NOVEL GUIDANCE AND NONLINEAR CONTROL SOLUTIONS IN PRECISION GUIDED
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N O VE L G U ID AN C E AN D
N O N L IN E AR C O N T R O L
S O L U T IO N S IN
P R E C IS IO N G U ID E D
P R O J E C T IL E S
R IC HA R D A . HUL L
T E C HNIC A L F E L L OW
C O L L INS A E R O S P A CE
JUNE 3 0 , 2 0 2 0
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This document does not contain any export controlled technical data.O U T L IN E :
I. Introduction to Precision Guided Projectiles
SGP 5” Projectile Flight Test Video
Guidance, Navigation, Control Block Diagram
Guidance, Navigation and Control Issues
II. Guidance Law – GENEX
III. Nonlinear Robust Control Law Design
Background – Dynamic Inversion, Feedback Linearization, Back-stepping, Robust Recursive
IV. Nonlinear Recursive Pitch-Yaw Autopilot Design for a Guided Projectile
Nonlinear Recursive Pitch-Yaw Controller Design
Aerodynamic Functions for Hypothetical Projectile
Pitch-Yaw Controller Equations
Mass Properties for Hypothetical Projectile
MATLAB Simulation Results for Hypothetical Pitch-Yaw Projectile Controller
V. Conclusions
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This document does not contain any export controlled technical data.5 -i n ch Mu l ti Servi ce Stan d ard Gu i d ed Proj ecti l e
•Video Courtesy of BAE Systems
https://www.bing.com/videos/search?q=Navy+5+inch+guided+projectile+video&view=detail&mid=E674
46D3A21E0F0E9D7FE67446D3A21E0F0E9D7F&FORM=VIRE
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This document does not contain any export controlled technical data.S YS T E M B L O C K D IAG R AM
Guidance, Navigation and Control
Target Steering Actuator Actuator
Location Commands Commands Deflections
Control Airframe/
Guidance
Autopilot Augmentation Propulsion
Law
System
Nav
Solution
Navigation Inertial
Filter Sensors
IMU output Projectile
Motion
GPS
Receiver
Antenna
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This document does not contain any export controlled technical data.G U ID AN C E IS S U E S :
Minimum Control Effort Optimal Control Problem with:
• Final Position (Miss Distance) Constraint
• Terminal Angle of Fall (Velocity Unit Vector) Constraint
• Control Action Constraint (Normal to Velocity Vector)
• Nonlinear Dynamics (True Optimal Solution is TPBV Problem)
• May be Final Time (Time on Target) Constraint
Several Well Known Sub-Optimal Solutions
• Modifications to Biased Proportional Nav
• Explicit Guidance
• Generalized Explicit Guidance (GENEX)
• Not well suited to Guided Projectiles of “Limited Maneuver Capability”
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This document does not contain any export controlled technical data.N AVIG AT IO N IS S U E S :
• Electronics Must Be Densely Packaged and Low Cost!
• Inertial Sensors Must Survive High “G” Gun Launch
Environment (~10,000 g’s)
• GPS Must Acquire Quickly, In-Flight, On A Rapidly Moving,
Spinning Projectile
• Navigation Filter Must Initialize and Self-Orient In Mid-Flight
• What to Do if GPS is Unavailable
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This document does not contain any export controlled technical data.C O N T R O L IS S U E S :
• Ballistic Portion of Flight Must Avoid Roll Resonance Issues
• Airframe Control Capture at Canard Deployment
• Airframe May Become Unstable at Canard Deploy
• IMU Sensors May Be Saturated at Canard Deploy
• High Roll Rate
• Deployment Shock
• Large Flight Envelope of Mach and Dynamic Pressure
• Rapidly Changing Flight Envelope
• Winds and Variations in Atmospheric Properties Cause Significant
Uncertainties in Gain Scheduled Parameters (No Air Data Probes)
• Highly Nonlinear and Uncertain Aerodynamics
• Canard “Vortex Shedding” Interactions with Tails
• Uncertain Tail “Clocking” Relative to Canards
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This document does not contain any export controlled technical data.E XP L IC IT G U ID AN C E
Solution to a Time Varying Linear Optimal Control Problem of the form:
tf
u2
minimize J dt
0
2
Subject to the following state constraints:
Where we define:
Which yields the optimal control:
u
1
2
6 x1 2 T x2
T
Note – we often augment this with an acceleration term
to include known effects of gravity and drag.
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This document does not contain any export controlled technical data.G E N E R AL IZ E D E XP L IC IT G U ID AN C E ( G E N E X)
“Generalized Vector Explicit Guidance”, Ernest J. Ohlmeyer and Craig A. Phillips, AIAA Journal of
Guidance, Control and Dynamics, Vol. 29, No. 2, March-April 2006
Solution to a Time Varying Linear Optimal Control Problem of the form:
tf
u2 Family of functions, parameterized by scalar n
minimize J n dt Higher n allows greater penalty weight on control
0
2T usage as T → 0
Subject to the following state constraints:
Where we define:
Which yields the optimal control:
u
1
2
k1 x1 k2 x2T
T
where :
Note that n = 0 results in k1 = 6 and k2 = -2 reducing to
k1 (n 2)( n 3)
k 2 (n 1)( n 2) the standard explicit guidance gains
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This document does not contain any export controlled technical data.G E N E X w i t h G r a v i t y Te r m ( N E W ) :
Note on Updated Derivation of Generalize Explicit Guidance to Include Gravity Acceleration Term from
Ernie Ohlmeyer, May 2016
Solution to a Time Varying Linear Optimal Control Problem of the form:
tf
u2 Family of functions, parameterized by scalar n
minimize J n dt Higher n allows greater penalty weight on control
0
2T usage as T → 0
Subject to the following state constraints: Where we define:
Which yields the optimal control:
u
1
2
k1 x1 k2 x2T k3 g
T
where : k1 ( n 2)( n 3)
k2 ( n 1)( n 2)
( n 2)( n 1)
k3 Setting k3 = 0 recovers original GENEX
2
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This document does not contain any export controlled technical data.E XAM P L E T R AJ E C T O R IE S U S IN G G E N E X
GENEX Guidance from Apogee N=
N=
0.5
1.0
GENEX Guided Projectile
V0 = 1000 m/s 3
N=
N=
1.5
2.0
N= 2.5
Launch Elevation = 50 deg 2
Mach
Final Gamma = -80 deg 1
Target Range = 50 km 0
0 50 100 150 200 250
GENEX Guided Projectile N= 0.5
14 50
N= 1.0
gamma - deg
N= 1.5 0
12 N= 2.0
N= 2.5 -50
10 -100
0 50 100 150 200 250
8 25
Altitude - km
20
AlphaT - deg
15
6
10
5
4 0
0 50 100 150 200 250
guidance cmds mag- gees
2 6
4
0
2
-2 0
0 10 20 30 40 50 60 0 50 100 150 200 250
time
Ground Range - km
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This document does not contain any export controlled technical data.R E C U R S IVE C O N T R O L D E S IG N :
“Recursivist” – one who views the world
as a giant opportunity to apply the
rigorous methods of recursive nonlinear
control design, one layer at a time!
See also “Integrator Backsteppinger”,“Dynamic
Inversionist” and “Feedback Linearizationist”
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This document does not contain any export controlled technical data.D YN AM IC IN VE R S IO N
Given a nonlinear system “affine” w.r.t. the control input:
If g(x) is invertible the control:
Will cause the system to track the desired dynamics :
Note that we are not actually inverting the dynamics of the
entire system, only the algebraic input connection matrix!
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This document does not contain any export controlled technical data.D YN AM IC IN VE R S IO N
The primary restrictions on Dynamic Inversion are:
1. The system must be scalar or square – that is it must
have the same number of inputs as states, and
2. The matrix g(x) must be non-singular over the entire
region of interest.
Invertibility of g(x) also insures stability of the zero dynamics, so that
the system is minimum phase.
In fact, it is a stronger condition, which guarantees that the system
can be decoupled into n independently controllable subsystems by the
n control inputs.
This condition is somewhat rare in real systems!
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This document does not contain any export controlled technical data.F E E D B AC K L IN E AR IZ AT IO N
• Not Conventional (Jacobi) Linearization
• Systematic Method for finding a state transformation that maps the
system to an “equivalent” linear system.
• Design Control for the Linear System
• Map the Control back to the original nonlinear system.
0 u u(x, v)
u x
v k T z
Linearization loop
Pole-Placement loop z
z z(x)
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This document does not contain any export controlled technical data.F E E D B AC K L IN E AR IZ AT IO N
Feedback Linearization involves a transformation of state variables in order to
make the control design and stability analysis of the system easier.
State
Transformation Linear
Nonlinear
System System
Control
Design
Stable
Stable Linear
Nonlinear
Inverse State System
System
Transformation
But, in order to guarantee “equivalence” of dynamics and transference of stability
properties between the linear and nonlinear systems, we must rely on concepts from set
theory, and differential geometry.
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This document does not contain any export controlled technical data.F E E D B AC K L IN E AR IZ AT IO N - C O M M E N T S
Input-Output Feedback Linearization is more desirable for the control
of missiles and aircraft, in which output tracking (command following)
is of interest.
Exact Output Tracking is not possible for non-minimum phase
systems, which are equivalent to systems with less than full
relative degree (r < n), and unstable zero dynamics.
Tail Control of Aircraft and Missiles exhibits minimum phase
response in alpha and beta, but non-minimum phase
characteristics in normal accelerations.
The machinery of State Feedback Linearization could be used to
find an alternative output for non-minimum phase acceleration
tracking – however the transformations are fairly intractable .
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This document does not contain any export controlled technical data.IN T E G R AT O R B AC K S T E P P IN G
Given:
u x
g (x) +
f (x)
Design a stabilizing “fictitious: control (x) for the output
subsystem:
u x
+ g (x) +
(x) f ( x) g ( x) ( x)
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This document does not contain any export controlled technical data.IN T E G R AT O R B AC K S T E P P IN G
Then “backstep” that control through the integrator of the previous
subsystem:
u z x
+ g (x) +
f ( x) g ( x) ( x)
It can be seen that a change of variable z (x) is
equivalent to adding zero to the original subsystem:
Resulting in an equivalent subsystem:
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This document does not contain any export controlled technical data.R E C U R S IVE D E S IG N
By recursive application of Integrator Backstepping, we
can extend the results to higher order systems provided
they have the strict feedback cascaded form:
At each step we define a
new state variable:
Until the control design
equation “pops” out:
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This document does not contain any export controlled technical data.R E C U R S IVE D E S IG N
Also at each step, we recursively accumulate the terms
of the Lyapunov function:
1 2 1 2 1 2
V z1 z 2 z n
2 2 2
For which it can be shown:
Advantages of Recursive Backstepping Design
over Feedback Linearization
1. Transformation is simple (no PDE’s to solve)
2. No need to cancel “beneficial” nonlinearities
3. No need to cancel “weak” nonlinearities
4. Can add robust stabilizing terms to overcome uncertainties
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This document does not contain any export controlled technical data.R O B U S T R E C U R S IVE D E S IG N
Model the system with uncertainties:
Which meet the Generalized Matching Conditions
Bound the uncertainty that appears in the equations:
f ( zi ) i
Add a robust fictitious control term to the
Such as:
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This document does not contain any export controlled technical data.D YN AM IC R E C U R S IVE D E S IG N
Dynamic Recursive Design extends the Recursive Process to
systems which do not meet the strict feedback cascaded form:
By differentiating u (n-r) times to
create additional state variables, such
that:
Until the control design that
“pops” out is a derivative of u:
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This document does not contain any export controlled technical data.6 - D O F R IG ID B O D Y E Q U AT IO N S O F M O T IO N
Recursive Design for Pitch/Yaw Acceleration Control
p
ω B q Θ B
r ( , , )
ωB ΘB
u MB
R
u P
Y y v x
v B v y x
z
vz x I y
vB X I z
fB
αB
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This document does not contain any export controlled technical data.6 - D O F R IG ID B O D Y E Q U AT IO N S O F M O T IO N
12 state equations in body coordinates:
fB External body forces
m mass
Using the Euler Rate equations: M B External body moments
J B Moment of Inertia Matrix
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This document does not contain any export controlled technical data.6 - D O F R IG ID B O D Y E Q U AT IO N S O F M O T IO N
The external body forces are computed as functions of:
Fx , , M , q , p , y , r FxBT , , M , q Fxi , , M , q , r
f B Fy , , M , q , p , y FyBT , , M , q Fyi , , M , q , y
Fz , , M , q , p , y Fz , , M , q Fz , , M , q , p
BT i
where :
Body Tail + Canard Control Increments
Angle of Attack
Angle of Side Slip
T
M Mach Number, 1 vx v
1 y
q dynmaic pressure
tan
v sin
v vT vx2 v y2 v z2
z T
cos cos cos sin sin sin cos sin sin cos sin cos
TBI1 TBI
T
cos sin cos cos sin sin sin sin cos cos sin sin
sin sin cos cos cos
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This document does not contain any export controlled technical data.Recursive Design for Coupled Pitch/Yaw Controller
Reference: Dynamic Robust Recursive Control: Theory and Applications, Ph.D.
Dissertation, Richard A. Hull, University of Central Florida, Orlando Florida, 1996.
First, Define Output Error States:
a z D desired pitch accelerati on
a y D desired yaw accelerati on
a z a z D
z1 a z pitch accelerati on feedback
a
y a yD a y yaw accelerati on feedback
Then, recursively define the state transformations:
Where 1 , 2 represent robust fictitious control terms to
overcome bounded uncertainties.
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This document does not contain any export controlled technical data.Recursive Design for Coupled Pitch/Yaw Controller
Choosing the desired dynamics to map to a stable linear system:
Design gains: k1 , k2
The Design Equation (without robust terms) is:
Where:
Assume that , , q , m, vm are known parameters,
and define the body normal accelerations to be:
a z 1 Fz
aB
a y m Fy
Where Fz and Fy are the aerodynamic forces normal to the body x axis.
We write the Recursive Design Equation for an acceleration controller:
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This document does not contain any export controlled technical data.Recursive Design for Coupled Pitch/Yaw Controller
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This document does not contain any export controlled technical data.Recursive Design for Coupled Pitch/Yaw Controller
Using just the pitch and yaw dynamic equations and assuming p = 0, we
can compute the rate of change in alpha and beta:
Then, assuming small angle approximations for alpha and beta:
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This document does not contain any export controlled technical data.Recursive Design for Coupled Pitch/Yaw Controller
Using approximations for the time derivatives of alpha and beta:
We will solve for the control input to give the required body angular accelerations …
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This document does not contain any export controlled technical data.Recursive Design for Coupled Pitch/Yaw Controller
Recalling the recursive design equation,
and using the pitch-yaw acceleration and body rate vectors:
a q
a B Z , ωB
a Y r
We can substitute the preceding results into the design equation
Fz Fz
F
Fy Fy
Where we have defined
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This document does not contain any export controlled technical data.Recursive Design for Coupled Pitch/Yaw Controller
Now, provided F is invertible, we can solve the recursive design
equation, for the required body angular accelerations:
Equating to the body moment equations:
We can solve for the body moments required to give the desired
body acceleration response:
Then, we “invert” the aerodynamic moment equations to find the
pitch and yaw control input deflections that will give the required
body moments above.
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This document does not contain any export controlled technical data.C o n v e n t i o n s f o r Ae r o d y n a m i c D e f i n i t i o n s
Body Rates Body Velocities,
Body Axes
& Forces
X
& Moments p, L u, Fx
X X
q, M
v, Fy
Y Y Y
r, N
w, Fz
Z Z Z
where :
Pitch Plane Yaw Plane Angle of Attack
Stability Stability Axes Angle of Side Slip
Axes X X
Canard Deflection,
V p, q, r Body Rates
V
Y u, v, w Body Velocities
Y
L, M, N Body Moments
Fx , Fy , Fz Body Forces
Z Z
© 2018 C ol lins Aerospac e, a U ni ted Technologies c ompany. All rights reserved.H y p o t h e t i c a l P r o j e c t i l e Ae r o d y n a m i c s M o d e l
Define aerodynamic forces and moments in the pitch and yaw axes in
terms of alpha, beta, and the pitch and yaw control deflections as:
Fz q Sref CZ ( , , P ) Aero Force along body z axis
Fy q Sref CY ( , , Y ) Aero Force along body y axis
m q Sref lref Cm ( , , P ) Aero Pitching Moment
n q Sref lref Cn ( , , Y ) Aero Yawing Moment
Where: q dynamic pressure, Sref reference area, lref reference length
Assume aerodynamic coefficients are the following functions of alpha,
beta, pitch and yaw control deflections (in degrees):
CZ ( , , P ) .000103 3 .00945 .017 .00055 2 .034 P
Coupled
CY ( , , Y ) .000103 3 .00945 .017 .00055 2 .034Y Nonlinear
Equations
Cm ( , , P ) .000215 3 .0195 .051 .015 .700 P
in Alpha
Cn ( , , Y ) .000215 3 .0195 .051 .015 .700Y and Beta
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This document does not contain any export controlled technical data.Aerodynamic Force and Moment Coefficients for a Hypothetical Projectile
as Function of Alpha, Beta for Delta = 0
Aero Moments Coefficients - delta = 0 Aero Force Coefficients - delta = 0
15
20
15
10
10
5
5
Cm
0
Cz
0
-5
-5
-10
-15 -10
-20
50
-40 -15
0 -20 50
0 40
20 0 20
-50 40 -20 0
-50 -40
beta - deg
alpha - deg beta - deg alpha - deg
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This document does not contain any export controlled technical data.Aerodynamic Force and Moment Coefficients for a
Hypothetical Projectile as Function of Alpha, Beta and Delta
-Cm at +alpha indicates
Neutrally stable with zero
stable airframe
delta around alpha = 0
Hypothetical Projectile Aero Coefficients, Beta =0
25
20 10 deg delta to trim at
15
15 deg alpha
10
5
Cm
0
-5
-10
-15 + delta = + moment
-20
-25
-25 -20 -15 -10 -5 0 5 10 15 20 25
Alpha -deg
delta = 20
delta = 15
6
delta = 10
delta = 5
delta = 0
4
delta = -5
delta = -10
2 delta = -15
delta = -20
+ Alpha = - Fz
Cz
0
-2
-4 -20 deg
+ delta = - accel
-6
-25 -20 -15 -10 -5 0 5 10 15 20 25 +20 deg
Alpha -deg
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This document does not contain any export controlled technical data.Recursive Pitch/Yaw Controller for the Hypothetical Projectile
Using the Force Coefficient Polynomials:
CZ ( , , P ) a1 3 a2 a3 a4 2 b1 P
CY ( , , Y ) a1 3 a2 a3 a4 2 b1Y
where : a1 0.000103, a2 0.00945, a3 0.017, a4 0.00055, b1 0.034
We can find the required “aero slopes” matrix:
Fz Fz C z C z
180
F q Sref
Fy Fy C y C y
because ( , ) are in degrees!
From the coefficient polynomials:
CZ CZ
3a1 2 2a2 sign( ) a3 2a4 , a4 2
CY CY
a4 2 , 3a1 2 2a2 sign( ) a3 2a4
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This document does not contain any export controlled technical data.Recursive Pitch/Yaw Controller for the Hypothetical Projectile
Using F, we compute the required body angular rates and moments:
Ignore – pitch and yaw
components are zero if p = 0
a q
where: a B Z , ω B are from the accelerometer and gyro feedbacks
a Y r
the following parameters are estimated in real-time:
, Angle of Attack and Angle of Side Slip
Note – in the autopilot we have let
q dynmaic pressure w_b(2) = -r , therefore reverse the
vm velocity of projectile sign of the required yaw moment!
m mass of projectile
J zz J yz
J B pitch/yaw moments of intetial
J yz J yy Inertia Coupling Included
And k1 , k2 are gains chosen by the designer to place the poles of the
feedback linearized system.
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This document does not contain any export controlled technical data.Recursive Design Applied to the Hypothetical Projectile
Finally, we use the aerodynamic moment equations:
Cm ( , , P ) c1 3 c2 c3 c4 d1 P Note: Pitch and Yaw Control
Contributions May Not be
Cn ( , , Y ) c1 3 c2 c3 c4 d1 Y De-coupled in the Real World!
where : c1 0.000215, c2 0.0195, c3 0.051, c4 0.015, d1 0.700
to solve for the pitch and yaw control deflections needed to
give the required pitch and yaw moments:
M pR C ( , , P )
M R q Sref lref m
R
Cn ( , , Y )
B
MY
so the pitch and yaw control deflection commands are :
1 1
P
d1 q Sref lref
M pR c1 3 c2 c3 c4 Note – in the autopilot we have let
w_b(2) = -r , therefore reverse the
sign of the required yaw moment!
1
M YR c1 3 c2 c3 c4
1
Y
d1 q Sref lref
where again the parameters : , , q , are estimated in real - time,
and Sref and lref are the known reference area and length.
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This document does not contain any export controlled technical data.Hypothetical Projectile Parameters
A totally fictitious configuration intended as a model to study
guidance and control issues for guided projectiles.
Model Parameters
Parameter Value Units
Diameter 140 mm
Mass 50 kg
Xcg (from nose) 1.0 m
Ixx 0.200 Kg-m^2
Iyy = Izz 18.00 Kg-m^2
Iyz = Izy 0.50 Kg-m^2
Sref (aero reference area) 0.0154 m^2
Lref (aero reference length) 1.0 m
MRC (aero model ref center) 1.0 m
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This document does not contain any export controlled technical data.MATLAB/Simulink Simulation
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This document does not contain any export controlled technical data.MATLAB/Simulink Simulation Results
Simultaneous Pitch and Yaw Step Commands
Full Coupled Nonlinear Aero Model with Inertia Coupling
Pitch Channel response Yaw Channel response
100 command 100 command
2
Pitch Accel - m/s
2
Yaw Accel - m/s
50 50
0 0
-50 -50
-100 -100
0 5 10 15 20 0 5 10 15 20
10 10
5 5
alpha - deg
beta - deg
0 0
-5 -5
-10 -10
0 5 10 15 20 0 5 10 15 20
100 100
Pitch Rate - deg/s
Yaw Rate - deg/s
50 50
0 0
-50 -50
-100 -100
0 5 10 15 20 0 5 10 15 20
4 5
delta Pitch - deg
delta Yaw - deg
2
0 0
-2
-4 -5
0 5 10 15 20 0 5 10 15 20
Time - sec Time - sec
© 2018 C ol lins Aerospac e, a U ni ted Technologies c ompany. All rights reserved.
43
This document does not contain any export controlled technical data.MATLAB/Simulink Simulation Results
Simultaneous Pitch and Yaw Step Commands at Different Frequencies
Full Coupled Nonlinear Aero Model with Inertia Coupling
Pitch Channel Yaw Channel response
response
100 100 command
command
2
Pitch Accel - m/s
2
Yaw Accel - m/s
50 50
0 0
-50 -50
-100 -100
0 5 10 15 20 0 5 10 15 20
10 10
5 5
alpha - deg
beta - deg
0 0
-5 -5
-10 -10
0 5 10 15 20 0 5 10 15 20
100 100
Pitch Rate - deg/s
Yaw Rate - deg/s
50 50
0 0
-50 -50
-100 -100
0 5 10 15 20 0 5 10 15 20
4 4
delta Pitch - deg
delta Yaw - deg
2 2
0
0
-2
-2
-4
-4 -6
0 5 10 15 20 0 5 10 15 20
Time - sec Time - sec
© 2018 C ol lins Aerospac e, a U ni ted Technologies c ompany. All rights reserved.
44
This document does not contain any export controlled technical data.Su mmary an d C on cl u si on s
• Precision Guided Projectiles Pose Challenging Guidance, Navigation and Control
Problems
• GENEX Guidance Law Provides an Analytical Solution to the Optimal Guidance
Problem
• Nonlinear Recursive Control Design Approach Demonstrated to Get Analytical Solution
for Pitch/Yaw Autopilot for a Hypothetical Projectile with Coupled Nonlinear
Aerodynamics and Off-Diagonal Inertia Coupling Terms.
• Method requires computation of “slopes” of aerodynamic force functions, and
inverse of aero moment functions
• Good tracking control can be achieved without addition of integrators to controller
• Good method for getting quick “simulation quality” controller
• Problems may occur if aerodynamic partial derivatives matrix becomes ill-
conditioned
• Additional terms can be added to provide robustness to bounded uncertainties and
un-modelled nonlinear dynamics
© 2018 C ol lins Aerospac e, a U ni ted Technologies c ompany. All rights reserved.
45
This document does not contain any export controlled technical data.REFERENCES
1. Ernest J. Ohlmeyer and Craig A. Phillips, “Generalized Vector Explicit Guidance”, AIAA Journal of Guidance, Control
and Dynamics, Vol. 29, No. 2, March-April 2006
2. Buffington, J. M., Enns, D. F., and Teel, A. R., “Control Allocation and Zero Dynamics, AIAA Guidance, Navigation, and
Control Conference, San Diego, CA, Aug. 1996.
3. Colgren, R., and Enns, D, “Dynamic Inversion Applied to the F-117A”, AIAA Modeling and Simulation Technologies
Conference, New Orleans, LA, Aug. 1997.
4. Hull, R. A., “Dynamic Robust Recursive Control Design and Its Application to a Nonlinear Missile Autopilot,” Proceedings
of the 1997 American Control Conference, Vol. I, pp. 833-837, 1997.
5. Buffington, J. M. , and Sparks, A. G., “Comparison of Dynamic Inversion and LPV Tailless Flight Control Law Designs”,
Proceedings of the American Control Conference, June, 1998.
6. Hull, R. A., Ham, C., and Johnson, R. W., Systematic Design of Attitude Control System for a Satellite in a Circular Orbit
with Guaranteed Performance and Stability, Proceedings of the 14th Annual AIAA/Utah State University Small Satellite
Conference, August, 2000.
7. McFarland, M. B. and Hogue, S. M., “ Robustness of a Nonlinear Missile Autopilot Designed Using Dynamic Inversion”,
AIAA 2000-3970.
8. Slotine, J.-J. E., and Li, W., Applied Nonlinear Control, Prentice Hall, 1991.
9. Krstic, M., Kanellakopoulos, I., Kokotivic, P., Nonlinear and Adaptive Control Design, John Wiley & Sons, 1995.
10. Qu, Z., Robust Control of Nonlinear Uncertain Systems, John Wiley & Sons, 1998.
11. Khalil, H. K., Nonlinear Systems, Third Edition, Prentice Hall, 2002.
© 2018 C ol lins Aerospac e, a U ni ted Technologies c ompany. All rights reserved.
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