|a 1 | ≤ ¯ μ 1 = 2.2, |a 2 | ≤ ¯ μ 2 = 1.1, |b| ≥ bmn = 1.8
(86)
|d| ≤ μd fd, μd = 0.22, fd = |y|
(87)
where fd is not constant and known, whereas bmn, ¯ μ 1, ¯ μ 2, μd, bmn are positive, constant and unknown to the controller. From (86), (87) it follows that assumptions Ai, Aiv, Av of section 2
are satisfied.
The procedure of section 4 is followed in order to establish the terms involved in the control
and update laws, mentioned in remark 5. Eq. (80) can be rewritten as:
˙ x 1 = x 2
(88)
˙ x 2 = γ 2 a + bu + d
(89)
x 1 = y, x 2 = ˙ y, n = 2
(90)
A Robust State Feedback Adaptive Controller
with Improved Transient Tracking Error Bounds for Plants with Unknown Varying Control Gain
93
A Robust State Feedback Adaptive Controller with Improved Transient Tracking Error Bounds for Plants with Unknown Varying Control Gain 15
since n = 2, the state transformation based on the backstepping procedure involves the steps
0, 1, 2.
Step 0. Let
z 1 = e = y − yd = x 1 − yd
(91)
as in (9).
Step 1. Differentiating (91) with respect to time and arranging, yields:
˙ z 1 = −c 1 z 1 + z 2
(92)
z 2 = x 2 + c 1 z 1 − ˙ yd
(93)
as in (11), (12).
Step 2. Since n = 2, the second step is the last one. Differentiating (93) with respect to time,
using (89) and arranging, yields:
˙ z 2 = ˙ x 2 + c 1 ˙ z 1 − ¨ yd
= γ 2 a + bu + d + c 1 ˙ z 1 − ¨ yd
= γ 2 a + bu + d + c 1( x 2 + ϕ 1) − ¨ yd
(94)
using the definitions (91), (93), yields:
˙ z 2 = γ 2 a + bu + d + ϕ 2
(95)
ϕ 2 = c 1( z 2 − c 1 z 1) − ¨ yd
(96)
notice that the form of (95), (96) is that of (17), (18), respectively. This completes the state
transformation based on the backstepping procedure.
The parameters defined above can be summarized as:
z 1 = y − yd
(97)
z 2 = x 2 + c 1 z 1 − ˙ yd
(98)
x 1 = y, x 2 = ˙ yd
(99)
ϕ 1 = − ˙ yd
(100)
ϕ 2 = c 1( z 2 − c 1 z 1) − ¨ yd
(101)
According to remark 5, it remains to define ¯
ϕ, Vz. From (81), definition (23) and n = 2, it
follows that
¯
ϕ = |γ 2[1] |, |γ 2[2] |, fd, |ϕ 2 + c 2 z 2 |
= [ | ˙ y|, |y|, fd, |ϕ 2 + c 2 z 2 |]
(102)
From (35) and n = 2 it follows that
Vz = (1/2)( z 2 +
)
1
z 22
(103)
94
16
Will-be-set-by-IN-TECH
Applications of Nonlinear Control
Expressions (97) to (103) allow to define the control and update law. From (32), (48), (82) and
n = 2 it follows that
sgn( b) = +1
(104)
u = − 1 z
3 C
2( ¯
ϕ ˆ θ)2
(105)
bvz
˙ˆ θ = Γ ¯ ϕ|z 2 | ∂ ¯ Vz
∂
(106)
Vz
the main parameters needed to compute u and ˆ θ are: ¯ ϕ (102), ϕ 2 (101), Cbvz (28), z 2 (98), z 1
(97), ∂ ¯
Vz/ ∂Vz (42), Vz (103). In addition, Γ is a diagonal matrix whose diagonal elements are
positive constants defined by the user.
3
2
d
y,y
1
0
0
20
40
60
80
100
time
0.1
e
0
−0.10
20
40
60
80
100
time
4
u 2
0
0
20
40
60
80
100
time
Fig. 2. Example 1, upper: output y (continuous line), desired output yd (dash–dot line);
middle: tracking error e; lower: control input u.
Since the aim is that y converges towards yd, with a threshold of 0.1, we set Cbe = 0.1.
We use the reference model (5) with yd( to) = y( to), ˙ yd( to) = 0, am,1 = 1, am, o = 1.
We use the following parameter values for the control and update laws: c 1 = 2, c 2 = 2,
Γ = diag { 1, 1, 1, 1 }.
The results are shown in figures 2 and 3. We have choosen yd( t 0) ≈ y( t 0) in order to obtain a
rapid convergence of y towards yd. Figure 2 shows that. i) the tracking error e converges
asymptotically towards Ω e = {e : |e| ≤ 0.1 }. ii) The output y converges towards yd with threshold 0.1 without large transient differences. Figure 3 shows that ˆ θ 1, ..., ˆ θ 4 are not
decreasing with respect to time. This occurs because ˙ˆ θ is non-negative. The procedure for the
sample plant (80) is simpler in comparison with adapive controllers that use the Nussbaum
gain method.
A Robust State Feedback Adaptive Controller
with Improved Transient Tracking Error Bounds for Plants with Unknown Varying Control Gain
95
A Robust State Feedback Adaptive Controller with Improved Transient Tracking Error Bounds for Plants with Unknown Varying Control Gain 17
0.1
^ θ 1 0.05
00
20
40
60
80
100
time
0.2
^ θ 2 0.1
00
20
40
60
80
100
time
0.2
^ θ 3 0.1
00
20
40
60
80
100
time
0.4
^ θ 4 0.2
00
20
40
60
80
100
time
Fig. 3. Example 1, entries of the updated parameter vector ˆ θ, from upper to lower: ˆ θ 1; ˆ θ 2, ˆ θ 3, ˆ θ 4.
9. Acknowledgements
A. Rincon acknowledges financial support provided by “Programa de becas para estudiantes
sobresalientes de posgrado”, Universidad Nacional de Colombia - vicerrectoría de
Investigación. This work was partially supported by Universidad Nacional de Colombia -
Manizales, project 12475, Vicerrectoría de Investigación, DIMA.
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6
A Robust Motion Tracking Control
of Piezo-Positioning Mechanism
with Hysteresis Estimation
Amir Farrokh Payam,
Mohammad Javad Yazdanpanah and Morteza Fathipour
Department of Electrical and Computer Engineering, University of Tehran, Tehran,
Iran
1. Introduction
Piezoelectric actuators are the most suited actuation devices for high precision motion
operations in the positioning tasks include miro/nano-positioning [1]. These actuators have
unlimited motion resolution and posses some advantages such as ignorable friction,
noiseless, zero backlash and easy maintenance, in comparison with the conventional
actuated systems which are based on the sliding or revolute lower pairs [2].
Producing large forces, fast response and high efficiency are major advantages of
piezoelectric actuators. But, it has some drawbacks such as hysteresis behavior, drift in time,
temperature dependence and vibration effects. Molecular friction at sites of materials
imperfections due to domain walls motion is the general cause of hysteresis in piezoelectric
materials [3]. The hysteresis is a major nonlinearity for piezo-actuators and often limits
system performance via undesirable oscillations or instability. Therefore, it is difficult to
obtain an accurate trajectory tracking control. Numerous mathematical methods have been
proposed to analyze the hysteresis behavior of piezoelectric actuators. These studies may be
categorized in asymmetrical and symmetrical methods. The asymmetrical types of hysteretic
models include polynomial model [4], Preisach’s model [5], neural network model [6] and
Karasnoselskii and Pokrovskii [7]. The symmetrical types of hysteretic model include
Duhem model [8], Bouc-Wen model [9] and Lugre model [9].
The asymmetrical methods establish the nonlinear relations between the input and output
based on the measured input/output data sets. Because superposition of a basic hysteresis
operator is a fundamental principle in these models, they are also called to operator based
model. Although, an operator based model may give a good match with experimental data,
the dynamics of the piezoelectric material is not formulated in these modeling methods and
model parameter identification and implementation is more difficult in this case. The
symmetrical methods employ nonlinear differential equations in order to describe
hysteresis. In this case, the dynamics of the piezoelectric materials are described but the non-
symmetric hysteresis is not modeled. However these models are more tractable for control
design. In order to include the hysteresis effect and compensating its effect, Lugr