Adaptive Control

Adaptation dynamic process by which the controlleradjusts its interaction with a systemin order to carry out an objective orreach a goal w o exact knowledge ofthe system

Some incomplete history19505 gain scheduling

early Model Reference Adaptive ControlMRAC

1958 R Kalman self tuning controllerregulator for the linear quadraticproblem

19605 stability of adaptive controllersLyapunov stabilityadaptation learning Feldbaum TsypkiD

1966 Parks Lyapunov redesign approach toMRAC

19705 Stability analysis Narendra Morse

19805 limitations Rohrs et al sensitivity toUnmodeled dynamics1983 Morse's conjecture X axtbu

a C IRb Fo

cannot stabilize w o knowledge of sign b1983 Nussbaum disproves Morse's conjecture1984 Willems Byrnes simplified Nussbaum's

constructionexploration us exploitation

Ex ample adaptive regulationin 0kt u x t C IR scalar state

ult EIR inputGEIR is unknown

Goali output regulation1 E o as e ault bounded

what if 0 is knownO o u 0 is optimalUEO x t x o eat 30 since Qso

0 0 U x

X x x t _eco e t o

O o u Otl x

X Ox Ot Dx

What if 0 is unknown

Adaptive control law

X Ox tu u Iti x

onfto.inestimate

tuning law for the controller

Closed loop system plant controller

ix ox to 1 x dsfaf.fiyachcireYalation72

Xlt 30 as e

Note the closed loop system is nonlinear eventhough the unknown plant is linearthe controller is dynamicstate

i.e has a

Ict o t ft x s ds

We will use Lyapunov stability theoryx 1

22 first attempt

t i TtVlxltl2 kit ItX t o Ect 1 ect

unknown but fixedI I 1 XtCan be 70

V x Lzx2 may not be the best choice

Try Vex 8 thx Icon ofdepends on both of the state variables tando

j e 2 it For

XX t J o

x o I 1 X t to 03 2

et t 0 8 2t 8 03 2

so

Is this enough to give xlt o as t

Review Lyapunov stabilityI f x f 107 0 Ceo is an

equilibriumIs x o a stable equilibrium

Stability V E o FS o s.t

4071 8 Htt E at o

i

Asymepthstability stability attractivityFS o s.tt KC 071 8 him t.lt o

C a

Global Asymptotic Stability GAS

V E o V x o xltll.SE after some toasymptotic stability

Thy Lyapunov stability criterionX f't

et V be a C and positive definite fan of stateX 70 VX and e o iff X o

Define its time derivative along X ft by

x TVC effleythen

if x Eo everywhere then 0 is a stableequilibriumif i e 0 for all O and i co o then0 is AD

if i a 0 for all e o and V 07 0 andis radially unbounded e or as

I i GAS

Backtanpleto 43 E k Kid e Cao

x f text's to of d p d

V x2

system is stable in the sense of Lyapunovnot enough info to get ASCx I 2

Lx 07 0 for all IX 8 with keo

Need better tools weak Lyapunov for

Keyt ays1 closed loop dynamics of plant t adaptivecontroller are generally nonlinear even if

the plant is linear2 the controller is dynamicstate dynamics

has a nontrivial

this enables tearing

3 standard Lyapunov stability theory not strongenough need to account for a

life B X indep of I

4 I need not converge to 0