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Nonlinear Dynamics of Adaptive Linear Systems: An Elementary Example
Copyright © IFA C :\onlinear COlltrol Snt e llls Design . Capri. It al, 1989
NONLINEAR DYNAMICS OF ADAPTIVE LINEAR SYSTEMS: AN ELEMENTARY EXAMPLE L. Praly C.A.l.
: \II!OIll(J!iqll l' .
c:!"ol,' dn .\fill l'.l . 35 Ril l' Fmll!"/,
Suill! 1-/ 0 11 or';.
11](J5
F UII!ai ll l'h/l'IIlI.
\Ye analyze the phase p ortrait and describe some non local non linear behavior of a dm;"d loop ,y,tem made of an adapt i\'C proportional controll er and a di sturbed first order lincar ,y,tClIl. Our rc"ults a re obtained applying p erturba tion methods -- P oincarc l11ethod. structural sta l,ility of lIortnally hyper bolic invariant sets. The conclusion is twofold: • An adapt ive linear s~'st('m in the idea l case is vpry critical. In pr('s('nce of ]H'rturll
Introduction
1
T ypicall y, adaptive linear controllers are designed from a linear point of vi ew:
This cont roller is known to give bounded solu tions when placed in feed back with a s ~' stem such that a sequence a exists to sa tisfy for each t. 00 - R :s: a(t) :s: 00 + Rand:
• First , a linear but parameterized li near controller is designed , applying linear system theory.
~ ~
• Second , a parameter adaptation law is designed . npplying linear es timation t heory.
q
O( t
+ 1)
+
{) ,, (t)
V(i) (U(i
00
=
u(i)
+
,,(1)2
+
10+(t~- Ool} ((1+(1) -
v(i)
R. 00
C2 q
(2)
;O
+
V
(3)
=
ay(t -1 )
+
u(t-1)+o
(5)
with a belonging to [flo - R. 00 + R] ~ I oreover. if 6 is zero and l' is COllst ant . (O( t). y( I )) cOlwcrges globally to ( a . 1') . in an 12 -;;I'nse an d l'xlOollenti:111y if l' is not zero ( Good"'in and Sin. 19S-l ). Ho\\,/'\'f'r. for l' zero . any arbitrarily small disturhnnc(, t lea/l> to an intermitt ent phe~ nomC'nllm (Pomeau and ~Ianvilk. 19S0 ). " 'hose characterist ics on t he v-component sllggested the name bursting (,.l.ncll'rson. 19S5: .J a·idane-Sai·dane and ~Iacchi. 19S5 ). From a linear point of view. a constant exogenous signal lending to such a wry high frequency output is \'ery disappointing. In l'ractirl'. s(' \'era l fixes ar" added - dead zone ( Egard t . 19, 9). int('rnal model principle ( Elliott and Goodwin. 19S.. ). filtering (.-\nder50n et a!.. 1%6 ). . . Howe\·er. if those fixes are not appropriately cho,en. a qualitatiwly simi la r bebavior may he oh,C'\'\'('d for these more intricate ca:;es (P raly. 19SS I. T hi" mot i\'ates our interest in the closed loop sy"tem (1). ( 5 ) and the objec ti\'e of tb s pa~ per is to anill~' zp its phase pOl'trait and to explain some non local nonl inear b"ha\' ior of it s solut ions. This will implici tely d emon>! rat e that lion lillear system theory is filII of very appropriate re:;tIl t :; and the lil!('ar point of \'icw can be a(h'alltagcousl~' compcnsated for.
( 1)
flu)
1' (1)
• the adapted pnrameter is looked for in t he intprvnl -
la(i) - a(i - l)I:S:
,=/+ 1
+A2
q
In particular. thi s result alOpliC's to the simplest of these sys tem s. object of the forthcoming ,tudy:
u' V and l' are the system input . sys tem output and ,et p oint signa l respectively. (s. O) is the controller state . () h a\'ing being gi\'en the interpret ation of an adap ted pa r ameter. R. 00 . f1 and A" a re design parameters. int erpreted as:
[00
r+q
L
Cl
(4)
v(t)2
+ 1) - ,,(I) - (I (t)U(I)) . An sup{l.s(t + 1l} + V(t)2
+ min {1.
- O(i)V(t)
+
<
where:
Howe\·er. an adaptive linear controller is a dYllamic nonlin ear controller. For exa mple . in this paper we \I'ill be concerned with the follo\l'ing proportional controlkr III it s robust adapt ive version: /,2"(1)
sup{l.~(I+l)}
q
;'\ot only design but analysis is m ade fr om a linear poin t of view. Doundedness of the solutions is established applying the robust li nea r stabi lity theory. T heir properti es ar(' st ud ied simil arly with, some times, the help of a\'E'raging theory.
+ 1)
[Y(i+1)-U(i) - a(i)Y(i)]2 1
• Finally, an adapti\'e li nenr controller is obtained from the parameteri zedlinear controller wi th those parameters gi\'en on line by the adaptation law.
s( t
i~l
+ R]
• the larger the posit i\'p A: I IS . the slower the adapta tion. i.e. the O-dynamics. is.
257
258
L Pralv
To appply these results. it is more appropriate to write the system in the so called standard form. This is done as follows: when restricted to the set: W=
{
(y.s.lI)
I
s~l.llI-lIo+y
<5-(II-a)y 11 + y
2
I ~R
}
(6)
with r constant and lett ing:
x = yl<5 :
1.'
= II-a :
This paper is a short versIon of a report written by ;"lartin E spana and the author (Espana and Praly, 1988). Due to space limit a tions. no simu lation results will be presented in this paper. They can be easily reproduced by the reader (take for example d = 0.01).
2 Q
=
rib:
the closed loop system is decribed by the map ~ from R2 into R2. giwn by:
The frozen system
Our analysis will be done by considering perturbation of the so called frozen system: ~ (v.l')
f
(8) The equation for the ,,-component is omited since it has no influence on the remaining part of the system. The standard form (8) puts in relief the role played , in the qualitative beha\'ior of the solutions, by the set point-todisturbance ratio Q and by the positiw disturbance-toadaptation speed ratio d. In particular this shows that d is the effective adaptation speed control. As most of the adaptive linear systems. this system belongs to the family of systems which can be described , with A. , I3 , C smooth functions, by the map ~g from IIr into Rn, given by:
=
1,: (
~
as a small
)
(10)
-1.'1'+1+<1
As mentioned in Introduct ion, it is a family of linear systems parameterized by ('. The linear system theory allows us to completely describe its phase portrait : Property 1 (Frozen system): A frozen sy.
1 +<1 .r = - 1+ 1,"
( ll )
• if 111'1 is stritciy larger than 1, it diverges exponentially to infinity from the point (1/-', 1') of the same graph.
(9)
• if~' is equal to 1, it is a period-2 solution.
For d small , this map appears locally as a perturbation of a map made of a family of linear maps. This remark motivates for applying perturbation methods - Poincare method , structural stability of normally hyperbolic invariant sets, averaging. By doing so. and for the general system (9), can be obtained:
• if ';' is f.qual to - l. its l'-component grows linf.arly towards +00.
As will become clear later, the impor tant fact s in this result are: • the set:
• existence of limit sets (Bodson et aI. , 1086: Praly and Pomet, 1087).
Sf
• existence of in\'ariant sets (Praly, 1985. 1089: Riedle and I\:okot ovic. 1(86). Their attractivity and I or repulsiv ity gives then information on the solution behavior.
To obtain the basis of our perturbation analysis. we start by studying the system obtained by taking d equal to zero and called the frozen system. Then. fixed points and periodic solutions of the map ~ are considered. In Section 4. we establish existence and properties of locally invariant sets. Critical elements and locally im'ariant se ts are combined in Section 5 to obtain theoretical results on the system global dynamics. These results are interpreted in Section 6.
~ ::. ' ~';i -1 }
which is a graph and is invariant under
(12) ~f'
• the frozen system stability boundary 11/,1 = 1 which separates this im'ariant set into normally hyperbolic locally invariant components, namely: - Sf n {( 1.' ..r) I IL-I > 1 } is exponentially repellent, - Sf n {(1.' , .r) 111.'1 < I } is exponentially attractive,
• description of the motion along these invariant sets (Ljung, Soderstrom. 1984: :'l.nderson et al.. 1986: RiedIe, 1986: Benwniste et al.. 1987: Praly, 1080). Here considering our disturbance-to-adaptation speed ratio d as a sma ll parameter for the perturbation analy sis. we apply these methods to our elementary example. From their generality, we expect that severa l of our conclusions extend to more general situations. Also. om analy sis can be completed to obtain more precise local results by a bifurcation analysis (see (Golden and Ydstie. 1(88) or (;" lareels and Bitmead. 1986. 1088) for example).
= {(';' , x) I x =
• the periocl-2 solutions which are critical.
3
Equilibrium point and period-2 solutions
Going back to the actual system. let us see how the limit se ts of the frozen system are disturbed. An elementary continuity argument shows (Praly and POlnet. 1087): Lemma 1 (Existence of periodic solutions): A neceswry condition for a ,'olution to be a period-T ,'01,,tion of ~ which remain.' bounded a,' d goes to O. i" that the accumulation point of it" mitial condition be one of the followmg 3 point .. : I. ' 0
<1 1 + <1±~ 2
}
(13)
:"Jonlinear Dynamics of Adaptive Linear Systems
259
Consequently, from the the frozen system equilibrium set SJ, only one point is of interest. Similarly from the set of critical period-2 solutions, only one persists by a continuous perturbation. To precise this necessary condition, we apply Poincare method (Lefschetz . 1977) and get:
Property 3 (Repellent locally invariant set):
Property 2 (Critical elements): i) The map I: has a unique fixed point for all
There exists a bounded Lipschitz continuous function H defined on {I~'I 2: 1 + f} and such that, with the function 9H defined by:
0 different from 0 or -1. It is exponentially stable for 0-' in (-1, - P,) and exponentially unstable for 0 -, outside [- 1, -P,] . with P, the unique solution of:
2(P+ 1) =
d
(14)
-2-
P +d
ii) For any 0 strictly smaller than 1 I7l absolute value, one can find a strictly positive con .. tant do , such that for all d, smaller than do. there exit two locally unique period-2 solutions wh,ch can be approximated by: 1 -
0 1 + 0 Of v'1='Q2 (I ----'----'----') .)
l+a±-~-
+
2
T
For any non-zero d, let
be the smallest positive root of:
f
' (l-l/-,H(JjJ)) 1 + dH(JjJ)2
= tP + dH (1,1)
1>fJ(1/: )
(20)
i) If: 1~'I2:
1 + f, I1>H(tC')1 2: 1 + f , sgn(1)H('';'')) '" sgn(1j: ) (2 1)
then
H( 1)l/('';')) '" 1 + 0
(22)
,;,H( ";',)
-
ii) There exists p positive such that: (<:>,y) = Z:;(,p,x) and (1,'),T) in {1..;..12: 1 + f} X R with,p1> positive implies:
(15)
Old)
Iy -
H("')I 2: (1
+ p)
Ix -
H:
(23)
H(JjJ)1
sup
{~\H( JjJ)
1
+ 0'\ }
The.
iii) Approximation of
T
Hence, in fact for any d, there exists an exponcnt ially repellent locally invariant set. It can be approximated by the frozen parameter invariant set SJ, for d sufficient ly smalL
=
27r j2d(1 - ( 2 )
( 16 )
+ Old)
From this result, it follows that the control objective: output = set point cannot be met asymptotically if the set point-ta-disturbance ratio is too small (X arendra and Annaswamy, 1986). Ew'n more, with this small ratio, there exists a period-2 solution lying close to the frozen system stabili ty boundary. This solution being a focus, it explains locally and at least transient Iy the intermittent behavior mentioned in Introduction. Also, being proportional to jd(l - 0'2) , the frequency of the corresponding bursts decreases as the disturbance-ta-adaptation speed ratio goes to zero or the set point-ta-disturbance ratio goes to 1. Though not established here, it is also noticeable that. for a equal to zero. the period-2 solution is:
= 1
~,
.T
= 0 or 1
(17)
It lies exactly on the frozen system stability boundary and is a critical attractor.
Locally invariant sets
l.(t
+ 1) -
1+
0
)
=
-l'(t) (.r(1) -
+ . ;.,
Property 4 (Attractive locally invariant set): For any d such that:
o
1
< d < d' =
211 + 0'1 2
( \1 ~1
+ +
311 211
+ 0'1 _ + 0'1
1)
(24)
let 1) be the smallest positive root of:
2>(1)) =
(
1) -
_dn2) _0 _ 1 + dn6
-
( 1 + 2no)nod
2
1 + dn~
(25)
where no and n, are defined by:
no
+ 0'1
11
=
(1
+ dn6) no
(26)
(1+ 2n o)d
1)
There exists a bounded continuou.o Lips chitz function G .o1lch that ) with t"c( <:» a function implicitly defined by:
+ d G(t·c(O))
1j'c(9)
(27)
1 + d G{t"c(<:»)2 (i) If:
In Section 2. we haw noticed that the graph SJ is im'ariant for th e frozen systf'm. For thc actual system. wc get (when this makes sense): (
1
d
is bounded.
" =
4
{1';'1"l+'l
~ ( )) (18)
1+l'(t+1) l+l' t +d _ _~(I_+~o~)_T(~t~)(_X~(t~)l~·(~t~)_-_l~)-~
(1 + L"( t) l( 1 + l'( t) + d.l'( 1)( 1 + J'( t))) "-ith d in the second term on the right hand. we see that SJ is close to be a locally invariant set of::. In fact. SJ being locally normally hyperbolic for the frozen system. for the actual system with d small enough. we can expect the existence of locally invariant sets. close to SJ. being repellent in the set {Iv! > I } and attracti"e in the set {Ivl < I} (Shub . 1978: Hirsch. Pugh and 51mb. 1976). To pro\'(' this existence we apply the graph transform technique (Shub. 1978).
10
I :s
1-
and
1)
Ice( <))1 :s
(28)
1 - ')
then:
G(o)
1 +0 -
l'
o (ii) Let
+
VG{L')
d G(v)(I-vG{L'))
1 + dG(L')2 ~
)
(29)
.oatisfy: (30)
there exists a. depending on ~ and strictly "maller than 1 such that: (o.y) = ::(l' , x) and (L',X) in { Iv! < 1 - III X {Ixl :S () implies:
L. Praly
260 ly-G(4))1 ~ a Ix-G(I/»)I
(31)
1.'.' (1)';'(1-1»0 and 1';·(I)12: 1 +f. VO:St
(34)
{~IG(0) _1+Q I }
Case 0< -1: For all I in (0, T). (~,(t) - l·(t -l)) ~·(t -1) is negative. With (i), this implies that 1~' (t)1 goes monotonically to and crosses 1 + E. Con,'eqlLently T is finite.
Consequently, for d sufficiently small, there exists an exponentially attractiye locally inyariant set which can be approximated by the froz en parameter inyariant set Sf.
Case a = -1: H(v) == 0 and any point in the repel/ent locally invariant set i.' a fixed point of~. T I,' infinite.
sup
(iii)ApproximationofG:
{Iwl«l-ry)}
d
1 + 1jJ
is bounded.
\Vith these two properties, we see that for d small enough the linear analysis made for the frozen system giws a good approximation of the behayior of the actual system as long as we are far enough from the frozen sys tem stability boundary 11,1' 1 = 1 and its im'ariant set Sf.
Case -1 < Q :S 0: If l'(O) is positive. ":'(1) is strictly increasiny and therefore converges to +:x: while x(l) goes to zero. Con.'eq1tently. T is infinite. If ,;.( 0) < min {I/o. -( 1 + d}. l'( t) is "trictly decreasing to -QC while .r( t) goe,' to zero. Consequently. T is infinite. If 1/0 < dO) < -( 1 + E) . dt) i.... trictl y increasing and craSH,' -(1 + d· Con$equ ently . T i" finite.
Behavior of the solutions: technical results
Case 0 < 0: 1..( t) is mono tonically going towards 1/ Q. Consequently. T i.< finite if and only if dO) is negative and/or 1/0 < 1 + E.
Knowing the existence of critical elements and locally invariant sets, we are now in position for studying the behavior of the solutions. \"ith E giw'n by (19), 71 giwn by (25), d* given by (24) and:
Pl'operty 7 (Solutions in the "strict instability" set outside the l'epellent locally inval·iant set, sets: A , I ).' For any non zero d: (i) If for some time 10. a solution ,
5
X .
1
> 1 - 71
A = {(I/>,x)11/> ~ -(1
Hence , there is no .. olution sali.sfying for ever xi- H( I/» and I~' I 2: 1
G
D In the following we state sewral properties which are proyed in (Espana and Praly, 1988 ). They arc giwn without any comment, their interpretation being the topic of the next Sectio11. Property 5 (Solution boundedness, sets: A to I ): (i) If Q lie~ in (- L 01, ~ has unbounded solutzon,'. (ii) If Q is not in [- L 01. all the ,'olution,' of ~ are bounded. Property 6 (Solutions in the repellent locally invariant set, sets: A , I ): For any non zero d: (i) The fixed point of~. belongs to the repel/ent locally 111' variant set if and only if 11/al:S 1 + f
.
x(O) = H( 1.'·( O))
and let T be the largest integer slLch that:
(if) if 10 I i.< .,tridly smaller than l. a.ll the .'ollltions in the locally invariant .
Property 9 (Solutions in the "strict stability" set outside the attractive locally invariant set, set: E): For any d. 0 < d < d* (i) Any ,'olution In { id:S 1 - 'd x R exponentially approache., the graph {rv. G ( v))}. MOreOl'eT a. sollJ.tion "ta.rting in {ld :S 1 - 'd x R remal11.' In tll1,< .. et a.< long IL,< it remain3 in the 3et {I·r ! 2: l~ r,}
(ii) if 10 I < 1 a.nd for .'ome I1me 10. a .. olltlion .
(11) Let (l.·(t),x(I)) be a solution such that: f
.
Property 8 (Solutions in the attractive locally invariant set, set: E ). For any d. 0 < d < d* (i) The 1tniqlLe cqlLilihrinm point of ~ i.< in the locally invllriant .
F
E
+E
(ii) Moreover, while the ,;·,H( J/·))} and cros.
or graphically:
B
(35)
(36)
I={(I/> ,x)1 1+E:SI/>}
C
H (1!·(to))
+ E)}
B={(I/>,x)1 -(1+E)<-(1-1])andx~x} C = {( I/>,x)1 - (1 + f) < I/> < -(1- 71) and Ixl < X} D={( I/>,x)1 -(l+f)<1/J<-(l-ry)andx:S-x} E={( I/>,x)1 -(l-ry):SI/>:Sl-ry} F = {(I/> , x) I 1 - 'r/ < I/> < 1 + f and X :S x} G = {(I/>,x)11 - 71 < I/> < 1 + f and Ixl < X} H = {( 1/>, x) I 1 - 71 < I/> < 1 + f and x :S - X }
A
i-
then there exi.'t.< a (finite) tim.e 11 -,u.ch that:
we decompose the plane (I/>, x) into nine subsets:
11,"( 0) 1 2: 1 +
.T(tO)
(32)
(33)
( 37)
then there exi.'t.' a (fin1te) time 11 ,
Iv(ll)1 > 1 - 'I
( 38)
:--1onlinear Dvn a mics o f Adapti\'e Linea r Systems H ence, for lal < 1, there ij no so lution satisfying for ever l,pl ~ 1-1) .
(iii) Moreover, while a so lution remainj in the set {d(1i~~~!)nl~ ,p ~ 1 -1) } x {lxl~O ( rejp . in the set {-(I -1)) ~ ,p ~ d( li~~~f)nl} X {lx l ~ 0 ), it crOjses the attractive locally invariant set at each time t (rcsp. it remains on the same side).
Property 10 (Solutions in the "critical stability" sets B , D , F , H ) : A s long as a solution remains in the jet Ixl ~x}. its':; component is exponentially decaying, H ence there is no solu tion satisfying for ever 1 - 1) < 11,1'1 < 1 + € and 1:1'1 ~ '(
{(,p,x)1 l -1)
Property 11 (Solutions in the "critical stability" set G ): If 11/al is larger than 1 + f and. for d small enough. period-2 sohttionj exist and are in the jtt : (39)
6
Behavior of the solutions: interpretation
The behavior of the solutions of l: can be roughly expla ined with the followinli remarks: According to Property 7, a solut ion in t he set A or I, but not in the repellent locally invariant set, diverges exponentially from a set which is the graph of a uniformly bounded fun ction of ,p . This explains an exponential growt h of the x-component which becomes and remains large. Moreover, for a solution in t he set I, at each time t , the x-component changes side with respec t to this graph. This ex plains a burst with a very high frequen cy content of this component. Conversely, for a solution in the set A. the x-component remains on the same side of the graph. It correspond s a burst without oscillations. This beha\'ior appears for any value of the disturbance and the set point. According to P roperty 10. a solut ion in the set A or I with a la rge I-component or in the set B . D . F or H has it s 1jo-component exponent ially decaying. This behavior exists for any value of the disturbance and the set point. According to Property 9. as soon as a soluti on enters the set E . it is exponentially attracted toward s a set ,,·hich is once again the gr aph of a un iformly bounded functi on of L·. This explains the exponential dec rease of the Icomponent and the fast decay of it s high frequency content if it were present . This beha\-ior happ en s for any \'alue of the set point -to-d ist urbance ratio and at least for small values of the disturbance-to-adaptation speed ratio. \\'hile a solu tion. in the set E. goes to the attracti\'e locally im-ariant set. its evoluti on is more and more similar to th is of the soluti on s in this set. According to P roperti es 4 and I . this explains a speed of the tJ'-component of the Jrder of d. :-'loreo\'er , accordi ng to Propert ies 2 and 8, if the set point-to-disturbance ratio la I is strictly larger than 1, the solutions converge t o the fixed point which corre-
261
sponds to the desired working conditions. But, if this ratio is strictly sma ller than 1, accord ing to Propert ies 8 and 9, the solu ti ons lea\'e the se t E and, very likely, ent er the set G. Among the condit ions for existence of this behavior are smallness of the di sturbance-to-adap tation speed ratio. Aft er entering the set G. a solu tion may either \cave it. going to the set I. F or H or remains in G. According to P ropert ies 2 and 11. for a set point -to-disturbance ratio stri ct ly smaller than 1 and for a disturbance-to-adaptat ion sp eed ratio sufficient ly smal l. there ex ist t,,·o per iod-2 solut ions in G . They are at tractive if set point and di sturbance have same signs and repellent in the opposite case. The former case explains why the solutions may remain in t he set G and why we ca n expec t the burst in g phenomenum to disappear asymptotically. :-'loreO\·el'. a lso from Property 2. as the set point -to-disturbance ratio is closer and closer t o (though smaller than) 1. the rotation of the solutions around these period-2 solutions is slower and sIO\\'er and , therefore. the bursts are less and less frequent. From si mulation s, it seems t hat the attractive locally invariant set and the repellent lo cally im'ariant set are smoothly connected through the set G . From a theoretical point of view, we know that if the fixed point lies in the set I , the repellent locally invar iant set is the stable manifold of thi s criti cal element. Its intersection with the boundary 1/., = 1 + € being trans\'erse, we expect that it extends in the set E . giving a candidate for an attractive locally invariant set. Using this conjecture as a work ing hypothesis, the more a solut ion approaches the in\'ariant set while it is in the set E. the more its evolution will be similar to the solut ions in this in\'ariant set even in the set I. But , accOl'ding to Property 6. for a set point-to-dist urbance ratio strictly sma ller than 1 in absolute value an d negat.i\·e (resp. positi\'e), the solutions in the set I and in the repellent locally im'ariant set are unbounded (resp. bounded) . On the other hand . t he bigger its "'-component is, the more the xcomponent of a solut ion in the set I but not in the repellent loc ally invariant se t. is pushed away (exponent ia lly) fr om this in\'1U'iant set. This reasoning explains the possibility of a \-ery high sensit ivity to initial conditions of solutions starting in the set E. close to the att ract i\'e locally invariant set or to remain simple and according to Property 4, close to the graph of the function: I
l+a = --. 1 + L'
(40)
According to Prop erty 5. for 0 in (- l. 0]. ~ has unb01lnded solutions lying on the rep ellent locally invariant se t. Hence. ewn though the set p oint is sufficiently excit ing to est imate a single p arame ter. if its lewl does not exceed t he disturbance. unbo1lnded solu t. ions arc possible (\'arendra and A. nnaswamy. 1986) and this for any value of the disturbance- t o-adaptation speed ratio. Al so. Properties 5 and G establish that all the solutions are bounded if these in the repellent locally in\'ariant set are also bounded. A.ccording to Property 2. a se t point-to-disturbance ratio st ri ctly larger than 1 corresponds to a exponcntially stable fixed point. _~ ccording to Property 5 each solut ion remains in a compact set. According to Lemma 1 and for a suffic iently sma ll disturbance-to-adaptation speed ratio. there is no periodic sol ution other than the fixed point. This sugges t s th at the fixed point is a global attractor. In this case. the bursti ng phenomenum should not take place. Qualitati\'ely speak ing. this case most resembles to
L. Pralv
262
the ideal case. Summarizing for the case of a small disturbance-to-ad3ptat ion speed ratio. according to the "alue of a, the se t point-to-disturbance ratio. three essentially different behaviors of the solut ions of ~ can be predicted:
• lal
» 1 ( high lewl excitation) : bounded solutions. no bursting, no periodic solution. a globally attractive fixed point is conjectured. beha"ior similar to the ideal case.
Lefschetz S. ( 1971): Differential equations: Geometric the, ory. Dm·er. Ljung L .. Soderstrom T. (1984): Th eory and practice oj recursiv e identification. :V!IT Press. ),Iaree ls 1.!\1.Y .. Bitmead R. ( 1986) : Nonlinear dynamics in adaptive control: chaotic and periodic stabilization. A utomatica 22 641 -655. ), Iareels 1. ),1.Y .. Bitmead R. (19S8 ): Nonlinear dynamic •• in adaptive control: periodic and chaotic stabilization - II. Analysis. Automatica 24 485-497 .
• 0 < a «1 (low le"el excitation) : bounded solutions. periodic solutions ex ist and are conjectured to be global attractors. the fixed point is a saddle. bursting is present but is conjectured to disappear asymptotically.
:\a.rendra K.S. , Annaswamy A. ( 1086): Robust adaptive control in the presence of bounded disturbance.!. IEEE Transactions on Automatic Control. April.
• - 1 « a ::; 0 (low le"el exc itation): unbounded solutions exist, periodic solut ions exist. bursting IS present, the fix ed point is an unst able node.
Pomeau Y .. )' laJl\'i lle P. (1980): Intermittent transition to inrb1,lence in di"s ipative dynamica l ,'ystems. Comm. :-- l ath. Phys., \ -01. 74.189- 19 •.
Since the set point-to-disturbance ratio a, is a relatiYe':' ~uantity, drastic qualitati,'e changes of the systems behav ior may be expected when both rand 6 are close to ) which is the natural working condition for an adapti"e linear cont roller.
Praly L. ( 1085): A geometric approa ch for the local analy.. i.. of a one -$tep,ahea.d adaptive control/er. Proc. of the 4th Yale Workshop on Applications of Adaptive Systems Theory. June.
7
References
Anderson B.D .O. ( 1985): Adaptive "ystems, lack of persi •• , tency of excitation and bur$ting phenomena. Automatica
21 3. Anderson B.D.O .. Bitmeao R.R... Johnson C.R .. Eokot o"ic P.V ., Kosut R.L ., l\ l areels I. l\ l.Y. , Praly 1. , Riedle n.D. )986): Stability of Adaptive SY$tem$: Pas$ivity and Averaging Analy.!i.!. :-'IIT Press . Bem'en iste A. , ), Iet ivier :-'1.. Priouret P. (1981): Algorithmes adaptatifs et approximation.' .,toc/w stique.': theorie et application.' . 1-.1asson. Bodson ),1.. Sast ry S .. Anderson B .D.O .. )'lareels I.),I.Y .. Bitmead R.R. (1986): Nonlinear averaging theorem.'. and the determination of parameter convergence rate .• in adap' tive control. Systems oS.: Control Letters. I. pp. 1-i5- 151. Egardt B. (I070): Stability of Adapl1 ve Controllen Springer \'erlag. Elliott H .. Goodwin G .c. (1984): Adapti"e implementation of the internal model principle. Proceed ings of the 23ro IEEE Conference on Decision and Control. Espat'ia ),1.. Praly L. (lOS8): On the global dY71amic .' oj lin ear adaptive sy,.tem., : an elcmentn.ry worked example. CAI Report. October. Golden ), 1.. Ydst ie E. (1988 ): Bifnrcation in model refer, ence adaptive control .'y.o/em.•. Systems oS.: Control Letters. \ '.11. :\. 5 . :\m·embcr. Goodwin G.c.. Sin E.S. (10S-i): Adaptil'e Filt ering. Pre , diction and Control. Prenticc-Hall. Hirsch )'l.\\' .. Pugh C.C., Shub)' l. (19,6 ): Invariant man, ifolds. Lecture :\otes in )'lat. 583 . Springer \'erlag. hidane-Sai·dane),1.. ),Iacchi O. (19SS ): Qlta .•i periodic selfstabilization of adaptive ARMA predicto r... International Journal of Ad aptive Control and Signal Processing. )' larch.
Pra ly 1. (1080): Topological orbital eq1!ivalence with a, .!ymptotic pha$ e for a two time scale.! di.!crete time sys' tern. To appear in )'l athematics of Control, Signals, and Systems. Springer Verlag. Praly 1. ( 1988): Oscillatory behavior and fixes in adaptive linear control: a worked example. Proceed. of the 1988 IFA C \Vork shop on Robu st Aclapti" e Control. !\ewcastle. Praly 1. , Pomet J.-B. (19S7): Periodic solutions in adap, tzve sy,'tems: the regular case. Xth IFAC \\'orld Congress. July. Riedle n. (1986) : Integral manifolds of slow adaptation . Ph. D. Dissert ation . Elect rical Engineering, Uniwrsity of Illinois at Crbana-Champaign. USA. Riedle B.D. , Kokotoyic P.V. ( 1086): Integral manifold oj ,.lowada.ptation. I.E .E.E. Transactions on .-\utomatic Control. April. SIl\lb )' l. (1078): Stabilite Globale des Sy,.temes Dynami , que ••. Asterisque 56. Societe )' l ath';matique de France.
NONLINEAR DYNAMICS OF ADAPTIVE LINEAR SYSTEMS: AN ELEMENTARY EXAMPLE L. Praly C.A.l.
: \II!OIll(J!iqll l' .
c:!"ol,' dn .\fill l'.l . 35 Ril l' Fmll!"/,
Suill! 1-/ 0 11 or';.
11](J5
F UII!ai ll l'h/l'IIlI.
\Ye analyze the phase p ortrait and describe some non local non linear behavior of a dm;"d loop ,y,tem made of an adapt i\'C proportional controll er and a di sturbed first order lincar ,y,tClIl. Our rc"ults a re obtained applying p erturba tion methods -- P oincarc l11ethod. structural sta l,ility of lIortnally hyper bolic invariant sets. The conclusion is twofold: • An adapt ive linear s~'st('m in the idea l case is vpry critical. In pr('s('nce of ]H'rturll
Introduction
1
T ypicall y, adaptive linear controllers are designed from a linear point of vi ew:
This cont roller is known to give bounded solu tions when placed in feed back with a s ~' stem such that a sequence a exists to sa tisfy for each t. 00 - R :s: a(t) :s: 00 + Rand:
• First , a linear but parameterized li near controller is designed , applying linear system theory.
~ ~
• Second , a parameter adaptation law is designed . npplying linear es timation t heory.
q
O( t
+ 1)
+
{) ,, (t)
V(i) (U(i
00
=
u(i)
+
,,(1)2
+
10+(t~- Ool} ((1+(1) -
v(i)
R. 00
C2 q
(2)
;O
+
V
(3)
=
ay(t -1 )
+
u(t-1)+o
(5)
with a belonging to [flo - R. 00 + R] ~ I oreover. if 6 is zero and l' is COllst ant . (O( t). y( I )) cOlwcrges globally to ( a . 1') . in an 12 -;;I'nse an d l'xlOollenti:111y if l' is not zero ( Good"'in and Sin. 19S-l ). Ho\\,/'\'f'r. for l' zero . any arbitrarily small disturhnnc(, t lea/l> to an intermitt ent phe~ nomC'nllm (Pomeau and ~Ianvilk. 19S0 ). " 'hose characterist ics on t he v-component sllggested the name bursting (,.l.ncll'rson. 19S5: .J a·idane-Sai·dane and ~Iacchi. 19S5 ). From a linear point of view. a constant exogenous signal lending to such a wry high frequency output is \'ery disappointing. In l'ractirl'. s(' \'era l fixes ar" added - dead zone ( Egard t . 19, 9). int('rnal model principle ( Elliott and Goodwin. 19S.. ). filtering (.-\nder50n et a!.. 1%6 ). . . Howe\·er. if those fixes are not appropriately cho,en. a qualitatiwly simi la r bebavior may he oh,C'\'\'('d for these more intricate ca:;es (P raly. 19SS I. T hi" mot i\'ates our interest in the closed loop sy"tem (1). ( 5 ) and the objec ti\'e of tb s pa~ per is to anill~' zp its phase pOl'trait and to explain some non local nonl inear b"ha\' ior of it s solut ions. This will implici tely d emon>! rat e that lion lillear system theory is filII of very appropriate re:;tIl t :; and the lil!('ar point of \'icw can be a(h'alltagcousl~' compcnsated for.
( 1)
flu)
1' (1)
• the adapted pnrameter is looked for in t he intprvnl -
la(i) - a(i - l)I:S:
,=/+ 1
+A2
q
In particular. thi s result alOpliC's to the simplest of these sys tem s. object of the forthcoming ,tudy:
u' V and l' are the system input . sys tem output and ,et p oint signa l respectively. (s. O) is the controller state . () h a\'ing being gi\'en the interpret ation of an adap ted pa r ameter. R. 00 . f1 and A" a re design parameters. int erpreted as:
[00
r+q
L
Cl
(4)
v(t)2
+ 1) - ,,(I) - (I (t)U(I)) . An sup{l.s(t + 1l} + V(t)2
+ min {1.
- O(i)V(t)
+
<
where:
Howe\·er. an adaptive linear controller is a dYllamic nonlin ear controller. For exa mple . in this paper we \I'ill be concerned with the follo\l'ing proportional controlkr III it s robust adapt ive version: /,2"(1)
sup{l.~(I+l)}
q
;'\ot only design but analysis is m ade fr om a linear poin t of view. Doundedness of the solutions is established applying the robust li nea r stabi lity theory. T heir properti es ar(' st ud ied simil arly with, some times, the help of a\'E'raging theory.
+ 1)
[Y(i+1)-U(i) - a(i)Y(i)]2 1
• Finally, an adapti\'e li nenr controller is obtained from the parameteri zedlinear controller wi th those parameters gi\'en on line by the adaptation law.
s( t
i~l
+ R]
• the larger the posit i\'p A: I IS . the slower the adapta tion. i.e. the O-dynamics. is.
257
258
L Pralv
To appply these results. it is more appropriate to write the system in the so called standard form. This is done as follows: when restricted to the set: W=
{
(y.s.lI)
I
s~l.llI-lIo+y
<5-(II-a)y 11 + y
2
I ~R
}
(6)
with r constant and lett ing:
x = yl<5 :
1.'
= II-a :
This paper is a short versIon of a report written by ;"lartin E spana and the author (Espana and Praly, 1988). Due to space limit a tions. no simu lation results will be presented in this paper. They can be easily reproduced by the reader (take for example d = 0.01).
2 Q
=
rib:
the closed loop system is decribed by the map ~ from R2 into R2. giwn by:
The frozen system
Our analysis will be done by considering perturbation of the so called frozen system: ~ (v.l')
f
(8) The equation for the ,,-component is omited since it has no influence on the remaining part of the system. The standard form (8) puts in relief the role played , in the qualitative beha\'ior of the solutions, by the set point-todisturbance ratio Q and by the positiw disturbance-toadaptation speed ratio d. In particular this shows that d is the effective adaptation speed control. As most of the adaptive linear systems. this system belongs to the family of systems which can be described , with A. , I3 , C smooth functions, by the map ~g from IIr into Rn, given by:
=
1,: (
~
as a small
)
(10)
-1.'1'+1+<1
As mentioned in Introduct ion, it is a family of linear systems parameterized by ('. The linear system theory allows us to completely describe its phase portrait : Property 1 (Frozen system): A frozen sy.
1 +<1 .r = - 1+ 1,"
( ll )
• if 111'1 is stritciy larger than 1, it diverges exponentially to infinity from the point (1/-', 1') of the same graph.
(9)
• if~' is equal to 1, it is a period-2 solution.
For d small , this map appears locally as a perturbation of a map made of a family of linear maps. This remark motivates for applying perturbation methods - Poincare method , structural stability of normally hyperbolic invariant sets, averaging. By doing so. and for the general system (9), can be obtained:
• if ';' is f.qual to - l. its l'-component grows linf.arly towards +00.
As will become clear later, the impor tant fact s in this result are: • the set:
• existence of limit sets (Bodson et aI. , 1086: Praly and Pomet, 1087).
Sf
• existence of in\'ariant sets (Praly, 1985. 1089: Riedle and I\:okot ovic. 1(86). Their attractivity and I or repulsiv ity gives then information on the solution behavior.
To obtain the basis of our perturbation analysis. we start by studying the system obtained by taking d equal to zero and called the frozen system. Then. fixed points and periodic solutions of the map ~ are considered. In Section 4. we establish existence and properties of locally invariant sets. Critical elements and locally im'ariant se ts are combined in Section 5 to obtain theoretical results on the system global dynamics. These results are interpreted in Section 6.
~ ::. ' ~';i -1 }
which is a graph and is invariant under
(12) ~f'
• the frozen system stability boundary 11/,1 = 1 which separates this im'ariant set into normally hyperbolic locally invariant components, namely: - Sf n {( 1.' ..r) I IL-I > 1 } is exponentially repellent, - Sf n {(1.' , .r) 111.'1 < I } is exponentially attractive,
• description of the motion along these invariant sets (Ljung, Soderstrom. 1984: :'l.nderson et al.. 1986: RiedIe, 1986: Benwniste et al.. 1987: Praly, 1080). Here considering our disturbance-to-adaptation speed ratio d as a sma ll parameter for the perturbation analy sis. we apply these methods to our elementary example. From their generality, we expect that severa l of our conclusions extend to more general situations. Also. om analy sis can be completed to obtain more precise local results by a bifurcation analysis (see (Golden and Ydstie. 1(88) or (;" lareels and Bitmead. 1986. 1088) for example).
= {(';' , x) I x =
• the periocl-2 solutions which are critical.
3
Equilibrium point and period-2 solutions
Going back to the actual system. let us see how the limit se ts of the frozen system are disturbed. An elementary continuity argument shows (Praly and POlnet. 1087): Lemma 1 (Existence of periodic solutions): A neceswry condition for a ,'olution to be a period-T ,'01,,tion of ~ which remain.' bounded a,' d goes to O. i" that the accumulation point of it" mitial condition be one of the followmg 3 point .. : I. ' 0
<1 1 + <1±~ 2
}
(13)
:"Jonlinear Dynamics of Adaptive Linear Systems
259
Consequently, from the the frozen system equilibrium set SJ, only one point is of interest. Similarly from the set of critical period-2 solutions, only one persists by a continuous perturbation. To precise this necessary condition, we apply Poincare method (Lefschetz . 1977) and get:
Property 3 (Repellent locally invariant set):
Property 2 (Critical elements): i) The map I: has a unique fixed point for all
There exists a bounded Lipschitz continuous function H defined on {I~'I 2: 1 + f} and such that, with the function 9H defined by:
0 different from 0 or -1. It is exponentially stable for 0-' in (-1, - P,) and exponentially unstable for 0 -, outside [- 1, -P,] . with P, the unique solution of:
2(P+ 1) =
d
(14)
-2-
P +d
ii) For any 0 strictly smaller than 1 I7l absolute value, one can find a strictly positive con .. tant do , such that for all d, smaller than do. there exit two locally unique period-2 solutions wh,ch can be approximated by: 1 -
0 1 + 0 Of v'1='Q2 (I ----'----'----') .)
l+a±-~-
+
2
T
For any non-zero d, let
be the smallest positive root of:
f
' (l-l/-,H(JjJ)) 1 + dH(JjJ)2
= tP + dH (1,1)
1>fJ(1/: )
(20)
i) If: 1~'I2:
1 + f, I1>H(tC')1 2: 1 + f , sgn(1)H('';'')) '" sgn(1j: ) (2 1)
then
H( 1)l/('';')) '" 1 + 0
(22)
,;,H( ";',)
-
ii) There exists p positive such that: (<:>,y) = Z:;(,p,x) and (1,'),T) in {1..;..12: 1 + f} X R with,p1> positive implies:
(15)
Old)
Iy -
H("')I 2: (1
+ p)
Ix -
H:
(23)
H(JjJ)1
sup
{~\H( JjJ)
1
+ 0'\ }
The.
iii) Approximation of
T
Hence, in fact for any d, there exists an exponcnt ially repellent locally invariant set. It can be approximated by the frozen parameter invariant set SJ, for d sufficient ly smalL
=
27r j2d(1 - ( 2 )
( 16 )
+ Old)
From this result, it follows that the control objective: output = set point cannot be met asymptotically if the set point-ta-disturbance ratio is too small (X arendra and Annaswamy, 1986). Ew'n more, with this small ratio, there exists a period-2 solution lying close to the frozen system stabili ty boundary. This solution being a focus, it explains locally and at least transient Iy the intermittent behavior mentioned in Introduction. Also, being proportional to jd(l - 0'2) , the frequency of the corresponding bursts decreases as the disturbance-ta-adaptation speed ratio goes to zero or the set point-ta-disturbance ratio goes to 1. Though not established here, it is also noticeable that. for a equal to zero. the period-2 solution is:
= 1
~,
.T
= 0 or 1
(17)
It lies exactly on the frozen system stability boundary and is a critical attractor.
Locally invariant sets
l.(t
+ 1) -
1+
0
)
=
-l'(t) (.r(1) -
+ . ;.,
Property 4 (Attractive locally invariant set): For any d such that:
o
1
< d < d' =
211 + 0'1 2
( \1 ~1
+ +
311 211
+ 0'1 _ + 0'1
1)
(24)
let 1) be the smallest positive root of:
2>(1)) =
(
1) -
_dn2) _0 _ 1 + dn6
-
( 1 + 2no)nod
2
1 + dn~
(25)
where no and n, are defined by:
no
+ 0'1
11
=
(1
+ dn6) no
(26)
(1+ 2n o)d
1)
There exists a bounded continuou.o Lips chitz function G .o1lch that ) with t"c( <:» a function implicitly defined by:
+ d G(t·c(O))
1j'c(9)
(27)
1 + d G{t"c(<:»)2 (i) If:
In Section 2. we haw noticed that the graph SJ is im'ariant for th e frozen systf'm. For thc actual system. wc get (when this makes sense): (
1
d
is bounded.
" =
4
{1';'1"l+'l
~ ( )) (18)
1+l'(t+1) l+l' t +d _ _~(I_+~o~)_T(~t~)(_X~(t~)l~·(~t~)_-_l~)-~
(1 + L"( t) l( 1 + l'( t) + d.l'( 1)( 1 + J'( t))) "-ith d in the second term on the right hand. we see that SJ is close to be a locally invariant set of::. In fact. SJ being locally normally hyperbolic for the frozen system. for the actual system with d small enough. we can expect the existence of locally invariant sets. close to SJ. being repellent in the set {Iv! > I } and attracti"e in the set {Ivl < I} (Shub . 1978: Hirsch. Pugh and 51mb. 1976). To pro\'(' this existence we apply the graph transform technique (Shub. 1978).
10
I :s
1-
and
1)
Ice( <))1 :s
(28)
1 - ')
then:
G(o)
1 +0 -
l'
o (ii) Let
+
VG{L')
d G(v)(I-vG{L'))
1 + dG(L')2 ~
)
(29)
.oatisfy: (30)
there exists a. depending on ~ and strictly "maller than 1 such that: (o.y) = ::(l' , x) and (L',X) in { Iv! < 1 - III X {Ixl :S () implies:
L. Praly
260 ly-G(4))1 ~ a Ix-G(I/»)I
(31)
1.'.' (1)';'(1-1»0 and 1';·(I)12: 1 +f. VO:St
(34)
{~IG(0) _1+Q I }
Case 0< -1: For all I in (0, T). (~,(t) - l·(t -l)) ~·(t -1) is negative. With (i), this implies that 1~' (t)1 goes monotonically to and crosses 1 + E. Con,'eqlLently T is finite.
Consequently, for d sufficiently small, there exists an exponentially attractiye locally inyariant set which can be approximated by the froz en parameter inyariant set Sf.
Case a = -1: H(v) == 0 and any point in the repel/ent locally invariant set i.' a fixed point of~. T I,' infinite.
sup
(iii)ApproximationofG:
{Iwl«l-ry)}
d
1 + 1jJ
is bounded.
\Vith these two properties, we see that for d small enough the linear analysis made for the frozen system giws a good approximation of the behayior of the actual system as long as we are far enough from the frozen sys tem stability boundary 11,1' 1 = 1 and its im'ariant set Sf.
Case -1 < Q :S 0: If l'(O) is positive. ":'(1) is strictly increasiny and therefore converges to +:x: while x(l) goes to zero. Con.'eq1tently. T is infinite. If ,;.( 0) < min {I/o. -( 1 + d}. l'( t) is "trictly decreasing to -QC while .r( t) goe,' to zero. Consequently. T is infinite. If 1/0 < dO) < -( 1 + E) . dt) i.... trictl y increasing and craSH,' -(1 + d· Con$equ ently . T i" finite.
Behavior of the solutions: technical results
Case 0 < 0: 1..( t) is mono tonically going towards 1/ Q. Consequently. T i.< finite if and only if dO) is negative and/or 1/0 < 1 + E.
Knowing the existence of critical elements and locally invariant sets, we are now in position for studying the behavior of the solutions. \"ith E giw'n by (19), 71 giwn by (25), d* given by (24) and:
Pl'operty 7 (Solutions in the "strict instability" set outside the l'epellent locally inval·iant set, sets: A , I ).' For any non zero d: (i) If for some time 10. a solution ,
5
X .
1
> 1 - 71
A = {(I/>,x)11/> ~ -(1
Hence , there is no .. olution sali.sfying for ever xi- H( I/» and I~' I 2: 1
G
D In the following we state sewral properties which are proyed in (Espana and Praly, 1988 ). They arc giwn without any comment, their interpretation being the topic of the next Sectio11. Property 5 (Solution boundedness, sets: A to I ): (i) If Q lie~ in (- L 01, ~ has unbounded solutzon,'. (ii) If Q is not in [- L 01. all the ,'olution,' of ~ are bounded. Property 6 (Solutions in the repellent locally invariant set, sets: A , I ): For any non zero d: (i) The fixed point of~. belongs to the repel/ent locally 111' variant set if and only if 11/al:S 1 + f
.
x(O) = H( 1.'·( O))
and let T be the largest integer slLch that:
(if) if 10 I i.< .,tridly smaller than l. a.ll the .'ollltions in the locally invariant .
Property 9 (Solutions in the "strict stability" set outside the attractive locally invariant set, set: E): For any d. 0 < d < d* (i) Any ,'olution In { id:S 1 - 'd x R exponentially approache., the graph {rv. G ( v))}. MOreOl'eT a. sollJ.tion "ta.rting in {ld :S 1 - 'd x R remal11.' In tll1,< .. et a.< long IL,< it remain3 in the 3et {I·r ! 2: l~ r,}
(ii) if 10 I < 1 a.nd for .'ome I1me 10. a .. olltlion .
(11) Let (l.·(t),x(I)) be a solution such that: f
.
Property 8 (Solutions in the attractive locally invariant set, set: E ). For any d. 0 < d < d* (i) The 1tniqlLe cqlLilihrinm point of ~ i.< in the locally invllriant .
F
E
+E
(ii) Moreover, while the ,
or graphically:
B
(35)
(36)
I={(I/> ,x)1 1+E:SI/>}
C
H (1!·(to))
+ E)}
B={(I/>,x)1 -(1+E)<-(1-1])andx~x} C = {( I/>,x)1 - (1 + f) < I/> < -(1- 71) and Ixl < X} D={( I/>,x)1 -(l+f)<1/J<-(l-ry)andx:S-x} E={( I/>,x)1 -(l-ry):SI/>:Sl-ry} F = {(I/> , x) I 1 - 'r/ < I/> < 1 + f and X :S x} G = {(I/>,x)11 - 71 < I/> < 1 + f and Ixl < X} H = {( 1/>, x) I 1 - 71 < I/> < 1 + f and x :S - X }
A
i-
then there exi.'t.< a (finite) tim.e 11 -,u.ch that:
we decompose the plane (I/>, x) into nine subsets:
11,"( 0) 1 2: 1 +
.T(tO)
(32)
(33)
( 37)
then there exi.'t.' a (fin1te) time 11 ,
Iv(ll)1 > 1 - 'I
( 38)
:--1onlinear Dvn a mics o f Adapti\'e Linea r Systems H ence, for lal < 1, there ij no so lution satisfying for ever l,pl ~ 1-1) .
(iii) Moreover, while a so lution remainj in the set {d(1i~~~!)nl~ ,p ~ 1 -1) } x {lxl~O ( rejp . in the set {-(I -1)) ~ ,p ~ d( li~~~f)nl} X {lx l ~ 0 ), it crOjses the attractive locally invariant set at each time t (rcsp. it remains on the same side).
Property 10 (Solutions in the "critical stability" sets B , D , F , H ) : A s long as a solution remains in the jet Ixl ~x}. its':; component is exponentially decaying, H ence there is no solu tion satisfying for ever 1 - 1) < 11,1'1 < 1 + € and 1:1'1 ~ '(
{(,p,x)1 l -1)
Property 11 (Solutions in the "critical stability" set G ): If 11/al is larger than 1 + f and. for d small enough. period-2 sohttionj exist and are in the jtt : (39)
6
Behavior of the solutions: interpretation
The behavior of the solutions of l: can be roughly expla ined with the followinli remarks: According to Property 7, a solut ion in t he set A or I, but not in the repellent locally invariant set, diverges exponentially from a set which is the graph of a uniformly bounded fun ction of ,p . This explains an exponential growt h of the x-component which becomes and remains large. Moreover, for a solution in t he set I, at each time t , the x-component changes side with respec t to this graph. This ex plains a burst with a very high frequen cy content of this component. Conversely, for a solution in the set A. the x-component remains on the same side of the graph. It correspond s a burst without oscillations. This beha\'ior appears for any value of the disturbance and the set point. According to P roperty 10. a solut ion in the set A or I with a la rge I-component or in the set B . D . F or H has it s 1jo-component exponent ially decaying. This behavior exists for any value of the disturbance and the set point. According to Property 9. as soon as a soluti on enters the set E . it is exponentially attracted toward s a set ,,·hich is once again the gr aph of a un iformly bounded functi on of L·. This explains the exponential dec rease of the Icomponent and the fast decay of it s high frequency content if it were present . This beha\-ior happ en s for any \'alue of the set point -to-d ist urbance ratio and at least for small values of the disturbance-to-adaptation speed ratio. \\'hile a solu tion. in the set E. goes to the attracti\'e locally im-ariant set. its evoluti on is more and more similar to th is of the soluti on s in this set. According to P roperti es 4 and I . this explains a speed of the tJ'-component of the Jrder of d. :-'loreo\'er , accordi ng to Propert ies 2 and 8, if the set point-to-disturbance ratio la I is strictly larger than 1, the solutions converge t o the fixed point which corre-
261
sponds to the desired working conditions. But, if this ratio is strictly sma ller than 1, accord ing to Propert ies 8 and 9, the solu ti ons lea\'e the se t E and, very likely, ent er the set G. Among the condit ions for existence of this behavior are smallness of the di sturbance-to-adap tation speed ratio. Aft er entering the set G. a solu tion may either \cave it. going to the set I. F or H or remains in G. According to P ropert ies 2 and 11. for a set point -to-disturbance ratio stri ct ly smaller than 1 and for a disturbance-to-adaptat ion sp eed ratio sufficient ly smal l. there ex ist t,,·o per iod-2 solut ions in G . They are at tractive if set point and di sturbance have same signs and repellent in the opposite case. The former case explains why the solutions may remain in t he set G and why we ca n expec t the burst in g phenomenum to disappear asymptotically. :-'loreO\·el'. a lso from Property 2. as the set point -to-disturbance ratio is closer and closer t o (though smaller than) 1. the rotation of the solutions around these period-2 solutions is slower and sIO\\'er and , therefore. the bursts are less and less frequent. From si mulation s, it seems t hat the attractive locally invariant set and the repellent lo cally im'ariant set are smoothly connected through the set G . From a theoretical point of view, we know that if the fixed point lies in the set I , the repellent locally invar iant set is the stable manifold of thi s criti cal element. Its intersection with the boundary 1/., = 1 + € being trans\'erse, we expect that it extends in the set E . giving a candidate for an attractive locally invariant set. Using this conjecture as a work ing hypothesis, the more a solut ion approaches the in\'ariant set while it is in the set E. the more its evolution will be similar to the solut ions in this in\'ariant set even in the set I. But , accOl'ding to Property 6. for a set point-to-dist urbance ratio strictly sma ller than 1 in absolute value an d negat.i\·e (resp. positi\'e), the solutions in the set I and in the repellent locally im'ariant set are unbounded (resp. bounded) . On the other hand . t he bigger its "'-component is, the more the xcomponent of a solut ion in the set I but not in the repellent loc ally invariant se t. is pushed away (exponent ia lly) fr om this in\'1U'iant set. This reasoning explains the possibility of a \-ery high sensit ivity to initial conditions of solutions starting in the set E. close to the att ract i\'e locally invariant set or to remain simple and according to Property 4, close to the graph of the function: I
l+a = --. 1 + L'
(40)
According to Prop erty 5. for 0 in (- l. 0]. ~ has unb01lnded solutions lying on the rep ellent locally invariant se t. Hence. ewn though the set p oint is sufficiently excit ing to est imate a single p arame ter. if its lewl does not exceed t he disturbance. unbo1lnded solu t. ions arc possible (\'arendra and A. nnaswamy. 1986) and this for any value of the disturbance- t o-adaptation speed ratio. Al so. Properties 5 and G establish that all the solutions are bounded if these in the repellent locally in\'ariant set are also bounded. A.ccording to Property 2. a se t point-to-disturbance ratio st ri ctly larger than 1 corresponds to a exponcntially stable fixed point. _~ ccording to Property 5 each solut ion remains in a compact set. According to Lemma 1 and for a suffic iently sma ll disturbance-to-adaptation speed ratio. there is no periodic sol ution other than the fixed point. This sugges t s th at the fixed point is a global attractor. In this case. the bursti ng phenomenum should not take place. Qualitati\'ely speak ing. this case most resembles to
L. Pralv
262
the ideal case. Summarizing for the case of a small disturbance-to-ad3ptat ion speed ratio. according to the "alue of a, the se t point-to-disturbance ratio. three essentially different behaviors of the solut ions of ~ can be predicted:
• lal
» 1 ( high lewl excitation) : bounded solutions. no bursting, no periodic solution. a globally attractive fixed point is conjectured. beha"ior similar to the ideal case.
Lefschetz S. ( 1971): Differential equations: Geometric the, ory. Dm·er. Ljung L .. Soderstrom T. (1984): Th eory and practice oj recursiv e identification. :V!IT Press. ),Iaree ls 1.!\1.Y .. Bitmead R. ( 1986) : Nonlinear dynamics in adaptive control: chaotic and periodic stabilization. A utomatica 22 641 -655. ), Iareels 1. ),1.Y .. Bitmead R. (19S8 ): Nonlinear dynamic •• in adaptive control: periodic and chaotic stabilization - II. Analysis. Automatica 24 485-497 .
• 0 < a «1 (low le"el excitation) : bounded solutions. periodic solutions ex ist and are conjectured to be global attractors. the fixed point is a saddle. bursting is present but is conjectured to disappear asymptotically.
:\a.rendra K.S. , Annaswamy A. ( 1086): Robust adaptive control in the presence of bounded disturbance.!. IEEE Transactions on Automatic Control. April.
• - 1 « a ::; 0 (low le"el exc itation): unbounded solutions exist, periodic solut ions exist. bursting IS present, the fix ed point is an unst able node.
Pomeau Y .. )' laJl\'i lle P. (1980): Intermittent transition to inrb1,lence in di"s ipative dynamica l ,'ystems. Comm. :-- l ath. Phys., \ -01. 74.189- 19 •.
Since the set point-to-disturbance ratio a, is a relatiYe':' ~uantity, drastic qualitati,'e changes of the systems behav ior may be expected when both rand 6 are close to ) which is the natural working condition for an adapti"e linear cont roller.
Praly L. ( 1085): A geometric approa ch for the local analy.. i.. of a one -$tep,ahea.d adaptive control/er. Proc. of the 4th Yale Workshop on Applications of Adaptive Systems Theory. June.
7
References
Anderson B.D .O. ( 1985): Adaptive "ystems, lack of persi •• , tency of excitation and bur$ting phenomena. Automatica
21 3. Anderson B.D.O .. Bitmeao R.R... Johnson C.R .. Eokot o"ic P.V ., Kosut R.L ., l\ l areels I. l\ l.Y. , Praly 1. , Riedle n.D. )986): Stability of Adaptive SY$tem$: Pas$ivity and Averaging Analy.!i.!. :-'IIT Press . Bem'en iste A. , ), Iet ivier :-'1.. Priouret P. (1981): Algorithmes adaptatifs et approximation.' .,toc/w stique.': theorie et application.' . 1-.1asson. Bodson ),1.. Sast ry S .. Anderson B .D.O .. )'lareels I.),I.Y .. Bitmead R.R. (1986): Nonlinear averaging theorem.'. and the determination of parameter convergence rate .• in adap' tive control. Systems oS.: Control Letters. I. pp. 1-i5- 151. Egardt B. (I070): Stability of Adapl1 ve Controllen Springer \'erlag. Elliott H .. Goodwin G .c. (1984): Adapti"e implementation of the internal model principle. Proceed ings of the 23ro IEEE Conference on Decision and Control. Espat'ia ),1.. Praly L. (lOS8): On the global dY71amic .' oj lin ear adaptive sy,.tem., : an elcmentn.ry worked example. CAI Report. October. Golden ), 1.. Ydst ie E. (1988 ): Bifnrcation in model refer, ence adaptive control .'y.o/em.•. Systems oS.: Control Letters. \ '.11. :\. 5 . :\m·embcr. Goodwin G.c.. Sin E.S. (10S-i): Adaptil'e Filt ering. Pre , diction and Control. Prenticc-Hall. Hirsch )'l.\\' .. Pugh C.C., Shub)' l. (19,6 ): Invariant man, ifolds. Lecture :\otes in )'lat. 583 . Springer \'erlag. hidane-Sai·dane),1.. ),Iacchi O. (19SS ): Qlta .•i periodic selfstabilization of adaptive ARMA predicto r... International Journal of Ad aptive Control and Signal Processing. )' larch.
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