new control methodology for nonlinear systems …...2016 biannual spring conference of koseabe, kic...

2016 KOSEAbe Biannual Conference, KIC, Brussels, April 30

New Control Methodology for Nonlinear Systems and Its Application to Astrodynamics

Hancheol Cho, Ph.D.

Marie-Curie COFUND Postdoctoral FellowSpace Structures and Systems Laboratory (S3L)

Department of Aerospace and Mechanical EngineeringUniversité de Liège, Belgium

2016 Biannual Spring Conference of KOSEAbe, KIC Europe, Brussels, April 30

Summary and Discussion

Application to Astrodynamics Problem (Formation-Keeping)

Derivation of Fundamental Equation of Motion

How Nature Is Controlling This Universe at Each Instant of Time

In this talk, a new method of reference controller design for a class of nonlinear systems is proposed.

ReferenceControllerDesign

Reference Control and Feedback Control


Why Is Control Important?

Atlantis meets Mir


Reference Control + Feedback Control

• Reference Control (Open-Loop Control)– Assume no perturbations/disturbances/uncertainties– Given (desired) constraints should be satisfied– In many cases, optimal control is simultaneously considered

?

Spacecraft Rendezvous


Reference Control + Feedback Control

• Feedback Control (Closed-Loop Control)– Perturbations/disturbances/uncertainties are considered– The error between the reference and the output is fed back– Ex) PID control, Sliding mode control, Adaptive control, etc.

SRP

Sensor

Actuator ProcessController ++Reference

inputActual Output

ErrorControlled

Signal

Disturbance

Manipulated Variable

Feedback Signal

+

-

++


Equation of Motion• No Control:• If we have desired output xd(t),

• If disturbances d(t) are considered,

• In this presentation, a new method to obtain the reference control Fref is proposed.

• The idea is rooted in how Nature (or God) controls this universe.

Ma F=

d refMx F F= +

d ref feed tMx F F F d=


Mechanics vs. Control• Philosophiæ Naturalis Principia Mathematica

(Sir Isaac Newton, 1687)


Motion vs. Force(Causationism vs. Teleology)

• Indeed, the problem that Newton famously solved was a control problem (not a mechanics problem):

Kepler’s first two laws → Newton’s law of universal gravitation(Newton used only geometry! (no calculus / algebra))

x F


How Is Nature (or God) Conducting This Universe?

• The desired constraints must be satisfied:

• There are infinitely many possible , but Nature always chooses only one that minimizes the following at each instant of time: TG q a M q a

↓

Gauss’s Principle of Least Action

(1829)qa

Aq b

q


Example: 2-D Pendulum• Unconstrained motion:

• Constraint:

• How to choose only one ?→ Gauss’s principle!

0 0, ,

0x my m g

q M a

2 2

2

0x xgy x yy g yL

q

2 2 2 2 2 x

x y L x y x yy

A bq

q


Solution in an Explicit Form

1. Constraints (Control Requirements)

Cf) Moore-Penrose Generalized Inverses

( : arbitrary)

+ +

Aq b

q A b I A A h h

1) 2)

3)

4)

T

T

+

+ + +

+ +

+ +

AA A AA AA A

AA AA

A A A A

Cf ) must be satisfied for a solution to exist.+AA b b



Ex) MP Generalized Inverses

1

21 2 2 2 2

1 2

1

1 ,

1 1 1 1 ,

0 1 0 1

1 0 0.2 0.4

2 0 0 0

nn

n

aa

a a aa a a

a

A A

A A A

A A



2. Gauss’s Principle (Optimal Control)

-1

ref

TT

Mq Ma F

Ma A AM A b Aa

& + +q A b I A A h 1

min T

Tref ref

G

q a M q a

F M F

Fundamental Equation of Constrained Motion (FECM)


What Are Good?

1. Explicit form2. Holonomic + nonholonomic constraints

(↔ Lagrange’s multiplier method)3. No linearization4. Global minimum (↔ Standard optimal control)

0) min ft T

tcf J d u Qu

-1

ref

TT

Mq Ma F

Ma A AM A b Aa


Kepler’s Laws and Newton’s Law of Gravitation

• Was Kepler lucky?



• Was Kepler lucky?

Kepler spent more than 5000 sheets of A4-sized paper to calculate the Martian orbit!



1. The orbit of a planet is an ellipse with the Sun at one of the two foci.

2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. (2nd was discovered before 1st)

Mathematically

2 2

2 2

,

/0

x yFD xy yx cDN x l

xx r y c ryy x



Then, the constraint force on the planet is given by

1. Fref(t) is directed along the focus of the ellipse.2. Fref(t) varies inversely with the square of the distance r

from the focus.

Cf) Kepler’s third law is redundant! Otherwise, what would happen?

2

12( ) T T

ref

xm crt

yl rr

F A AM A b Aa


Why Formation Flying?

• Cheap• Reliability• New technologies (ex. Interferometry)


: Chief (uncontrolled): Deputy (controlled)

Formation-Keeping


• Earth-Centered Inertial (ECI) frame• Unconstrained motion in the ECI frame

→ Newton’s gravitational law

Unconstrained Motion in the ECI frame

3/22 2 2ECI

X XGMY Y

X Y ZZ Za


• Local-Vertical, Local-Horizontal (LVLH) frame• Two constraints in the LVLH frame:

Two Constraints (Control Requirements)

2 2 2

2 2

2 2

, 2 , 2

0

2 0 1 0

x y x zxx yy y z x z

xy z y z

yz

Aq b


• Coordinate transformation (LVLH → ECI)

Coordinate Transformation (LVLH → ECI)

2 2

3/22 2 2

0

2 0 1 0ECI

X X xy z y zGMY Y y

X Y ZZ Z za +

-1 ?Tref

TF A AM A b Aa

?

Lx r Xy Yz Z

= R

R


• Two different methods to represent the state of a satellite:

1. Position (3) and velocity (3) in the ECI frame

2. 6 orbital elements (5 except for ν are constants)

Orbital Elements

X aY eZ iXYZ


• Coordinate transformation (LVLH → ECI)

Coordinate Transformation (LVLH → ECI)

3 1 3

Lx r Xy Yz Z

i

= R

R R R R

2 211 12 13 1

21 22 23 2

0

2 0 1 0

x XA A A by z y zy YA A A b

z Z


• Finally, we obtain the control force for the deputy in an explicit form:

Reference Control Fref

1

11 12 1313/22 2 2

21 22 232

( ) ( )

Tref

XA A Ab GMmm YA A Ab X Y Z Z

+T

+ + + + + +1 1

F = A AM A b Aa

= A c c A c c

11 21 1111 21 12 22 13 23

12 22 122 2 2 2 2 2 2 2 2 2 2 2 211 12 13 11 12 13 21 22 23 11 12 13

13 23 13

1 1where , , ( )

A A AA A A A A AA A A

A A A A A A A A A A A AA A A

+ +1A = c


• Parameters

• Uncontrolled motion

Numerical Simulations

0 0 0

0 0 0

7000 km, 0, 80 , 30 , 0 , 50 km, 13.067 km, 42.646 km, 26.134 km,

22.9784 m/s, 28.1764 m/s, 45.9568 m/s

a e ix y z

x y z


• Controlled motion

• Errors



• Control forces (Fref)



1. Control → Reference Control + Feedback Control2. Mechanics ↔ Control3. Nature is the best controller in this Universe

a) Given constraintsb) Gaussianc) Absolutely no error!

4. Fundamental equation of constrained motion5. Application to formation-keeping problem

Okay, we could have done many things with this controller,but it usually fails in a real life system…

Summary


• What we have NOT considered…1. Elliptical orbit2. Perturbations or disturbances (nonuniform gravity, air drag, SRP, …)3. Uncertain parameters (sensor noise or measurement error in mass,

position, velocity, …)4. Attitude constraint5. Constraint priority6. Control force saturation7. Time delay8. And more ……

In practice, things are not so simple!


• Rendezvous using differential drag (without thrust)

• A novel adaptive sliding mode controller is being developed and will be tested using a real satellite ‘QARMAN’ (rendezvous with another QB50 CubeSat) later this year.

My Research at ULg

Sd

x

y

Chaser Target

new control methodology for nonlinear systems …...2016 biannual spring conference of koseabe, kic...

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