A Unified Approach to High-Gain Adaptive Controllers

0.19 MB·English

by  Ian A. Gravagne

A Unified Approach to High-Gain Adaptive
9 Controllers
∗
0
0
2 Ian A. Gravagne†, John M. Davis‡, Jeffrey J. DaCunha§

n
a
J

5 Abstract
2
It has been known for some time that proportional output feedback

] willstabilizeMIMO,minimum-phase,lineartime-invariantsystemsifthe
C
feedback gain is sufficiently large. High-gain adaptive controllers achieve
O stability by automatically driving up the feedback gain monotonically.
. Morerecently,itwasdemonstratedthatsample-and-holdimplementations
h
ofthehigh-gainadaptivecontrolleralsorequireadaptationofthesampling
t
a rate. In this paper, we use recent advances in the mathematical field of
m dynamicequationsontimescalestounifyandgeneralizethediscreteand
[ continuous versions of the high-gain adaptive controller. We prove the
stability of high-gain adaptivecontrollers on a wide class of time scales.

1 Keywords: time scales, hybrid system, adaptivecontrol
v
3
7 1 Introduction
8
3
The concept of high-gainadaptive feedback arose froma desire to stabilize cer-
.
1
tain classes of linear continuous systems without the need to explicitly identify
0
theunknownsystemparameters. Thistypeofadaptivecontrollerdoesnotiden-
9
0 tifysystemparametersatall,butratheradaptsthefeedbackgainitselfinorder
: toregulatethesystem. Anumberofpapersexaminethedetailsofvariouskinds
v
of high-gain adaptive controllers [11, 15, 18, 23], among others. More recently,
i
X several papers have discussed one particularly practical angle on the high-gain
r adaptive controller, namely how to cope with input/output sampling. In par-
a
ticular, Owens [17] showed that it is not generally possible to stabilize a linear
systemwithadaptivehigh-gainfeedbackunderuniformsampling. Thus,Owens,
et. al., develop a mechanism to adapt the sampling rate as well as the gain, a
notion subsequently improved upon by Ilchmann and Townley [12, 13, 14], and
Logemann [16].
∗ThisworkwassupportedbyNSFgrants#EHS-0410685andCMMI#726996aswellasa
BaylorUniversityResearchCouncilgrant.
†DepartmentofElectricalandComputerEngineering,BaylorUniversity,Waco,TX76798,
Email:Ian [email protected]
‡Department of Mathematics, Baylor University, Waco, TX 76798, Email:
John M [email protected]
§LufkinAutomation, Houston,TX77031, Email: Jeffrey [email protected]
1
In this paper, we employ results from the burgeoning new field of mathe-
matics called dynamic equations on time scales to accomplish three principal
objectives. First, we use time scales to unify the continuous and discrete ver-
sionsofthe high-gaincontroller,whichhavepreviouslybeentreatedseparately.
Nextwegiveanupperboundonthesystemgraininesstoguaranteestabilizabil-
ityforamuchwiderclassoftimescalesthanpreviouslyknown,includingmixed
continuous/discrete time scales. Third, the paper represents the first applica-
tionofseveralveryrecentadvancesinstabilitytheoryandLyapunovtheoryfor
systems on time scales, and two new lemmas are presented in that vein. We
also give a simulation of a high-gain controller on a mixed time scale.
2 Background
We first state two assumptions that are required in the subsequent text.
(A1) Thesystemmodelandfeedbacklawaregivenbythelinear,time-invariant,
minimum phase system
x˙(t)=Ax(t)+Bu(t), y(t)=Cx(t), x(0)=x , (1)
0
u(t)= k(t)y(t), (2)
−
for t 0. System parameters A Rn×n, B,CT Rn×m, x Rn and
0
n are≥unknown. The feedback gai∈n k : R R+ is∈piecewise co∈ntinuous,
→
and nondecreasing as t . By minimum phase, we mean that the
→ ∞
polynomial
A λI B
det −
C 0
(cid:20) (cid:21)
with λ C is Hurwitz (zeros in open left hand plane).
∈
(A2) Furthermore,
(CB)T +(CB)>0, (3)
i.e. (CB)T +(CB) is positive definite. (In [18] it is pointed out that a
nonsingular input/output transformation T always exists such that B˜ =
BT−1 and C˜ =TC give (C˜B˜)T +(C˜B˜)>0.)
Under these conditions it has been known for some time (e.g. [11]) that
there are a wide class of gain adaptation laws k(t)=f(y(t)), f :Rm R, that
→
can asymptotically stabilize system (1) in the sense that
y L2[t , ), lim k(t)< .
0
∈ ∞ t→∞ ∞
Subsequently, various authors [11, 12, 17, 19] assumed that the output is ob-
tained via sample-and-hold, i.e. y := y(t ) with k := k(t ) and i N . Thus
i i i i 0
∈
it becomes necessary also to adapt the sample period h := t t so the
i i+1 i
−
closed-loop control objectives are
y ℓ2[i , ), lim k < , lim h >0.
i 0 i i
∈ ∞ i→∞ ∞ i→∞
2
Though severalvariations on these results exist, these remain the basic control
results for continuous and discrete high-gain adaptive controllers. The contin-
uous and discrete cases have previously been treated quite differently, but we
now construct a common framework for both using time scale theory.
3 A Time Scale Model
The system of (A1) can be replaced by
x∆(t)=Aˆ(t)x(t)+Bˆ(t)u(t), y(t)=Cx(t), x(0)=x , (4)
0
u(t)=k(t)y(t), t T, (5)
∈
whereTisanytimescaleunboundedabovewith0 T. Withaseriesexpansion
∈
similar to [11], we see that
Aˆ(t):=expc(µ(t)A)A, Bˆ(t)=expc(µ(t)A)B, (6)
where expc is the matrix power series function
1 1 1
expc(X):=I + X + X2+ + Xn−1+ (7)
2 6 ··· n! ···
and µ is the time scale graininess. Implementing control law (2) then gives
x∆(t)= (t)x(t), (t):=expc(µ(t)A)(A k(t)BC). (8)
A A −
Note that Aˆ,Bˆ and may all be time-varying, but we will henceforth drop
A
the explicit reference to t for these variables. For future reference, we also note
that if sup µ(t) < , then Aˆ and Bˆ are also bounded (c.f. Appendix,
t∈T ∞ || || || ||
Lemma 7.4).
The design objectives are to find graininess µ and feedback gain k as func-
tions of the output y,
µ(t)=g(y(t)), g :Rm [0, ), (9)
→ ∞
k(t)=f(y(t)), f :Rm R+, (10)
→
with t T and k C (T) nondecreasing, such that
rd
∈ ∈
∞
y L2[t0, )T := w : w(t)2∆t< , lim k(t)< .
∈ ∞ | | ∞ t→∞ ∞
(cid:26) Zt0 (cid:27)
It is important to keep in mind the generality of the expressions above. A
greatdealofmathematicalmachinerysupportstheexistenceofdeltaderivatives
onarbitrarytimesscales,aswellastheexistenceandcharacteristicsofsolutions
to (8). See, for example, e.g. [2, 3].
3
4 Stability Preliminaries
We begin this sectionwith a definitionandtheoremfromthe workofPo¨tzsche,
Siegmund, and Wirth [20]:
Definition 4.1. The set of exponential stability for the time-varying scalar
equation z∆(t)=λ(t)z(t), z :=z(t ) C with λ:T C and z,λ C (T), is
0 0 rd
∈ → ∈
given by
(T):= C(T) R(T)
S S ∪S
where
1 t log 1+µ(t)η(t)
C(T):= η(t) C:α= limsup || ||∆t>0 ,
S (cid:26) ∈ − t→∞ t−t0 Zt0 µ(t) (cid:27)
(11)
R(T):= η(t) R: t T, τ >t with τ T such that 1+µ(t)η(t)=0 ,
S { ∈ ∀ ∈ ∃ ∈ }
with η C (T) and arbitrary t T.
rd 0
∈ ∈
Theorem 4.2. [20] Solutions of the scalar equation z∆(t)=λ(t)z(t) are expo-
nentially stable on an arbitrary T if and only if λ(t) (T).
∈S
WenoteherethatPo¨tzche,Siegmund,andWirthdidnotexplicitlyconsider
scenarioswhereηistime-varying,buttheirstabilityanalysisremainsunchanged
for η(t).
The set R(T) contains nonregressiveeigenvalues λ(t), and a loose interpre-
tationof CS(T) suggeststhatit is necessaryfora regressiveeigenvalueto reside
S
in the area of the complex plane where 1+µ(t)λ(t) <1 “most” of the time.
|| ||
The contour 1+µ(t)λ(t) =1 is termed the Hilger Circle. Since the solution
of z∆(t)=λ(|t|)z(t) is x(t)|=| z e (t,t ), Theorem 4.2 states that, if λ(t) (T)
0 λ 0
∈S
then some K(t )>1 exists such that
0
z(t) = z e (t,t ) z Ke−α(t−t0), (12)
0 λ 0 0
|| || || |||| ||≤|| ||
where e is a generalized time scale exponential. The Hilger Circle will be
λ
important in the upcoming Lyapunov analysis, as will the following lemmas.
Lemma 4.3. Let z C (T) be a function which is known to satisfy the in-
rd
equality z∆(t) λ(t)∈z(t), z(0) = z R with λ : T R and λ C (T). If
0 rd
z(t)>0 for all≤t t T, then λ(t) ∈ +. → ∈
0
≥ ∈ ∈R
Proof. Defining v(t)=z∆(t) λ(t)z(t) gives rise to the initial value problem
−
z∆(t) λ(t)z(t)=v(t), z(0)=z , t T, (13)
0
− ∈
where v(t) 0.
≤
First, suppose λ(t) is negatively regressive for t t , i.e. λ(t) < 1/µ(t)
0
≥ −
with µ(t)>0. Then (13) yields zσ =µλz+z+µv =z(1+µλ)+µv <0.
4
On the other hand, suppose λ(T) is nonregressive for some T > t , T T.
0
∈
If λ(t) < 1/µ(t) over T > t t , then invoke the preceding argument. If
0
− ≥
λ(t)> 1/µ(t) over T >t t , then solve (13) to get
0
− ≥
t
z(t)=e (t,t )z + e (t,σ(τ))v(τ)∆τ, T >t t . (14)
λ 0 0 λ 0
≥
Zt0
t
Since e (t,t ) > 0 for t < T, we see that e (t,σ(τ))v(τ)∆τ < 0. However,
λ 0 t0 λ
for t T, (14) becomes
≥ R
ρ(T)
z(t)=0+ e (t,σ(τ))v(τ)∆τ, t T (15)
λ
≥
Zt0
Thus,z(t)<0forallt t forbothnegativelyregressiveandnonregressiveλ(t),
0
≥
acontradictionoftheLemma’spresupposition. Thisleavesonlyλ(t) +.
∈R
At this point we pause briefly to discuss Lyapunov theory on time scales.
DaCunha produced two pivotal works [5, 6] on solutions P of the generalized
time scale Lyapunov equation,
A(t)TP(t)+P(t)A(t)+µ(t)AT(t)P(t)A(t)= Q(t), t T, (16)
− ∈
where A(t),P(t),Q(t) Rn×n, A and Q are known and Q(t) > 0. Though it
∈
will not be necessary to solve (16) in this work1, we will see that the form of
(16) leads to an upper bound on the graininess that is generally applicable to
MIMO systems, an advancementbeyond previous works which gavean explicit
bound only for SISO systems.
Before the next lemma, we define
:= k :T R+, k C (T), k∆(t) 0 t T, lim k(t)= .
rd
K { → ∈ ≥ ∀ ∈ t→∞ ∞}
The next lemma follows directly.
Lemma 4.4. Given assumptions (A1) and (A2) and k , there exists a
nonzero graininess µ¯(t) and a time t∗ such that, for all µ(t∈) Kµ¯(t) and t>t∗,
the matrix kCBˆ satisfies a time scale Lyapunov equation w≤ith P =I, Q(t)
ε for smal−l ε >0, and Bˆ from (6). ≥
2 2
1DaCunhaprovesthatapositivedefinitesolutionP(t)tothetimescaleLyapunovequation
withpositivedefiniteQ(t)existsifandonlyiftheeigenvalues ofA(t)areintheHilgercircle
for all t ∈ T. Furthermore, P(t) is unique. As with the well known result from continuous
systemtheory(c.f. [21]),thesolutionisconstructive, with
P(t)=ZStΦAT(t)(s,0)Q(t)ΦA(t)(s,0)∆s,
whereΦA(t,t0)denotesthetransitionmatrixforthelinearsystemx∆(t)=A(t)x(t),x(t0)=
I. The correct interpretation of this integral is crucial: for each t ∈ T, the time scale over
which the integration is performed is St := µ(t)N0, which has constant graininess for each
fixedt.
5
Proof. We construct µ¯(t) as
1 λ CB+(CB)T
µ¯(t):= min{ } ε , t T, (17)
k(t) λ (CB)TCB − 1 ∈
(cid:18) max{ } (cid:19)
with ε >0 sufficiently small so that µ¯(t)>0 on T. This holds if
1
(CB)T +CB µ¯(t)k(t)(CB)TCB =ε I. (18)
1
−
Multiplying (18) by k(t) gives
−
( k(t)CB)T +( k(t)CB)+µ¯(t)( k(t)CB)T( k(t)CB)= k(t)ε I.
1
− − − − −
(We now drop the explicit time-dependence for readability.) Set
Σ(µ¯):=expc(µ¯A) I,
−
so that Bˆ(µ¯)=[I +Σ(µ¯)]B, yielding
( kCBˆ)T +( kCBˆ)+µ¯( kCBˆ)T( kCBˆ)= k(ε I Z),
1
− − − − − −
whereeachtermofZ isaproductofcontantstimesΣ(µ¯). Since Σ(µ¯) 0as
µ¯ 0 (c.f. Appendix, Lemma 7.4), there exists a time t∗ T w|h|en Z||→<ε .
1
→ ∈ || ||
Because the preceding arguments admit any graininess µ(t) µ¯(t), it follows
≤
that
( kCBˆ)T +( kCBˆ)+µ( kCBˆ)T( kCBˆ)= kQ, t>t∗, µ(t) µ¯(t),
− − − − − ≤
(19)
where Q(t) ε and ε :=ε Z >0.
2 2 1
≥ −|| ||
We comment briefly on the intuitive implication of Lemma 4.4. Equation
(19) shows that there exist positive definite Q(t) and P =I to satisfy an equa-
tion of the form (16). According to DaCunha, this implies that the eigenvalues
of kCBˆ lie strictly within the Hilger circle for t>t∗.
−
5 System Stability
We now come to the three central theorems of the paper. If BC is not known
to be full rank, or cannot be full rank because of the input/output dimensions,
then it must be assumed (or determined a priori) that the eigenvalues of A
−
k(t)BC attain negative real parts at some point in time. This phenomenon
is investigated in-depth by other authors [11, 23]. We then make use of the
observationby Owens,et. al. [18], that there must exist some k∗ >0 such that
if k(t) k∗ the system of (A1) has a positive-real realization. This, together
≡
withthe Kalman-YakubovichLemma [21],implies existenceofP,Q>0sothat
(A k∗BC)TP +P(A k∗BC)= Q and PB =CTCB. (20)
− − −
6
Theorem 5.1 (Exponential Stability). In addition to (A1) and (A2), suppose
(i) t T where T is a time scale which is unbounded above but with µ(t)
∈ ≤
µ¯(t),
(ii) k (implying from (17) that µ(t) 0, but not necessarily monotoni-
∈ K →
cally),
(iii) BC is not necessarily full rank, but there exists a time t∗ T such that
∈
the eigenvalues of (A k(t)BC) are strictly in the left-hand complex plane
for t t∗. −
≥
Then the system (4), (5) is exponentially stable in the sense that there exists
time t T and constants K(t ) 1, α 0, such that
0 0
∈ ≥ ≥
x(t) x(t0) Ke−21α(t−t0), t t0.
|| ||≤|| || ≥
Proof. Set k∗ := k(t∗). Then assumption (iii) is the prerequisite for equation
(20). Again, we suppress the time-dependence of x, k, µ, Aˆ and Bˆ. Similarly
toLemma 4.4,termscontainingΣ(µ)maybe addedtothe firstequalityin(20)
to obtain
(Aˆ k∗BˆC)TP +P(Aˆ k∗BˆC) ε t t∗∗, (21)
3
− − ≤− ≥
for some small ε >0. Note t=t∗∗ is the point at which terms involving Σ(µ)
3
become small enough for (21) to hold. Defining Z := CTCΣB PΣB, the
1
−
second equality in (20) gives
PBˆ =CTCBˆ+Z .
1
Consider the Lyapunov function V = x(t)TPx(t) with P from (20). Then,
using (20), (21) and Lemma 4.4,
V∆ =xT∆Px+xσTPx∆
=xT[ TP +P +µ TP ]x
A A A A
=xT (Aˆ k∗BˆC)TP +P(Aˆ k∗BˆC) (k k∗)(CTBˆTP +PBˆC)
− − − −
h
+µ(Aˆ kBˆC)TP(Aˆ kBˆC) x
− −
xT[ ε (k k∗)(CTBˆTCTCi+CTCBˆC+2Z C)
3 1
≤ − − −
+µAˆTPAˆ 2µkAˆTPBˆC+µk2CTBˆTCTCBˆC+µk2CTBˆTZ C]x
1
−
=xT[ ε +µAˆTPAˆ]x
3
−
+yT[ k(CBˆ)T kCBˆ+µk2(CBˆ)TCBˆ+µk2BˆTZ +k∗((CBˆ)T +CBˆ)]y
1
− −
+xT[ (k k∗)2Z 2µkAˆTPBˆ]y
1
− − −
(ε µ AˆTPAˆ )xTx
3
≤− − || ||
(kε µk Bˆ kZ 2k∗ CBˆ )yTy
2 1
− − || || ||− || ||
+(2 kZ +2µk AˆTPBˆ )xTy.
1
|| || || ||
At this point we observe the following:
7
AˆTPAˆ is bounded because Aˆ is bounded. Set γ :=sup AˆTPAˆ .
• || || 1 t∈T|| ||
µk is bounded by assumption (i) and (17).
•
kZ is bounded because k is proportional to 1 and 1 Σ(µ) const
• || 1|| µ µ|| || →
as µ 0. Set γ :=sup 2 kZ .
→ 2 t∈T || 1||
Set γ :=sup µk Bˆ γ .
• 3 t∈T || || 2
Set γ :=sup 2k∗ CBˆ .
• 4 t∈T || ||
Set γ :=sup 2µk AˆTPBˆ .
• 5 t∈T || ||
Recalling the standard inequality xTy βxTx+ 1yTy for any β > 0, we
≤ β
then have
1 1
V∆ (ε µγ βγ βγ )xTx (kε γ γ γ γ )yTy.
3 1 2 5 2 3 4 2 5
≤− − − − − − − − β − β
By assumption (ii), there exists a time t max t∗,t∗∗ such that, for suffi-
0
≥ { }
ciently small β,
(ε µγ βγ βγ )
V∆ − 3− 1− 2− 5 V :=η(t)V, t t ,
0
≤ γ (P) ≥
min
with η(t) < 0. Then, by [2, Theorem 6.1], Theorem 4.2, and Lemma 4.3 it
follows that there exists K(t ) 1 so that
0
≥
x(t) K x(t0) e−12α(t−t0), t t0,
|| ||≤ || || ≥
where α is defined in (11).
We pointoutthat,whenBC isfullrank,assumption(iii)aboveisnolonger
necessaryas there alwaysexists a k∗ suchthat the eigenvaluesof (A k(t)BC)
are strictly real-negative for k(t) k∗. One more lemma is required−before the
≥
next theorem.
Lemma5.2. IfTisatimescalewithboundedgraininess(i.e. µ∞ :=supt∈Tµ(t)<
), then
∞ ∞ ∞ ∞
c eαtdt eαt∆t c eαtdt,
1 2
≤ ≤
Zt0 Zt0 Zt0
where c , c , α R and c , c >0.
1 2 1 2
∈
Proof. Consider the case when α > 0. The process of time scale integration
is akin to the approximation of a continuous integral via a left-endpoint sum
of (variable width) rectangles. If the function to be summed is increasing (as
in this case), the sum of rectangular areas will be less than the continuous
integral,meaningc =1,c <1. Oneestimateofthelowerbound,then,follows
2 1
by simply increasing c until c eαt meets one of the rectangle right endpoints
1 1
which are given by eαρ(t). Thus c eαt eαρ(t), or equivalently, c eασ(t) eαt.
1 1
≤ ≤
8
This in turn yields c e−αµ(t). Therefore, the most conservative bound is
1
given by c e−αµ∞. ≤The case for α < 0 can be argued similarly, leading to
1
≤
the lemma’s conclusion:
e−αµ∞, α>0, 1, α 0,
c = c = ≥
1 (1, α 0, 2 (e−αµ∞, α<0.
≤
We are now in a position to state the main theorem of the paper.
Theorem 5.3. In addition to (A1) and (A2), assume the prototypical update
law, k∆(t) = y(t) 2 with k0 := k(0) > 0. Then limt→∞k(t) < and y
|| || ∞ ∈
L2[t0, )T.
∞
Proof. For the sake of contradiction,assume k(t) as t . Then k .
→∞ →∞ ∈K
Theorem5.1yieldsx L∞[t0, )T andthereforey L∞[t0, )T. Thesolution
∈ ∞ ∈ ∞
for k(t) is (by [2, Theorem 2.77]),
t t
k(t)=k(t )+ y(τ) 2∆τ k(t )+ C 2 x(τ) 2∆τ, t t
0 0 0.
|| || ≤ || || || || ≥
Zt0 Zt0
In conjunction with Lemma 5.2, this allows
∞
lim k(t) k(t )+ C 2 x(t) 2∆t
0
t→∞ ≤ || || || ||
Zt0
∞
k(t )+ C 2 x(t ) 2K2 e−α(t−t0)∆t
0 0
≤ || || || ||
Zt0
∞
=k + C 2 x(t ) 2K2 e−αt∆t
0 0
|| || || ||
Z0
< .
∞
This contradicts the assumption, so it must be that k(t)< for t T. It also
∞ ∞ ∈
immediately follows that y 2∆t< .
t0 || || ∞
ItseemspossiblethatTRheorem5.3maybeimprovedtoshowthattheoutput
is convergent, i.e. y(t) 0 as t . This is left as an open problem.
→ →∞
6 Discussion
We remark here that there is a great amount of freedom in the choice of the
update law for k(t) (c.f. [17]). We use the simplest choice for convenience (as
do most authors); the essential arguments remain unchanged for other choices.
There is also freedominthe choice ofupdate for µ(t). The expression(18), can
be simplified to
(CB)T +CB µ(t)k(t)(CB)TCB >0, t T,
− ∈
9
for any µ(t) µ¯(t). In the SISO case, this further reduces to the expres-
≤
sion derived by Owens [17], that µkCB < 2. It requires the graininess (which
may interpreted as the system sampling step size for sample-and-holdsystems)
to share at least an inverse relationship with the gain, but is otherwise quite
unrestrictive. Ilchmann and Townley [13] note that µ(t) = 1 meets
k(t)logk(t)
µkCB < 2 after sufficient time without knowledge of CB. While previous
works have always constructed a monotonically decreasing step size, the time
scale-based arguments in this paper reveal even greater freedom: µ(t) may ac-
tually increase,jump betweencontinuousanddiscrete intervals,orevenexhibit
bounded randomness. Two examples of the usefulness of this freedom come
next.
Forthefirstexample,wepointoutthatthenotationintheprevioussections
somewhat belies the fact that the system’s time domain (its time scale) may
be fully or partially discrete, and thus there is no guarantee in Theorem 5.3
that the output has stabilized between samples. As pointed out in [13] and
elsewhere, a sampled system with period µ is detectable if and only if
λ λ
k− lµ / Z for any λ =λ ; j =√ 1, (22)
k l
2πj ∈ 6 −
where λ ,λ 0,spec(A) . We next comment on how to circumvent the in-
k l
∈ { }
trasamplestabilizationproblem. Bothofthe followingmethods essentiallyper-
mitthe graininessµto“wiggle”abitsothatanoutputsamplemusteventually
occur away from a zero-crossing. Recalling that µ(t) µ¯(t) from Theorem
≤
5.1(i), let µ(t)=µ¯(t)v(t), where 0<v 1 is one of the sequences below:
≤
1.) Letvbeconsistofaninfinitelyrepeatedsubsequencewithn!+1elements
thatarerandomnumbersbetween0and1. Lettheseelements,labeledv ,v ,...,
1 2
beirrationalmultiplesofeachother. Assumeµ(t)convergestoµ¯v >0suchthat
1
y(t)=0butthetruecontinuousoutputisnonzero. Thisimpliesthatthereexist
integersk,l such that (λk−λl)µ¯v1 Z. As sequence v advances, there may be at
2πj ∈
worst n! combinations of j,k such that (λj−λk)µ¯vr Z for r =1...n!. However,
2πj ∈
at the next instant in time, (λk−λl)µ¯vn!+1 must be irrational and therefore not
2πj
in Z. The controller will detect a nonzero output and continue to adapt k(t)
and µ(t). In practice, of course, it is not possible to obtain a sequence of
truly irrational numbers, but most modern computer controllers have enough
accuracytorepresentthe ratiooftwoverylargeintegers,sothatthis technique
would only fail for impractically high magnitudes of λ λ .
k l
|| − ||
2.) Let v be a sequence of random numbers in the specified range. Even in
a computer with only 8-bit resolution for v, the probability of (λ1−λ2)µ¯v Z
2πi ∈
drops drastically after a few sample periods.
We remark here that, if (22) holds then ( ,C) is detectable because (A,C)
A
isdetectable. Thus,thestabilityofy impliesthestabilityofx. Wedonotdwell
on this here, but see e.g. [13] for a similar argument.
For the second example, we consider a problem posed by distributed con-
trol networks (c.f. [4, 8]). Here, one communication network supports many
10