Station Keeping with Range Only Sensing {Adaptive Control}

1. Supelec EECI Graduate School in Control Station Keeping with Range Only Sensing {Adaptive Control} A. S. Morse Yale University Gif – sur - Yvette May 24, 2012 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAAAAA

2. Roadmap 1. A Simple Example: SWITCHING BETWEEN TWO MODELS 2. An Application: STATION KEEPING

3. A Little Background A Little Background In the early days {say before the 1980s} the main type of switching controls studied in control theory were relay {eg. bang bang} controls. Although some relay controls used hysteresis {which is a simple form of memory logic}, for the most part “logic-based” control algorithms had not been studied. In the early 1980s at a meeting on Belle Isle in France it was conjectured that it is impossible to “stabilize” the uncertain linear system with a smooth controller without knowledge of the sign of the “high-frequency gain” b Shortly thereafter Roger Nussbaum {a math analyst at Rutgers with no background in control} proved the conjecture false by actually constructing a smooth nonlinear {but not rational} stabilizing control. This work motivated a number of people, notably BengtMartensson at Lund, to rethink adaptive control and in particular to ask just how much information is needed about a linear system in order to control it.

4. Bengt jolted conventional wisdom in adaptive control by constructively demonstrating that all one needs to know about a SISO linear system in order to “stabilize” it, is an upper bound on its dimension! Martensson accomplished this by using a logic-based switching control which keeps stepping through a countable family of linear controllers until the integral inequality is satisfied for some integeriwhere ti is the ith time a controller switch takes place. If there is a next switch after ti, it occurs at the smallest value of t ¸ti at which the above inequality fails to hold. Bengt’s work motivated a great deal of research on switching control. Meanwhile, in the late 1980s, during a short course at DLVFR {German Test and Research Institute for Aviation and Space Flight} in Oberpfaffenhofen Graham Goodwin lectured about a new and clever type of switching control.

5. Hysteresis Switching Problem: Given a finite set of scalar valued monotone non-decreasing signals ¹1, ¹2, ....., ¹m with the property that at least one signal in the set has a finite limit, develop a real-time algorithm for finding a member of the set with a finite limit. initialize ¾ hysteresis switching ..... Pick a positive constant h: ¾ = ¾* There is a finite time T at which switching stops and is has a finite limit as t!1. n y ¹¾ * + h < ¹¾

6. Lets look at a very simple example But before we start we are going to need the Bellman-Gronwall Lemma

7. Lemma: If some numbers tbta, some constant c> 0 and some nonnegative, piecewise-continuous function ®: [ta, tb] !IR, w: [ta, tb] ! IR is a continuous function satisfying then Bellman-Gronwall Lemma

8. Now on to the very simple example.....

9. un-modeled dynamic operator with small L2 norm nominal model noise with finite L1 norm Hey, I thought you said simple! Model 1: p = 1 b = 1 Model 2: p = 2 b = -1

10. un-modeled dynamic operator with small L2 norm nominal model noise with finite L1 norm multi-controller: Pick a positive number ¿D multi-estimator: initialize With ¾ frozen, have detectable through e1 verify this! output estimation errors: If p = 1 and ± and n are zero then monitor: y n dwell time switching logic hysteresis switching logic y n

11. multi-controller: multi-estimator: output estimation errors: Pick ¿D to stabilize this system Assume p = 1 injected system ± - -

12. For any T>0 there is a piecewise constant signal à : [0,1) ! {0, 1} such that where S = set of all dwell time switching signals with dwell time ¿D output estimation errors: h verify! injected system ± - -

13. For any T>0 there is a piecewise constant signal à : [0,1) ! {0, 1} such that where Verify that if a = b + c then a2·2b2 + 2c2 via the Bellman –Gronwall Lemma: For 0 ·t·T injected system ± - - OK Minimize g w/r ¿D

14. Let t* be the last time ·T at which ¾(t*) = 2. For any T>0 there is a piecewise constant signal à : [0,1) ! {0, 1} such that where initialize monitor: y n dwell time switching logic y n

15. Let t* be the last time ·T at which ¾(t*) = 2. ¿D ¿D ¿D ¿D ¿D T T T ¾ = 2 ¾ = 1 ¾ = 2 ¾ = 1 Ã = 0 Ã = 1 Ã = 0 ta t* t* t* Ã = 0 Ã = 0

16. Let t* be the last time ·T at which ¾(t*) = 2.

17. Let t* be the last time ·T at which ¾(t*) = 2. For any T>0 there is a piecewise constant signal à : [0,1) ! {0, 1} such that OK where OK Ip is the union of the time intervals on which ¾ = p

18. Response to Sinusoidal Noise ¿D = 0.1 ¿D = 0.5

19. Remarks Described a problem specific hybrid system and outlined a simple way to analyze it. The preceding illustrates that in some situations, the block diagram which describes a system can be completely different than the block diagram needed to analyze the system.

20. A Simple Example: SWITCHING BETWEEN TWO MODELS An Application: STATION KEEPING

21. 3 agents moving in formation at constant velocity V agent 0 FORMATION CONTROL Objective is for agent 0 is to join the formation at the position using only range sensing.

22. FORMATION CONTROL using range – only sensing

23. sensing error

24. sensing error alignment error Station keeping problem: Devise a strategy for moving agent 0 from x0 to x* 1. which depends only on the ri(t) and di 2. whose performance degrades gracefully with increasing sensing and alignment errors. Remarks: The data r1(0), r2(0), r3(0), d1, d2, d3 does not uniquely determine x*(0) No open loop solution exists, even if x*(t) = constant!

25. Error Model sensing error alignment error verify!

27. G = constant Leaders not co-linear G = nonsingular verify! control If e = 0, ¹ = 0, and ² =0 then x0 = x*

28. sensor errors Points 1, 2, 3 are not co-linear sensor & alignment errors

29. e u switched adaptive control Points 1, 2, 3 are not co-linear error system

30. observable controllable e u e u switched adaptive control verify! error system

31. Multi-Controller Multi-Estimator Dwell Time Switching Logic Monitor SUPERVISOR e u switched adaptive control e u

32. 2£2 matrices Multi-Controller Dwell Time Switching Logic Monitor e u Multi-Estimator

33. 2£2 matrices Multi-Controller Dwell Time Switching Logic Monitor e u A- bf = stable

34. 2£2 matrices Dwell Time Switching Logic Monitor e u Multi-Controller A+kc = stable

35. compact set of 2£2 nonsingular matrices Dwell Time Switching Logic Monitor e u A+kc = stable

36. compact set of 2£2 nonsingular matrices Dwell Time Switching Logic Monitor e u Monitor W

37. compact set of 2£2 nonsingular matrices Dwell Time Switching Logic Pth output estimation error exponentially weighted 2 –norm: e u W Dwell Time Switching Logic {quadratic in entries of P}

38. dwell time Initialize G P*is the value of P which minimizes M(Q,P) over G = 0 computation time = D - C Æ G = P* Sample Q = W and minimize M(Q,P) = D = D - C Æ M(Q,P*) < M(Q,G) Æ n y n y n y

39. ANALYTICAL PROPERTIES

40. 0 Dwell Time Switching Logic ANALYTICAL PROPERTIES e u W If no sensor or alignment errors, then e and the position error x0 – x* tend to zero exponentially fast.

41. Simulations Straight Trajectory Curved Trajectory

42. UPENN GRASP LAB TESTS Straight Trajectory Curved Trajectory

43. symmetric positive definite non-convex optimization problem ……. with respect to P2P strictly lower triangular Sample Q = W and minimize M(Q,P) from a finite set Convexity Issues While M(Q, P) is quadratic in the entries of P, P is typically non-convex because of the constraint that it elements must be nonsingular. Reparameterize: Each n£n nonsingular matrix B can be written as Bn£n = Un£n(I + Ln£n)Sn£n This fact can be used to embed the preceding in a set of 8 convex semi-definite programming problems; see papers.

44. Remarks We’ve talked a little bit about the basics of parameter adaptive control and about dwell time switching. We’ve outlined a provably correct solution to the range-only station keeping problem and demonstrated its feasibility via simulation and experimentation. While the solution is limited to agent descriptions which are simple kinematics point models, use of the concept of a “virtual shell” enables one to apply the solution to more general dynamic models. Experience has shown that the trial and error design of the multi-estimator, multi-controller, and monitor gains can be extremely tedious, since at present there are no tools or performance guideline for carry out any of these design processes. This is a fundamental shortcoming of all existing adaptive control design methodologies.