## 行政院國家科學委員會專題研究計畫 成果報告

### 基於 T-S 模糊模型之隨機非線性韌性適應控制 研究成果報告(精簡版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 98-2221-E-216-032-

執 行 期 間 ： 98 年 08 月 01 日至 99 年 07 月 31 日 執 行 單 位 ： 中華大學電機工程學系

計 畫 主 持 人 ： 李柏坤

計畫參與人員： 碩士班研究生-兼任助理人員：張毅平

報 告 附 件 ： 出席國際會議研究心得報告及發表論文

處 理 方 式 ： 本計畫可公開查詢

中 華 民 國 99 年 10 月 31 日

## Stochastic Nonlinear Robust Adaptive Control Based on T-S Fuzzy Models

Project Leader: Bore-Kuen Lee

Department of Electrical Engineering, Chung Hua University Hsinchu, Taiwan 300, R.O.C., Email: bklee@chu.edu.tw

Project Duration: August 2009 to July 2010 Project ID: NSC 98-2221-E-216 -032

**Abstract—In the field of adaptive fuzzy control, there****has been a severe deficiency by assuming the premise**
**variables will usually stay within the universe of dis-**
**course in the derivation of stability of the adaptive**
**control system. To overcome this deficiency, we develop**
**a switching adaptive control scheme using only essential**
**qualitative information of the plant to attain asymptotical**
**stability of the adaptive control system for a typical first-**
**order nonlinear system without imposing the mentioned**
**severe assumption. The switching adaptive control system**
**consists of an adaptive VSS controller for coarse control,**
**an adaptive fuzzy controller for fine control, and a**
**hysteresis switching mechanism. An adaptive VSS control**
**scheme is proposed to force the state to enter the universe**
**of discourse in finite time. While the premise variable is**
**within the universe of discourse, an adaptive fuzzy control**
**is proposed to learn the capability to stabilize the plant.**

**At the boundary of the universe of discourse, a hysteresis**
**switching scheme between the two controllers will be**
**proposed. We show that after finite times of switching, the**
**premise variables of the fuzzy system will remain within**
**the universe of discourse and stability of the closed-**
**loop system can be attained by applying Lyapunov direct**
**method.**

**In the current year, we focus on robust adaptive**
**control for deterministic nonlinear systems. Based on the**
**developed results, we shall attack the same problem for**
**nonlinear stochstic systems in the next year.**

**Index Terms: Adaptive fuzzy control, switching**
control, T-S fuzzy model

I. INTRODUCTION

There are many deterministic fuzzy adaptive control systems which are proposed in the literature since 2000. Generally speaking, the main difficulty for adap- tive fuzzy control systems arises from system uncer- tainty and disturbances. In the presence of these two uncertain terms, the first problem is how to guarantee uniform boundedness of parameter estimates, and the second one is how to design adaptive control law so as to guarantee system stability. In [1], it is assumed that the uncertainty term, which is also a function of plant input and system states, has an known upper bound to design a stabilizing control law. However, this assumption is unreasonable due to the following two problems.

P1. First, it is unreasonable to impose an upper bound of uncertainty term since plant input and

system states may diverge before guaranteeing system stability. Especially, the upper bound is hard to know in an adaptive control scenario.

P2. Second, we can not guarantee that the premise variables will be confined in a compact universe of discourse so that the uniform approximation property holds in the analysis of the stability of the adaptive fuzzy control system.

In [2], where an adaptive control of time delay nonlinear systems is considered, problems P1 and P2 also occurred. The same situation also took place in [3]

and [4]. In [5], [6], and [7], problem P1 is avoided, but problem P2 is also not considered in the analysis of the closed-loop system stability.

In [8], fuzzy systems are introduced to approximate system nonlinear functions and Lyapunov-based design techniques are employed to design stabilizing adaptive controllers to attain asymptotical stability of the state and the boundedness of the estimated parameters for regulation control problem. In their adaptive fuzzy control schemes, an essential deficiency is that the universe of discourse should depends on unknown system parameters, which is hard to define in advance.

Basically, problem P2 is also not overcome in this literature.

Based on the literature survey discussed above, in this study, we shall construct a robust fuzzy adap- tive control for nonlinear affine systems to overcome problems P1 and P2. We shall only use minimum information about modeling error of system uncertain terms, because adaptive controller should have the ability to learn the information of the modeling error by itself. We shall not assume that the trajectory of premise variables is limited to the universe of discourse of the fuzzy system. Without this assumption, it will be more difficult to design an stabilizing adaptive controller.

To attain our goals, we shall develop a switching adaptive control scheme to attain stability of the adap- tive control system for a typical first-order nonlinear system. We shall only make some essential qualitative assumptions of the plant, instead of requiring some quantitative information of the plant, to construct an adaptive controller. The proposed switching adaptive

control system consists of an adaptive VSS controller for coarse control, an adaptive fuzzy controller for fine control, and a hysteresis switching mechanism for switching of the previous two controllers. The adaptive VSS controller is used to force the premise variable to enter the universe of discourse in finite time.

While the premise variable is kept within the universe of discourse, the adaptive fuzzy controller will tune its parameters and gradually learn the capability to stabilize the plant. At the boundary of the universe of discourse, a hysteresis switching scheme between the adaptive VSS control law and the adaptive fuzzy control law will be proposed. We shall show that after finite times of switching, the premise variable of the fuzzy system will remain in the universe of discourse and stability of the adaptive control system will be attained by applying the Lyapunov direct method.

The remainder of this work is organized as fol- lows. The problem to be attacked and the hysteresis switching adaptive control scheme are described in Section 2. The adaptive VSS controller is proposed and analyzed in Section 3. Then, the considered adaptive fuzzy control is presented in Section 4. Analysis of the switching control system is made in Section 5 together with a simulation example. Finally, conclusions and discussions are given in Section 6.

**Notations**

For a vector x = [ x1 x_{2} · · · x_{n} ]^{T}, the asso-
ciated swap operation is defined as

swap(x) = [ xn xn−1 · · · x1 ]^{T}

For a vector x, we write x≥ 0 if every entry of x is greater than or equal to zero.

II. PROBLEMFORMULATION ANDTHE

HYSTERESISSWITCHINGADAPTIVECONTROL

Consider the plant

˙x = f (x) + u (1)

where f (x) is a scalar nonlinear continuous function
of the scalar variable x and u∈ R^{1} is the input. For
the nonlinear continuous function f (x), we make the
following assumptions.

*Assumption 1: f (x) is a continuous function and*
admits its maximum fmax on the compact connected
set Ωx with

fmax= max

x∈Ω^{x}|f(x)| (2)

where fmax is an unknown positive number.

*Assumption 2: The function f (x) satisfies*

¯¯

¯¯df (x) dx

¯¯

¯¯ ≤ κ^{f} (3)

for x∈ Ωx where κf is an unknown positive number.

*Assumption 3: For x /*∈ Ω^{x}, there is a least upper
bound ψ(x) of f (x) satisfying

|f(x)| ≤ c^{∗}1|x| + c^{∗}2|x|^{2}= ψ(x) for x /∈ Ω^{x} (4)

*u**VSS*

*u**VSS*

*fuzzy*

*u*

*x*
*1 h*−

(1 *h*)

− −

−1 1

Fig. 1. Illustration of the hysteresis switching control.

where c^{∗}_{1} and c^{∗}_{2} are unknown positive parameters.

*Assumption 4: We assume that*

½ xf (x) > 0, if x6= 0

f (x) = 0, if x = 0 (5)
and f (x) is a convex function for x ∈ Ω^{x}. Also to
simplify system analysis, we shall assume f (x) is an
odd function, i.e.,

f (−x) = −f(x) (6)

An example of such a function f (x) is given by
f (x) = ^{∗}x |x| + μ^{∗}x (7)
with ^{∗}> 0 and μ^{∗}> 0 under which the equilibrium
point 0 of the system dynamics (1) is unstable. In this
study, we shall consider the case that the nonlinear
function f (x) is unknown and a fuzzy approximator
F (x|θ) will be used to approximate an ideal nonadap-
tive stabilizing controller in the universe of discourse
Ω_{x} = [−1, 1] where x is the only premise variable.

Basically, when the state trajectory x(t) is outside the
universe of discourse Ωx, by utilizing the structure
information of f (x) given in (4) in Assumption 3,
we shall develop an adaptive VSS control u_{V SS}(t) to
force the state trajectory entering Ωx. On the other
hand, if the state trajectory x(t) is staying within Ωx,
an adaptive fuzzy control uf uzzy(t) will be applied
to further ensure that the system will be ultimately as-
ymptotically stable. Since switching between these two
control laws with infinite frequency at the boundary of
the region Ωx may happen, we shall use a hysteresis
switching control as described in the following to
avoid this problem. Let h, with 0 < h < 1, be the
hysteresis size and define the hysteresis zone Ωh as
Ω_{h} = {x| 1 − h ≤ |x| ≤ 1}. The hysteresis switching
control structure, as shown in Fig. 1, is described as
follows. At t = 0, the control structure is defined as

u(0) =

½ uV SS(0), if |x(0)| > 1 − h

uf uzzy(0), if |x(0)| ≤ 1 − h (8) For t > 0, while x(t) is outside the hysteresis zone Ωh, the control input u(t) is defined as

u(t) =

½ uV SS(t), if |x(t)| > 1

uf uzzy(t), if |x(t)| < 1 − h (9) and on the contrary, while x(t) is inside the hysteresis zone Ωh, u(t) is defined as

u(t) =

½ u_{V SS}(t), if u(t_{−}) = u_{V SS}(t_{−})
u_{f uzzy}(t), if u(t_{−}) = u_{f uzzy}(t_{−}) (10)

We note that while applying the adaptive VSS control law uV SS, the tuning parameters in the adaptive fuzzy controller will be kept invariant. On the other hand, while applying the adaptive fuzzy control law uf uzzy, the tuning parameters in the adaptive VSS controller will be frozen.

The problem to be attacked is formulated as follows.

**For the plant in (1) under assumptions Assumption 1-**
**Assumption 4, we shall construct an adaptive VSS**
controller and an adaptive fuzzy controller together
with the above hysteresis switching mechanism so that
the tuning parameters in the two adaptive controllers
are bounded and x(t)→ 0 as t → ∞.

III. DESIGN AND ANALYSIS OF THE ADAPTIVEVSS

CONTROL

In this section, an adaptive VSS control will be proposed and the system behavior will be analyzed.

Recall that the system function f (x) has an least
upper bound ψ(x) with the structural information as
indicated in (4) for x /∈ Ω^{x}. Here, we shall develop
an adaptive VSS control uV SS(t) to force the state
trajectory entering Ωx when the state trajectory is
outside the region Ωx. To attain this goal, we shall
construct estimates ˆc1and ˆc2of c^{∗}_{1}and c^{∗}_{2}, respectively,
so that the following inequality

|f(x)| ≤ ˆc1|x| + ˆc2|x|^{2} for x /∈ Ωx

can be attained. Based on the estimates ˆc_{1} and ˆc_{2}, the
proposed adaptive VSS control law will be defined as
uV SS= −(ˆc^{1}|x| + ˆc^{2}|x|^{2}+ r |x|)sign(x) (11)
where r is a given positive constant. The tuning laws
of ˆc1 and ˆc2 are given as

.

ˆ

c1 = Γ1|x|^{2}, ˆc1(0) = 0 (12)

.

ˆ

c2 = Γ1|x|^{3}, ˆc2(0) = 0 (13)
To analyze the system response when applying the
adaptive VSS control law defined in (11), (12), and
(13), we consider the Lyapunov function candidate

Va= 1
2x^{2}+1

2Γ^{−1}_{1} ˜c^{2}_{1}+1
2Γ^{−1}_{1} ˜c^{2}_{2}
where

˜

c1 = ˆc1− c^{∗}1

˜

c_{2} = ˆc_{2}− c^{∗}2

The following lemma, adopted from [8], is required for further analysis.

*Lemma 1: If V (t, x) is positive definite and ˙*V ≤

−k^{1}V + k2 where k1 > 0 and k2 ≥ 0 are bounded
constants, then

V (t, x) ≤ k2

k_{1} + (V (0) −k2

k_{1})e^{−k}^{1}^{t}
for all t. Also it is obvious that

tlim→∞V (t, x) ≤ k2

k_{1}

*Lemma 2: Consider the adaptive VSS control sys-*
tem defined by (1), (11), (12), and (13). The trajectories
of x(t), ˆc1(t), and ˆc2(t) are bounded over the time
interval (t0, ∞) where t^{0} is an arbitrary initial time,
and x(t) converges to the origin. Moreover, there is a
finite time t1such that x(t1) = 1 − h if x(t0) > 1 − h
or x(t1) = −(1 − h) if x(t0) < −(1 − h) where t1 is
a time instant with t1≤ t0+ T_{1} and

T1= Va(t0)

r(1 − h)^{2} (14)

**Proof: The time derivative of V**a along the system
trajectory of the adaptive VSS control system can be
evaluated as

.

Va

= x[f (x) + u_{V SS}] + ˜c_{1}|x|^{2}+ ˜c_{2}|x|^{3}

= xf (x) − ψ(x) |x| + (c^{∗}1|x| + c^{∗}2|x|^{2}) |x|

−(ˆc1|x| + ˆc2|x|^{2}+ r |x|) |x| + ˜c1|x|^{2}+ ˜c_{2}|x|^{3}

≤ −r |x|^{2}≤ 0 (15)

The above inequality implies that the trajectories of
x(t), ˆc1(t), and ˆc2(t) are bounded over the time
interval (t0, ∞) and V^{a}(t) is a non-increasing function
of t. From (12) and (13), it is obvious that both ˆc1(t)
and ˆc_{2}(t) are non-decreasing functions of t. Therefore
ˆ

c_{1}(t) and ˆc_{2}(t) both converge to some finite values as
t → ∞. On the other hand, we have

..

ˆ

c1= 2Γ1xh

f (x) − (ˆc^{1}|x| + ˆc^{2}|x|^{2}+ r |x|)sign(x)i
which is bounded, and we have ˙c1 is uniformly con-
tinuous. Then, Barbalat’s lemma [9], we have

tlim→∞˙c1(t) = 0

Consequentially, with ˙c1= Γ1|x|^{2}defined in (12), we
can conclude

tlim→∞x(t) = 0

From (15), there exists a positive function Z(t)≥ 0 such that

.

Va+ Z = −r |x|^{2}

Integrating the last differential equation, one can get
V_{a}(t) = V_{a}(t_{0}) +

Z t t0

[−Z(τ) − r |x|^{2}(τ )]dτ

= V_{a}(t_{0}) − r
Z t

t0

|x|^{2}(τ )dτ −
Z t

t0

Z(τ )dτ which implies

1/2x^{2}≤ V^{a}(t) ≤ V^{a}(t0) − r
Z t

t0

|x|^{2}(τ )dτ (16)
Now define a function y(t) as

y(t) = Z t

t0

|x|^{2}(τ )dτ

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x Fuzzy sets

μF1 μF2 μF3 μF4 μF5

Fig. 2. A typical case of the fuzzy sets in the rule base.

which is equivalent to the following differential equa- tion

dy/dt = x^{2}(t), y(t0) = 0
Then, from (16), we have

˙y + 2ry ≤ 2V^{a}(t0)
By Lemma 1, we can conclude that

y(t) ≤ 2V^{a}(t0)
Z t

t0

e^{−2r(t−τ)}dτ

= (1 − e^{−2r(t−t}^{0}^{)}
r )Va(t0)
and thus

Z t t0

x^{2}(τ )dτ ≤ (1 − e^{−2r(t−t}^{0}^{)}

r )Va(t0) <V_{a}(t_{0})
r

(17)
If x(t0) > 1 − h, we can show that there is a finite
time t1 such that x(t1) = 1 − h with t^{1}− t^{0} ≤ T^{1}
where T1is defined in (14). By contradiction, assume
that x(t) > 1− h > 0 for t ∈ [t^{0}, t0+ T1] . Then, we
have

Z t0+T1

t0

x^{2}(τ )dτ > (1 − h)^{2}T_{1}= V_{a}(t_{0})
r

which contradicts inequality (17). This verifies the
assertion. Similarly, if x(t0) < −(1 − h), there is
a finite time t1 such that x(t1) = −(1 − h) with
t_{1}− t0≤ T1. This ends the proof. ¥

IV. DESIGN OF THE ADAPTIVE FUZZY CONTROL

For the plant in (1), the only premise variable of the fuzzy system is x and the universe of discourse Ωxis chosen as Ωx= [−1, 1]. The rule base of the T-S fuzzy system is defined as: for 1≤ l ≤ L,

Rule l : If x is Fl, then y = θl.

where Fl is the fuzzy set with membership function
μ_{F}_{l}(x) and θl is the value specified in the antecedent

part of the l−th rule. The number L, which is the total number of rules, will be chosen as an odd number. A typical case is shown in Fig. 2 where the set of IF- THEN rules is complete, consistent, and continuous [10]. Based on the above rule base, the T-S fuzzy sys- tem, consisting of the singleton fuzzyifier, the product inference engine, and the center average defuzzifier [10], can be expressed as

F (x, θ) = ξ^{T}(x)θ (18)
where

θ = [θ1, ..., θL]^{T},
ξ_{l}(x) = μ_{F}_{l}(x)

PL i

μ_{F}_{l}(x)
,

ξ(x) = [ξ_{1}(x), ..., ξ_{L}(x)]^{T} (19)
From Fig. 2, we can observe that

XL i

μ_{F}_{l}(x) = 1,
for any x∈ Ωx and

ξ(x) =£

μ_{F}_{1}(x), ..., μ_{F}_{L}(x)¤T

(20)
with L = 5. From the triangular membership functions
shown in Fig. 2, we have, for any x∈ Ω^{x},

kξ(x)k^{2}= μ^{2}_{F}_{i}(x) + μ^{2}_{F}_{i+1}(x) (21)
for 1≤ i ≤ L − 1 and

μ_{F}_{i}(x) + μ_{F}_{i+1}(x) = 1 (22)
From (21) and (22), it is obvious that

1

2 ≤ kξ(x)k^{2}≤ 1 (23)

Note that since Ωx= [−1, 1] is symmetric with respect to the origin, the rule base will be chosen to symmetric in the sense that

ξ(−x) = ¯ξ(x) (24)

where ¯ξ(x) =swap(ξ(x)).

Now let Ai, for 1 ≤ i ≤ L, be the support of the
membership function μ_{F}_{i}(x), i.e.,

A_{i} =©
x ∈ Ωx

¯¯μ_{F}_{i}(x) > 0ª

Denote di as the center of the membership function
μ_{F}_{i}(x) for 1 ≤ i ≤ L and γi as the point such
that μ_{F}_{i}(γ_{i}) = μ_{F}_{i+1}(γ_{i}) for 1 ≤ i ≤ L − 1. For
the convenience of further analysis, now partition the
universe of discourse Ωx as Ωx= ∪^{2L}i=1^{−2}Ωx,i where

Ω_{x,2i}_{−1} = [d_{i}, γ_{i}), for 1 ≤ i ≤ L − 1
2
Ω_{x,2i} = [γ_{i}, d_{i+1}), for 1 ≤ i ≤ L − 1

2 − 1
Ω_{x,2i}_{−1} = (d_{i}, γ_{i}], for L + 3

2 ≤ i ≤ L − 1
Ω_{x,2i} = (γ_{i}, d_{i+1}], for L + 1

2 ≤ i ≤ L − 1

and

Ωx,L−1 = [γL−1
2 , d^{L+1}

2 )
Ωx,L = [d^{L+1}

2 , γ^{L+1}

2 ]

We make a final note that the fuzzy system F (x, θ) in (18) admits a linear approximator structure with respect to the parameter vector θ and

F (x, θ1) − F (x, θ^{2}) = ξ^{T}(x)(θ1− θ^{2}) (25)
For x∈ Ω^{x}, we can approximate the system function
f (x) by the fuzzy system F (x, θ) = ξ^{T}(x)θ so that

minθ kf(x) − F (x, θ)k_{∞}= W

for some W > 0 due to the universal approximation property of the constructed fuzzy system [10] and the infinite norm is defined as

kg(x)k_{∞}= sup

x∈Ωx

|g(x)|

Let’s denote a best fitted parameter θ^{∗} as
θ^{∗}∈ arg min

θ kf(x) − F (x, θ)k_{∞}
For x∈ Ω^{x}, we then have

|f(x) − F (x, θ^{∗})| ≤ W (26)
Finally, with respect to the membership functions
shown in Fig. 2, the hysteresis size h defined in (9)
will be chosen such that

0 < h ≤ 1

4− ε^{h} (27)

where εh is a small positive constant.

With the above definitin of the fuzzy system, the adaptive controller is defined as

.

ˆθ = Γ2ξx (28)

u(t) = −ˆθ^{T}ξ(x) (29)

V. A HYSTERESISSWITCHINGROBUSTFUZZY

ADAPTIVECONTROL

Based on the adaptive VSS controller and the adap- tive fuzzy controller, we shall study the proposed hys- teresis switching robust adaptive control defined as in (9) and (10). While applying adaptive VSS controller, the closed-loop dynamics is given

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

˙x = f (x) − (ˆc1|x| + ˆc2|x|^{2}+ r |x|)sign(x),

.

ˆ

c1= Γ1|x|^{2}, c1(0) = 0,

.

ˆ

c2= Γ1|x|^{3}, c2(0) = 0

.

ˆθ = 0,

(30) On the other hand, by letting Γ2 = I in (28), the closed-loop dynamics adaptive fuzzy control system is

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

˙x = f (x) − ˆθ^{T}ξ(x)

.

ˆθ = ξx, ˆθ(0) = 0

.

ˆ c1= 0

.

ˆ
c_{1}= 0

(31)

Note that the value of the fuzzy approximator is given by

F (x|ˆθ) = ˆθ^{T}ξ(x)

According to the tuning law of ˆ**θ defined in (??) and**
the definition of the vector ξ(x) in (20), some further
properties of ˆθ can be discovered.

*Lemma 3: Due to the structure of the fuzzy system*
and the tuning law of ˆθ(t) defined in (31), we have
the following results. (i) If Ai ⊂ [0, 1], then ˆθi(t) ≥ 0
and ˆθ_{i}(t) is a monotone increasing function of time.

On the other hand, if Ai ⊂ [−1, 0], then ˆθi(t) ≤ 0
and ˆθ_{i}(t) is a monotone decreasing function of time.

(ii) For Ai ⊂ [0, 1] or Ai ⊂ [−1, 0], if there is time
t_{0} such that x(t0) ∈ Ai, then ˆθ_{i}(t) > 0 for t ≥ t0.
Similarly, if there is time t0 such that x(t0) ∈ Ai,
then ˆθ_{i}(t) < 0 for t ≥ t0. (iii) If x(t) ∈ [1 − h, 1],
then ˆθ^{T}(t)ξ(x(t)) ≥ 0. On the other hand, if x(t) ∈
[−1, −(1 − h)], then ˆθ^{T}(t)ξ(x(t)) ≤ 0.

**Proof: (i) Denote A**i be the closure of Ai. If x ∈
A_{i}∩ Ai+1 ⊂ [0, 1], only ˆθi and ˆθ_{i+1} will be updated
according to

.

ˆθi = μ_{F}_{i}(x)x ≥ 0 (32)

.

ˆθi+1 = μ_{F}_{i+1}(x)x ≥ 0 (33)
and ˆθj will be kept fixed for j6= i and j 6= i + 1. On
the contrary, if x∈ Ai∩ Ai+1⊂ [−1, 0], only ˆθi and
ˆθ_{i+1} will be updated according to

.

ˆθi = μ_{F}_{i}(x)x ≤ 0 (34)

.

ˆθi+1 = μ_{F}_{i+1}(x)x ≤ 0 (35)
and ˆθj will be kept fixed for j 6= i and j 6= i +
1. Similarly, Since the initial guest of ˆθ is chosen as
ˆθ(0) = 0, equations (34)-(33) imply that if A_{i}⊂ [0, 1]

(Ai ⊂ [−1, 0]), then ˆθi ≥ 0 and ˆθi(t) is monotone
increasing (ˆθ_{i}(t) ≤ 0 and ˆθi(t) is monotone decreasing
). In summary, if Ai ⊂ [−1, 0] or Ai ⊂ [0, 1], then

¯¯

¯ˆθ^{i}(t)¯¯¯ is monotone increasing.

(ii) If there is a time t0such that x(t0) ∈ Ai⊂ [0, 1],
then, due to the continuity of the trajectory while using
the adaptive control law, there is an interval (ta, tb)
with t0∈ (t^{a}, tb) such that x(t) > 0 and μ_{F}_{i}(x(t)) >

0 for t_{0} ∈ (ta, t_{b}) . By (32), we have

.

ˆθ_{i}(t) > 0 for
t_{0} ∈ (ta, t_{b}) and thus ˆθ_{i}(t) > 0 for t ≥ t0. Proof of
the similar case for A_{i} ⊂ [−1, 0] is omitted.

(iii) If x(t)∈ [1−h, 1], then x(t) ∈ A4∩A5⊂ [0, 1]

and

ˆθ^{T}(t)ξ(x(t)) = μ_{F}_{4}(x(t))ˆθ_{4}(t) + μ_{F}_{5}(x)ˆθ_{5}(t) ≥ 0
On the other hand, x(t)∈ [−1, −(1 − h)], then x(t) ∈
A_{1}∩ A2⊂ [−1, 0] and

ˆθ^{T}(t)ξ(x(t)) = μ_{F}_{1}(x(t))ˆθ_{1}(t) + μ_{F}_{2}(x)ˆθ_{2}(t) ≤ 0

This completes the proof. ¥

*Remark 1: Suppose that the membership functions*
are specified as shown in Fig. 2. Then, according to
Lemma 3, we have both ˆθ1(t) and ˆθ2(t) are of non-
positive values and monotone decreasing. On the other
hand, ˆθ_{4}(t) and ˆθ_{5}(t) are of non-negative values and
monotone increasing.

*Lemma 4: The response x(t) of the hysteresis*
switching robust adaptive control defined as in (9),
**(10), (??), and (??) is symmetric in the sense that if**
n

x(t), ˆθ(t)o and©

y(t), ˇθ(t)ª

are the system responses corresponding to the initial states x(0) and −x(0), respectively, then y(t) = −x(t) and ˇθ(t) = −¯θ(t) where ¯θ(t) =swap(ˆθ(t)).

**Proof : (i) First, we shall show that the response of**
the adaptive VSS control system is symmetric. Since
both f (x) and (ˆc1|x| + ˆc^{2}|x|^{2}+ r |x|)sign(x) are odd
functions of x, so is their sum. Let y(t) =−x(t). Then,
by multiplying -1 to both sides of the first equation in
(30), we have

− ˙x = −f(x) + (ˆc^{1}|x| + ˆc^{2}|x|^{2}+ r |x|)sign(x)

= f (−x) − (ˆc^{1}|x| + ˆc^{2}|x|^{2}+ r |x|)sign(−x)
where we have used the property that f (x) is an odd
function, i.e., f (−x) = −f(x). The above differential
equation implies that y(t) = −x(t) is the solution
to the closed-loop dynamics defined as, with y(0) =

−x(0)

⎧⎨

⎩

˙y = f (y) − (ˆc^{1}|y| + ˆc^{2}|y|^{2}+ r |y|)sign(y)

˙c_{1}= Γ_{1}|x|^{2}, c_{1}(0) = 0

˙c2= Γ1|x|^{3}, c2(0) = 0

(36)
Note that the trajectories c1(t) and c_{2}(t) are the same
for the systems in (30) and (36).

(ii) Next, we shall show that the response of the adaptive fuzzy system defined in (31) is symmetric.

Now define ¯θ =swap(ˆθ) and ¯ξ =swap(ξ). Then it follows that

¯θ^{T}¯ξ(x) = ˆθ^{T}ξ(x)

Note that due to the symmetric structure of the fuzzy sets in the rule base, we have

¯ξ(x) = ξ(−x)

Let y(t) = −x(t) and ˇθ = −¯θ. Then, by multiplying -1 to both sides of the first equation in (31), we get

− ˙x = −f(x) + ˆθ^{T}ξ(x)

= f (−x) + ˆθ^{T}ξ(x)
which implies

˙y = f (y) + ¯θ^{T}¯ξ(x)

= f (y) + ¯θ^{T}ξ(−x)

= f (y) − ˇθ^{T}ξ(y)

Similarly, by the swapping operation and the symmet- ric property of the vector ξ(x) in (24), the second

equation in (31) can be expressed as

¯.

θ = ¯ξ(x)(x) = ξ(−x)(x)

Now multiplying -1 to both sides of the last equation, one has

−

¯.

θ = ξ(−x)(−x) = ξ(y)(y)

and by the definition of the vector ˇθ, it can be con-

cluded that _{.}

ˇθ = ξ(y)(y)

Therefore, the responses y(t) = −x(t) and ˇθ = −¯θ consist of the solution to the closed-loop dynamics of the adaptive fuzzy control system in (31)

( ˙y = f (y) − ˇθ_{.} ^{T}ξ(y), y(0) = −x(0)

ˇθ = ξ(y)(y), ˇθ(0) = 0 (37) (iii) Finally, we note that the switching mechanism defined in (9) and (10) is symmetric to the origin x = 0.

By combining the results in parts (i), (ii), and (iii), the assertion can be concluded. This completes the

proof. ¥

Due to the symmetry of the responses of the switch-
ing control system as described in Lemma 4, we shall
assume x(0) > 0 in the analysis of the dynamics of
the switching control system. If x(0) > 1− h, then
the adaptive VSS control law in (30) will ensure that
there is a finite time t1 such that x(t1) = 1 − h and
x(t) > 1 − h for t ∈ [0, t1). At t = t_{1}, the adaptive
fuzzy control law in (31) will then be applied.

*A. Analysis of switching behavior*

In this section, we shall focus on discuss switching behavior of the switching control law at the boundaries of the hysteresis zone Ωh= [1 − h, 1]∪[−1, −(1−h)].

For further analysis, we shall need some definitions.

*Definition 1: We say that continuous switching of*
*N times at the positive boundary x = 1 − h happens*
at t = ti for 1 ≤ i ≤ N with t^{i} < ti+1 if there are
finite time instants tN +1 and ¯ti with ti < ¯ti < ti+1

for 1 ≤ i ≤ N such that (i) the adaptive VSS
controller is applied in (t0, t1) for some t0 < t1, (ii)
the adaptive fuzzy controller is used for t ∈ [ti, ¯t_{i}]
with x(t)∈ [1 − h, 1] for 1 ≤ i ≤ N, (iii) the adaptive
VSS controller is applied within the interval (¯t_{i}, t_{i+1})
for 1≤ i ≤ N, and (iv) the adaptive fuzzy control law
is used after t = tN +1such that there is no time instant
t¯_{N +1}such that{x(t) |tN +1≤ t ≤ ¯tN +1} ⊂ [1 − h, 1],
x(¯tN +1) = 1, and the adaptive VSS control law is
applied after t = ¯tN +1. For the above situation, we
also say that continuous switching of N times at the
positive boundary x = 1− h happens since t = t^{1}. If
N = 1, we say a switching at the positive boundary
x = 1 − h happens at t = t^{1}. Similarly, we may
let N → ∞, in this case we shall say continuous
switching of infinite times at the positive boundary
x = 1 − h happens at t = t^{i} for 1 ≤ i < ∞ with

ti < ti+1 or continuous switching of infinite times at
the positive boundary x = 1− h happens since t = t^{1}.
By the switching mechanism defined in (9) and (10),
if a continuous switching of N times at the positive
boundary x = 1− h happens at t = t^{i} for 1≤ i ≤ N,
then we should have x(ti) = 1 − h, x(¯t^{i}) = 1, and
x(t) ∈ [1 − h, 1] for t ∈ [t^{i}, ¯ti] for 1 ≤ i ≤ N.

*Definition 2: We say that continuous switching of*
N times at the positive boundary x = −(1 − h)
happens at t = ti for 1 ≤ i ≤ N with t^{i} < ti+1

if there are finite time instants tN +1 and ¯ti with
ti < ¯ti < ti+1 for 1 ≤ i ≤ N such that (i)
the adaptive VSS control is applied in (t0, t_{1}) for
some t0 < t_{1}, (ii) the adaptive fuzzy control law is
used for t ∈ [ti, ¯t_{i}] with x(t) ∈ [−1, −(1 − h)] for
1 ≤ i ≤ N, (iii) the adaptive VSS control law is
applied within the interval (¯t_{i}, t_{i+1}) for 1 ≤ i ≤ N,
and (iv) the adaptive fuzzy control law is used after
t = t_{N +1} such that there is no time instant ¯t_{N +1}
such that{x(t) |t^{N +1}≤ t ≤ ¯t^{N +1}} ⊂ [−1, −(1 − h)],
x(¯tN +1) = −1, and the adaptive VSS control law is
applied after t = ¯tN +1. For the above situation, we
also say that continuous switching of N times at the
positive boundary x =−(1 − h) happens since t = t^{1}.
If N = 1, we say a switching at the positive boundary
x = −(1 − h) happens at t = t^{1}. Similarly, we may
let N → ∞, in this case we shall say continuous
switching of infinite times at the positive boundary
x = −(1 − h) happens at t = ti for 1≤ i < ∞ with
t_{i} < t_{i+1} or continuous switching of infinite times at
the positive boundary x = −(1 − h) happens since
t = t_{1}.

*Definition 3: We say that a switch at the positive*
boundary x = 1− h (at the negative boundary x =

−(1 − h)) happens N times at t = ti for 1≤ i ≤ N
with ti < t_{i+1} if a switching at the positive boundary
x = 1 − h (or at the negative boundary x = −(1 −
h)) happens at t = ti for 1 ≤ i ≤ N. Similarly, as
N → ∞, we shall say a switch at the positive boundary
x = 1 − h (or at the negative boundary x = −(1 − h))
happens infinite times at t = ti for 1≤ i < ∞ with
ti< ti+1or a switch at the positive boundary x = 1−h
(or at the negative boundary x =−(1 − h)) happens
infinite times since t = t1.

*Definition 4: We say that there is no switching at*
the boundary x = 1− h (or at the boundary x =

−(1−h)) happened since t = t^{1}if (i) the adaptive VSS
control is applied in (t0, t1) for some t0< t1, (ii) the
adaptive fuzzy control law is applied after t = t1, and
(iii) there is not a switching at the boundary x = 1− h
(or at the boundary x =−(1 − h)) happens at t = t^{0}1

fro any t^{0}_{1}≥ t1.

Since the system response is symmetric as stated in Lemma 4, we shall focus on analyzing the switching property at the boundary x = 1− h.

*Lemma 5: Assume that there is no switching hap-*
pened at the boundary x = 1− h since t = t^{1}. Then
it is impossible that {x(t) |t ≥ t^{1}} ⊂ [1 − h, 1] .

**Proof: By contradiction, assume that we have**
x(t1) = 1−h and the adaptive fuzzy control law in (31)
is applied for t≥ t^{1} and{x(t) |t ≥ t^{1}} ⊂ [1 − h, 1] .
From Lemma 3, we have ˆθ4(t1) ≥ 0 and ˆθ^{5}(t1) ≥ 0.

Note also that for x(t) ∈ [1 − h, 1], we have 0 ≤
μ_{F}_{4}(x(t)) ≤^{1}2 and ^{1}_{2} ≤ μF5(1 − h) ≤ μF5(x(t)) ≤ 1.

Then, by the tuning law of ˆθ, we have ˆθ4(t) = ˆθ4(t1) +

Z t t1

μ_{F}_{4}(x(τ ))x(τ )dτ ≥ 0 (38)
and

ˆθ5(t) = ˆθ5(t1) + Z t

t1

μ_{F}_{5}(x(τ ))x(τ )dτ

≥ (t − t^{1}) μ_{F}_{5}(1 − h)(1 − h) (39)
With (38) and (39), the value of ˆθ^{T}(t)ξ(x(t)) can be
evaluated as

ˆθ^{T}(t)ξ(x(t)) = ˆθ4(t)μ_{F}_{4}(x(t)) + ˆθ5(t)μ_{F}_{5}(x(t))

≥ (t − t1) (1 − h)μ^{2}F5(1 − h)
Now define a constant tM to satisfy

tM > 2fmax

(1 − h)μ^{2}F5(1 − h) (40)
Then from the first equation of the closed-loop dynam-
ics of the adaptive fuzzy system in (31), we have

x(t1+ tM)

≤ 1 − h + f^{max}tM−1

2(1 − h)μ^{2}F5(1 − h)t^{2}M

By the definition of tM given in (40), we lead to a contradiction that

x(t_{1}+ t_{M}) < 1 − h

Therefore, the assumed situation is impossible. ¥ In the following, we shall first investigate the learn- ing capability of the adaptive tuning law when a switching at the boundary happens.

*Lemma 6: Suppose that a switching at the boundary*
x = 1 − h happens at t = t1 and the adaptive fuzzy
control is applied during the interval [t1, ¯t1]. Let t2

be the next time when the adaptive fuzzy control law
is applied. Then, the time difference ¯t1− t^{1} can be
estimated as

¯t_{1}− t1≥ h
fmax

(41) Moreover, we have

F (x(t_{2}), ˆθ(t_{2}))

≥ F (x(t^{1}), ˆθ(t1)) + ∆F (42)
where

∆F =1 2

h(1 − h) fmax

Similarly, if a switching at the boundary x =−(1 −h) happens at t = t1, then we have

F (x(t2), ˆθ(t2)) ≤ F (x(t^{1}), ˆθ(t1)) − ∆^{F} (43)

**Proof: Since a switching at the boundary x = 1**−h
happens at t = t1, by Definition 1, we have x(t2) =
x(t1) = 1 − h, x(¯t^{1}) = 1, x(t) ∈ [1 − h, 1] for t ∈
[t1, ¯t1], and the adaptive control law is applied during
the interval [t1, ¯t_{1}]. From the switching control law
defined in (30) and (31), we see that the parameter
vector ˆθ(t) is frozen during the interval (¯t_{1}, t_{2}) and
thus according to the second equation of (31), we have

ˆθ(t2) = ˆθ(¯t1) = ˆθ(t1) +
Z t^{¯}1

t1

ξ(x(τ ))x(τ )d(τ ) (44) The value of the fuzzy approximator F (x(t), ˆθ(t)) evaluated at t = t2 is given by

F (x(t_{2}), ˆθ(t_{2})) = ξ^{T}(x(t_{2}))ˆθ(t_{2}) = ξ^{T}(x(t_{1}))ˆθ(¯t_{1})
Now, by applying (44), we get

F (x(t_{2}), ˆθ(t_{2}))

= ξ^{T}(x(t1))

"

ˆθ(t1) +
Z t^{¯}1

t1

ξ(x(τ ))x(τ )d(τ )

#

= F (x(t1), ˆθ(t1)) +

Z ^{¯}t1

t1

ξ^{T}(t1) [ξ(x(τ )) − ξ(x(t^{1}))] x(τ )d(τ )
+ξ^{T}(x(t1))ξ(x(t1))

Z ^{¯}t1

t1

x(τ )d(τ ) (45)
Since x(t) ∈ [1 − h, 1] for t ∈ [t1, ¯t_{1}], from the
membership functions defined in Fig. 2, we have

ξ(x) =£

0 0 0 2 − 2x 2x − 1 ¤T

and

ξ(x(τ )) − ξ(x(t1))

= £

0 0 0 −2 2 ¤T

(x(τ ) − x(t^{1}))
for t∈ [t^{1}, ¯t1] . Note that x(t1) = 1 − h. Therefore, it
follows from inequality (27) that

ξ^{T}(x(t1)) [ξ(x(τ )) − ξ(x(t^{1}))]

= [8(1 − h) − 6] (x(τ) − x(t^{1}))

≥ 0

where the fact x(τ )− x(t1) = x(τ ) − (1 − h) ≥ 0 for
t ∈ [t1, ¯t_{1}] has been used. Using (45) and (23), we can
obtain

F (x(t_{2}), ˆθ(t_{2}))

≥ F (x(t^{1}), ˆθ(t1)) + ξ^{T}(x(t1))ξ(x(t1))
Z t^{¯}1

t1

x(τ )d(τ )

≥ F (x(t1), ˆθ(t_{1})) +1

2(1 − h)(¯t1− t1)

Since x(t)∈ [1 − h, 1] for t ∈ [t^{1}, ¯t1] , following from
Lemma 3, we have

˙x(t) = f (x(t)) − ˆθ^{T}(t)ξ(x(t)) ≤ f^{max}

Therefore, with x(t1) = 1 − h and x(¯t^{1}) = 1, we have

¯t_{1}− t1≥ h
fmax

This shows inequality (42). Analysis of the case x =

−(1 − h) is omitted. ¥

*Remark 2: We shall note that when the adaptive*
control law is applied beginning from t = t1, we have

˙x(t1) = f (x(t1)) − ˆθ^{T}(t1)ξ(x(t1))

= f (1 − h) − ˆθ^{T}(t1)ξ(x(t1))
Therefore, if f (1 − h) is greatly larger than
ˆθ^{T}(t1)ξ(x(t1)), then a switch of control law at x =
1 − h may happens due to instability of the adaptive
control system under the current parameter setting. To
avoid continuous switching behavior at x = 1− h, the
only way is to increase the value of F (x(t), ˆθ(t)) =
ˆθ^{T}(t)ξ(x(t)). Lemma 6 shows that if the adaptive
control law is applied beginning from t = t2 just after
a switch of control law at x = 1−h happened at t = t^{1},
then with the learning capability of ˆθ^{T}(t), ˙x(t1) will
be decreased by an amount ∆F as indicated in (42) so
that the adaptive control system at t = t2 has a better
chance to avoid continuous switching at the boundary
x = 1 − h.

*Remark 3: Since we shall spend at least* _{f}_{max}^{h} time
length to complete a switching, this implies that if
continuous switching of infinite times at the positive
boundary x = 1− h happens since t = t^{1}, then such
a infinite-times switching can not be completed in a
finite interval.

*Lemma 7: Suppose that a switching at the boundary*
x = 1 − h happens at t = t1 and the adaptive fuzzy
control is applied during the interval [t1, ¯t_{1}]. In this
case, ˆθ_{5}(t)− ˆθ4(t) is a monotone increasing function
of t in the intervals [t1, ¯t_{1}] . Particularly, at t = ¯t_{1}, we
have

ˆθ_{5}(¯t_{1})−ˆθ4(¯t_{1}) ≥ ˆθ5(t_{1})−ˆθ4(t_{1})+4ε_{h}(1 − h)h
f_{max} (46)
Moreover, if ˆθ_{5}(t_{1}) ≥ ˆθ4(t_{1}), then ˆθ_{5}(t) ≥ ˆθ4(t) for
t ∈ [t1, ¯t_{1}] and

F (ξ(x(t)), ˆθ(t)) ≥ F (ξ(x(t^{1})), ˆθ(t1) (47)
for t ∈ [t^{1}, ¯t1]. In addition, if x(t) is also non-
decreasing, then F (ξ(x(t)), ˆθ(t)) is a monotone in-
creasing function of t in the intervals [t1, ¯t1] .

**Proof: For t**∈ [t1, ¯t_{1}], we have
ˆθ5(t) = ˆθ5(t1) +

Z t t1

μ_{F}_{5}(x(τ ))x(τ )dτ

= ˆθ_{5}(t_{1}) +
Z t

ti

[2x(τ ) − 1] x(τ)dτ (48)

and

ˆθ_{4}(t) = ˆθ_{4}(t_{1}) +
Z t

t1

μ_{F}_{4}(x(τ ))x(τ )dτ

= ˆθ_{4}(t_{1}) +
Z t

t1

[2 − 2x(τ)] x(τ)dτ (49)

Therefore, the difference ˆθ5(t)−ˆθ^{4}(t) can be evaluated
as

ˆθ5(t)−ˆθ^{4}(t) = ˆθ5(t1)−ˆθ^{4}(t1)+

Z t ti

[4x(τ ) − 3] x(τ)dτ
(50)
Since x(t) ∈ [1 − h, 1] for t ∈ [t^{1}, ¯t1] and 0 < h ≤

1

4− ε^{h} as defined in (27), the integrand of the integral
in (50) is greater than zero. Therefore, ˆθ_{5}(t)−ˆθ4(t) is a
monotone increasing function for t∈ [t1, ¯t_{1}]. Actually,
a lower bound of ˆθ_{5}(t) − ˆθ4(t) can be evaluated as

ˆθ5(t) − ˆθ^{4}(t) ≥ ˆθ^{5}(t1) − ˆθ^{4}(t1) + 4εh(1 − h)(t − t^{1})
(51)
Particularly, at t = ¯t1, with (41), one can lead to

ˆθ5(¯t1) − ˆθ^{4}(¯t1)

≥ ˆθ5(t_{1}) − ˆθ4(t_{1}) + 4ε_{h}(1 − h)(¯t1− t1)

≥ ˆθ^{5}(t1) − ˆθ^{4}(t1) +4εh(1 − h)h
f_{max}

Moreover, from (51), it follows that if ˆθ5(t1) ≥ ˆθ^{4}(t1),
then ˆθ5(t) ≥ ˆθ^{4}(t) for any t ∈ [t^{1}, ¯t1] .

Now we show that F (ξ(x(t)), ˆθ(t)) is a monotone
increasing function of t in the intervals [t1, ¯t_{1}] . Using
**the definitions of the membership functions in (??), we**
can get

ˆθ^{T}(t)ξ(x(t))

= [2x(t) − 1]h

ˆθ_{5}(t) − ˆθ4(t)i
+ ˆθ_{4}(t)
Since both ˆθ5(t) −ˆθ^{4}(t) as well as ˆθ4(t) are monotone
increasing under the assumption ˆθ5(t1) ≥ ˆθ^{4}(t1) and
x(t) ≥ x(t^{1}) = 1 − h, it follows that inequality (47)
holds. In addition, if x(t) is also non-decreasing, then
F (ξ(x(t)), ˆθ(t)) is a monotone increasing function of
t in the intervals [t1, ¯t1] . This completes the proof.¥
*Lemma 8: Assume that a continuous switching of*
infinite times at the positive boundary x = 1 − h
happens since a finite time t = t1. Then there is an
index I1 such that ˆθ5(t) ≥ ˆθ^{4}(t) for t ≥ t^{I}1 .

**Proof: Assume that there are two time sequences**
{t^{i}}^{∞}i=1 and {¯t^{i}}^{∞}i=1 with ti < ¯ti < ti+1 such that
the adaptive fuzzy control law is used for t ∈ [t^{i}, ¯ti]
with x(t) ∈ [1 − h, 1]. It is noted that x(t^{i}) = 1 − h
and x(¯ti) = 1 for any i. While keeping switching at
the boundary x = 1− h, we shall repeatedly use the
result in Lemma 7 and we shall show that there is
a time instant tI1 such that ˆθ5(tI1) ≥ ˆθ^{4}(tI1). In the
following, we shall identify ¯t0as t1. By using (46) for

t ∈ [t^{i}, ¯ti], we have
ˆθ5(¯ti) − ˆθ^{4}(¯ti)

≥ ˆθ5(t_{i}) − ˆθ4(t_{i}) +4ε_{h}(1 − h)h
fmax

= ˆθ_{5}(¯t_{i}_{−1}) − ˆθ4(¯t_{i}_{−1}) +4εh(1 − h)h
f_{max} (52)
for i≥ 1. By using the recursive inequality (52), one
can get

ˆθ_{5}(t_{N +1}) − ˆθ4(t_{N +1})

= ˆθ5(¯tN) − ˆθ^{4}(¯tN)

≥ ˆθ5(t_{1}) − ˆθ4(t_{1}) + N 4ε_{h}(1 − h) h
fmax

(53) Therefore, if we choose

N =

⎡

⎢⎢

⎢ max³

0, ˆθ4(t1) − ˆθ^{5}(t1)´
fmax

4εh(1 − h)h

⎤

⎥⎥

⎥
wheredxe is the smallest integer with x ≤ dxe, then,
following from (53), ˆθ5(tI1) ≥ ˆθ^{4}(tI1) where we
define I1 = N + 1. Consequentially, from (52), we
ˆθ5(ti) ≥ ˆθ^{4}(ti) for i ≥ I^{1}. Therefore, by repeatedly
using Lemma 7, we can conclude that ˆθ5(t) ≥ ˆθ^{4}(t)
for t≥ t^{I}1. Note that if ˆθ4(t1) ≤ ˆθ^{5}(t1), then I1= 1.

*Lemma 9: It is impossible that a continuous switch-*
ing of infinite times at the positive boundary x = 1−h
happens since a finite time t = t1.

**Proof: By contradiction, assume that a continuous**
switching of infinite times at the positive boundary x =
1 − h happens since a time instant t = t^{1}. By Lemma
8, there is an index I1 such that ˆθ5(ti) ≥ ˆθ^{4}(ti) for
i ≥ I^{1}. Then, by inequality (47) in Lemma 7, we have
F (x(t), ˆθ(t)) = ˆθ^{T}(t)ξ(x(t)) ≥ F (x(t^{i}), ˆθ(ti)) for t ∈
[ti, ¯ti] with i ≥ I^{1}. Therefore, for any positive integer
N , we have

x(¯t_{N +I}_{1})

= x(t_{N +I}_{1}) +
Z t^{¯}_{N +I1}

t_{N+I1}

hf (x(τ )) − ˆθ^{T}(τ )ξ(x(τ ))i
dτ

≤ 1 − h +

Z ^{¯}t_{N+I1}
t_{N+I1}

hf_{max}− ˆθ^{T}(t_{N +I}_{1})ξ(x(t_{N +I}_{1}))i
dτ(54)
Now, following from inequality (42) in Lemma 6 and
inequality (41), for any positive integer N , we have

F (x(tN +I1), ˆθ(tN +I1))

≥ F (x(tI1), ˆθ(t_{I}_{1})) + N ∆_{F}

≥ N∆^{F}

and thus inequality (54) implies
x(¯t_{N +I}_{1})

≤ 1 − h +
Z ^{¯}t_{N+I1}

t_{N+I1}

[fmax− N∆^{F}] dτ (55)

= 1 − h + (fmax− N∆F) (¯t_{N +I}_{1}− tN +I1)
(56)

Suppose that we choose N such that N >

»fmax

∆F

¼

and thus fmax−N∆F < 0. Then, following from (41) and (56), we have

x(¯tN +I1) ≤ 1 − h + (f^{max}− N∆^{F}) h

fmax < 1 − h
However, the above inequality contradicts the assump-
tion that a continuous switching of infinite times at the
positive boundary x = 1−h happens since t = t^{1}. This

completes the proof. ¥

*Lemma 10: It is impossible that a switching at the*
positive boundary x = 1 − h (or at the negative
boundary x =−(1 − h)) happens infinite times since
any finite time t = t1.

**Proof: By contradiction, assume that a switch at**
the positive boundary x = 1− h (or at the negative
boundary x =−(1 − h)) happens infinite times at t =
t_{i} for 1 ≤ i < ∞ with ti < t_{i+1}. Denote a time
sequence ¯t_{i} for 1≤ i < ∞ with ti < ¯t_{i} < t_{i+1} such
that the adaptive fuzzy control law is applied in [ti, ¯t_{i}] .
Also define a time sequence ˇt_{i}for 1≤ i < ∞ such that
the adaptive VSS control law is applied in¡¯ti, ˇti¢

and
ti < ¯ti < ˇti ≤ t^{i+1} for 1 ≤ i < ∞. Then, since the
adaptive fuzzy control is applied in [ti, ¯ti], following
from Lemma 6, we have

ˆθ5(¯ti) − ˆθ^{5}(ti) =
Z t^{¯}i

ti

μ_{F}_{5}(x(τ ))x(τ )dτ

≥ μF5(1 − h)(1 − h) (¯t^{i}− t^{i})

≥ μF5(1 − h)(1 − h) h fmax

Since ˆθ_{5}(t) is monotone increasing, i.e., ˆθ_{5}(t_{i+1}) ≥
ˆθ_{5}(¯t_{i}) as t_{i+1}≥ ¯ti, we have

ˆθ_{5}(t_{i+1}) − ˆθ5(t_{i}) ≥ ˆθ5(¯t_{i}) − ˆθ5(t_{i})

≥ μF5(1 − h)(1 − h) h
f_{max}
Therefore, recursively using the above inequality, one
can obtain

ˆθ_{5}(t_{N +1}) ≥ ˆθ5(t_{1}) + N μ_{F}_{5}(1 − h)(1 − h) h
f_{max}
Since μ_{F}_{5}(x) is monotone increasing in the interval
[1 − h, 1], for t ∈ [t^{N +1}, ¯tN +1] , we have

ˆθ^{T}(t)ξ(x(t))

= ˆθ4(t)μ_{F}_{4}(x(t)) + ˆθ5(t)μ_{F}_{5}(x(t))

≥ ˆθ^{5}(t)μ_{F}_{5}(x(t))

≥ ˆθ^{5}(tN +1)μ_{F}_{5}(1 − h)

≥ μF5(1 − h)

∙

ˆθ5(t1) + (1 − h) h

f_{max}N μ_{F}_{5}(1 − h)

¸

Suppose that we choose N such that N >

⎡

⎢⎢

⎢

fmax

μ_{F5}(1−h) − ˆθ5(t_{1})
h(1 − h)μF5(1 − h)f_{max}

⎤

⎥⎥

⎥

and thus

fmax< μ_{F}_{5}(1−h)

∙

ˆθ5(t1) + N μ_{F}_{5}(1 − h)(1 − h) h
fmax

¸

Then one can lead to x(¯tN +1)

= x(t_{N +1}) +
Z ^{¯}tN+1

tN+1

hf (x(τ )) − ˆθ^{T}(τ )ξ(x(τ ))i
dτ

< 1 − h

However, the above inequality contradicts the assump- tion that a switching at the positive boundary x = 1−h happens infinite times since t = t1. This completes the

proof. ¥

*Lemma 11: Under the specified switching mecha-*
nism in (8)-(10) including the adaptive VSS control in
(30) and the adaptive fuzzy control in (31), we have
the following results.

(i) there is a finite time tf0 such that x(t)∈ Ω^{x}and
the adaptive fuzzy control is used for t ≥ t^{f}0

and

(ii) the parameters ˆc1(t) and ˆc2(t) in the adaptive VSS control are bounded for t∈ [0, ∞).

**Proof: (Part i) Suppose that x(0) > 1**− h. Then the
adaptive VSS control in (30) will ensure that there is
a finite time t1 such that x(t1) = 1 − h and x(t) >

1 − h for t ∈ [0, t^{1}) as stated in Lemma 2. At t = t1,
the adaptive fuzzy control law in (31) will then be
applied. When applying the adaptive fuzzy control law
at some time instant t = ts, the case that a continuous
switching of infinite times at the positive boundary x =
1 − h since t = ts has been excluded by Lemma 9.

Therefore, there are three possible cases at t = t_{s}.
(A1) There is no control law switching happened at

the boundary x = 1− h since t = ts, i.e., {x(t) |t ≥ ts} ⊂ Ωx= [−1, 1].

(A2) There is no control law switching happened at
the boundary x = 1− h at t = t^{s} and there are
switching operations after t = ts.

(A3) A continuous switching of finite times happens
at the positive boundary x = 1− h since t = t^{s}.
The above three cases are also applied to the situ-
ation that x(0) < −(1 − h). If |x(0)| ≤ 1 − h, then
the adaptive fuzzy control will be applied at t = 0 and
there are two possible cases.

(B1) There is no switching of control laws for t≥ 0,
i.e.,{x(t) |t ≥ 0 } ⊂ Ω^{x}= [−1, 1].

(B2) There are switching operations after t = 0.

Combining the situations (A1)-(A3) and (B1)-(B2), we can conclude that there are two possibilities for the trajectory {x(t), t ≥ 0} .

(C1) There is a finite time tf0such that x(t)∈ Ω^{x}and
the adaptive fuzzy control is used for t≥ t^{f}0.
(C2) A switching of control law at either x = 1− h

or x =−(1 − h) happens infinite times.

Note that Cases (C1) and (C2) are mutually exclu- sive, since the time to complete a switching is greater