Station Keeping with Range Only Sensing {Adaptive Control}

A. S. Morse

Yale University

Gif – sur - Yvette May 24, 2012

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 controlsstudied in control theory were relay {eg. bang bang} controls.

Although some relay controls used hysteresis {which is a simple form of memorylogic}, 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

_y = 3y + bu; b2 f ¡ 1;1g

Shortly thereafter Roger Nussbaum {a math analyst at Rutgers with nobackground in control} proved the conjecture false by actually constructinga smooth nonlinear {but not rational} stabilizing control.

This work motivated a number of people, notably Bengt Martensson at Lund, torethink adaptive control and in particular to ask just how much information is needed about a linear system in order to control it.

A Little BackgroundA Little Background

Bengt jolted conventional wisdom in adaptive control by constructivelydemonstrating that all one needs to know about a SISO linear systemin 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 ti is the ith time a controller switch takes place. If there is a next switch after ti, it occurs at the smallest valueof t ¸ ti 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 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.

Problem:

There is a finite time T at whichswitching stops and is has a finite limit as t ! 1.

initialize ¾

¾ = ¾*

n y¹¾ * + h < ¹¾

Lets look at a very simple example

But before we start we are going to need the Bellman-Gronwall Lemma

Lemma: If some numbers tb ta, some constant c > 0 and some nonnegative,piecewise-continuous function ®: [ta, tb] ! IR, w: [ta, tb] ! IRis a continuous function satisfying

then

Bellman-Gronwall Lemma

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

p2 f1;2g

_y = 3y + (3¡ 2p)u+ n + ±±y

noise with finite L1 norm

un-modeled dynamic operatorwith small L2 norm

_y = 3y + bu b2 f ¡ 1;1g

_y = 3y + (3¡ 2p)u

nominal model

Model 1: p = 1 b = 1Model 2: p = 2 b = -1

Hey, I thought you said simple!

p2 f1;2g

multi-estimator:

_x1 = ¡ x1 + 4y + u_x2 = ¡ x2 + 4y ¡ uoutput estimation errors:

e1 = x1 ¡ y; e2 = x2 ¡ y

hysteresisswitching

logic ¹ 2

monitor: _¹ 1 = e2

1; _¹ 2 = e22; ¹ 1(0) = ¹ 2(0) = 0

initialize

yn

ny

dwell time switching

logic ¹ 2

_¹ i = ¡ ¸¹ i + e2i

multi-controller: u = ¡ 4(3¡ 2¾)y; ¾: [0;1 ) ! f1;2g_y = ¡ y +4y+ (3¡ 2p)u+ n + ±±y

¹¾= argminf ¹ 1;¹ 2g

Pick a positive number ¿D

If p = 1 and ± and n are zero then _e1 = ¡ e1

With ¾ frozen, havedetectable through e1

_y = 3y + (3¡ 2p)u+ n + ±±y

noise with finite L1 norm

un-modeled dynamic operatorwith small L2 norm

nominal model

verify this!

_y = ¡ y +4y+ (3¡ 2p)u+ n + ±±y_y = 3y + (3¡ 2p)u+ n + ±±y

multi-estimator:

_x1 = ¡ x1 + 4y + u_x2 = ¡ x2 + 4y ¡ u

_x1 = ¡ x1 ¡ 8(1 ¡ ¾)y_x2 = ¡ x2 + 8(2 ¡ ¾)y

x¾ = (2¡ ¾)x1 + (¾¡ 1)x2

output estimation errors:

e1 = x1 ¡ y; e2 = x2 ¡ y

injectedsystem

±

--

Assume p = 1

jjzjj =q R1

0 z2(t)dt

Pick ¿D to stabilize this system

p2 f1;2g

multi-controller: u = ¡ 4(3¡ 2¾)y; ¾: [0;1 ) ! f1;2g

injectedsystem

±

--

output estimation errors:

e1 = x1 ¡ y; e2 = x2 ¡ y

e¾¡ e1 = (1¡ ¾)(x1 ¡ x2)

For any T>0 there is a piecewise constantsignal à : [0,1) ! {0, 1} such that

where

h

x¾= (2¡ ¾)x1 + (¾¡ 1)x2

S = set of all dwell time switching signals with dwell time ¿D

verify!

injectedsystem

±

--

For any T>0 there is a piecewise constantsignal à : [0,1) ! {0, 1} such that

where

For 0 · t · T

OK

Minimize g w/r ¿D

Verify that if a = b + cthen a2 · 2b2 + 2c2via the Bellman –Gronwall Lemma:

monitor:

initialize

yn

ny

For any T>0 there is a piecewise constantsignal à : [0,1) ! {0, 1} such that

where

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

dwell time switching

logic

¹¾= argminf ¹ 1;¹ 3g

¿D ¿D ¿D ¿D ¿D

¾ = 2¾ = 2 ¾ = 1¾ = 1

TT

t*t* t*

T

ta

à = 0 à = 0à = 1

à = 0 à = 0

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

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

For any T>0 there is a piecewise constantsignal à : [0,1) ! {0, 1} such that

where

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

jje¾¡ Ã(e¾¡ e1)jj2T = jj(1¡ Ã)e¾jj2T + jjÃe1jj2T

· jje1jj2t¤ + jje1jj2T

OK

OK

Ip is the union of the timeintervals 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 MODELSAn 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

FORMATION CONTROL

using range – only sensing

x1 x3

x0

x2

ri ¼jjxi ¡ x0jj; i 2 f1;2;3g

r3r2;r1;

r1

r2

r3

f

sensing error

x1 x3

x0

x2

d3d1

d2

ri ¼jjxi ¡ x0jj; i 2 f1;2;3gdi ¼jjxi ¡ x¤jj; i 2 f1;2;3g

f r1; r2; r3

sensing error

f r1; r2; r3; d1; d2; d3g

x¤

alignment error

Station keeping problem: Devise a strategy formoving 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 doesnot uniquely determine x*(0)

No open loop solution exists, even ifx*(t) = constant!

Error Model

alignment errorsensing error

verify!

G = constant

Leaders not co-linear G = nonsingular

If e = 0, ¹ = 0, and² =0 then x0 = x*

controlverify!

x1 x3

x0

x2

d3d1

d2

ri ¼jjxi ¡ x0jj; i 2 f1;2;3gdi ¼jjxi ¡ x¤jj; i 2 f1;2;3g

f r1; r2; r3f r1; r2; r3; d1; d2; d3g

x¤

_xi = constant; i 2 f1;2;3gPoints 1, 2, 3 are not co-linear

_x0 = u0 _u0 = uei = jjri jj2 ¡ d2

i ; i 2 f1;2;3g

e =" e1 ¡ e3

e2 ¡ e3

#

e = x + ¹ jjG¡ 1xjj + ²Äx = Gu

G2£ 2 = 2"

x03 ¡ x01x03 ¡ x02

#= constant & nonsingular

sensor errors sensor & alignment errors

x1 x3

x0

x2

d3d1

d2

ri ¼jjxi ¡ x0jj; i 2 f1;2;3gdi ¼jjxi ¡ x¤jj; i 2 f1;2;3g

f r1; r2; r3f r1; r2; r3; d1; d2; d3g

x¤

_xi = constant; i 2 f1;2;3gPoints 1, 2, 3 are not co-linear

_x0 = u0 _u0 = uei = jjri jj2 ¡ d2

i ; i 2 f1;2;3g

e =" e1 ¡ e3

e2 ¡ e3

#

error system

u ee = x + ¹ jjG¡ 1xjj + ²Äx = Gu

f ¹ ; ²g

switched adaptive control

error system

u ee = x + ¹ jjG¡ 1xjj + ²Äx = Gu

f ¹ ; ²g

e = X b+ ¹ jjG¡ 1X bjj + ²_X = X A + Guc

u e

f ¹ ; ²gX = [x _x ]2£ 2 c = [0 1 ] A =

" 0 01 0

#b=

" 10

#

switched adaptive control

controllableobservable

verify!

SUPERVISOR

e = X b+ ¹ jjG¡ 1X bjj + ²_X = X A + Guc

u e

f ¹ ; ²g

switched adaptive control

Multi-Controller Multi-Estimator

u e

Monitor

Dwell Time Switching Logic

e = X b+ ¹ jjG¡ 1X bjj + ²_X = X A + Guc

u e

f ¹ ; ²g

Multi-Controller Multi-Estimator

Monitor

Dwell Time Switching Logic

f Z1;Z2g

2£2 matrices

(Z1 + GZ2 ¡ X ) ! 0e ¼(Z1 + GZ2)b

e = X b+ ¹ jjG¡ 1X bjj + ²_X = X A + Guc

u e

f ¹ ; ²g

Multi-Controller

Monitor

Dwell Time Switching Logic

f Z1;Z2g _Z1 = Z1(A ¡ bf ) + ef_Z2 = Z2(A ¡ bf ) + uc

A - bf = stable

2£2 matrices

(Z1 + GZ2 ¡ X ) ! 0e ¼(Z1 + GZ2)b

e = X b+ ¹ jjG¡ 1X bjj + ²_X = X A + Guc

u e

f ¹ ; ²g

Multi-Controller

Monitor

Dwell Time Switching Logic

f Z1;Z2g _Z1 = Z1(A ¡ bf ) + ef_Z2 = Z2(A ¡ bf ) + uc

A + kc = stable

u = G¡ 1X k _X = X (A + kc)

2£2 matrices

(Z1 + GZ2 ¡ X ) ! 0e ¼(Z1 + GZ2)b

e = X b+ ¹ jjG¡ 1X bjj + ²_X = X A + Guc

u e

f ¹ ; ²g

Monitor

Dwell Time Switching Logic

f Z1;Z2g _Z1 = Z1(A ¡ bf ) + ef_Z2 = Z2(A ¡ bf ) + uc

A + kc = stable

u = G¡ 1X k _X = X (A + kc)

u = bG¡ 1(Z1 + bGZ2)k

bG ¼ G

bG 2 G = compact set of 2£2 nonsingular matrices

(Z1 + GZ2 ¡ X ) ! 0e ¼(Z1 + GZ2)b

e = X b+ ¹ jjG¡ 1X bjj + ²_X = X A + Guc

u e

f ¹ ; ²g

Monitor

Dwell Time Switching Logic

f Z1;Z2g _Z1 = Z1(A ¡ bf ) + ef_Z2 = Z2(A ¡ bf ) + uc

u = bG¡ 1(Z1 + bGZ2)k

Monitor

W

bG 2 G = compact set of 2£2 nonsingular matrices

e ¼(Z1 + GZ2)b

e = X b+ ¹ jjG¡ 1X bjj + ²_X = X A + Guc

u e

_Z1 = Z1(A ¡ bf ) + ef_Z2 = Z2(A ¡ bf ) + uc

u = bG¡ 1(Z1 + bGZ2)k

Dwell Time Switching Logic

W

f Z1;Z2g

f ¹ ; ²gbG 2 G = compact set of 2£2 nonsingular matrices M (Q;P ) = tracef [ I P ]Q [ I P ]0g

Q 2 IR4£ 4; P 2 G

_W = ¡ 2¸W +" Z1b¡ e

Z2b

#" Z1b¡ eZ2b

#0

M (W(t);P ) = e¡ 2¸tjje¡ (Z1+P Z2)bjj2t ; t ¸ 0; P 2 G

e ¼(Z1 + GZ2)b

Pth output estimation error

exponentially weighted 2 –norm: jj ¢jjt ¢=s Z t

0f e sjj ¢jj2gds

Dwell Time Switching Logic{quadratic in entries of P}

M(Q,P*) < M(Q,G)Æ

ny

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

Ædwell time

computation time

G = P*Æ

M (Q;P ) = tracef [ I P ]Q [ I P ]0gQ 2 IR4£ 4; P 2 G

ANALYTICAL PROPERTIES

e = X b+ ¹ jjG¡ 1X bjj + ²_X = X A + Guc

u e

_Z1 = Z1(A ¡ bf ) + ef_Z2 = Z2(A ¡ bf ) + uc

u = bG¡ 1(Z1 + bGZ2)k

Dwell Time Switching Logic

W

bG

f Z1;Z2g

f ¹ ; ²g

_W = ¡ 2¸W +" Z1b¡ e

Z2b

#" Z1b¡ eZ2b

#0

If no sensor or alignment errors, then e and the position error x0 – x* tend to zero exponentially fast.

0ANALYTICAL PROPERTIES

If jj¹ jj < 1jjG ¡ 1 jj ; then theinduced L 1 and L 2 gains from² to x0 ¡ x¤ are¯nite

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 ofthe constraint that it elements must be nonsingular.

Sample Q = W and minimize M(Q,P) ……. with respect to P 2 P

non-convexoptimization

problem

Each n£n nonsingular matrix B can be written as

Bn£n = Un£n(I + Ln£n)Sn£n

Reparameterize:

This fact can be used to embed the preceding in a set of 8convex semi-definite programming problems; see papers.

strictly lower triangular

from a finite set

Convexity Issues M (Q;P ) = tracef [ I P ]Q [ I P ]0gQ 2 IR4£ 4; P 2 Gsymmetric positive

definite

Remarks

We’ve outlined a provably correct solution to the range-only station keeping problemand demonstrated its feasibility via simulation and experimentation.

While the solution is limited to agent descriptions which are simple kinematicspoint models, use of the concept of a “virtual shell” enables one to applythe 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 thereare 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 andabout dwell time switching.