Adaptive control for a discrete-time first-order nonlinear system with both parametric and non-parametric uncertainties

Abstract

A simple first-order discrete-time nonlinear system, which has both parametric uncertainty and non- parametric uncertainty, is studied in this paper. The uncertainty of non-parametric part is characterized by a Lipschitz constant L, and the nonlinearity of parametric part is characterized by an exponent index b. An adaptive controller is constructed for this model in both cases of b = 1 and b > 1, and its closed-loop stability is established under some conditions. When b = 1, the conditions given reveal the magic number 3/2 + radic2 which appeared in previous study on capability and limitations of the feedback mechanism.

Full text

Adaptive Control for a Discrete-time First-order Nonlinear System with
Both Parametric and Non-parametric Uncertainties
Hongbin Ma, Kai-Yew Lum and Shuzhi Sam Ge
Abstract— A simple first-order discrete-time nonlinear sys-
tem, which has both parametric uncertainty and non-
parametric uncertainty, is studied in this paper. The uncertainty
of non-parametric part is characterized by a Lipschitz constant
L, and the nonlinearity of parametric part is characterized by
an exponent index b. An adaptive controller is constructed for
this model in both cases of b= 1 and b > 1, and its closed-loop
stability is established under some conditions. When b= 1,
the conditions given reveal the magic number 3
2+√2which
appeared in previous study on capability and limitations of the
feedback mechanism.
Index Terms— Feedback mechanism, Capability and lim-
itation, Parametric uncertainty, Non-parametric uncertainty,
Adaptive control
I. INTRODUCTION
Feedback, a fundamental concept in automatic control,
aims to reduce the effects of the plant uncertainty on the
desired control performance. Because of the essence of the
feedback control, there has been much effort devoted to
the control of uncertain dynamical systems in the history
of control, particularly in the areas of adaptive control and
robust control (e.g. [1]–[6]). In these areas, considerable
progress has been made in dealing with uncertainties of
dynamical systems; however, systematic and quantitative
characterization of the maximum capability and limitation
of the whole feedback mechanism only appeared in the
last decade. The term “feedback mechanism” refers to all
possible feedback control laws and hence it is not restricted
in a class of special control laws. A brief survey on this
challenging topic can be found in the plenary lecture [7] by
Guo in International Congress of Mathematicians, 2002.
The first step towards this direction was made in [8],
where Guo attempted to answer the following question for
a nontrivial example of discrete-time nonlinear polynomial
plant model with parametric uncertainty: What is the largest
nonlinearity that can be dealt with by feedback? More
specifically, in [8], for the following nonlinear uncertain
system
yt+1 =θφt+ut+wt+1, φt=O(yb
t), b > 0
(I.1)
where θis the unknown parameter, bcharacterizes the
nonlinear growth rate of the system, and {wt}is the Gaus-
Hongbin Ma and Kai-Yew Lum are with the Temasek
Laboratories, National University of Singapore, Singapore 117508.
Email: {tslmh,tsllumky}@nus.edu.sg
Shuzhi Sam Ge is with the Department of Electrical and Computer
Engineering, National University of Singapore, Singapore 117576; the Inter-
active Digital Media Institute, National University of Singapore, Singapore
117576. Email: samge@nus.edu.sg
sian noise sequence, a critical stability result is found —
system (I.1) is not a.s. globally stabilizable if and only
if b≥4. This result indicates that there exist limitations
of the feedback mechanism in controlling the discrete-
time nonlinear adaptive systems, which is not seen in the
corresponding continuous-time nonlinear systems (see [8],
[9]). The “impossibility” result has been extended to some
classes of uncertain nonlinear systems with unknown vector
parameters in [10], [11] and a similar result for system (I.1)
with bounded noise is obtained in [12].
Stimulated by the pioneering work in [3], a series of
efforts ([13]–[16]) have been made to explore the maximum
capability and limitations of feedback mechanism. Among
these work, a breakthrough for non-parametric uncertain
systems was made by Xie and Guo in [13], where a class of
first-order discrete-time dynamical control systems
yt+1 =f(yt) + ut+wt+1, f (·)∈ F(L)(I.2)
is studied and another interesting critical stability phe-
nomenon is proved by using new techniques which are totally
different from those in [8]. More specifically, in [13], F(L)is
a class of nonlinear functions satisfying Lipschitz condition,
hence the Lipschitz constant Lcan characterize the size of
the uncertainty set F(L). Xie and Guo obtained the following
results: if L < 3
2+√2, then there exists a feedback control
law such that for any f∈ F(L), the corresponding closed-
loop control system is globally stable; and if L≥3
2+√2,
then for any feedback control law and any y0∈ R1, there
always exists some f∈ F(L)such that the corresponding
closed-loop system is unstable. So for system (I.2), the
“magic” number 3
2+√2characterizes the capability and
limits of the whole feedback mechanism. The impossibility
part of the above results has been generalized to similar high-
order discrete-time nonlinear systems with single Lipschitz
constant [14] and multiple Lipschitz constants [11].
From the work mentioned above, we can see two different
threads: one is focused on parametric nonlinear systems
and the other one is focused on non-parametric nonlinear
systems. By examining the techniques in these threads, we
find that different difficulties exist in the two threads, differ-
ent controllers are designed to deal with the uncertainties and
completely different methods are used to explore the capa-
bility and limitations of the feedback mechanism. Motivated
by these work, we want to explore the following problems:
When both parametric and non-parametric uncertainties are
present in the system, what is the maximum capability of
feedback mechanism in dealing with these uncertainties?
And how to design feedback control laws to deal with both

kinds of internal uncertainties? These problems stimulate our
research in this paper, and we shall study a discrete-time first-
order nonlinear system with both two kinds of uncertainties,
which sheds light on the aforementioned problems.
The remainder of this paper is organized as follows.
Problem formulation together with some assumptions is
given first in Section II, then an adaptive controller to be
studied is designed in Section III, which aims at dealing
with both non-parametric and parametric uncertainties. The
closed-loop stability of the proposed controller is presented
and rigorously proved in Section IV, which reveals again the
magic number 3
2+√2in case where the uncertain parametric
part is of linear growth rate. And then effectiveness of the
designed adaptive controller is shown by some simulation
examples in Section V. Finally Section VI provides a sum-
mary with some concluding remarks.
II. PROB LEM FORMULATION
Consider the following system
yt+1 =f(yt) + θφt+ut+wt+1 (II.1)
where yt,utand wtare the output, input and noise, respec-
tively; f(·)∈ F(L)is an unknown function (the set F(L)
will be defined later) and θis an unknown parameter.
In this system, two kinds of uncertainties exist: (i) Internal
uncertainties are embodied in both non-parametric part f(yt)
and θφt; (ii) External uncertainties are embodied in the
noise part wt+1. Previous exploration on capability and
limitation of feedback mechanism motivates the following
interesting question: How to quantify the uncertainties which
can be dealt with the feedback mechanism when both non-
parametric and parametric uncertainties are present in the
system?
To make further study, the following assumptions are used
throughout this paper:
Assumption 2.1: The unknown function f:R → R
belongs to the following uncertainty set
F(L) = {f:|f(x)−f(y)| ≤ L|x−y|+c}(II.2)
where cis an arbitrary non-negative constant.
Assumption 2.2: The noise sequence {wt}is bounded,
i.e.
|wt| ≤ w
where wis an arbitrary positive constant.
Assumption 2.3: The tracking signal {y∗
t}is bounded, i.e.
|y∗
t| ≤ S, ∀t≥0(II.3)
where Sis a positive constant.
Assumption 2.4: In the parametric part θφt, we have no
any a priori information of the unknown parameter θ, but
φt=g(yt)is measurable and satisfies
M0≤ |g(x1)−g(x2)
xb
1−xb
2| ≤ M(II.4)
for any x16=x2, where M0≤Mare two positive constants
and b≥1is a constant.
Remark 2.1: Assumption 2.4 implies that function g(·)
has linear growth rate when b= 1. Especially when g(x) =
x, we can take M=M0= 1. Condition (II.4) need only
hold for sufficiently large x1and x2, however we require it
holds for all x16=x2to simplify the proof.
Remark 2.2: Assumption 2.4 excludes the case where g(·)
is a bounded function, which can be handled easily by
previous research. In fact, in that case w0
t+1 =θφt+wt+1
must be bounded, hence by the result of [13], system (II.1)
is stabilizable if and only if L < 3
2+√2.
III. ADAP TIV E CONT ROLL ER DESIGN
In this section, we construct a unified adaptive controller
for both cases of b= 1 and b > 1.
For convenience, we introduce some notations which are
used in later parts. Let I= [a, b]be an interval, then
m(I)∆
=1
2(a+b)denotes the center point of interval I,
and r(I)∆
=1
2|b−a|denotes the radius of interval I. And
correspondingly, we let I(x, δ)=[x−δ, x +δ]denote a
closed interval centered at x∈ R with radius δ≥0.
A. Estimate of Parametric Part
At time t, we can use the following information:
y0, y1,··· , yt,u0, u1,··· , ut−1and φ0, φ1,· · · , φt.
Define
zj
∆
=yj+1 −uj(III.1)
and
It
∆
=T
(i,j)∈Jt
I(zj−zi
φj−φi,L|yj−yi|
|φj−φi|+2w+c
|φj−φi|)(III.2)
where
Jt
∆
={(i, j)∈ N :i < j < t, φi6=φj}(III.3)
then, we can take
ˆ
θt=m(It), δt=r(It)(III.4)
as the estimate of parameter θat time tand corresponding
estimate error bound, respectively. With ˆ
θtand δtdefined
above, ¯
θt=ˆ
θt+δtand θt=ˆ
θt−δtare the estimates
of the upper and lower bounds of the unknown parameter
θ, respectively. According to Eq. (III.2), obviously we can
see that {¯
θt}is a non-increasing sequence and {θt}is non-
decreasing.
B. Estimate of Non-parametric Part
Since the non-parametric part f(yt)may be unbounded
and the parametric part is also unknown, generally speaking
it is not easy to estimate the non-parametric part directly. To
resolve this problem, we choose to estimate
gt
∆
=θφt+f(yt)
as a whole part rather than to estimate f(yt)directly. In this
way, consequently, we can obtain the estimate of f(yt)by
removing the estimate of parametric part from the estimate
of gt.

Define it= arg min
i<t |yt−yi|(III.5)
then, we get
gt=gt−zit+zit
= [θφt+f(yt)] −[θφit+f(yit) + wit+1 ] + zit
= [θ(φt−φit) + zit]+[f(yt)−f(yit)−wit+1]
(III.6)
Thus, intuitively, we can take
ˆgt
∆
=ˆ
θt(φt−φit) + zit=ˆ
θt(φt−φit)+(yit+1 −uit)
(III.7)
as the estimate of gtat time t.
C. Design of Control ut
Let
¯
bt
∆
=max
i≤tyi= max(¯
bt−1, yt)
bt= min
i≤tyi= min(bt−1, yt).(III.8)
Under Assumptions 2.1-2.4, we can design the following
control law
ut=−ˆgt+y∗
t+1 if |yt−yit| ≤ D
−ˆgt+1
2(¯
bt+bt)if |yt−yit|> D (III.9)
where Dis an appropriately large constant, which will be
addressed in the proof later.
Remark 3.1: The controller designed above is different
from most traditional adaptive controllers in its special form,
information utilization and computational complexity. To
reduce its computational complexity, the interval Itgiven by
Eq. (III.2) can be calculated recursively based on the idea in
Eq. (III.8). More implementation details are omitted here to
save space.
IV. STAB ILITY OF CLOSED-LOO P SYST EM
In this section, we shall investigate the closed-loop sta-
bility of system (II.1) using the adaptive controller given
above. We only discuss the case that the parametric part
is of linear growth rate, i.e. b= 1. For the case where the
parametric part is of nonlinear growth rate, i.e. b > 1, though
simulations show that the constructed adaptive controller
can stabilize the system under some conditions, we have
not rigorously established corresponding theoretical results;
further investigation is needed in the future to yield deeper
understanding.
A. Main Results
The adaptive controller constructed in last section has the
following property:
Theorem 4.1: When b= 1,M L
M0<3
2+√2,the controller
defined by (III.1)— (III.9) can guarantee that the output {yt}
of the closed-loop system is bounded. More precisely, we
have
lim sup
t→∞ |yt−y∗
t| ≤ 2w+c. (IV.1)
Based on Theorem 4.1, we can classify the capability and
limitations of feedback mechanism for the system (II.1) in
case of b= 1 as follows:
Corollary 4.1: For the system (II.1) with both parametric
and non-parametric uncertainties, the following results can
be obtained in case of b= 1:
(i) If ML
M0<3
2+√2, then there exists a feedback
control law guaranteeing that the closed-loop system
is stabilized.
(ii) When φt=yt(i.e. g(x) = x), the presence of
uncertain parametric part θφtdoes not reduce the
critical value 3
2+√2of the feedback mechanism which
is determined by the uncertainties of non-parametric
part.
Proof of Corollary 4.1: (i) This result follows from The-
orem 4.1 directly. (ii) When g(x) = x, we can take M=
M0= 1. In this case, the sufficiency can be immediately
obtained via Theorem 4.1; on the other hand, the necessity
can be obtained by the “impossibility” part of Theorem 1
in [13]. In fact, if L≥3
2+√2, for any given control law
{ut}, we need only take the parameter θ= 0, then by [13,
Theorem 2.1], there exists a function fsuch that system
(II.1) cannot be stabilized by the given control law. 2
Remark 4.1: As we have mentioned in the introduction
part, system (I.2), a special case of system (II.1), has been
studied in [13]. Comparing system (II.1) and system (I.2),
we can see that system (II.1) has also parametric uncertainty
besides non-parametric uncertainty and noise disturbance.
Hence intuitively speaking, it will be more difficult for the
feedback mechanism to deal with uncertainties in system
(II.1) than those in system (I.2). Noting that M0≤M, we
know this fact has been partially verified by Theorem 4.1.
And Corollary 4.1 (ii) indicates that in the special case of
φt=yt, since the structure of parametric part is completely
determined, the uncertainty in non-parametric part becomes
the main difficulty in designing controller, and the parametric
uncertainty has no influence on the capability of the feedback
mechanism, that is to say, the feedback mechanism can still
deal with the non-parametric uncertainty characterized by the
set F(L)with L < 3
2+√2.
Remark 4.2: Theorem 4.1 is also consistent with classic
results on adaptive control for linear systems. In fact, when
L= 0, the non-parametric part f(yt)vanishes, consequently
system (II.1) becomes a linear-in-parameter system
yt+1 =θφt+ut+wt+1 (IV.2)
where θis the unknown parameter, and φt=g(yt)can
have arbitrary linear growth rate because by Theorem 4.1,
we can see that no restrictions are imposed on the values
of Mand M0when L= 0. Based on the knowledge from
existing adaptive control theory [3], system (IV.2) can be
always stabilized by algorithms such as minimum-variance
adaptive controller no matter how large the θis. Thus the
special case of Theorem 4.1 reveals again the well-known
result in a new way, where the adaptive controller is defined
by Eq. (III.9) together with Eqs. (III.1)—(III.8).
Corollary 4.2: If b= 1,M L
M0<3
2+√2,c=w= 0,
then the adaptive controller defined by (III.1)— (III.9) can
asymptotically stabilize the corresponding noise-free system,

i.e.
lim
t→∞ |yt−y∗
t|= 0.(IV.3)
B. Proof of Theorem 4.1
To prove Theorem 4.1, we need the following Lemmas:
Lemma 4.1: Assume {xn}is a bounded sequence of real
numbers, then we must have
lim
n→∞ min
i<n |xn−xi|= 0.(IV.4)
Proof: It is a direct conclusion of [13, Lemma 3.4]. It
can be proved by argument of contradiction. 2
Lemma 4.2: Assume that L∈(0,3
2+√2),d≥0,n0≥0.
If non-negative sequence {hn, n ≥0}satisfies
hn+1 ≤ Lmax
i≤nhi−1
2
n
X
i=0
hi+d!+
,∀n≥n0(IV.5)
where x+∆
=max(x, 0),∀x∈ R, then we must have
lim
n→∞
n
X
i=0
hi<∞.(IV.6)
Proof: See [13, Lemma 3.3]. 2
Proof of Theorem 4.1: We divide the proof into four steps.
In Step 1, we deduce the basic relation between yt+1 and
˜
θt, and then a key inequality describing the upper bound of
|yt−yit|is established in Step 2. Consequently, in Step 3,
we prove that |yt−yit|→∞as t→ ∞ if ytis not bounded,
and hence the boundedness of output sequence {yt}can be
guaranteed. Finally, in the last step, the bound of tracking
error can be further estimated based on the stability result
obtained in Step 3.
Step 1: Let
˜
θt
∆
=θ−ˆ
θt
y#
t+1
∆
=θφt+f(yt) + wt+1 −ˆgt
(IV.7)
then, by definition of utand Eq. (III.9), obviously we get
yt+1 =y#
t+1 +y∗
t+1 if |yt−yit| ≤ D
y#
t+1 +1
2(¯
bt+bt)if |yt−yit|> D (IV.8)
Now we discuss y#
t+1. By Eq. (III.7) and Eq. (II.1), we get
y#
t+1 =θφt+f(yt) + wt+1 −ˆgt
=θφt+f(yt) + wt+1 −ˆ
θt(φt−φit)
−(θφit+f(yit) + wit+1)
= (θ−ˆ
θt)(φt−φit)
+[f(yt)−f(yit)] + (wt+1 −wit+1)
=˜
θt(φt−φit)+[f(yt)−f(yit)] + (wt+1 −wit+1)
(IV.9)
In case of φt=φit, i.e. yt=yit, obviously we get
|y#
t+1|=|wt+1 −wit+1 | ≤ 2w;(IV.10)
otherwise, we get
y#
t+1 = ( ˜
θt+Dt,it)(φt−φit)+(wt+1 −wit+1)
(IV.11)
where
Di,j
∆
=f(yi)−f(yj)
φi−φj
.
Obviously Dij =Dji. In the latter case, i.e. when φt6=φit,
for any (i, j)∈Jt, noting that
zj−zi= (yj+1 −uj)−(yi+1 −ui)
=θ(φj−φi)+[f(yj)−f(yi)]
+[wj+1 −wi+1]
(IV.12)
we obtain that
θ−zj−zi
φj−φi
=−Di,j −wj+1 −wi+1
φj−φi
.(IV.13)
Therefore
˜
θt+Dt,it= (θ−zj−zi
φj−φi)−(ˆ
θt−zj−zi
φj−φi) + Dt,it
=Dt,it−Di,j −wj+1−wi+1
φj−φi+ ∆i,j (t)
(IV.14)
where
∆i,j (t) = zj−zi
φj−φi−ˆ
θt.(IV.15)
Step 2: Since ML
M0<3
2+√2, there exists a constant
 > 0such that M L
M0+ < 3
2+√2. Let
Bt
∆
=[bt,¯
bt],∆Bt=Bt−Bt−1(IV.16)
and consequently
|Bt|=¯
bt−bt,|∆Bt|=|Bt|−|Bt−1|(IV.17)
By the definition of bt,¯
btand Bt, we obtain that
|Bt+1|=|Bt|if yt+1 ∈Bt
1
2|Bt|+|yt+1 −1
2(bt+¯
bt)|if yt+1 6∈ Bt
(IV.18)
By the definition of it, obviously we get
|yt−yit|=|∆Bt|if yt6∈ Bt−1
≤ |∆Bi|if yt∈∆Bi⊂Bt−1(IV.19)
Step 3: Based on Assumption 2.4, for any fixed  > 0,
we can take constants Dand D0such that |φi−φj|> D0>
4M(2w+c)
when |yi−yj|> D. Now we are ready to show
that for any s > 0, there always exists t>ssuch that
|yt−yit| ≤ D.
In fact, suppose that it is not true, then there must exist
s > 0such that |yt−yit|> D for any t>s, correspondingly
|φt−φit|> D0. Consequently, by the definition of D, for
sufficiently large tand j < t, we obtain that
|wj+1 −wij+1
φj−φij|≤|2w
D0|<1
4M;(IV.20)
together with the definition of ˆ
θt, we know that for any s <
i < j < t,
|∆i,j (t)|=|zj−zi
φj−φi−ˆ
θt| ≤ L
M0+2w+c
|φj−φi|.(IV.21)
hence for s < j < t, i =ij, we get
|∆j,ij(t)|=|∆ij,j (t)| ≤ L
M0+2w+c
D0≤L
M0+1
4M.
(IV.22)

Now we consider Dt,it−Dj,ij.
Let dn=Dn,in, then, by the definition of Di,j , noting
that |yj−yi|≥|yj−yij|> D for any j > s, we obtain that
|Di,j |=|f(yi)−f(yj)
yi−yj|·|yi−yj
φi−φj| ≤ L
M0,(IV.23)
so we can conclude that {dn, n > s}is bounded. Then, by
Lemma 4.1, we conclude that
lim
t→∞ min
s<j<t |dt−dj|= 0.(IV.24)
Consequently there exists s0> s such that for any t > s0,
we can always find a corresponding j=j(t)satisfying
|Dt,it−Dj,ij|=|dt−dj|<1
4M. (IV.25)
Summarizing the above, for any t>s0, by taking j=j(t),
we get
|˜
θt+Dt,it|=|Dt,it−Dij,j −wj+1−wi+1
φj−φij
+ ∆ij,j (t)|
≤ |Dt,it−Dij,j |+|wj+1−wi+1
φj−φij|+|∆ij,j (t)|
≤1
4M+1
4M+ ( L
M0+1
4M)
=L
M0+3
4M
∆
=L.
(IV.26)
Therefore
|y#
t+1|=|(˜
θt+Dt,it)φt−φit
yt−yit(yt−yit)+(wt+1 −wit+1)|
≤LM|yt−yit|+ 2w.
(IV.27)
Since |yt−yit|> D, we know that
yt+1 =y#
t+1 +1
2(¯
bt+bt).(IV.28)
From Eq. (IV.26) together with the result in Step 2, we obtain
that
|Bt|≤|Bt+1 | ≤ max{|Bt|,1
2|Bt|+|yt+1 −1
2(bt+¯
bt)|}
= max{|Bt|,1
2|Bt|+|y#
t+1|} (IV.29)
Thus noting (IV.27), we obtain the the following key inequal-
ity:
|∆Bt| ≤ (LM|yt−yit|+ 2w−1
2|Bt|)+(IV.30)
where
LM= ( L
M0+3
4M)M=ML
M0+3
4 < 3
2+√2.
(IV.31)
Considering the arbitrariness of t>s0, together with
Lemma 4.2, we obtain that
P
j>s0|∆Bj|<∞,(IV.32)
and consequently {|Bt|} must be bounded. By applying
Lemma 4.1 again, we conclude that
|yt−yit| ≤ min
i<t |yt−yi| → 0(IV.33)
which contradicts the former assumption!
Step 4: According to the results in Step 3, for any s > 0,
there always exists t > s such that |yt−yit| ≤ D. Then,
we can easily obtain that {|˜
θt|} is bounded, say |˜
θt| ≤ L0.
Considering that
y#
t+1 =˜
θt(φt−φit)+[f(yt)−f(yit)] + (wt+1 −wit+1)
(IV.34)
we can conclude that
|yt+1|≤|y#
t+1 +y∗
t+1|
≤L0|φt−φit|+ (L|yt−yit|+c)+2w
≤Y
(IV.35)
where Y=L0MD +LD +c+ 2w+S.
The proof below is similar to that in [13]. Let
t0= inf
t>0{t:|yt| ≤ Y}, tn= inf
t>tn−1{t:|yt| ≤ Y}.
(IV.36)
Because of the result obtained above, we conclude that for
any n≥1,tnis well-defined and tn<∞.
Let vn=ytn, then obviously {vn}is bounded. Then, by
applying Lemma 4.1, we get
min
i<n |vn−vi| → 0(IV.37)
as n→ ∞. Thus for any ε > 0, there exists an integer n0
such that for any n>n0,
min
i<n |vn−vi|< ε. (IV.38)
So
|ytn−yitn|= min
i<tn|ytn−yi| ≤ min
i<n |ytn−yti|< ε. (IV.39)
By taking εsufficiently small, we obtain that
|ytn+1| ≤ L0M ε +Lε +c+ 2w+S≤Y(IV.40)
for any n>n0.
Thus based on definition of tn, we conclude that tn+1 =
tn+ 1! Therefore for any t≥tn0,
|yt| ≤ Y(IV.41)
which means that the sequence {yt}is bounded.
Finally, by applying Lemma 4.1 again, for sufficiently
large t,|yt−yit| ≤ ε, consequently
|yt+1 −y∗
t+1|=|y#
t+1| ≤ L0M ε +Lε +c+ 2w. (IV.42)
Because of arbitrariness of ε, Theorem 4.1 is true. 2
V. SIMULATION STU DY
In this section, two simulation examples will be given
to illustrate the effects of the adaptive controller designed
above. In both simulations, the tracking signal is taken as
y∗
t= 10 sin t
10 and the noise sequence is i.i.d. randomly
taken from uniform distribution U(0,1). The simulation
results for two examples are depicted in Figure 1 and Figure
2, respectively. In each figure, the output sequence ytand the
reference sequence y∗
tare plotted in the top-left subfigure;
the tracking error sequence et
∆
=yt−y∗
tis plotted in the
bottom-left subfigure; the control sequence utis plotted in
the top-right subfigure; and the parameter θtogether with its
upper and lower estimated bounds is plotted in the bottom-
right subfigure.

Fig. 1. Simulation example 1: (g(x) = x, b = 1, M =M0= 1)
Simulation Example 1: This example is for case of b=
1, and the unknown plant is
yt+1 =f(yt) + θg(yt) + wt+1 , f (·)∈ F(L)(V.1)
with L= 2.9<3
2+√2,g(x) = x(i.e. b= 1, M =M0= 1),
and
f(x)=1.4xsin log(|x|+ 1).(V.2)
For this example, we can verify that
|f0(x)|= 1.4|sin log(|x|+ 1) ±xcos log(|x|+1)
|x|+1 | ≤ 2.8< L
(V.3)
consequently |f(x)−f(y)|< L|x−y|, i.e. f(·)∈ F(L).
Simulation Example 2: This example is for case of b >
1, and the unknown plant is
yt+1 =f(yt) + θg(yt) + wt+1 , f (·)∈ F(L)(V.4)
with L= 2.9,g(x) = x2(i.e. b= 2, M =M0= 1), and
f(x)=2x+ sin x2.(V.5)
For this example, we can verify that |f(x)−f(y)|< L|x−
y|+ 2, i.e. f(·)∈ F(L).
From the simulation results, we can see that in both ex-
amples, the adaptive controller can track the reference signal
successfully. The simulation study verified our theoretical
result and indicate that under some conditions, the adaptive
control law constructed in this paper can deal with both
parametric and non-parametric uncertainties, even in some
cases when the parametric part is of nonlinear growth rate.
In case of b= 1, the stabilizability criteria have been
completely characterized by a simple algebraic condition;
however, in case of b > 1, it is very difficult to give complete
theoretical characterization. Note that usually more accurate
estimate of parameter can be obtained in case of b > 1than
in case of b= 1, however, worse transient performance may
be encountered.
VI. CONCLUSION
In this paper we investigate a simple first-order nonlinear
system with both non-parametric uncertainties and paramet-
ric uncertainties, where these two kinds of uncertainties are
both taken into consideration the first time in the exploration
Fig. 2. Simulation example 2: (g(x) = x2, b = 2, M =M0= 1)
of the capability and limitations of the feedback mechanism.
We have constructed a unified adaptive controller which can
be used in both cases of b= 1 and b > 1. When the
parametric part is of linear growth rate (b= 1), we have
proved the closed-loop stability under some assumptions
and a simple algebraic condition M L
M0<3
2+√2, which
reveals essential connections with the known magic number
L=3
2+√2discovered in recent work [13] on the study of
feedback mechanism capability.
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