中 華 大 學

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中 華 大 學 碩 士 論 文

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Abstract

In the conventional adaptive control field, parameter tuning law based on σ−modification has been widely used to construct an adaptive controller. However, using the σ−modification parameter tuning law, usually only the UUB stability can be attained for the adaptive control system. In this study, we simplify the parameter tuning law and the adaptive VSS control law to construct the proposed adaptive VSS control systems for typical first-order and second-order systems. Even though the σ−modification term in the parameter tuning law and the stabilizing control term in the adaptive control law are omitted, we shall show that asymptotical stability of the state and boundedness of the parameter estimates can be guaranteed. Particularly, in the proposed adaptive control schemes, only the structural information of the plant is needed. Several simulation examples are given to verify the proposed adaptive VSS control algorithms.

Keywords: Adaptive Control , VSS Control

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Contents

1 Introduction 1

1.1 Literature survey and Motivation . . . 1

1.2 Organization of the Thesis . . . 2

2 Problem formulation and controller design for first-order systems 3 2.1 Problem formulation . . . 3

2.2 Basic tuning laws of ˆc1 and ˆc2 . . . 4

2.3 Tuning laws of ˆc1 and ˆc2 by letting σ = 0 . . . 11

2.4 Tuning laws of ˆc1 and ˆc2 by letting σ = 0 with g(x)=g(x)¯c . . . 16

3 Problem formulation and controller design for second-order systems 26 3.1 Case 1 . . . 26

4 Conclusion and Discussion 35

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List of Figures

2-1 Comparison of log(x(t)), log(ˆc1(t)), and log(ˆc2(t)) at the transcient stage

for 0 ≤ t ≤ 10−4 seconds. . . 7

2-2 The responses of x(t) in Example 1. . . 8

2-3 The responses of ˆc1(t) and ˆc2(t) in Example 1. . . 8

2-4 The control input u(t) in Example 1. . . 9

2-5 The mutually hunting phenomenon between ˆψ(x) and f (x).in Example1 . 9 2-6 The response of the Lyapunov function Va(t)in Example1 . . . 10

2-7 Zooming of the state x(t) at the transient stage in Example 2. . . 16

2-8 The response of x(t) in Example 2. . . 17

2-9 The control input u(t) in Example 2. . . 17

2-10 The responses of ˆc1(t) and ˆc2(t) in Example 2. . . 18

2-11 The response of the Lyapunov function Va(t)in Example 2 . . . 18

2-12 The state x(t) in Example 3. . . 23

2-13 The control input u(t) in Example 3. . . 24

2-14 Evolution of parameters ˆca(t) and ˆcb(t)in Example 3. . . 24

2-15 The Lyapunov function Va(t) in Example 3. . . 25

3-1 The responses of x1(t) and x2(t) in Example 4. . . 31

3-2 The response of s(t) in Example 4. . . 32

3-3 The control input u(t) in Example 4. . . 32

3-4 The responses of ˆci(t) for 1 ≤ i ≤ 5 in Example 4. . . 33

3-5 The response of the Lyapunov function Va(t)in Example 4. . . 33

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3-6 The responses of x1, x2 approach to the sliding surface. Example 4 . . . 34

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Chapter 1 Introduction

1.1 Literature survey and Motivation

The variable structure system (VSS) theory has been widely studied and used to controller design in the last two decades. This switching control theory was originally developed in Russia [1, 2]. It is a useful tool to deal with system uncertainties and external disturbance.

Earlier results of VSS control were based on state feedback assuming that the states are available[3, 4, 2]. Most VSS control methods with output feedback control used a state observer to design the switching controller [5, 6, 7, 8, 9]. VSS control using static output feedback was considered in [10, 11, 12]. In the literature [5, 6, 10, 8, 12, 11], the output VSS control is limited to those systems with relative degree one. The case that the system is of higher relative degree was attacked in [13, 7, 9]. In the presence of uncertainty and disturbance, the so called matching condition or invariance condition is assumed in order to attain the robustness of the VSS control system[6, 13, 2, 7, 9, 12, 14].

In the conventional adaptive control field, parameter tuning law based on σ−modification has been widely used to construct an adaptive controller. However, using the σ−modification parameter tuning law, usually only the UUB stability can be attained for the adaptive control system. In this study, we simplify the parameter tuning law and the adaptive VSS control law to construct the proposed adaptive VSS control systems for typical first-

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order and second-order systems. Even though the σ−modification term in the parameter tuning law and the stabilizing control term in the adaptive control law are omitted, we shall show that asymptotical stability of the state and boundedness of the parameter estimates can be guaranteed. Particularly, in the proposed adaptive control schemes, only the structural information of the plant is needed. In the study of adaptive control of second-order systems, the problem of mismatched uncertainty is also attacked. Several simulation examples are given to verify the proposed adaptive control algorithms.

1.2 Organization of the Thesis

The remainder of this thesis is organized as follows. In Chapter 2, adaptive VSS con- trol for typical first-order systems will be studied. In Section 2.1, problem formulation for adaptive VSS control of the first-order system is made. In Section 2.2, conventional adaptive VSS control based on the σ−modification parameter tuning law and the stan- dard VSS control design is reviewed. In Section 2.3, a simplified adaptive VSS control by omitting the σ−modification term in the parameter tuning law and the stabilizing control term in the adaptive control law is proposed and analyzed. In section 2.4, adaptive VSS control for a more complex first-order system is studied.

In Chapter 3, adaptive VSS control for typical second-order systems will be studied.

In Section 3.1, problem formulation for adaptive VSS control of the second-order system is made. In Section 3.2, a proposed adaptive VSS control by omitting the σ−modification term in the parameter tuning law is proposed and analyzed.

Conclusions and discussions are given in Chapter 4.

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Problem formulation and controller design for first-order systems

2.1 Problem formulation

Now we consider the plant

˙x = f (x) + u (2.1)

where f (x) is a scalar nonlinear continuous function of the scalar variable x and u ∈ R1 is the system input. An example of such a function f (x) is given by

f (x) = μx + x|x| (2.2)

for which the equilibrium point 0 is unstable. In this case, we assume that a least upper bound of f (x) satisfies

|f(x)| ≤ c1|x| + c2|x|2 = ψ(x) (2.3) where c1 and c2 are unknown positive parameters. For the example in (2.2), we can set

c1 = μ, c2 =

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We shall develop an adaptive VSS control uV SS(t) to ensure asymptotical stability of the closed-loop system. To attain this goal, we shall construct ˆc1 and ˆc2, which are the estimates of c1 and c2,so that

|f(x)| ≤ ˆc1|x| + ˆc2|x|2 for x /∈ Ωx

Based on the estimates ˆc1 and ˆc2, the adaptive VSS control law will be defined as

uV SS =−ˆψ(x)sign(x) (2.4)

where

ψ(x) = ˆˆ c1|x| + ˆc2|x|2+ r|x|

and r is a positive constant. In the following, we shall discuss diﬀerent tuning laws of the ˆc1, and ˆc2, and analyze the system stability.

1

and ˆ c

2

Here, we shall consider the following basic tuning laws of ˆc1 and ˆc2 as motivated by the works in [15]

.

ˆ

c1 = Γ1(|x|2− σ(ˆc1− c01)), (2.5)

.

ˆ

c2 = Γ2(|x|3− σ(ˆc2− c02)) (2.6)

where c01 and c02 are the initial guests of ˆc1 and ˆc2, respectively. To analyze the system behaviour of the adaptive VSS control system defined by (2.1), (2.4), (2.5), and (2.6), we consider the Lyapunov function candidate

Va= 1

2x2+ 1

−1121+1 2Γ−12 ˜c22

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where

˜

c1 = ˆc1− c1

˜

c2 = ˆc2− c2

Lemma 1 [15] 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−k1t

for all t. Also it is obvious that

t→∞limV (t, x)≤ k2

k1

Lemma 2 Consider the adaptive VSS control system defined by (2.1), (2.4), (2.5), and (2.6). The trajectories of x(t), ˆc1(t), and ˆc2(t) are bounded over the time interval (t0,∞).

Moreover, by choosing Γ1 = Γ2 = 2rσ, the steady-state value of x(t) obeys

kx(∞)k ≤ Γ

1 2

1

°°

°°

°°

c1− c01

c2− c02

°°

°°

°° (2.7)

Proof: By using the function ψ(x) in (2.3), the time derivative of Va along the system trajectory can be evaluated as

.

Va = xf (x)− ψ(x) |x| + (c1|x| + c2|x|2)|x| − (ˆc1|x| + ˆc2|x|2+ r|x|) |x|

−11 ˜c1 .

˜

c1+ Γ−12 ˜c2 .

˜ c2

Then, using inequality (2.3), and the tuning laws of ˆc1 and ˆc2 in (2.5) and (2.6), respec-

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tively, we can obtain

.

Va≤ −r |x|2− ˜c1σ(ˆc1− c01)− ˜c2σ(ˆc2− c02)

Furthermore, by letting ˆc1− c01 = ˆc1− c1+ c1− c01 and ˆc2− c02 = ˆc2− c2+ c2− c02,one can lead to

.

Va ≤ −r |x|2− σ˜c21− ˜c1σ(c1− c01)− σ˜c22− ˜c2σ(c2− c02) and by completing the square, we get

.

Va ≤ −r |x|2− σ

2c˜21− σ 2˜c22

2

¯¯c1− c01

¯¯2+σ 2

¯¯c2− c02

¯¯2

≤ −r |x|2− σ

2c˜21− σ 2˜c22

2

¯¯c1− c01

¯¯2+σ 2

¯¯c2− c02

¯¯2

Now choose Γ−11 = Γ−12 = 2rσ so that Γ1 = Γ2 = 2rσ. Then the above inequality can be reformed as

.

Va ≤ −2rVa+σ 2

°°

°°

°°

c1− c01

c2− c02

°°

°°

°°

2

which implies that the trajectories of x(t), ˆc1(t), and ˆc2(t) are bounded over the time inter- val [t0,∞). Furthermore, by invoking Lemma 1 and using the definition of the Lyapunov function candidate Va, it follows that

1

2kx(∞)k2 ≤ Va(∞) ≤

σ 2

°°

°°

°°

c1− c01

c2− c02

°°

°°

°°

2

2r = Γ−11 2

°°

°°

°°

c1− c01

c2− c02

°°

°°

°°

2

(2.8)

and thus inequality (2.7) can be concluded. ¥

In order to minimize kx(∞)k , ineqlities (2.8) and (2.7) suggest that we can decrease the value of σ or increase the amplitude of Γ1.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10−4

−5

−4

−3

−2

−1 0 1

t in sec.

log(x(t))

log(x(t)) log(c1(t)) log(c2(t))

Figure 2-1: Comparison of log(x(t)), log(ˆc1(t)), and log(ˆc2(t)) at the transcient stage for 0≤ t ≤ 10−4 seconds.

Example 1 In the example, we consider the case

f (x) = 4x + 4x|x|

and thus

c1 = 4, c2 = 4

We note that the plant in this example is a highly unstable system. Therefore, with the initial state x(0) = 5, the state x(t) usually grows very fast when there a suitable control is not applied.

Since the constant r in the control law u(t) is set as 2 and the constant σ is set as 0.01, the parameters Γ1 and Γ2 in the tuning law of ˆc1and ˆc2, respectively, are given as Γ1 = Γ2 = 2r/σ = 400. Therefore, the σ−modification laws in (2.5) and (2.6) become

.

ˆ

c1 = Γ1(|x|2− σ(ˆc1− c01)) = 400|x|2− 4ˆc1,

.

ˆ

c2 = Γ2(|x|3− σ(ˆc2− c02)) = 400|x|3− 4ˆc1

Simply speaking, ˆc1 and ˆc2 are the filtered versions of 100 |x|2 and 100 |x|3 with the filter

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0 1 2 3 4 5 6 7 8 9 10 0

1 2 3 4 5 6

t in sec.

x(t)

Figure 2-2: The responses of x(t) in Example 1.

0 1 2 3 4 5 6 7 8 9 10

0 5 10 15 20 25 30

t in sec.

c1(t)

0 1 2 3 4 5 6 7 8 9 10

0 20 40 60 80 100

t in sec.

c2(t)

Figure 2-3: The responses of ˆc1(t) and ˆc2(t) in Example 1.

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0 1 2 3 4 5 6 7 8 9 10

−1200

−1000

−800

−600

−400

−200 0

t in sec.

u(t)

Figure 2-4: The control input u(t) in Example 1.

0 1 2 3 4 5 6

−3

−2

−1 0 1 2 3 4

t in sec.

f(x) and ψ(t)

log(ψ(t)) log(f(x))

Figure 2-5: The mutually hunting phenomenon between ˆψ(x) and f (x).in Example1

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0 1 2 3 4 5 6 7 8 9 10 0

2 4 6 8 10 12 14

t in sec.

Va(t)

Figure 2-6: The response of the Lyapunov function Va(t) in Example1 being as s/4+11 , i.e.,

ˆ

c1 = 400

s + 4|x|2 = 1

s/4 + 1100|x|2 ˆ

c2 = 400

s + 4|x|3 = 1

s/4 + 1100|x|3

The above two equations mean ˆc1 and ˆc2 will follow 100 |x|2 and 100 |x|3with a lag time about 1 second.

Since the initial conditions are given as x1(0) = 5, ˆc1(0) = 0, and ˆc2(0) = 0, we have

˙x(0) > 0. However, the terms 400|x|2 and 400 |x|3 in the tuning laws of ˆc1(t) and ˆc2(t), respectively, are very large compared with f (x) so that the growing speed of ˆc1(t) and ˆ

c2(t) are much faster than that of x(t) as shown in Fig. 2-1 at the transient stage for 0≤ t ≤ 10−4 seconds. With the rapid growth of ˆc1(t) and ˆc2(t), the amplitude ˆψ(x) of the control signal u(t) will also has a rapid growth so as to stabilizes the system. Therefore, it will be very soon that ˙x(t) < 0 , and x(t) will decay very fast shown in Fig. 2-2.

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The amplitude ˆψ(x) of the control signal u(t) with will then also decay very fast as x(t) does as shown in Fig. 2-4.

When ˆc1(t) and ˆc2(t) are decreased so that f (x) > ˆψ(x), then x(t) begins to slowly increase when x(t) = 10−3 at t = 0.679 seconds. The increasing rate of x(t) at this stage is very low since x(t) is about the order of 10−3 at this stage. While x(t) is increasing and ˆc1(t) as well as ˆc2(t) are decreasing, from the tuning laws of ˆc1(t) and ˆc2(t) in (2.5) and (2.6), ˆc1(t) and ˆc2(t) will also starts to increase after some time. Now with the increasing of x(t), ˆc1(t) and ˆc2(t), ˆψ(x) will also stop decreasing and start to increase.

This results in ˆψ(x) > f (x) at about t = 3.76 seconds and x(t) begins to slowly decrease once again. This mutually hunting phenomenon between ˆψ(x) and f (x), which can be observed in Fig. 2-5, will last forever and f (x) − ˆψ(x) → 0 as t → ∞. However, only boundedness of x(t) can be observed as indicated in Lemma 2 and it is not necessarily that x(t) → 0 as t → ∞. We note that the value x(∞) depends on the initial value x(0). For example, if x(0) > 0, then x(∞) = 0.1605. On the other hand, if x(0) < 0, then x(∞) = −0.1605. The Lyapunov function Va(t) is actually monotone decreasing as shown in Fig. 2-6.

1

2

by letting σ = 0

As observed in the subsection 2.2, we now try to construct the tuning laws of ˆc1 and ˆc2

by letting σ = 0 to ensure that kx(t)k → ∞ as t → ∞. By setting σ = 0 in (2.5) and (2.6), the tuning laws of ˆc1 and ˆc2 becomes

˙c1 = Γ1x2, Γ1 > 0 (2.9)

˙c2 = Γ1|x|3 (2.10)

The adaptive VSS control law uV SS is given in (2.4). To analyze the system response when applying the adaptive VSS control law defined in (2.4), (2.9), and (2.10), we consider the

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Lyapunov function candidate

Va= 1

2x2+ 1

−1121+1 2Γ−11 ˜c22

Lemma 3 For the adaptive VSS control system defined by (2.1), (2.9), and (2.10) with the control law u(t) defined in (2.4) or , the trajectories of x(t), ˆc1(t), and ˆc2(t) are bounded over the time interval (t0,∞), and x(t) converges to the origin. Moreover, there is a finite time t1 such that x(t1) = 1− h for x(t0) > 1− h or x(t1) = −(1 − h) for x(t0) <−(1 − h) where t1 is a time instant with t1 ≤ t0 + T1 and

T1 = Va(t0)

r(1− h)2 (2.11)

Proof: By using the control law (2.4), the time derivative of Va along the adaptive control system trajectory can be evaluated as

.

Va = x[f (x) + uV SS] + ˜c1|x|2+ ˜c2|x|3

= xf (x)− ψ(x) |x| + (c1|x| + c2|x|2)|x|

−(ˆc1|x| + ˆc2|x|2+ r|x|) |x| + ˜c1|x|2+ ˜c2|x|3

≤ −r |x|2 ≤ 0 (2.12)

where ˜c1 = ˆc1− c1 and ˜c2 = ˆc2− c2. The above inequality implies that the trajectories of x(t), ˆc1(t), and ˆc2(t) are bounded over the time interval (t0,∞) and Va(t) is a non- increasing function of t. From (2.9) and (2.10), it is obvious that both ˆc1(t)and ˆc2(t) are non-decreasing functions of t. Therefore ˆc1(t)and ˆc2(t) both converge to finite values as t→ ∞. On the other hand, we have

..

ˆ

c1 = 2Γ1x£

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

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which shows that ..ˆc1is uniformly bounded over (t0,∞). This implies that ˙c1 is uniformly continuous. Then, by Barbalat’s lemma [16], we have

t→∞lim˙c1(t) = 0

Consequentially, with ˙c1 = Γ1|x|2 defined in (2.9), we can conclude

t→∞limx(t) = 0

Now consider the case that the control law (2.4) is applied. From (2.12), there exists a function Z(t) ≥ 0 such that

.

Va+ Z =−r |x|2 The solution to the above diﬀerential equation is given as

Va(t) = Va(t0) + Z t

t0

[−Z(τ) − r |x|2(τ )]dτ

= Va(t0)− r Z t

t0

|x|2(τ )dτ − Z t

t0

Z(τ )dτ

By the definition of Va(t)and the fact that Z(t) ≥ 0, the above equation implies

1/2x2 ≤ Va(t)≤ Va(t0)− r Z t

t0

|x|2(τ )dτ (2.13)

For further analysis, we define a signal y(t) as

y(t) = Z t

t0

|x|2(τ )dτ

which is equivalent to

dy/dt = x2(t), y(t0) = 0

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Then, from (2.13), we have

˙y + 2ry≤ 2Va(t0) and then by Lemma 1, we can conclude

y(t) ≤ 2Va(t0) Z t

t0

e−2r(t−τ)

= (1− e−2r(t−t0)

r )Va(t0)

and thus Z t

t0

x2(τ )dτ ≤ (1− e−2r(t−t0)

r )Va(t0) < Va(t0)

r (2.14)

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 T1 is defined in (2.11). 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 = Va(t0) r

which contradicts the inequality (2.14). This concludes the assertion. Similarly, if x(t0) <

−(1 − h), there is a finite time t1 such that x(t1) =−(1 − h) with t1− t0 ≤ T1. ¥ Remark 1 >From the proof of Lemma 3, it is obvious that if the adaptive VSS control law uV SS in (2.4) is now simplified as

uV SS =−ˆψ1(x)sign(x) (2.15)

where

ψˆ1(x) = ˆc1|x| + ˆc2|x|2

then we also have the results that the trajectories of x(t), ˆc1(t), and ˆc2(t) are bounded over the time interval (t ,∞), and x(t) converges to the origin.

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Example 2 In the example, we consider the case

f (x) = 4x + 4x|x|

and thus

c1 = 4, c2 = 4

We note that the plant in this example is a highly unstable system. Therefore, with the initial state x(0) = 5, the state x(t) usually grows very fast when there a suitable control is not applied. The parameter Γ in the tuning law is chosen as Γ = 1.

Since the initial conditions are given as x1(0) = 5, ˆc1(0) = 0, and ˆc2(0) = 0, we have f (x(0)) > ˆψ1(x(0)). Since ˙x(0) > 0, there is a very short time interval such that x(t) is increasing as shown in Fig. 2-7. However, the terms |x|2 and |x|3 in the tuning laws of ˆc1(t) and ˆc2(t), respectively, are large so that the growing speed of ˆc1(t) and ˆc2(t) will be very fast. With the rapid growth of ˆc1(t) and ˆc2(t), the amplitude ˆψ(x) of the control signal u(t) will also has a rapid growth so as to stabilizes the system. Therefore, it will be very soon that ˙x(t) = f (x) − ˆψ(x) < 0 , and x(t) will decay very fast. The evolution of the state x(t) and the control input u(t) are shown in Fig. 2-8 and Fig. 2-9, respectively.

However, as x(t) is decreasing, so is the increasing rate of ˆc1(t) and ˆc2(t) as shown in Fig.

2-3. The amplitude ˆψ(x) of the control signal u(t) will then also decay very fast as x(t) decreases as shown in Fig. 2-9. Since f (x) is a monotone increasing function of x, f (x) decreases as x(t) decrease. However, ˆc1(t) and ˆc2(t) are monotone increasing function of t, in spite of increasing or decreasing of x(t). In this example, we have f (x) − ˆψ(x) < 0 after x(t) starts to decrease. Therefore, x(t) is monotone decreasing and is convergent to zero at the steady state as shown in Fig. 2-8. Meanwhile, ˆc1(t) and ˆc2(t) converge to some finite values as shown in Fig. 2-3. In Fig. 2-11, it is also shown that the Lyapunov function Va(t) is actually monotone decreasing as shown .

We note that the tuning laws in (2.9) and (2.10) are simplified versions of (2.5) and (2.6), respectively, by setting σ = 0. With by σ = 0, ˆc1(t) and ˆc2(t) are monotone

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0

1 2 3 4 5 6 7

t in sec.

x(t)

Figure 2-7: Zooming of the state x(t) at the transient stage in Example 2.

increasing function of t and the hunting phenomenon between ˆψ(x) and f (x) as observed in Example 1 can be avoided. Moreover, once if x(t) is convergent to some finite value x(∞), then we shall have x(∞) = 0. Otherwise, ˆc1(t) and ˆc2(t) will monotonously diverge to +∞, which is impossible.

1

2

by letting σ = 0 with g(x)=

g(x)¯c

Now we consider the following plant

˙x = f (x) + g(x)u (2.16)

Suppose that

g(x) = g(x)¯

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0 1 2 3 4 5 6 7 8 9 10 0

1 2 3 4 5 6 7

t in sec.

x(t)

Figure 2-8: The response of x(t) in Example 2.

0 1 2 3 4 5 6 7 8 9 10

−300

−250

−200

−150

−100

−50 0

t in sec.

u(t)

Figure 2-9: The control input u(t) in Example 2.

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0 1 2 3 4 5 6 0

0.5 1 1.5 2 2.5

t in sec.

c1(t)

0 1 2 3 4 5 6

0 2 4 6 8 10 12

t in sec.

c2(t)

Figure 2-10: The responses of ˆc1(t) and ˆc2(t) in Example 2.

0 1 2 3 4 5 6 7 8 9 10

24 24.5 25 25.5 26 26.5 27 27.5 28 28.5

t in sec.

Va(t)

Figure 2-11: The response of the Lyapunov function Va(t) in Example 2

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where ¯g(x) ≥ ε¯g > 0 is a known function and c > 0 is an unknown constant. Assume that the least upper bound of f (x)/¯g(x)satisfies

¯¯

¯¯f (x)

¯ g(x)

¯¯

¯¯ ≤ c1|x| + c2|x|2 = ψ(x) (2.17)

where c1 and c2 are unknown positive parameters and there is an unknown positive constant cg¯ such that ¯¯¯¯

d¯g(x) dx

¯¯

¯¯ ≤ cg¯ <∞ (2.18)

for x ∈ (−∞, ∞). Now define

ca = cc1 cb = cc2

We now try to construct the tuning laws of ˆca and ˆcb to estimate ca and cb, respectively, so as to ensure asymptotical stability of the closed-loop system. Particularly, we shall demandkx(t1)k = 1 − h at some finite time t1 when the initial state x(t0) is set as

|x(t0)| > 1. By setting σ = 0 in (2.5) and (2.6), the tuning laws of ˆca and ˆcb becomes

˙ca = Γ1|x|2g(x)¯ (2.19)

˙cb = Γ2|x|3g(x)¯ (2.20)

The adaptive VSS control law uV SS is now given as

uV SS =−(ˆca|x| + ˆcb|x|2+ r|x|)sign(x) (2.21)

To analyze the system response when applying the adaptive VSS control law defined in (2.21), (2.19), and (2.20), we consider the Lyapunov function candidate

Va= 1

2cx2+1

−11 ˜c2a+1 2Γ−12 ˜c2b

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where ˜ca = ˆca− ca and ˜cb = ˆcb− cb.

Lemma 4 Consider the adaptive VSS control system defined by (2.16), (2.21), (2.19), and (2.20). The trajectories of x(t), ˆca(t), and ˆcb(t) are bounded over the time interval (t0,∞), and x(t) converges to the origin. Moreover, there is a finite time t1 such 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 (2.22)

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

.

Va = cx ˙x + Γ−11 ˜ca .

˜

ca+ Γ−12 ˜cb .

˜ cb

= cx[f (x) + g(x)uV SS] + Γ−11 ˜ca .

˜

ca+ Γ−12 ˜cb .

˜ cb

= x[cf (x) + ¯g(x)uV SS] + Γ−11 ˜ca .

˜

ca+ Γ−12 ˜cb .

˜ cb

= x[cf (x)− ¯g(x)(ˆca|x| + ˆcb|x|2+ r|x|)sign(x)] + Γ−11 ˜ca .

˜

ca+ Γ−12 Γ−12 c˜b .

˜ cb

= x¯g(x)[cf (x)

¯

g(x) − (ˆca|x| + ˆcb|x|2+ r|x|)sign(x)] + Γ−11 c˜a .

˜

ca+ Γ−12 Γ−12 ˜cb .

˜ cb

Therefore it follows from the last equation that

.

Va = x¯g(x)[cf (x)

¯

g(x) − c(c1|x| + c2|x|2)sign(x)]

+x¯g(x)[c(c1|x| + c2|x|2)sign(x) − (ˆca|x| + ˆcb|x|2+ r|x|)sign(x)]

−11 c˜a .

˜

ca+ Γ−12 Γ−12 ˜cb .

˜ cb

= c¯g(x)[f (x)

¯

g(x)x− (c1|x| + c2|x|2)|x|]

+¯g(x)[c(c1|x| + c2|x|2)|x| − (ˆca|x| + ˆcb|x|2+ r|x|) |x|]

−11 c˜a .

˜

ca+ Γ−12 ˜cb .

˜ cb

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Since xf (x)/¯g(x)− (c1|x| + c2|x|2)|x| ≤ 0 and c¯g(x) > 0, the above equation implies

.

Va ≤ ¯g(x)[c(c1|x| + c2|x|2)|x| − (ˆca|x| + ˆcb|x|2+ r|x|) |x|] + Γ−11 c˜a .

˜

ca+ Γ−12 ˜cb .

˜ cb

= −¯g(x)[(˜ca|x|2 + ˜cb|x|3) + r|x|2] + Γ−11 c˜a .

˜

ca+ Γ−12 ˜cb .

˜ cb

= −r |x|2¯g(x)≤ 0

The above inequality implies that the trajectories of x(t), ˆca(t), and ˆcb(t) are bounded over the time interval (t0,∞) and Va(t) is a non-increasing function of t. >From (2.19) and (2.20), it is obvious that both ˆca(t) and ˆcb(t) are non-decreasing functions of t.

Therefore ˆca(t) and ˆcb(t) both converge to finite values as t → ∞. On the other hand, we have

..

ˆ

ca = 2Γ1x£

f (x)− (ˆca|x| + ˆcb|x|2+ r|x|)sign(x)¤ +Γ1|x|2 d¯g(x)

dx

£f (x)− (ˆca|x| + ˆcb|x|2+ r|x|)sign(x)¤

Since x(t), ˆca(t),and ˆcb(t) are bounded, together with the boundedness of g(x)dx given in (2.18), we can conclude that

..

ˆ

cais bounded. Therefore, ˙ca is uniformly continuous. Then, Barbalat’s lemma [16], we have

t→∞lim˙ca(t) = 0

Consequentially, with ˙ca = Γ1|x|2 ¯g(x) defined in (2.9) and ¯g(x) ≥ εg¯ > 0 , we can conclude

t→∞limx(t) = 0

The remaining proof concerning the reach time of x(t) = 1 − h or x(t) = −(1 − h) is as the same as in the proof of Lemma 3. This completes the proof. ¥

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Example 3 In the example, we consider the case

f (x) = 4x + 4x|x|

g(x) = 1 2

¡1 + x2cos2

and thus

¯

g(x) = 1 + x2cos2x≥ 1 c = 2, c1 = 4, c2 = 4, ca = cc1 = 8,

cb = cc2 = 8

We note that the plant in this example is a highly unstable system. Therefore, with the initial state x(0) = 5, the state x(t) usually grows very fast when there a suitable control is not applied. The parameters Γ1 and Γ2 in the tuning law are chosen as Γ1 = Γ2 = 10.

Since the initial conditions are given as x1(0) = 5, ˆc1(0) = 0, and ˆc2(0) = 0, we have f (x(0)) > ˆψ1(x(0)). Since ˙x(0) > 0, there is a very short time interval such that x(t) is increasing. However, the terms|x|2 and |x|3 in the tuning laws of ˆc1(t) and ˆc2(t), respectively, are large so that the growing speed of ˆc1(t) and ˆc2(t) will be very fast. With the rapid growth of ˆc1(t) and ˆc2(t), the amplitude ˆψ(x) of the control signal u(t) will also has a rapid growth so as to stabilizes the system. Therefore, it will be very soon that

˙x(t) = f (x)− ˆψ(x) < 0 , and x(t) will decay very fast. The evolution of the state x(t) and the control input u(t) are shown in Fig. 2-12 and Fig. 2-13, respectively. However, as x(t) is decreasing, so is the increasing rate of ˆc1(t) and ˆc2(t) as shown in Fig. 2-14. The amplitude ˆψ(x) of the control signal u(t) will then also decay very fast as x(t) decreases as shown in Fig. 2-13. Since f (x) is a monotone increasing function of x, f (x) decreases as x(t) decrease. However, ˆc1(t) and ˆc2(t) are monotone increasing function of t, in spite of increasing or decreasing of x(t). In this example, we have f (x) − ˆψ(x) < 0 after x(t)

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

1 2 3 4 5 6

t in sec.

x(t)

Figure 2-12: The state x(t) in Example 3.

starts to decrease. Therefore, x(t) is monotone decreasing and is convergent to zero at the steady state as shown in Fig. 2-12. Meanwhile, ˆc1(t) and ˆc2(t) converge to some finite values as shown in Fig. 2-14. In Fig. 2-15, it is also shown that the Lyapunov function Va(t) is actually monotone decreasing as shown .

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−350

−300

−250

−200

−150

−100

−50 0

t in sec.

u(t)

Figure 2-13: The control input u(t) in Example 3.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 2 4 6 8

t in sec.

ca(t)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 5 10 15 20 25 30

t in sec.

cb(t)

Figure 2-14: Evolution of parameters ˆca(t) and ˆcb(t) in Example 3.

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 16

18 20 22 24 26 28 30 32

t in sec.

Va(t)

Figure 2-15: The Lyapunov function Va(t) in Example 3.

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Problem formulation and controller design for second-order systems

3.1 Case 1

Now we consider the Second-order plant

˙x1 = x2 (3.1)

˙x2 = f2(x) + u (3.2)

where f2(x)is a scalar nonlinear continuous function of the vector x = [x1 x2]T and u ∈ R1 is the system input. We assume that the least upper bound of f2(x)satisfies

|f2(x)| ≤ ψ2(x) (3.3)

ψ2(x) = c21+ c22|x1| + c23|x1|2+ c24|x2| + c25|x2|2

An example of such a function f2(x) is given by

f2(x) = c21sin(x1) + c22|x1| + c23x1|x1| + c24|x2| + c25x2|x2| (3.4)

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where for which the equilibrium point 0 is unstable. We can express ψ(x) in a more compact way as follows

ψ2(x) = φT2(x)θ2 θ2 = h

c21 c22 c23 c24 c25 iT

φ2(x) = h

1 |x1| |x1|2 |x2| |x2|2 iT

where c21, c22, c23, c24, and c25 are unknown positive parameters. We shall develop an adaptive VSS control u(t) to ensure asymptotical stability of the closed-loop system. To attain this goal, we shall construct ˆc21, ˆc22, ˆc23, ˆc24 and ˆc25 which are the estimates of c21, c22, c23, c24, and c25 respectively, so that

|f2(x)| ≤ ˆψ2(x) ψˆ2(x) = φT2(x)ˆθ2

ˆθ2 = h ˆ

c21 ˆc22 ˆc23 ˆc24 ˆc25

iT

Then the diﬀerence between ˆψ2(x)and ψ2(x) can be expressed as

ψˆ2(x)− ψ2(x) = φT2(x)˜θ2 (3.5)

where the parametric error vector ˜θ2 is defined as

˜θ2 = ˆθ2− θ2

Based on the estimate ˆθ2, we shall develop an adaptive VSS control law. Define a sliding surface as

s(x) = 0

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with

s = r1x1+ x2 (3.6)

where r is a positive constant. For the plant in (??)-(??), the proposed adaptive VSS controller is defined as

⎧⎨

.

ˆθ2 =|s| Γ2φT2(x) u =−(x1+ rx2)−³

ψ(x) + rˆ 2|s|´

sign(s)

(3.7)

The detailed expressions of the tuning laws of the parameters ˆc2i(t) for 1≤ i ≤ 5 are

given as ⎧

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩

.

ˆ

c1 = γ1|s|

.

ˆ

c2 = γ2|x1| |s|

.

ˆ

c3 = γ3x21|s|

.

ˆ

c4 = γ4|x2| |s|

.

ˆ

c5 = γ5x22|s|

(3.8)

Note that

˙s = r1˙x1+ ˙x2

= r1x2+ f2+ u (3.9)

Lemma 5 Consider the plant in (3.1)-(3.2) and the VSS control law in (3.7). Then ˆ

ci(t) is a non-decreasing function of t, ˆci(t) converges to a finite value as t→ ∞, and

t→∞limx1(t) = 0

t→∞limx2(t) = 0

Proof: Consider a Lyapunov function candidate

Va= 1

x21 +1

s2+ φT2(x)˜θ2

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where ˜ci = ˆci− ci. Then, the time derivative of Va along the system trajectory can be evaluated as

V˙a = x1˙x1+ s ˙s + ˜θ−12 Γ−12

.

ˆθ2

= x1x2+ s (r1x2+ f2+ u) + ˜θ−12 Γ−12

.

ˆθ2

= x1(s− r1x1) + s (r1x2+ f2+ u) + ˜θ−12 Γ−12

.

ˆθ2

= −r1x21+ s (x1+ r1x2+ f2+ u) + ˜θ−12 Γ−12

.

ˆθ2

Then using the control law u and the tuning law of ˆθ2 defined in (3.7), we have V˙a = −rx21+ s³

f2 −³

ψˆ2(x) + r2|s|´

sign(s)´

+ ˜θ−12 Γ−12

.

ˆθ2

= −rx21+ s (f2− ψ2(x)sign(s)) −³

ψˆ2(x)− ψ2(x)´

|s| − r2|s|2+ ˜θ−12 Γ−12

.

ˆθ2

≤ −rx21− r2|s|2

where we have use the properties in (3.3) and (3.5). The last inequality implies that Va(t) is non-increasing and is bounded from below so that the trajectories of x(t) and ˆ

ci(t) for 1 ≤ i ≤ 5 are bounded over the time interval (t0,∞). From the tuning laws of the parameters ˆc2i(t) for m1≤ i ≤ 5 in (3.8), it is obvious that ˆci(t) is a non-decreasing function of t. Therefore ˆci(t) converges to a finite value as t → ∞.

(*** Some problem still in this part) On the other hand, from (3.9), we have

¯¯

¯¯

¯ lim

∆t→0, ∆t>0 .

ˆ

c1(t + ∆t)−ˆc.1(t)

∆t

¯¯

¯¯

¯ = γ1

¯¯

¯¯ lim

∆t→0, ∆t>0

|s(t + ∆t)| − |s(t)|

∆t

¯¯

¯¯

≤ γ1| ˙s(t)|

= γ1¯¯¯f − x1−³

ψ(x) + rˆ |s|´

sign(s)¯¯¯

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which is a bounded function over t ∈ [0, ∞). This shows that ˙c1 is uniformly continuous.

Then, by Barbalat’s lemma [16], we have

t→∞lim˙c1(t) = 0

Consequentially, with ˆc.1(t) = γ1|s| defined in (2.9), we can conclude

t→∞lims(t) = 0

Moreover, from equation (??), we have

˙x1 = x2 =−rx1+ s

which implies that

t→∞limx1(t) = 0

t→∞limx2(t) = 0

This completes the proof. ¥

Example 4 In the example, we consider the case

f (x) = sin(x1) + 2x1+ 2x1|x1| + x2+ 2x2|x2|

and thus

h

c1 c2 c3 c4 c5 i

=h

1 2 2 1 2 i

The constant r in the control law u(t) is set as 5 and the parameters γi in the tuning law of ˆci is chosen as γi = 1. The initial conditions are given as x1(0) = 2, x2(0) =−1, and ˆ

ci(0) = 0 for 1≤ i ≤ 5. The response of x(t) is shown in Fig. 3-1 where the asymptotical

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.5 1 1.5 2

t in sec.

x1(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−10

−8

−6

−4

−2 0

t in sec.

x2(t)

Figure 3-1: The responses of x1(t) and x2(t) in Example 4.

shown in Fig. 3-2 where the system state trajectory hits the sliding surface s(t) = 0 in finite time and stays on the sliding surface afterward. The control input u(t) is shown in Fig. 3-3 where the chattering phenomenon happens while the sliding motion takes place.

The behaviour of tuning parameters ˆci(t) for 1 ≤ i ≤ 5 are shown in Fig. 3-4. It can be observed that ˆci(t) is monotone increasing until sliding motion, i.e., s(t) = 0, happens.

The Lyapunov function Va(t) is actually monotone decreasing as shown in Fig. 3-5.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−1 0 1 2 3 4 5 6 7 8 9

t in sec.

s(t)

Figure 3-2: The response of s(t) in Example 4.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−400

−300

−200

−100 0 100 200 300 400 500

t in sec.

u(t)

Figure 3-3: The control input u(t) in Example 4.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.5 1

c1(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 1 2

c2(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 2 4

c3(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 1 2

θ4−hat(t)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 5

t in sec.

θ5−hat(t)

Figure 3-4: The responses of ˆci(t) for 1 ≤ i ≤ 5 in Example 4.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 5 10 15 20 25 30 35 40 45 50

t in sec.

Va(t)

Figure 3-5: The response of the Lyapunov function Va(t) in Example 4.

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−2 −1.5 −1 −0.5 0 0.5

−1 0 1 2 3 4 5 6 7 8

x1(t) x2(t)

x1(t) vs. x 2(t)

Figure 3-6: The responses of x1, x2 approach to the sliding surface. Example 4

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Conclusion and Discussion

In the conventional adaptive control field, parameter tuning law based on σ−modification has been widely used to construct an adaptive controller. However, using the σ−modification parameter tuning law, usually only the UUB stability can be attained for the adaptive control system. In this study, we simplify the parameter tuning law and the adaptive VSS control law to construct the proposed adaptive VSS control systems for typical first- order and second-order systems. Even though the σ−modification term in the parameter tuning law and the stabilizing control term in the adaptive control law are omitted, we shall show that asymptotical stability of the state and boundedness of the parameter estimates can be guaranteed. Particularly, in the proposed adaptive control schemes, only the structural information of the plant is needed. In the study of adaptive control of second-order systems, the problem of mismatched uncertainty is also attacked. Several simulation examples are given to verify the proposed adaptive control algorithms.

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