New Control Methodology for Nonlinear Systems and Its Application to Astrodynamics

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1 New Control Methodology for Nonlinear Systems and Its Application to Astrodynamics Hancheol Cho, Ph.D. Marie-Curie COFUND Postdoctoral Fellow Space Structures and Systems Laboratory (S3L) Department of Aerospace and Mechanical Engineering Université de Liège, Belgium 2016 KOSEAbe Biannual Conference, KIC, Brussels, April 30

Reference Control and Feedback Control How Nature Is Controlling This Universe at Each

2 In this talk, a new method of reference controller design for a class of nonlinear systems is proposed. Reference Control and Feedback Control How Nature Is Controlling This Universe at Each Instant of Time Reference Controller Design Derivation of Fundamental Equation of Motion Application to Astrodynamics Problem (Formation-Keeping) Summary and Discussion

3 Why Is Control Important? Atlantis meets Mir

4 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

5 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. Disturbance Error SRP Reference input Controlled Signal Controller Feedback Signal Manipulated Variable Actuator + + Sensor + Process Actual Output

6 No Control: Equation of Motion Ma If we have desired output x d (t), = If disturbances d(t) are considered, F Mx = F + F d In this presentation, a new method to obtain the reference control F ref is proposed. The idea is rooted in how Nature (or God) controls this universe. ref Mx F F F d d = ref feed t

7 Mechanics vs. Control Philosophiæ Naturalis Principia Mathematica (Sir Isaac Newton, 1687)

8 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

possible, but Nature always chooses only one that minimizes the

9 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: G q a T Mq a q Aq b q Gauss s Principle of Least Action (1829) a

10 Example: 2-D Pendulum Unconstrained motion: x m 0 0 q,, y M 0 m a g Constraint: A b x x y L x y x y y How to choose only one? Gauss s principle! q q q 2 2 x 0 gy x y x 2 y g L y

11 Solution in an Explicit Form 1. Constraints (Control Requirements) Aq + + q A b I A A h ( h : arbitrary) Cf) Moore-Penrose Generalized Inverses 1) 2) 3) 4) Cf ) + AA A A AA A b AA + T + + T + A A + AA b b must be satisfied for a solution to exist. A AA A A

12 Solution in an Explicit Form Ex) MP Generalized Inverses A a a a A 1 2 n a1 a2 an , A 0 1 A A A 2 0 A a a a 1 2 n,

13 Solution in an Explicit Form 2. Gauss s Principle (Optimal Control) + + q A b I A A h & min G T q a M q a T 1 Fref M Fref Mq Ma F ref T Ma A AM A b Aa -1 T Fundamental Equation of Constrained Motion (FECM)

14 Mq Ma F What Are Good? ref T -1 T Ma A AM A b Aa 1. Explicit form 2. Holonomic + nonholonomic constraints ( Lagrange s multiplier method) 3. No linearization 4. Global minimum ( Standard optimal control) T cf ) min J u Qu d t f t 0

15 Was Kepler lucky? Kepler s Laws and Newton s Law of Gravitation

16 Was Kepler lucky? Kepler s Laws and Newton s Law of Gravitation Kepler spent more than 5000 sheets of A4-sized paper to calculate the Martian orbit!

17 Kepler s Laws and Newton s Law of Gravitation 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. (2 nd was discovered before 1 st ) Mathematically 2 2 FD x y, xy yx c DN x l 2 2 x r y x c / r y x y 0

18 Kepler s Laws and Newton s Law of Gravitation Then, the constraint force on the planet is given by x T 1 T m c ref ( t) r F A AM A b Aa l y r r F ref (t) is directed along the focus of the ellipse. 2. F ref (t) varies inversely with the square of the distance r from the focus. Cf) Kepler s third law is redundant! Otherwise, what would happen?

19 Why Formation Flying? Cheap Reliability New technologies (ex. Interferometry)

20 Formation-Keeping : Chief (uncontrolled) : Deputy (controlled)

21 Unconstrained Motion in the ECI frame Earth-Centered Inertial (ECI) frame Unconstrained motion in the ECI frame Newton s gravitational law a ECI X X GM Y Y X 2 Y 2 Z 2 3/2 Z Z

22 Two Constraints (Control Requirements) Local-Vertical, Local-Horizontal (LVLH) frame Two constraints in the LVLH frame: x y, 2x z 2 2 xx yy y z, 2x z x y z y z y z Aq b

23 Coordinate Transformation (LVLH ECI) X X x 2 2 GM 0 y z y z Y Y y /2 a ECI + X Y Z Z Z z T Fref A AM A b Aa -1 T? Coordinate transformation (LVLH ECI) x rl X y Y = R z Z R?

24 Orbital Elements 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) X a Y e Z i X Y Z

25 Coordinate Transformation (LVLH ECI) Coordinate transformation (LVLH ECI) x rl X y Y = R z Z i R R R R x X y z y z A11 A12 A13 b1 y Y A21 A22 A 23 b 2 z Z

26 Coordinate Transformation (LVLH ECI) Coordinate transformation (LVLH ECI) x rl X y Y = R z Z i R R R R x X y z y z A11 A12 A13 b1 y Y A21 A22 A 23 b 2 z Z

27 Reference Control F ref Finally, we obtain the control force for the deputy in an explicit form: 1 T + T F ref = A AM A b Aa X b1 A11 A12 A + GMm 13 m ( + + ) ( ) Y /2 b + 2 X Y Z + + = A 1 c c A1 c c A21 A22 A 23 Z A11 A21 A A11 A21 A12 A22 A13 A where A 1 = A ,, A A12 A11 A12 A c 13 A11 A12 A13 A21 A22 A23 ( A11 A12 A13 ) A 13 A 23 A 13

28 Numerical Simulations Parameters a 7000 km, e 0, i 80, 30, 0, 50 km, x km, y km, z km, x m/s, y m/s, z m/s Uncontrolled motion

29 Controlled motion Numerical Simulations Errors

30 Control forces (F ref ) Numerical Simulations

31 Summary 1. Control Reference Control + Feedback Control 2. Mechanics Control 3. Nature is the best controller in this Universe a) Given constraints b) Gaussian c) Absolutely no error! 4. Fundamental equation of constrained motion 5. Application to formation-keeping problem Okay, we could have done many things with this controller, but it usually fails in a real life system

32 In practice, things are not so simple! What we have NOT considered 1. Elliptical orbit 2. Perturbations or disturbances (nonuniform gravity, air drag, SRP, ) 3. Uncertain parameters (sensor noise or measurement error in mass, position, velocity, ) 4. Attitude constraint 5. Constraint priority 6. Control force saturation 7. Time delay 8. And more

33 My Research at ULg Rendezvous using differential drag (without thrust) ˆx S d ŷ Chaser Target 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.

Spacecraft Dynamics and Control

Spacecraft Dynamics and Control Matthew M. Peet Arizona State University Lecture 1: In the Beginning Introduction to Spacecraft Dynamics Overview of Course Objectives Determining Orbital Elements Know