行政院國家科學委員會專題研究計畫成果報告

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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

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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 adaptive fuzzy control systems arises from system uncertainty 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 adaptive 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 adaptive 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

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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 follows. 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 x₂ · · · 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∈ R¹ 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)

uVSS

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)

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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^∗₁and 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 ˆc₁ and ˆc₂, the proposed adaptive VSS control law will be defined as uV SS= −(ˆc¹|x| + ˆc²|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 2x²+1

2Γ⁻¹₁ ˜c²₁+1 2Γ⁻¹₁ ˜c²₂ where

˜

c1 = ˆc1− c^∗1

˜

c₂ = ˆc₂− c^∗2

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

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

−k¹V + k2 where k1 > 0 and k2 ≥ 0 are bounded constants, then

V (t, x) ≤ k2

k₁ + (V (0) −k2

k₁)e^−k¹^t for all t. Also it is obvious that

tlim→∞V (t, x) ≤ k2

k₁

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 t⁰ 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+ T₁ and

T1= Va(t0)

r(1 − h)² (14)

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

.

Va

= x[f (x) + u_{V SS}] + ˜c₁|x|²+ ˜c₂|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+ ˜c₂|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 ˆc₂(t) are non-decreasing functions of t. Therefore ˆ

c₁(t) and ˆc₂(t) both converge to some finite values as t → ∞. On the other hand, we have

..

ˆ

c1= 2Γ1xh

f (x) − (ˆc¹|x| + ˆc²|x|²+ r |x|)sign(x)i which is bounded, and we have ˙c1 is uniformly continuous. Then, Barbalat’s lemma [9], we have

tlim→∞˙c1(t) = 0

Consequentially, with ˙c1= Γ1|x|²defined 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(t₀) +

Z t t0

[−Z(τ) − r |x|²(τ )]dτ

= V_a(t₀) − r Z t

t0

|x|²(τ )dτ − Z t

t0

Z(τ )dτ which implies

1/2x²≤ 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

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−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 equation

dy/dt = x²(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−t⁰⁾ r )Va(t0) and thus

Z t t0

x²(τ )dτ ≤ (1 − e^−2r(t−t⁰⁾

r )Va(t0) <V_a(t₀) r

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

Z t0+T1

t0

x²(τ )dτ > (1 − h)²T₁= V_a(t₀) 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 t₁− 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 system, 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) = [ξ₁(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_F₁(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)k²= μ²_F_i(x) + μ²_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)k²≤ 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= ∪^2Li=1⁻²Ωx,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

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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, θ²) = ξ^T(x)(θ1− θ²) (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^Tξ(x) (29)

V. A HYSTERESISSWITCHINGROBUSTFUZZY

ADAPTIVECONTROL

Based on the adaptive VSS controller and the adaptive fuzzy controller, we shall study the proposed hysteresis 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^Tξ(x)

.

ˆθ = ξx, ˆθ(0) = 0

.

ˆ c1= 0

.

ˆ c₁= 0

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Note that the value of the fuzzy approximator is given by

F (x|ˆθ) = ˆθ^Tξ(x)

According to the tuning law of ˆθ 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 t₀ 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 Ai 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

¯¯

¯ˆθⁱ(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 t₀ ∈ (ta, t_b) . By (32), we have

.

ˆθ_i(t) > 0 for t₀ ∈ (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_F₄(x(t))ˆθ₄(t) + μ_F₅(x)ˆθ₅(t) ≥ 0 On the other hand, x(t)∈ [−1, −(1 − h)], then x(t) ∈ A₁∩ A2⊂ [−1, 0] and

ˆθ^T(t)ξ(x(t)) = μ_F₁(x(t))ˆθ₁(t) + μ_F₂(x)ˆθ₂(t) ≤ 0

This completes the proof. ¥

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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, ˆθ₄(t) and ˆθ₅(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| + ˆc²|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) + (ˆc¹|x| + ˆc²|x|²+ r |x|)sign(x)

= f (−x) − (ˆc¹|x| + ˆc²|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) − (ˆc¹|y| + ˆc²|y|²+ r |y|)sign(y)

˙c₁= Γ₁|x|², c₁(0) = 0

˙c2= Γ1|x|³, c2(0) = 0

(36) Note that the trajectories c1(t) and c₂(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^Tξ(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^Tξ(x)

= f (−x) + ˆθ^Tξ(x) which implies

˙y = f (y) + ¯θ^T¯ξ(x)

= f (y) + ¯θ^Tξ(−x)

= f (y) − ˇθ^Tξ(y)

Similarly, by the swapping operation and the symmetric 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_. ^Tξ(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 switching 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 = t₁, 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 tⁱ < 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 = t¹. If N = 1, we say a switching at the positive boundary x = 1 − h happens at t = t¹. 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

(8)

ti < ti+1 or continuous switching of infinite times at the positive boundary x = 1− h happens since t = t¹. 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 = tⁱ for 1≤ i ≤ N, then we should have x(ti) = 1 − h, x(¯tⁱ) = 1, and x(t) ∈ [1 − h, 1] for t ∈ [tⁱ, ¯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 tⁱ < 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, t₁) for some t0 < t₁, (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 = t¹. If N = 1, we say a switching at the positive boundary x = −(1 − h) happens at t = t¹. 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 = t₁.

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 = t¹if (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 = t⁰1

fro any t⁰₁≥ 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 = t¹. Then it is impossible that {x(t) |t ≥ t¹} ⊂ [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≥ t¹ and{x(t) |t ≥ t¹} ⊂ [1 − h, 1] . From Lemma 3, we have ˆθ4(t1) ≥ 0 and ˆθ⁵(t1) ≥ 0.

Note also that for x(t) ∈ [1 − h, 1], we have 0 ≤ μ_F₄(x(t)) ≤¹2 and ¹₂ ≤ μF5(1 − h) ≤ μF5(x(t)) ≤ 1.

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

Z t t1

μ_F₄(x(τ ))x(τ )dτ ≥ 0 (38) and

ˆθ5(t) = ˆθ5(t1) + Z t

t1

μ_F₅(x(τ ))x(τ )dτ

≥ (t − t¹) μ_F₅(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_F₄(x(t)) + ˆθ5(t)μ_F₅(x(t))

≥ (t − t1) (1 − h)μ²F5(1 − h) Now define a constant tM to satisfy

tM > 2fmax

(1 − h)μ²F5(1 − h) (40) Then from the first equation of the closed-loop dynamics of the adaptive fuzzy system in (31), we have

x(t1+ tM)

≤ 1 − h + f^maxtM−1

2(1 − h)μ²F5(1 − h)t²M

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

x(t₁+ t_M) < 1 − h

Therefore, the assumed situation is impossible. ¥ In the following, we shall first investigate the learning 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− t¹ can be estimated as

¯t₁− t1≥ h fmax

(41) Moreover, we have

F (x(t₂), ˆθ(t₂))

≥ F (x(t¹), 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(t¹), ˆθ(t1)) − ∆^F (43)

(9)

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(¯t¹) = 1, x(t) ∈ [1 − h, 1] for t ∈ [t1, ¯t1], and the adaptive control law is applied during the interval [t1, ¯t₁]. From the switching control law defined in (30) and (31), we see that the parameter vector ˆθ(t) is frozen during the interval (¯t₁, t₂) 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(t₂), ˆθ(t₂)) = ξ^T(x(t₂))ˆθ(t₂) = ξ^T(x(t₁))ˆθ(¯t₁) Now, by applying (44), we get

F (x(t₂), ˆθ(t₂))

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

ξ^T(x(t1)) [ξ(x(τ )) − ξ(x(t¹))]

= [8(1 − h) − 6] (x(τ) − x(t¹))

≥ 0

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

F (x(t₂), ˆθ(t₂))

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

t1

x(τ )d(τ )

≥ F (x(t1), ˆθ(t₁)) +1

2(1 − h)(¯t1− t1)

Since x(t)∈ [1 − h, 1] for t ∈ [t¹, ¯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(¯t¹) = 1, we have

¯t₁− 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 = t¹, 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 = t¹, 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, ¯t₁]. In this case, ˆθ₅(t)− ˆθ4(t) is a monotone increasing function of t in the intervals [t1, ¯t₁] . Particularly, at t = ¯t₁, we have

ˆθ₅(¯t₁)−ˆθ4(¯t₁) ≥ ˆθ5(t₁)−ˆθ4(t₁)+4ε_h(1 − h)h f_max (46) Moreover, if ˆθ₅(t₁) ≥ ˆθ4(t₁), then ˆθ₅(t) ≥ ˆθ4(t) for t ∈ [t1, ¯t₁] and

F (ξ(x(t)), ˆθ(t)) ≥ F (ξ(x(t¹)), ˆθ(t1) (47) for t ∈ [t¹, ¯t1]. 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] .

Proof: For t∈ [t1, ¯t₁], we have ˆθ5(t) = ˆθ5(t1) +

Z t t1

= ˆθ₅(t₁) + Z t

ti

[2x(τ ) − 1] x(τ)dτ (48)

(10)

and

ˆθ₄(t) = ˆθ₄(t₁) + Z t

t1

μ_F₄(x(τ ))x(τ )dτ

= ˆθ₄(t₁) + Z t

t1

[2 − 2x(τ)] x(τ)dτ (49)

Therefore, the difference ˆθ5(t)−ˆθ⁴(t) can be evaluated as

ˆθ5(t)−ˆθ⁴(t) = ˆθ5(t1)−ˆθ⁴(t1)+

Z t ti

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

1

4− ε^h as defined in (27), the integrand of the integral in (50) is greater than zero. Therefore, ˆθ₅(t)−ˆθ4(t) is a monotone increasing function for t∈ [t1, ¯t₁]. Actually, a lower bound of ˆθ₅(t) − ˆθ4(t) can be evaluated as

ˆθ5(t) − ˆθ⁴(t) ≥ ˆθ⁵(t1) − ˆθ⁴(t1) + 4εh(1 − h)(t − t¹) (51) Particularly, at t = ¯t1, with (41), one can lead to

ˆθ5(¯t1) − ˆθ⁴(¯t1)

≥ ˆθ5(t₁) − ˆθ4(t₁) + 4ε_h(1 − h)(¯t1− t1)

≥ ˆθ⁵(t1) − ˆθ⁴(t1) +4εh(1 − h)h f_max

Moreover, from (51), it follows that if ˆθ5(t1) ≥ ˆθ⁴(t1), then ˆθ5(t) ≥ ˆθ⁴(t) for any t ∈ [t¹, ¯t1] .

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

ˆθ^T(t)ξ(x(t))

= [2x(t) − 1]h

ˆθ₅(t) − ˆθ4(t)i + ˆθ₄(t) Since both ˆθ5(t) −ˆθ⁴(t) as well as ˆθ4(t) are monotone increasing under the assumption ˆθ5(t1) ≥ ˆθ⁴(t1) and x(t) ≥ x(t¹) = 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) ≥ ˆθ⁴(t) for t ≥ t^I1 .

Proof: Assume that there are two time sequences {tⁱ}^∞i=1 and {¯tⁱ}^∞i=1 with ti < ¯ti < ti+1 such that the adaptive fuzzy control law is used for t ∈ [tⁱ, ¯ti] with x(t) ∈ [1 − h, 1]. It is noted that x(tⁱ) = 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) ≥ ˆθ⁴(tI1). In the following, we shall identify ¯t0as t1. By using (46) for

t ∈ [tⁱ, ¯ti], we have ˆθ5(¯ti) − ˆθ⁴(¯ti)

≥ ˆθ5(t_i) − ˆθ4(t_i) +4ε_h(1 − h)h fmax

= ˆθ₅(¯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

ˆθ₅(t_{N +1}) − ˆθ4(t_{N +1})

= ˆθ5(¯tN) − ˆθ⁴(¯tN)

≥ ˆθ5(t₁) − ˆθ4(t₁) + N 4ε_h(1 − h) h fmax

(53) Therefore, if we choose

N =

⎡

⎢⎢

⎢ max³

0, ˆθ4(t1) − ˆθ⁵(t1)´ fmax

4εh(1 − h)h

⎤

⎥⎥

⎥ wheredxe is the smallest integer with x ≤ dxe, then, following from (53), ˆθ5(tI1) ≥ ˆθ⁴(tI1) where we define I1 = N + 1. Consequentially, from (52), we ˆθ5(ti) ≥ ˆθ⁴(ti) for i ≥ I¹. Therefore, by repeatedly using Lemma 7, we can conclude that ˆθ5(t) ≥ ˆθ⁴(t) for t≥ t^I1. Note that if ˆθ4(t1) ≤ ˆθ⁵(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 = t¹. By Lemma 8, there is an index I1 such that ˆθ5(ti) ≥ ˆθ⁴(ti) for i ≥ I¹. Then, by inequality (47) in Lemma 7, we have F (x(t), ˆθ(t)) = ˆθ^T(t)ξ(x(t)) ≥ F (x(tⁱ), ˆθ(ti)) for t ∈ [ti, ¯ti] with i ≥ I¹. 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_I₁)) + 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)

(11)

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 assumption that a continuous switching of infinite times at the positive boundary x = 1−h happens since t = t¹. 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_ifor 1≤ i < ∞ such that the adaptive VSS control law is applied in¡¯ti, ˇti¢

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

ˆθ5(¯ti) − ˆθ⁵(ti) = Z t^¯i

ti

≥ μF5(1 − h)(1 − h) (¯tⁱ− tⁱ)

≥ μF5(1 − h)(1 − h) h fmax

Since ˆθ₅(t) is monotone increasing, i.e., ˆθ₅(t_i+1) ≥ ˆθ₅(¯t_i) as t_i+1≥ ¯ti, we have

ˆθ₅(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

ˆθ₅(t_{N +1}) ≥ ˆθ5(t₁) + N μ_F₅(1 − h)(1 − h) h f_max Since μ_F₅(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_F₄(x(t)) + ˆθ5(t)μ_F₅(x(t))

≥ ˆθ⁵(t)μ_F₅(x(t))

≥ ˆθ⁵(tN +1)μ_F₅(1 − h)

≥ μF5(1 − h)

∙

ˆθ5(t1) + (1 − h) h

f_maxN μ_F₅(1 − h)

¸

Suppose that we choose N such that N >

⎡

⎢⎢

⎢

fmax

μ_F5(1−h) − ˆθ5(t₁) h(1 − h)μF5(1 − h)f_max

⎤

⎥⎥

⎥

and thus

fmax< μ_F₅(1−h)

∙

ˆθ5(t1) + N μ_F₅(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 assumption 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^xand the adaptive fuzzy control is used for t ≥ t^f0

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, t¹) 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 situation 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^xand the adaptive fuzzy control is used for t≥ t^f0. (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

行政院國家科學委員會專題研究計畫 成果報告