Finite Set Control Transcription for Optimal Control Applications

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BackgroundApplicationsConclusions

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, 2009Savannah, Georgia

1 Capt, USAF; Ph.D. Candidate2 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

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Outline

BackgroundSystem DescriptionFSCT Method Overview

ApplicationsLinear Switched SystemLunar LanderSmall Spacecraft Attitude Control

Conclusions


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System DescriptionFSCT Method Overview

System Description

I Hybrid System Dynamics

y = f(t,y,u)

I Continuous States

y =ˆ

y1 · · · yny

˜T

yi ∈ R

I Discrete Controls

u = [u1 · · · unu ]T

ui ∈ Ui = {ui,1, . . . , ui,mi}

I ExamplesI Switched SystemsI Task Scheduling and Resource Allocation ModelsI On-Off Control SystemsI Control Systems with Saturation Limits


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Solving an Optimal Control Problem Numerically

Minimize J = φ(t0,y0, tf ,yf ) +R tft0

L(t,y,u) dt

subject toy = 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

cTy (x) cTψ0(x) cTψf

(x) cTβ (x)iT

= 0

↘X

NLP Solver


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FSCT Method Overview

I Parameter vector consists only of states and times

x = [· · · yi,j,k · · · · · · ∆ti,k · · · t0 tf ]T

I Control history is completely defined byI Pre-specified control sequenceI Control value time durations, ∆ti,k , between switching points

I Key parameterization factors

ny Number of Statesnu Number of Controlsnn Number of Nodesnk Number of Knotsns Number of Segments (ns = nunk + 1)


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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

tt0 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


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Linear Switched SystemLunar LanderSmall 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

y 2

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

y 2

y =

[−1 100−10 −1

]y

(b) u = 2

Figure: Individually Stable Systems


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Two Stable Linear SystemsI Several switching laws

(a) Unstable u =

1, y1y2 < 02, otherwise

(b) Stable u =

1, y1 > y2

2, otherwise

(c) Stable u =

1, yTP 1y < yP 2y2, otherwise

where P uAu + ATuP u = −I

−8000 −6000 −4000 −2000 0 2000−8000

−7000

−6000

−5000

−4000

−3000

−2000

−1000

0

1000

y1

y 2

y = Auy

u =

{1, y1y2 < 02, otherwise

(a)

−35 −30 −25 −20 −15 −10 −5 0 5 10 15

−30

−25

−20

−15

−10

−5

0

5

10

y1

y 2

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

y 2

y = Auy

u =

{1, yTP1y < y

TP2y2, otherwise

(c)

Figure: Three Switching Laws


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Two Stable Linear Systems

I FSCT OptimizationJ = F (x) = tf − t0

yTf 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

y 2

(a)

−20 −15 −10 −5 0 5 10 15 20

−30

−25

−20

−15

−10

−5

0

5

10

y1

y 2

(b)

Figure: FSCT Locally Optimal Switching Trajectories

I Optimization implies the switching law

u =

1, − 1m

≤ y2y1

≤ m

2, otherwise


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2-Dimensional Lunar Lander

I 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

,

I Controls

u1 ∈ {−50, 0, 50} m/s2,

u2 ∈ {−20, 0, 20} m/s2

I Initial and FinalConditions

r0 = [200 15]T km

v0 = [−1.7 0]T km/s

rf = 0

vf = 0

v0

r0, t0

rf ,vf , tf


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2-Dimensional Lunar Lander

−20 0 20 40 60 80 100 120

0

100

200

Minimum Timer

km

rrrrrrrrrrrrrrrrrrrrrrrrrrrr

−20 0 20 40 60 80 100 120−4

−2

0

2v

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


(a)

−20 0 20 40 60 80 100 120 140 160

0

100

200

Minimum Fuelr

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


−20 0 20 40 60 80 100 120 140 160−20

0

20 u2

Time (s)

m/s2


(b)

Figure: Optimal Solutions for the Minimum-Time (a) and Minimum-Fuel (b)Lunar Lander Problem


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Small Spacecraft Attitude Control: Fixed ThrustI Fixed thrust cold gas propulsion for

arbitrary attitude trackingI Reference trajectory defined by rqi0

and rωi(t)

I Minimize deviations between body frameand reference frame with minimumpropellant mass consumption

J = β1pf − β2mpf

pf−p0 =

Z tf

t0

p dt =

Z tf

t0

“

rqvb”T “

rqv

b”

dt.

l3

↗Thuster Pair

l1

r

l2

↙Thuster Pair

y =

2

6

6

6

6

6

6

6

6

6

4

bqi

bωi

mp

rqi

p

3

7

7

7

7

7

7

7

7

7

5

= f (t,y,u)

ui ∈ U = {−1, 0, 1}

I where ui indicates for each principalaxis whether the positive-thrustingpair, the negative-thrusting pair, orneither is in the on position


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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


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Small Spacecraft Attitude Control: Variable Thrust

I Variable thrust cold gas propulsionI Valve rod modifies nozzle throat area

I Include additional states to modelvariable thrust

I Resulting dynamics are still hybrid

Nozzle Throat

30o

30o

29.5o

Valve Core Rod

Valve Core Motion

I States and Controls

y =

2

6

6

6

6

6

6

6

6

4

bqi

bωi

mp

d

vrqi

p

3

7

7

7

7

7

7

7

7

5

u =

»

w

a

–

wi ∈ {0, 1}ai ∈ {−1, 0, 1}

I wi indicates whether the ith thruster pair is on or offI ai indicates the acceleration of the valve core rods of the ith thruster pair


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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


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Conclusions

I This investigation explores the range of applications of the FSCTmethod

I The applicability of the method extends to all engineering disciplines

I FSCT vs. Multiple Lyapunov FunctionsI Optimal control laws may be extracted whose performance exceeds

those derived using a Lyapunov argument

I Multiple independent decision inputs managed simultaneouslyI Solutions derived via the FSCT method are utilized in conjunction

with a hybrid system model predictive control schemeI Optimized control schedules can be realized in the context of potential

perturbations or other unknowns

I Some continuous control input systems may be more accuratelydescribed as systems ultimately relying on discrete decision variables

I Continuous control variables may often be extended into a set ofcontinuous state variables and discrete inputs


Finite Set Control Transcription for Optimal Control Applications

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Transcript of Finite Set Control Transcription for Optimal Control Applications