Talk charts from the 19th AAS/AIAA Space Flight Mechanics Meeting, February 8-12, 2009 Savannah, Georgia

1. university-log
Background
Applications
Conclusions
Finite Set Control Transcription for Optimal Control
Applications
Stuart A. Stanton1
Belinda G. Marchand2
Department of Aerospace Engineering and Engineering Mechanics
The University of Texas at Austin
19th AAS/AIAA Space Flight Mechanics Meeting, February 8-12, 2009
Savannah, Georgia
1 Capt, USAF; Ph.D. Candidate
2 Assistant Professor
The views expressed here are those of the author and do not reflect the official policy or position of
the United States Air Force, Department of Defense, or the U.S. Government.
Stanton, Marchand Finite Set Control Transcription 1

2. university-log
Background
Applications
Conclusions
Outline
Background
System Description
FSCT Method Overview
Applications
Linear Switched System
Lunar Lander
Small Spacecraft Attitude Control
Conclusions
Stanton, Marchand Finite Set Control Transcription 2

3. university-log
Background
Applications
Conclusions
System Description
FSCT Method Overview
System Description
Hybrid System Dynamics
˙y = f(t, y, u)
Continuous States
y =
ˆ
y1 · · · yny
˜T
yi ∈ R
Discrete Controls
u = [u1 · · · unu ]T
ui ∈ Ui = {˜ui,1, . . . , ˜ui,mi }
Examples
Switched Systems
Task Scheduling and Resource Allocation Models
On-Off Control Systems
Control Systems with Saturation Limits
Stanton, Marchand Finite Set Control Transcription 3

4. university-log
Background
Applications
Conclusions
System Description
FSCT Method Overview
Solving an Optimal Control Problem Numerically
Minimize J = φ(t0, y0, tf , yf ) +
R tf
t0
L(t, y, u) dt
subject to
˙y = f(t, y, u),
0 = ψ0(t0, y0),
0 = ψf (tf , yf ),
0 = β(t, y, u)
?
Minimize J = F(x)
subject to
c(x) =
h
cT
˙y (x) cT
ψ0
(x) cT
ψf
(x) cT
β (x)
iT
= 0
NLP Solver
Stanton, Marchand Finite Set Control Transcription 4

5. university-log
Background
Applications
Conclusions
System Description
FSCT Method Overview
FSCT Method Overview
Parameter vector consists only of states and times
x = [· · · yi,j,k · · · · · · ∆ti,k · · · t0 tf ]T
Control history is completely defined by
Pre-specified control sequence
Control value time durations, ∆ti,k, between switching points
Key parameterization factors
ny Number of States
nu Number of Controls
nn Number of Nodes
nk Number of Knots
ns Number of Segments (ns = nunk + 1)
Stanton, Marchand Finite Set Control Transcription 5

6. university-log
Background
Applications
Conclusions
System Description
FSCT Method Overview
FSCT Method Overview
x = [· · · yi,j,k · · · · · · ∆ti,k · · · t0 tf ]T
u1 ∈ U1 = {1, 2, 3},
u2 ∈ U2 = {−1, 1}.
u∗
=
»
1 2 3 1 2 3
−1 1 −1 1 −1 1
–
1
2
3
1
2
-1
1
-1
1
-1
y1
y2
Seg.
u1
u2
t
t0 tf
ny = 2
nu = 2
nn = 4
nk = 5
ns = 11
∆t1,1 ∆t1,2 ∆t1,3 ∆t1,4 ∆t1,5 ∆t1,6
∆t2,1 ∆t2,2 ∆t2,3 ∆t2,4 ∆t2,5 ∆t2,6
1 2 3 4 5 6 7 8 9 10 11
Stanton, Marchand Finite Set Control Transcription 6

7. university-log
Background
Applications
Conclusions
Linear Switched System
Lunar Lander
Small Spacecraft Attitude Control
Two Stable Linear Systems
˙y = f(y, u) = Auy,
u ∈ {1, 2} ,
where
A1 =
»
−1 10
−100 −1
–
, A2 =
»
−1 100
−10 −1
–
−30 −20 −10 0 10 20 30
−30
−20
−10
0
10
20
y1
y2
˙y =
−1 10
−100 −1
y
(a) u = 1
−20 −10 0 10 20 30
−20
−15
−10
−5
0
5
10
15
20
y1
y2
˙y =
−1 100
−10 −1
y
(b) u = 2
Figure: Individually Stable Systems
Stanton, Marchand Finite Set Control Transcription 7

8. university-log
Background
Applications
Conclusions
Linear Switched System
Lunar Lander
Small Spacecraft Attitude Control
Two Stable Linear Systems
Several switching laws
(a) Unstable u =

1, y1y2 < 0
2, otherwise
(b) Stable u =

1, y1 > y2
2, otherwise
(c) Stable u =

1, yT P 1y < yP 2y
2, otherwise
where P uAu + AT
u P u = −I
−8000 −6000 −4000 −2000 0 2000
−8000
−7000
−6000
−5000
−4000
−3000
−2000
−1000
0
1000
y1
y2
˙y = Auy
u =
1, y1y2 < 0
2, otherwise
(a)
−35 −30 −25 −20 −15 −10 −5 0 5 10 15
−30
−25
−20
−15
−10
−5
0
5
10
y1
y2
˙y = Auy
u =
1, y1 > y2
2, otherwise
(b)
−40 −30 −20 −10 0 10 20 30 40
−30
−20
−10
0
10
20
30
y1
y2
˙y = Auy
u =
1, yT
P1y < yT
P2y
2, otherwise
(c)
Figure: Three Switching Laws
Stanton, Marchand Finite Set Control Transcription 8

9. university-log
Background
Applications
Conclusions
Linear Switched System
Lunar Lander
Small Spacecraft Attitude Control
Two Stable Linear Systems
FSCT Optimization
J = F(x) = tf − t0
yT
f yf = 1
u∗
k =
3
2
+
1
2
(−1)k
−20 −15 −10 −5 0 5 10 15 20
−30
−25
−20
−15
−10
−5
0
5
10
y1
y2
(a)
−20 −15 −10 −5 0 5 10 15 20
−30
−25
−20
−15
−10
−5
0
5
10
y1
y2
(b)
Figure: FSCT Locally Optimal Switching Trajectories
Optimization implies the switching law
u =

1, − 1
m
≤ y2
y1
≤ m
2, otherwise
Stanton, Marchand Finite Set Control Transcription 9

10. university-log
Background
Applications
Conclusions
Linear Switched System
Lunar Lander
Small Spacecraft Attitude Control
2-Dimensional Lunar Lander
Dynamics
˙y =
2
6
6
4
˙r1
˙r2
˙v1
˙v2
3
7
7
5 =
2
6
6
4
v1
v2
u1
−g + u2
3
7
7
5 ,
Controls
u1 ∈ {−50, 0, 50} m/s2
,
u2 ∈ {−20, 0, 20} m/s2
Initial and Final
Conditions
r0 = [200 15]T
km
v0 = [−1.7 0]T
km/s
rf = 0
vf = 0
v0
r0, t0
rf , vf , tf
Stanton, Marchand Finite Set Control Transcription 10

11. university-log
Background
Applications
Conclusions
Linear Switched System
Lunar Lander
Small Spacecraft Attitude Control
2-Dimensional Lunar Lander
−20 0 20 40 60 80 100 120
0
100
200
Minimum Time
r
km
rrrrrrrrrrrrrrrrrrrrrrrrrrrr
−20 0 20 40 60 80 100 120
−4
−2
0
2
v
km/s
vvvvvvvvvvvvvvvvvvvvvvvvvvvv
−20 0 20 40 60 80 100 120
−50
0
50 u1
m/s2
u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1
−20 0 20 40 60 80 100 120
−20
0
20 u2
Time (s)
m/s2
u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2
(a)
−20 0 20 40 60 80 100 120 140 160
0
100
200
Minimum Fuel
r
km
rrrrrrrrrrrrrrrrrrrrrrrrrrrr
−20 0 20 40 60 80 100 120 140 160
−2
−1
0
1 v
km/s
vvvvvvvvvvvvvvvvvvvvvvvvvvvv
−20 0 20 40 60 80 100 120 140 160
−50
0
50 u1
m/s2
u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1u1
−20 0 20 40 60 80 100 120 140 160
−20
0
20 u2
Time (s)
m/s2
u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2u2
(b)
Figure: Optimal Solutions for the Minimum-Time (a) and Minimum-Fuel (b)
Lunar Lander Problem
Stanton, Marchand Finite Set Control Transcription 11

12. university-log
Background
Applications
Conclusions
Linear Switched System
Lunar Lander
Small Spacecraft Attitude Control
Small Spacecraft Attitude Control: Fixed Thrust
Fixed thrust cold gas propulsion for
arbitrary attitude tracking
Reference trajectory defined by rqi
0
and rωi(t)
Minimize deviations between body frame
and reference frame with minimum
propellant mass consumption
J = β1pf − β2mpf
pf −p0 =
Z tf
t0
˙p dt =
Z tf
t0
“
r
qv
b
”T “
r
qv
b
”
dt.
l3
Thuster Pair
l1
r
l2
Thuster Pair
˙y =
2
6
6
6
6
6
6
6
6
6
4
b
˙qi
b
˙ωi
˙mp
r
˙qi
˙p
3
7
7
7
7
7
7
7
7
7
5
= f(t, y, u)
ui ∈ U = {−1, 0, 1}
where ui indicates for each principal
axis whether the positive-thrusting
pair, the negative-thrusting pair, or
neither is in the on position
Stanton, Marchand Finite Set Control Transcription 12

13. university-log
Background
Applications
Conclusions
Linear Switched System
Lunar Lander
Small Spacecraft Attitude Control
Small Spacecraft Attitude Control: Fixed Thrust
Actual Trajectory
Desired Trajectory
0 2 4 6 8 10 12 14 16 18 20
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (s)
Quaternions: Actual vs. Desired
(a)
0 2 4 6 8 10 12 14 16 18 20
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time (s)
rad/s
Angular Velocity: Actual vs. Desired
(b)
Figure: Fixed Thrust Attitude Control
Stanton, Marchand Finite Set Control Transcription 13

14. university-log
Background
Applications
Conclusions
Linear Switched System
Lunar Lander
Small Spacecraft Attitude Control
Small Spacecraft Attitude Control: Variable Thrust
Variable thrust cold gas propulsion
Valve rod modifies nozzle throat area
Include additional states to model
variable thrust
Resulting dynamics are still hybrid
Nozzle Throat
30o
30o
29.5o
Valve Core Rod
Valve Core Motion
States and Controls
y =
2
6
6
6
6
6
6
6
6
4
b
qi
b
ωi
mp
d
v
r
qi
p
3
7
7
7
7
7
7
7
7
5
u =
»
w
a
–
wi ∈ {0, 1}
ai ∈ {−1, 0, 1}
wi indicates whether the ith thruster pair is on or off
ai indicates the acceleration of the valve core rods of the ith thruster pair
Stanton, Marchand Finite Set Control Transcription 14

15. university-log
Background
Applications
Conclusions
Linear Switched System
Lunar Lander
Small Spacecraft Attitude Control
Small Spacecraft Attitude Control: Variable Thrust
Actual Trajectory
Desired Trajectory
0 2 4 6 8 10 12 14 16 18 20
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (s)
Quaternions: Actual vs. Desired
(a)
0 2 4 6 8 10 12 14 16 18 20
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time (s)
rad/s
Angular Velocity: Actual vs. Desired
(b)
Figure: Variable Thrust Attitude Control
Stanton, Marchand Finite Set Control Transcription 15

16. university-log
Background
Applications
Conclusions
Conclusions
This investigation explores the range of applications of the FSCT
method
The applicability of the method extends to all engineering disciplines
FSCT vs. Multiple Lyapunov Functions
Optimal control laws may be extracted whose performance exceeds
those derived using a Lyapunov argument
Multiple independent decision inputs managed simultaneously
Solutions derived via the FSCT method are utilized in conjunction
with a hybrid system model predictive control scheme
Optimized control schedules can be realized in the context of potential
perturbations or other unknowns
Some continuous control input systems may be more accurately
described as systems ultimately relying on discrete decision variables
Continuous control variables may often be extended into a set of
continuous state variables and discrete inputs
Stanton, Marchand Finite Set Control Transcription 16