
show that the body-fixed angular velocities and the Euler angle tracking errors observed by both
flight controllers during ascent are of comparable peak magnitude, yet the results generated by
the PID controller (particularly the roll rate p and eφ ) are far more oscillatory in nature than
those generated by the hybrid adaptive controller. This is due to the previously discussed
difference in the strength of oscillation of the control signals issued by each flight controller.
Figure 24 also reveals that the smoother hybrid adaptive control signal commands serve to lessen
the Euler angle tracking error by mediating a dip in θe around the 20 second mark, alleviating a
peak in θe that occurs around 90 seconds, and removing a peak from ψe just before 80 seconds
have elapsed. Figure 33 presents a comparison of the magnitude of the tracking error generated
by each flight controller under nominal ascent flight conditions. This plot shows that the hybrid
adaptive controller is able to maintain a tracking error magnitude that is roughly equivalent to the
error magnitude observed by the PID flight controller for a large portion of ascent, with the
hybrid controller’s elevated tracking error just before the 20 second mark being balanced out by
its significantly better performance after roughly 70 seconds have passed.
49
The first two off-nominal rigid body ascent simulations consider a reduction of the thrust that
is generated by the first stage solid rocket motor. This substantially reduces the amount of
control authority available to the ascent guidance system about the pitch and yaw axes. Such a
circumstance could be brought on by abnormalities in the burn characteristics of the solid rocket
fuel, or bias in the predicted thrust values that the gain-scheduled PID flight controller is
calibrated to. When assessing the performance of each flight controller in this scenario, the
discussion focuses on the behavior of θe and ψe since these two components measure the ability
of the launch vehicle to track the desired trajectory in the pitch and yaw planes.
Figures 25 and 26 illustrate the effects that a 5% thrust reduction has on the ascent dynamics
when the Ares I is guided by either PID or hybrid adaptive flight control. Figure 25 shows that
the engine gimbal angle command signals generated by the PID and hybrid adaptive controllers
are consistently of comparable magnitude (albeit the PID command is more oscillatory). This
indicates that both flight controllers are exerting similar levels of control effort in the event of a
5% thrust reduction. However, as shown in Fig. 26, the PID flight controller struggles to mitigate
the resulting Euler angle tracking error. This is particularly evident in the behavior of the θe
component since it spends a significant portion of the simulation well above 0.02 radians, peaks
near 0.06 radians at roughly 90 seconds, and climbs rapidly past 0.07 radians as ascent under
PID control concludes at the 120 second mark. Additionally, the ψe component of the tracking
error exhibits a peak near 0.02 radians at roughly 80 seconds when the PID controller is utilized.
The enhanced ability of the hybrid adaptive controller to compensate for the modified vehicle
dynamics is depicted in the lower half of Fig. 26, where the θe component of the tracking error is
maintained at or below approximately 0.02 radians for the duration of ascent and ψe remains
close to zero after the 20 second mark. The superior trajectory tracking capability of the hybrid
50
adaptive flight controller in the presence of a 5% thrust reduction is also shown in Fig. 34, where
the tracking error magnitude observed by the PID flight controller is notably elevated (indicating
poorer PID performance) after approximately 40 seconds have elapsed.
Figures 27 and 28 depict the dynamic response of the Ares I when it is guided by either PID
or hybrid adaptive flight control and there is a 10% reduction in the first stage rocket motor’s
thrust. Figure 27 shows that both the PID and hybrid adaptive controllers dictate engine gimbal
commands of comparable magnitude, and thus demand similar levels of control effort. However,
Fig. 28 shows that the PID flight controller allows θe to grow to 0.05 radians just before the 60
second mark, peak at a value approaching 0.08 radians around the 100 second mark, and diverge
rapidly past 0.15 radians as ascent draws to a close. It also shows that, when under PID control,
ψe exhibits a notable peak just before 80 seconds have passed. In contrast, Fig. 28 also shows
that the hybrid adaptive flight controller is able to prevent θe from exceeding roughly 0.03
radians for the majority of the ascent process, and remove the peak from ψe. The θe component of
the tracking error also climbs to just 0.04 radians at the end of ascent when the hybrid adaptive
controller is instituted. The magnitude of the tracking error generated by each flight controller
when the launch vehicle experiences a 10% thrust reduction is compared in Fig. 35. Given the
compromised vehicle dynamics, the hybrid adaptive controller is able to maintain a far smaller
tracking error magnitude than the PID controller after roughly 30 seconds have elapsed. In
addition, the error magnitude at the end of ascent is nearly 5 times greater when under PID
control (0.19 rad) than when under hybrid adaptive control (0.04 rad). The large guidance
command tracking error seen by the PID controller at the end of the ascent phase could also pose
a threat to the successful transition between the launch vehicle’s 1st and 2nd stages.
51
The final two off-nominal rigid body ascent simulations consider an elevation of the
aerodynamic loads via an increase in the aerodynamic force and moment coefficients.
Unforeseen atmospheric conditions or uncertainty in the aerodynamic model that was used in the
design of the PID controller could cause such a circumstance. Figures 29 and 30 summarize the
ascent dynamics when under PID or hybrid adaptive control, and a 10% increase in the
aerodynamic force and moment coefficients is introduced. As before, the engine gimbal
commands generated by both the PID and hybrid adaptive flight controllers are of similar
magnitude throughout ascent (see Fig. 29). Figure 30 exposes the ability of the hybrid adaptive
controller to reduce the tracking error by decreasing the oscillation in eφ , moderating a dip in θe
at the 20 second mark, and removing distinct peaks from θe and ψe that occur at roughly 90 and
80 seconds, respectively. Figure 36 presents the time history of the tracking error magnitude
observed by each flight controller during ascent. Observe that (as shown in Fig. 36) the
significantly better performance of the hybrid adaptive system after 70 seconds have passed
outweighs its poor tracking performance before the 20 second mark. Figure 38 reveals that the
performance of the PID flight controller in the presence of a 10% increase in the aerodynamic
loads is only marginally different from the benchmark established by the PID controller in the
nominal simulation. The performance of the hybrid adaptive controller under a 10% increase in
the aerodynamic loads is virtually identical to its own performance benchmark set in the nominal
simulation (see Fig. 38). This suggests that the sensitivity of the performance of the ascent flight
control system to changes in the aerodynamic loading is small.
Figures 31 and 32 show the simulation results when under PID or hybrid adaptive flight
controllers, and the aerodynamic coefficients are subjected to a 20% increase. Both flight
controllers demand similar levels of control effort in this off-nominal case as well (see Fig. 31).
52
Figure 32 shows that the hybrid adaptive controller is again able to outperform the PID controller
by greatly reducing the oscillation in eφ , lessening the amplitude of the dip in θe at 20 seconds,
and removing peaks from θe and ψe that occur just before 80 and 90 seconds, respectively. The
enhanced trajectory tracking capability of the hybrid adaptive ascent flight controller is also
illustrated in Fig. 37, where its far superior performance after the 70 second mark offsets the
overshoot that occurs before 20 seconds have passed. Figure 38 shows that the tracking error
magnitude for the PID and hybrid adaptive controllers in this off-nominal case (i.e., 120%
aerodynamic loads) is only slightly perturbed and nearly identical, respectively, to the
corresponding error magnitude observed in nominal simulations. The similarity of the results
under both nominal conditions and a 20% increase in aerodynamic coefficients further supports
the theory that the sensitivity of the tracking error performance of either flight controller to
changes in the aerodynamic loads is limited.
53
21
0 20 40 60 80 100 120-0.02
-0.01
0
0.01
0.02
0.03
0.04
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
PID
0 20 40 60 80 100 120-0.02
-0.01
0
0.01
0.02
0.03
0.04
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
Notes: • Gimbal commands from both controllers
are of comparable magnitude. • PID signals oscillate more, particularly
during first 30 seconds. Hybrid
Figure 21. Nominal rigid Ares I – gimbal angle commands.
54
22
0 20 40 60 80 100 120-4
-3
-2
-1
0
1
2
3
4x 104
Time (s)
Rol
l Tor
que
Com
man
d (ft
-lb)
PID
0 20 40 60 80 100 120-4
-3
-2
-1
0
1
2
3
4x 104
Time (s)
Rol
l Tor
que
Com
man
d (ft
-lb)
Hybrid
Notes: • Roll torque command from both controllers
is of same order of magnitude. • PID signal oscillates much more.
Figure 22. Nominal rigid Ares I – roll torque command.
55
23
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
Time (s)
Bod
y-Fi
xed
Ang
ular
Vel
ocity
(rad
/s)
pqr
PID
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
Time (s)
Bod
y-Fi
xed
Ang
ular
Vel
ocity
(rad
/s)
pqr
Notes: • Body-fixed roll rate p oscillates much
more when under PID control than when under hybrid adaptive control (due to strength of oscillation in control input).
Hybrid
Figure 23. Nominal rigid Ares I – angular velocities.
56
24
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
PID
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
Notes: • Advantages of hybrid control instead
of PID: o Φe – much less oscillation. o θe – smaller dip at 20s, no peak
at 90s. o ψe – no peak near 80s.
Hybrid
Figure 24. Nominal rigid Ares I – tracking errors.
57
25
0 20 40 60 80 100 120-0.02
0
0.02
0.04
0.06
0.08
0.1
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
PID
0 20 40 60 80 100 120-0.02
0
0.02
0.04
0.06
0.08
0.1
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
Notes: • Gimbal commands from both controllers are
of comparable magnitude throughout ascent.
Hybrid
Figure 25. 95% thrust, rigid Ares I – gimbal angle commands.
58
26
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
PID
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
Notes: • Advantages of hybrid control instead
of PID: o θe – maintained below ~0.02 rad,
smaller dip at 20s, no peak at 90s.
o ψe – no peak near 80s. Hybrid
Figure 26. 95% thrust, rigid Ares I – tracking errors.
59
27
0 20 40 60 80 100 120-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
PID
0 20 40 60 80 100 120-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
Notes: • Gimbal commands from both controllers
are of comparable magnitude throughout ascent. Hybrid
Figure 27. 90% thrust, rigid Ares I – gimbal angle commands.
60
28
0 20 40 60 80 100 120-0.05
0
0.05
0.1
0.15
0.2
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
PID
0 20 40 60 80 100 120-0.05
0
0.05
0.1
0.15
0.2
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
Notes: • Advantages of hybrid control instead
of PID: o θe – maintained far below 0.05
rad (diverges rapidly at 120s when under PID control), no peak at 100s.
o ψe – no peak near 80s.
Hybrid
Figure 28. 90% thrust, rigid Ares I – tracking errors.
61
29
0 20 40 60 80 100 120-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
PID
0 20 40 60 80 100 120-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
Notes: • Gimbal commands from both controllers are
of comparable magnitude throughout ascent. Hybrid
Figure 29. 110% Aerodynamic load, rigid Ares I – gimbal angle commands.
62
30
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
PID
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
Notes: • Advantages of hybrid control instead
of PID: o Φe – much less oscillation. o θe – smaller dip at 20s, no peak
at 90s. o ψe – no peak near 80s.
Hybrid
Figure 30. 110% Aerodynamic load, rigid Ares I – tracking errors.
63
31
0 20 40 60 80 100 120-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
PID
0 20 40 60 80 100 120-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
Notes: • Gimbal commands from both controllers are of
comparable magnitude throughout ascent. Hybrid
Figure 31. 120% Aerodynamic load, rigid Ares I – gimbal angle commands.
64
32
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
PID
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
Notes: • Advantages of hybrid control instead
of PID: o Φe – much less oscillation. o θe – smaller dip at 20s, no peak
at 90s. o ψe – no peak near 80s.
Hybrid
Figure 32. 120% Aerodynamic load, rigid Ares I – tracking errors.
65
33
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (s)
||Ee||
(rad)
PIDHybrid
Notes: • PID controller performs better
for a brief period before 20s. • Roughly equivalent performance
between 20s and 70s. • Hybrid controller performs
better from 70s onward (offsets poor performance before 20s).
Figure 33. Nominal rigid Ares I – Tracking error norm.
34
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Time (s)
||Ee||
(rad)
PIDHybrid
Notes: • Brief period of better PID
performance before 20s. • PID error is notably elevated
after 40s, while the hybrid controller steadily maintains smaller error.
Figure 34. 95% thrust, rigid Ares I – Tracking error norm.
66
35
0 20 40 60 80 100 1200
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Time (s)
||Ee||
(rad)
PIDHybrid
Notes: • PID error is significantly elevated
after 30s and diverges rapidly as the simulation concludes (posing a threat to successful 1st/2nd stage transition).
• Hybrid controller steadily maintains smaller error (~5x smaller at 120s).
Figure 35. 90% thrust, rigid Ares I – Tracking error norm.
36
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (s)
||Ee||
(rad)
PIDHybrid
Notes: • Similar trends to nominal case. • PID briefly better before 20s,
comparable performance during the middle of the simulation, hybrid much better after 70s.
Figure 36. 110% Aerodynamic load, rigid Ares I – Tracking error norm.
67
37
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (s)
||Ee||
(rad)
PIDHybrid
Notes: • Similar trends to nominal case. • PID briefly better before 20s,
comparable performance during the middle of the simulation, hybrid much better after 70s.
Figure 37. 120% Aerodynamic load, rigid Ares I – Tracking error norm.
38
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (s)
||Ee||
(rad)
PID, NominalHybrid, NominalPID, 110% AeroHybrid, 110% AeroPID, 120% AeroHybrid, 120% Aero
Notes: • Sensitivity of tracking
performance of either controller to changes in aerodynamic loading is small.
Figure 38. Rigid Ares I – Influence of aerodynamic load on tracking error norm.
68
4.4 Flexible Body Performance The second round of performance studies is conducted with structural flex dynamics enabled
within SAVANT. This also activates the flex filters (whose transfer function HF (z) is given in
Section 3.4) for all subsequent flexible body ascent simulations. Recall that the pitch gimbal
angle θp tends to diverge at the end of all ascent simulations because the 1st stage rocket motor is
nearly exhausted at 120 seconds and the flight control system is compensating for the resulting
loss of thrust. Figures 39 through 42 display the results of nominal flexible body ascent
simulations when the Ares I utilizes either PID or hybrid adaptive flight control. Figures 39 and
40 reveal that both flight controllers generate control signal commands of similar magnitude
throughout ascent. This indicates that the PID and hybrid adaptive flight controllers demand
similar levels of performance from the engine gimbal actuators and the roll control system.
However, the control signals generated by the PID controller (particularly the roll torque
command, and the pitch engine gimbal command θp during the first 30 seconds of ascent) are
much more oscillatory than those issued by the hybrid adaptive controller. Figures 41 and 42
show that the significant oscillation of the PID control input causes the body-fixed angular
velocity of the launch vehicle, and thus the tracking error, to oscillate as well. These fluctuations
are strongest in the body-fixed roll rate p and the eφ component of the tracking error. The ability
of the hybrid adaptive controller to reduce these oscillations is also illustrated by Figs. 41 and 42,
where eφ and p are notably smoother when under hybrid control. In addition, Fig. 42 uncovers
the presence of peaks in θe and ψe around the 80 second mark when the vehicle is under PID
control, whereas the hybrid adaptive flight controller steadily maintains smaller tracking errors
during that time. The PID controller also experiences a larger dip in θe (it becomes nearly -0.02
rad) at the 20 second mark than the hybrid adaptive controller (for which θe is roughly 0.005 rad
69
at that time). Figure 51 presents a comparison of the magnitude of the tracking error that is
generated by each flight controller in this nominal case. The results show that the hybrid adaptive
controller is able to maintain an equivalent or slightly smaller tracking error than its PID
counterpart for the majority of ascent, with the worse tracking performance of the hybrid system
just before the 20 second mark being balanced out by its significantly better performance after
roughly 70 seconds have elapsed.
The first two off-nominal flexible body ascent simulations consider a reduction of the first
stage rocket motor’s thrust. When comparing the trajectory tracking performance of each flight
controller under such a circumstance, the discussion focuses on the behavior of θe and ψe since
the thrusters of the roll control system remain unaltered. Consequently, the roll tracking
dynamics (characterized by eφ ) are largely unchanged from the nominal flexible body
simulation. Figures 43 and 44 summarize the dynamic response of the flexible launch vehicle
when it is guided by either PID or hybrid adaptive flight control, and there is a 5% reduction in
thrust from the first stage rocket motor. Given this reduction of control authority about the body-
fixed pitch and yaw axes, both flight controllers require similar levels of control effort by issuing
engine gimbal commands of comparable magnitude throughout ascent (see Fig. 43). However, as
shown in Fig. 44, the performance of the hybrid adaptive controller is significantly better. The
hybrid adaptive flight controller is able to reduce the dip in θe that occurs at 20 seconds, remove
a peak from ψe just before the 80 second mark, and alleviate a peak in θe at 90 seconds which
approaches 0.06 radians when under PID control. Also, θe is climbing rapidly past 0.06 radians
as ascent under PID control concludes. In contrast, θe grows to just 0.02 radians at the end of the
simulation which utilizes hybrid adaptive control. Figure 52 presents a comparison of the
tracking error magnitude for the two flight controllers. While the tracking performance of the
70
PID controller is slightly better for a short period before the 20 second mark, the hybrid adaptive
controller generates a markedly lower tracking error magnitude after 30 seconds have elapsed.
Figures 45 and 46 show the effects that a 10% thrust reduction has on the flexible launch
vehicle’s ascent dynamics when guidance corrections are provided by either the PID or hybrid
adaptive flight controller. Figure 45 shows that, as before, the hybrid adaptive and classical PID
flight controllers place similar demands on the control actuators by dictating engine gimbal
commands which are of comparable magnitude for the duration of the simulation. The tracking
error performance, on the other hand, varies greatly between the two flight controllers. Figure 46
shows that the PID controller allows θe to grow to 0.05 radians before the 60 second mark, peak
near 0.08 radians around 100 seconds, and climb rapidly towards 0.2 radians as the ascent
simulation is ending. A considerable peak also appears in ψe just before the 80 second mark. In
stark contrast, the lower half of Fig. 46 shows that the hybrid adaptive controller is able to keep
θe, well below 0.03 radians for the vast majority of the simulation. In addition, θe grows to just
0.04 radians at the end of ascent and ψe remains close to zero when the launch vehicle is guided
by the hybrid controller. Figure 53 presents a comparison of the overall tracking error magnitude
when there is a 10% thrust reduction. As shown in Fig. 53, the PID controller performs slightly
better during a short interval before the 20 second mark, but the hybrid adaptive controller
clearly exhibits superior tracking performance after the 30 second mark. Furthermore, the
tracking error magnitude at the end of the ascent phase is more than four times greater when
under PID control (0.2 rad) than when under hybrid adaptive control (>0.05 rad). Such
considerable tracking error could pose a threat to the successful execution of the complex
procedures which must occur during the transition between the 1st and 2nd stages.
71
The final two off-nominal flexible body ascent simulations explore the effects of elevated
aerodynamic loads which are generated by an increase in the aerodynamic force and moment
coefficients. Figures 47 and 48 show the effects of a 10% increase in the aerodynamic force and
moment coefficients on the ascent dynamics of the flexible Ares I when the vehicle utilizes PID
or hybrid adaptive flight control. Figure 47 illustrates the similarity of the control signals issued
by each flight controller, which serves to verify the feasibility of the hybrid adaptive controller as
an alternative to PID control. The tracking error performance of the two flight controllers,
however, is visibly different. As shown in Fig. 48, the PID flight controller causes significant
oscillations to appear in eφ , θe dips to nearly -0.02 radians at the 20 second mark, and distinct
peaks form in θe and ψe just before and after the 80 second mark. Alternatively, the lower half of
Fig. 48 shows that eφ is much smoother, θe dips to just 0.005 radians, and the peaks in θe and ψe
around the 80 second mark are eliminated when hybrid adaptive control is implemented. The
superior tracking performance of the hybrid adaptive controller in this off-nominal case is clearly
shown in Fig. 54, where the equivalent or substantially smaller error magnitude observed by the
hybrid adaptive controller after the 20 second mark far outweighs the better performance of the
PID controller that is seen in a narrow window just before the 20 second mark. The similarity of
the tracking error magnitude observed by either flight controller under nominal conditions and a
10% increase in the aerodynamic coefficients (see Fig. 56) suggests that uncertainty in the
aerodynamic model has only a limited effect on the performance of the flight control system.
Figures 49 and 50 contain results of ascent simulations which consider a 20% increase in the
aerodynamic coefficients when the vehicle is guided by the PID or hybrid adaptive flight
controller. Figure 49 shows that both the PID and hybrid adaptive flight controller dictate engine
gimbal commands of comparable magnitude throughout ascent. This indicates that both flight
72
controllers demand similar levels of performance from the control actuators. However, even
though the strength of the control input from both flight controllers is alike, the tracking error
performance is markedly different. Figure 50 shows that (while under PID control) significant
oscillations appear in eφ , θe dips to -0.02 radians at the 20 second mark, and prominent peaks
appear in θe and ψe around the 80 second mark. Conversely, the hybrid adaptive controller
generates a much smoother eφ , limits the dip in θe at the 20 second mark to 0.005 radians, and
eliminates the peaks from θe and ψe around the 80 second mark (see Fig. 50). Figure 55 shows
that, though the PID controller exhibits slightly better performance for a short period just before
the 20 second mark, the hybrid adaptive controller maintains an equivalent or considerably
smaller tracking error magnitude for the remainder of the ascent simulation. Also, the similarity
of the tracking error magnitude for either flight controller under nominal conditions and a 20%
increase in the aerodynamic coefficients (see Fig. 56) further supports the hypothesis that
variations in the aerodynamic loads have only a marginal effect on the performance of either
flight control system.
73
39
0 20 40 60 80 100 120-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
PID
0 20 40 60 80 100 120-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
Notes: • Gimbal commands from both controllers
are of comparable magnitude. • PID signals oscillate more, particularly
during first 30 seconds of ascent. Hybrid
Figure 39. Nominal flexible Ares I – gimbal angle commands.
74
40
0 20 40 60 80 100 120-3
-2
-1
0
1
2
3
4x 104
Time (s)
Rol
l Tor
que
Com
man
d (ft
-lb)
PID
0 20 40 60 80 100 120-3
-2
-1
0
1
2
3
4x 104
Time (s)
Rol
l Tor
que
Com
man
d (ft
-lb)
Hybrid
Notes: • Roll torque command from both controllers
is of same order of magnitude. • PID signal oscillates much more.
Figure 40. Nominal flexible Ares I – roll torque command.
75
41
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
Time (s)
Bod
y-Fi
xed
Ang
ular
Vel
ocity
(rad
/s)
pqr
PID
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
Time (s)
Bod
y-Fi
xed
Ang
ular
Vel
ocity
(rad
/s)
pqr
Notes: • Body-fixed roll rate p oscillates much
more when under PID control than when under hybrid adaptive control (due to strength of oscillation in control input).
Hybrid
Figure 41. Nominal flexible Ares I – angular velocities.
76
42
0 20 40 60 80 100 120-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
PID
0 20 40 60 80 100 120-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
Notes: • Advantages of hybrid control instead
of PID: o Φe – much less oscillation. o θe – smaller dip at 20s, no peak
at 90s. o ψe – no peak near 80s.
Hybrid
Figure 42. Nominal flexible Ares I – tracking errors.
77
43
0 20 40 60 80 100 120-0.02
0
0.02
0.04
0.06
0.08
0.1
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
PID
0 20 40 60 80 100 120-0.02
0
0.02
0.04
0.06
0.08
0.1
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
Notes: • Gimbal commands from both controllers are
of comparable magnitude throughout ascent.
Hybrid
Figure 43. 95% thrust, flexible Ares I – gimbal angle commands.
78
44
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
PID
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
Notes: • Advantages of hybrid control instead
of PID: o θe – maintained below ~0.02 rad,
smaller dip at 20s, no peak at 90s.
o ψe – no peak near 80s. Hybrid
Figure 44. 95% thrust, flexible Ares I – tracking errors.
79
45
0 20 40 60 80 100 120-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
PID
0 20 40 60 80 100 120-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
Notes: • Gimbal commands from both controllers
are of comparable magnitude throughout ascent. Hybrid
Figure 45. 90% thrust, flexible Ares I – gimbal angle commands.
80
46
0 20 40 60 80 100 120-0.05
0
0.05
0.1
0.15
0.2
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
PID
0 20 40 60 80 100 120-0.05
0
0.05
0.1
0.15
0.2
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
Notes: • Advantages of hybrid control instead
of PID: o θe – maintained far below 0.05
rad (diverges rapidly at 120s when under PID control), no peak at 100s.
o ψe – no peak near 80s.
Hybrid
Figure 46. 90% thrust, flexible Ares I – tracking errors.
81
47
0 20 40 60 80 100 120-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
PID
0 20 40 60 80 100 120-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
Notes: • Gimbal commands from both controllers are
of comparable magnitude throughout ascent. Hybrid
Figure 47. 110% Aerodynamic load, flexible Ares I – gimbal angle commands.
82
48
0 20 40 60 80 100 120-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
PID
0 20 40 60 80 100 120-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
Notes: • Advantages of hybrid control instead
of PID: o Φe – much less oscillation. o θe – smaller dip at 20s, no peak
at 90s. o ψe – no peak near 80s.
Hybrid
Figure 48. 110% Aerodynamic load, flexible Ares I – tracking errors.
83
49
0 20 40 60 80 100 120-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
PID
0 20 40 60 80 100 120-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0.04
0.05
Time (s)
Eng
ine
Gim
bal A
ngle
Com
man
d (ra
d)
θp
θy
Notes: • Gimbal commands from both controllers are of
comparable magnitude throughout ascent. Hybrid
Figure 49. 120% Aerodynamic load, flexible Ares I – gimbal angle commands.
84
50
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
PID
0 20 40 60 80 100 120-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time (s)
Eul
er A
ngle
Tra
ckin
g E
rror (
rad)
φe
θe
ψe
Notes: • Advantages of hybrid control instead
of PID: o Φe – much less oscillation. o θe – smaller dip at 20s, no peak
at 90s. o ψe – no peak near 80s.
Hybrid
Figure 50. 120% Aerodynamic load, flexible Ares I – tracking errors.
85
51
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (s)
||Ee||
(rad)
PIDHybrid
Notes: • PID controller performs better for
a brief period before 20s. • Equal or slightly better hybrid
performance between 20s and 70s.• Hybrid controller performs far
better from 70s onward (offsets poor performance before 20s).
Figure 51. Nominal flexible Ares I – Tracking error norm.
52
0 20 40 60 80 100 1200
0.02
0.04
0.06
0.08
0.1
0.12
Time (s)
||Ee||
(rad)
PIDHybrid
Notes: • Brief period of better PID
performance before 20s. • PID error is notably elevated
after 30s, while the hybrid controller steadily maintains smaller error.
• PID error is ~5x greater than hybrid error at 120s.
Figure 52. 95% thrust, flexible Ares I – Tracking error norm.
86
53
0 20 40 60 80 100 1200
0.05
0.1
0.15
0.2
0.25
Time (s)
||Ee||
(rad)
PIDHybrid
Notes: • PID error is significantly elevated
after 30s, diverges rapidly as the simulation concludes, and is ~5x greater than the hybrid error at 120s (posing a threat to successful 1st/2nd stage transition).
• Hybrid controller steadily maintains smaller error.
Figure 53. 90% thrust, flexible Ares I – Tracking error norm.
54
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (s)
||Ee||
(rad)
PIDHybrid
Notes: • Similar trends to nominal case. • PID briefly better before 20s,
comparable performance from 20s to 70s, hybrid much better after 70s.
Figure 54. 110% Aerodynamic load, flexible Ares I – Tracking error norm.
87
55
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (s)
||Ee||
(rad)
PIDHybrid
Notes: • Similar trends to nominal case. • PID briefly better before 20s,
comparable performance from 20s to 70s, hybrid much better after 70s.
Figure 55. 120% Aerodynamic load, flexible Ares I – Tracking error norm.
56
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (s)
||Ee||
(rad)
PID, NominalHybrid, NominalPID, 110% AeroHybrid, 110% AeroPID, 120% AeroHybrid, 120% Aero
Notes: • Sensitivity of tracking
performance of either controller to changes in aerodynamic loading is small.
Figure 56. Flexible Ares I – Influence of aerodynamic load on tracking error norm.
88
4.5 Rigid vs. Flexible Performance Comparison To conclude the analysis of results, the influence of structural flexibility on the tracking error
magnitude under both nominal and off-nominal flight conditions is examined. For this purpose,
the performance of each flight controller under a prescribed set of flight conditions with
structural flexibility disabled (i.e., rigid vehicle dynamics) within SAVANT is compared to its
performance under the same flight conditions with structural flexibility (i.e., flexible vehicle
dynamics) enabled. Figure 57 shows that, under nominal conditions, enabling flexibility within
SAVANT causes the PID controller to have brief periods of slightly shifted (more commonly
worse) performance at times when the tracking error magnitude passes through a local minimum
or maximum, generates a notable reduction in the peak error magnitude that occurs just before 20
seconds when under hybrid adaptive control, and causes the hybrid controller to perform slightly
better between 20 and 100 seconds. Figure 58 illustrates the influence of structural flexibility on
the tracking performance of each flight control system when there is a 10% reduction in thrust
from the 1st stage rocket motor. As shown in Fig. 58, enabling flexibility causes the error
magnitude observed by the PID controller to shift slightly (becoming mostly worse) near a few
local maxima, whereas the peak error magnitude before 20 seconds is clearly lowered and
performance between roughly 20 and 100 seconds is slightly better when the vehicle is guided by
the hybrid adaptive controller. Figure 59 depicts the effects of enabling structural flexibility in
SAVANT when there is a 20% increase in the aerodynamic loads. Incorporating flexible vehicle
dynamics in this case causes the PID controller to again see predominantly worse performance at
local minima and maxima of the tracking error magnitude, while the hybrid adaptive controller
reduces the peak error magnitude that occurs before 20 seconds and experiences better
performance between approximately 20 and 100 seconds.
89
57
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (s)
||Ee||
(rad)
PID, RigidHybrid, RigidPID, FlexibleHybrid, Flexible
Notes: • Effects of flexibility:
o PID – mostly worse performance at local minima/maxima.
o Hybrid – lower peak error before 20s, better performance from 20s to 100s.
Figure 57. Nominal Ares I – Influence of flexibility on tracking error norm.
58
0 20 40 60 80 100 1200
0.05
0.1
0.15
0.2
0.25
Time (s)
||Ee||
(rad)
PID, RigidHybrid, RigidPID, FlexibleHybrid, Flexible
Notes: • Effects of flexibility:
o PID – slight shift in performance near local maxima.
o Hybrid – lower peak error before 20s, better performance from 20s to 100s.
Figure 58. 90% thrust – Influence of flexibility on tracking error norm.
90
59
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Time (s)
||Ee||
(rad)
PID, RigidHybrid, RigidPID, FlexibleHybrid, Flexible
Notes: • Effects of flexibility:
o PID – mostly worse performance at local minima/maxima.
o Hybrid – lower peak error before 20s, better performance from 20s to 100s.
Figure 59. 120% Aerodynamic load – Influence of flexibility on tracking error norm.
91
5
Chapter 5
Concluding Remarks
Chapter 5 Concluding Remarks
In this research, a hybrid adaptive ascent flight control system was developed for use on
launch vehicles. Though the generic hybrid adaptive control approach has been previously
applied to flight control of a rigid aircraft (for the purpose of recovering stability after the aircraft
suffers some form of non-catastrophic in-flight damage),19-21 a number of unique challenges
were encountered while designing the hybrid adaptive launch vehicle flight controller. These
issues included compensating for guidance command input in quaternion form, integrating flex
filters which prevent harmful interaction between the flight controller and the structural bending
modes, and providing the dynamic inversion with a model of the launch vehicle’s ascent flight
dynamics. In addition, the examination of different online parameter identification methods and
neural network weight update laws proved essential to obtaining smooth and stable control
signals.
5.1 Methods, Results, & Conclusions To initiate development of the hybrid adaptive flight control system, the ascent dynamics of
the Ares I launch vehicle are modeled and linearized in Chapter 2 so the flight controller can
92
generate control signals via a dynamic inversion. The structure of the hybrid adaptive ascent
flight controller is then described in detail in Chapter 3, including alterations necessary for
implementation aboard the launch vehicle. These modifications include the conversion of
guidance command input from quaternion form to body-fixed angular rates, investigation of RLS
and multidimensional Newton’s method online parameter update methods, development of a
neural network structure that is capable of reproducing the nonlinearities of ascent flight,
examination of tracking-error-based and modeling-error-based neural network weight update
laws, and integration of the flex filters into the output feedback loop. Chapter 4 contains a
description of the high fidelity Ares I ascent flight simulator (entitled SAVANT) that was
obtained from NASA Marshall to analyze the performance of the hybrid controller.23 SAVANT
models the first 120 seconds of ascent, or until shortly before the 1st stage solid rocket motor is
expended and jettisoned. Structural flexibility of the vehicle can be enabled or disabled within
the simulator environment. Recall that the feedback flex filters are bypassed in rigid body ascent
simulations, and activated for flexible body simulations. A classical gain-scheduled PID ascent
flight controller was also acquired from NASA Marshall to serve as a performance benchmark.
Results of ascent simulations which model the rigid or fully flexible launch vehicle under both
nominal and off-nominal flight conditions are presented in Chapter 4.
Before the performance of the PID and hybrid adaptive flight controllers could be compared,
nominal rigid body ascent simulations were conducted for the purpose of assessing and selecting
the adaptive laws that the hybrid controller utilizes (see Section 4.2). The data from these
simulations showed that the RLS parameter identification algorithm caused the engine gimbal
commands to oscillate considerably. Such oscillations in the control input have the potential to
destabilize the dynamics by exciting flexible structural modes. As a result, the RLS method was
93
eliminated from consideration as part of the final controller design. The multidimensional
Newton’s method parameter update, on the other hand, provided smooth control signals
throughout the entire ascent phase. Consequently, it was employed by the hybrid adaptive flight
controller for all subsequent performance testing. When analyzing the neural network weight
update laws, only the modeling-error-based law was able to produce stable network output.
Regardless of the adaptive learning rate, the tracking-error-based weight update law forced the
adaptive signal uad to diverge rapidly. In addition, the modeling-error-based weight update was
designed to minimize the contribution to the control signal from the neural network in a least
squares fashion. As a result, all further performance testing of the hybrid controller was
conducted with the modeling-error-based neural network weight update law.
As illustrated in Section 4.3, results of nominal rigid body ascent simulations show that both
the PID and hybrid adaptive flight controllers issue control signals of comparable magnitude, and
thus place similar demands on the control actuator hardware. However, the PID control signals
(particularly the roll torque command) are more oscillatory in nature than the hybrid adaptive
control signals. This oscillation of the PID control input causes the body-fixed angular rates of
the launch vehicle, and thus the guidance command tracking error, to fluctuate as well. The
smoother hybrid adaptive control signals serve to diminish any unnecessary rotational motion of
the launch vehicle. A comparison of the tracking error magnitudes reveals that the hybrid
adaptive controller is able to maintain a level of performance that is nearly equivalent to the
benchmark set by the classical PID controller for the majority of the ascent phase. The PID
controller does perform better for a short period just before the 20 second mark, but the
significantly better performance of the hybrid controller after the 70 second mark more than
compensates for this.
94
To highlight the more robust guidance command tracking capability of the hybrid adaptive
flight control system, rigid body ascent simulations are conducted with several combinations of
off-nominal vehicle dynamics. The first two scenarios subject the Ares I launch vehicle model to
reductions in the 1st stage rocket motor’s thrust of 5% and 10%, respectively. In each case, the
magnitude of the engine gimbal commands issued (and thus the amount of control effort
demanded) by both flight control systems is similar. In the event of a 5% thrust reduction, the
PID controller causes the θe component of the tracking error to be significantly elevated and
diverge quickly as ascent concludes. A 10% thrust reduction serves to exacerbate the already
poor performance of the PID flight controller, with θe climbing rapidly towards 0.2 radians as
ascent draws to a close. In stark contrast, the hybrid adaptive control system is able to
consistently maintain low levels of tracking error for the duration of both off-nominal
simulations. In both cases, the tracking error magnitude plot shows a brief interval of better PID
performance just before the 20 second mark. Nevertheless, the far smaller tracking error
magnitude generated by the hybrid adaptive controller after the 20 second mark clearly illustrates
its superior performance. It is also important to note that the tracking error magnitude seen by the
PID controller at the end of the 1st stage ascent phase (under both 5% and 10% thrust reductions)
is nearly 5 times greater than that seen by the hybrid adaptive controller. Such a large guidance
tracking error could pose a threat to the successful transition between the 1st and 2nd stages.
The final two off-nominal rigid body ascent simulations consider a 10% and 20% increase of
the aerodynamic force and moment coefficients, respectively. As before, the gimbal commands
issued by both flight controllers in each scenario are of comparable magnitude throughout ascent.
This indicates a similar level of control effort being put forth. Plots of the tracking error
magnitude reveal that a 10% or 20% increase in the aerodynamic loads does little to alter the
95
performance of either flight controller from the results obtained under nominal conditions (i.e.,
the PID controller performs slightly better for a short period of time before the 20 second mark,
but the hybrid adaptive controller maintains an equivalent or notably smaller error magnitude
after that). In addition to verifying the supremacy of the hybrid adaptive controller under a wide
range of flight conditions, this suggests that the sensitivity of the performance of either ascent
flight control system to variations in the aerodynamic loading is small.
Following the rigid body performance studies, ascent simulations are conducted with
structural flexibility of the launch vehicle (and consequently the flex filters) enabled. Section 4.4
presents the following results: considering the flexible vehicle under nominal flight conditions,
the hybrid adaptive flight controller maintains an equivalent tracking error magnitude for the
majority of the ascent phase, and the slightly better performance of the PID controller just before
the 20 second mark is outweighed by the hybrid controller’s significantly lower tracking error
magnitude after the 70 second mark. Results from ascent simulations that consider a 10% and
20% increase in the aerodynamic coefficients are nearly identical to the nominal results. The
superior robustness of the hybrid adaptive controller is most clearly demonstrated in the presence
of a 5% or 10% thrust reduction. In such circumstances, the PID controller experiences
significantly elevated tracking error that diverges rapidly as ascent draws to a close whereas the
hybrid adaptive controller steadily maintains much smaller errors (often 50% or less of the
corresponding PID value). The flexible body simulation results are very similar to the results
gathered from the rigid body studies (see Section 4.5). The most visible difference in all cases is
a reduction of the peak in the tracking error magnitude that is observed by the hybrid adaptive
controller just before the 20 second mark. In addition, the tracking performance of the hybrid
adaptive controller becomes slightly better between 20 and 100 seconds.
96
5.2 Contributions of this Work The main objectives of this research were to modify a hybrid adaptive flight control scheme
(which has not been previously considered for application to spacecraft) so that it can be utilized
by the next generation of launch vehicles during ascent, analyze the performance of the hybrid
adaptive ascent flight controller through high-fidelity numerical simulations, and compare these
results to the performance benchmark established by a classical PID ascent flight control system
under a range of different flight conditions. To satisfy the goal of making the hybrid adaptive
controller compatible with a launch vehicle, the following tasks were performed:
• The ascent dynamics of a launch vehicle were characterized in fully nonlinear form, and
subsequently linearized for use in the dynamic inversion. This insures that the dynamic
inversion accurately captures the dependence of the control signals δ on all relevant states
(i.e., the variables contained in ω and σ).
• A command conversion operation was added to transform the predetermined quaternion
guidance command (as supplied by NASA) to the body-fixed angular rate commands
needed by the hybrid adaptive controller. This makes the hybrid controller a drop-in
replacement for its classical PID counterpart.
• Two different online parameter identification algorithms were analyzed: an established
RLS method, and a multidimensional Newton’s method update that was modified to
include an adjustable step size which regulates the rate of change in the parameters.
Results of a nominal ascent simulation revealed that the RLS update generated high
frequency oscillations in the rocket nozzle gimbal commands. Such oscillations are
highly undesirable since they have the potential to excite flexible structural modes of
long and slender launch vehicles. In contrast, the Newton’s method-based parameter
97
update generated smooth gimbal commands for the duration of the ascent simulation, and
thus was selected for use in the hybrid adaptive ascent flight controller.
• The input structure of the neural network was altered so that the nonlinear terms
contributing to the dynamic response of the launch vehicle can be recreated at the
network output node. In addition, two different approaches to updating the network
weights were examined: tracking-error-driven vs. modeling-error-driven update laws.
Nominal ascent simulations revealed that only the modeling-error-driven weight update
law produced stable network output, so it was selected for all subsequent testing.
• ‘Flex filters’ were integrated into the feedback loop for the purpose of preventing
interaction between flexible structural modes and the flight control system.
To validate the performance of the hybrid adaptive ascent flight controller, a high-fidelity Ares I
ascent flight simulator (entitled SAVANT) was obtained from NASA Marshall. Merging the
hybrid controller with SAVANT required the following:
• Creation of the hybrid adaptive control architecture in Simulink model form. This model
file has the proper input/output structure which enables direct placement into SAVANT.
Ascent simulations were then conducted which compared the performance of the hybrid adaptive
and classical PID flight controllers under both nominal and off-nominal flight conditions with
structural flexibility of the launch vehicle either enabled or disabled. The following trends were
observed:
• When considering only rigid body dynamics under nominal flight conditions, tracking
performance of the two flight controllers is very comparable for a large portion of the
ascent phase (with offsetting periods of better performance for each control system
occurring at the beginning and end of ascent).
98
• When considering rigid body dynamics and off-nominal flight conditions, the superior
tracking capability of the hybrid adaptive controller is clearly illustrated by a reduction of
the 1st stage rocket motor’s thrust. In this case, the PID controller allowed the tracking
error to become significantly elevated and diverge rapidly at the end of the simulation,
whereas the hybrid controller steadily maintained smaller error.
• Similar trends were observed by both flight controllers under the same sets of flight
conditions with structural flexibility enabled within SAVANT. Also, the hybrid adaptive
controller performed slightly better with flexibility enabled than in the rigid body cases.
5.3 Future Work The classical linear feedback ascent flight control systems currently employed by launch
vehicles must satisfy a myriad of specifications which address the ability of the flight controller
to reject disturbances in a robust manner. A number of these requirements quantify robustness
via the use of phase and gain margins. These parameters characterize the attributes of the range
of signals which can be stabilized by the flight control system. Developing methods of extending
these linear system stability margin metrics to nonlinear adaptive systems remains as the highly
crucial final step towards obtaining an adaptive flight controller that is certifiable for testing in
hardware.29 A method is currently under development which considers bounding the closed-loop
adaptive system with a linear time-invariant (LTI) system in some finite local time window, and
using classical methods on this LTI approximation to extract phase and gain margins which
provide a local measure of stability.21,22 Another approach uses the same LTI bounding
approximation over a finite interval, reformulates this LTI system as a function of input time-
delay, and computes the time delay margin for the purpose of quantifying local stability.30 As
applied to spacecraft flight control systems, the time delay margin gauges how much time delay
99
the guidance command input can tolerate while the vehicle maintains a stable closed-loop
response. Any future supplements to the work contained in this dissertation should first examine
the aforementioned methods for developing stability margin metrics for the adaptive system.
In addition to verifying the compliance of the ascent flight control system with typical
specifications regarding stability, a number of refinements could potentially be made to the
hybrid adaptive launch vehicle flight controller. First, to further accelerate acceptance within the
aerospace community, the adaptive elements could be disabled during times where their input is
deemed unnecessary. Considering the online parameter identification, the nominal values for the
dynamic inversion matrices could be stored in lookup tables and sensed feedback could be used
to adjust these values only when the response deviates sufficiently from the nominal model. The
neural network could also be disabled when the linear feedback portion of the flight controller is
adequately handling the tracking error. Determining the thresholds at which to disable the
adaptive elements is a vital aspect of such a modification. The natural frequencies and damping
ratios of the reference model (and thus, the gains of the PI portion of the hybrid adaptive
controller) could be scheduled to account for evolution of the desired handling characteristics of
the launch vehicle as ascent progresses. Finally, since the potential for interaction between the 1st
stage bending modes and the ascent flight control system of the Ares I has drawn such attention,
the attenuation properties of the feedback flex filters could be reexamined for the purpose of
maximizing the amount of bandwidth available to the flight controller. One approach would be
to use flex filters with wider pass bands in conjunction with adaptive laws that track the shift in
frequency of the flexible modes as propellant is consumed, and alter the limits of the pass band
so that only the current flex mode frequencies are filtered out.31 Consequently, this gives more
bandwidth to the flight controller so it can effectively correct guidance tracking errors.
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Appendix A – Matlab Code & Simulink Diagrams
*.m file code used to convert quaternion command to body-fixed angular rate command ============================================================== % Eq. 31 from "Review of Attitude Representations Used for Aircraft Kinematics" % by Phillips, Hailey, and Gebert. n = size(guidance_prop.Guid_Qi2b,1); omega_c = zeros(n,4); for i = 2:n dt = guidance_prop.Guid_Qi2b(i,1) - guidance_prop.Guid_Qi2b(i-1,1); eDot = (guidance_prop.Guid_Qi2b(i,2:5) - guidance_prop.Guid_Qi2b(i-1,2:5))./dt; e0 = guidance_prop.Guid_Qi2b(i,2); ex = guidance_prop.Guid_Qi2b(i,3); ey = guidance_prop.Guid_Qi2b(i,4); ez = guidance_prop.Guid_Qi2b(i,5); A = [-ex -ey -ez; e0 -ez ey; ez e0 -ex; -ey ex e0]; omega_c(i,1) = omega_c(i-1,1) + dt; omega_c(i,2:4) = A\(2*eDot'); end load Initial_Inertia save('HybridInfo', 'Inertia_0','omega_c') *.m file code used to initialize the Simulink blocks in the hybrid adaptive controller =========================================================== IDinterval = 10; % discrete sample interval between online parameter estimate updates NMval = 0.005; % state variable threshold for Newton's method parameter update NMscale = 10000; % 1/NMscale = fractional step size taken in Newton's method parameter update %Initializing Dynamic Inversion Matrices mdot = -1.17e4/32.174048; % lbm/s => (lb/s)/g0 (ft/s^2)] l_jet = 42; % (ft) F_rkt = 2850000; % (lb) x_gim = -42; % (ft) load HybridInfo % Contents: Inertia_0 (lb*ft*s^2), omega_c (rad/s) f1 = diag([0 mdot*l_jet^2 mdot*l_jet^2]); f2 = zeros(3,3); gdel = diag([1 -F_rkt*x_gim -F_rkt*x_gim]); F1 = inv(Inertia_0)*f1; F2 = inv(Inertia_0)*f2; Gdel = inv(Inertia_0)*gdel;
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%Initialization of RLS online parameter update theta = [(eye(size(F1,2))+dt*F1)'; (dt*F2)'; (dt*Gdel)']; c = 1000*eye(size(F1,2)+size(F2,2)+size(Gdel,2)); ro = 0.99; fi = 1; la = 0.001; ny = 1e-6; %Reference Model Parameters: %Natural Frequencies = diag([wp, wq, wr]) omega_n = diag([1.2 0.8 0.8]); %Damping Ratios = diag([zeta_p, zeta_q, zeta_r]) %%% 0 < zeta < 1 %%% zeta = diag([1/sqrt(2) 1/sqrt(2) 1/sqrt(2)]); %Ki and Kp Gain Matrices for PI Controller K_i = omega_n.^2; K_p = 2*zeta*omega_n; %Calculations for NN Weight Update Law numInpts = 27; % number of scalar inputs to NN linear_basis_fcn = eye(numInpts); % Tracking error based weight update law A = [zeros(3) eye(3); -K_i -K_p]; P = lyap(A',eye(6)); B = [zeros(3); eye(3)]; Gamma = 1; %learning rate, no greater than 1/dt for stability mu = 0; %e-modification term %Modeling error based weight update law R = 1*eye(numInpts); Screenshot of SAVANT Simulink model (with PID flight controller installed) on following page ====================================================================
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May the road rise up to meet you,
may the wind be always at your back,
may the sun shine warm upon your face,
the rains fall softly upon your fields,
and until we meet again,
may God hold you in the hollow of His hand.
In loving memory of
Arthur & Mary Helen Machell
June 25th, 1921 – November 17th, 2008 & July 18th, 1921 – July 9th, 2006
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