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.: A A A A A A A A AA A A AAAA A Supelec EECI Graduate School in Control

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

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 Bengt Martensson 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. A Little Background

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 integer i where t i is the ith time a controller switch takes place. If there is a next switch after t i, it occurs at the smallest value of t ¸ t i at which the above inequality fails to hold. 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. Bengt’s work motivated a great deal of research on switching control.

Pick a positive constant h: hysteresis switching..... Hysteresis Switching Given a finite set of scalar valued monotone non-decreasing signals ¹ 1, ¹ 2,....., ¹ m 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. Problem: There is a finite time T at which switching stops and is has a finite limit as t ! 1. initialize ¾ ¾ = ¾ * ny ¹ ¾ * + h < ¹ ¾

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

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

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

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

multi-estimator: output estimation errors: hysteresis switching logic monitor: initialize y n ny dwell time switching logic multi-controller: Pick a positive number ¿ D If p = 1 and ± and n are zero then With ¾ frozen, have detectable through e 1 noise with finite L 1 norm un-modeled dynamic operator with small L 2 norm nominal model verify this!

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

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

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

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

¿D¿D ¿D¿D ¿D¿D ¿D¿D ¿D¿D ¾ = 2 ¾ = 1 T T t*t* t*t* t*t* T tata à = 0 à = 1 à = 0 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 Let t * be the last time · T at which ¾ (t * ) = 2. OK I p is the union of the time intervals on which ¾ = p

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

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.

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

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

using range – only sensing

sensing error

alignment error Station keeping problem: Devise a strategy for moving agent 0 from x 0 to x * 1. which depends only on the r i (t) and d i 2. whose performance degrades gracefully with increasing sensing and alignment errors. Remarks: The data r 1 (0), r 2 (0), r 3 (0), d 1, d 2, d 3 does not uniquely determine x * (0) No open loop solution exists, even if x * (t) = constant!

Error Model alignment error sensing error verify!

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

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

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

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

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

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

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

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

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

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

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

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

ANALYTICAL PROPERTIES

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

Simulations Straight Trajectory Curved Trajectory

UPENN GRASP LAB TESTS Straight Trajectory Curved Trajectory

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. Sample Q = W and minimize M(Q,P) ……. with respect to P 2 P non-convex optimization problem Each n £ n nonsingular matrix B can be written as B n £ n = U n £ n (I + L n £ n )S n £ n Reparameterize: This fact can be used to embed the preceding in a set of 8 convex semi-definite programming problems; see papers. strictly lower triangular from a finite set Convexity Issues symmetric positive definite

Remarks 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. We’ve talked a little bit about the basics of parameter adaptive control and about dwell time switching.