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IET Control Theory & Applications
Research Article
Finite-time tracking control for pneumatic
servo system via extended state observer
ISSN 1751-8644
Received on 25th March 2017
Revised 10th July 2017
Accepted on 3rd August 2017
E-First on 8th September 2017
doi: 10.1049/iet-cta.2017.0327
www.ietdl.org
Ling Zhao1 , Bin Zhang1, Hongjiu Yang2, Yingjie Wang2
1Hebei Provincial Key Laboratory of Heavy Machinery Fluid Power Transmission and Control, The College of Mechanical Engineering, Yanshan
University, Qinhuangdao 066004, People's Republic of China
2Department of Automation, Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, People's Republic of China
E-mail: zhaoling@ysu.edu.cn
Abstract: In this study, a non-singular fast terminal sliding mode finite-time tracking control strategy is presented to improve
response rapidity and control precision of a pneumatic servo system via an extended state observer. The extended state
observer is designed to estimate total disturbances of the system. Moreover, the proposed controller is established to ensure
good performances of the closed-loop system. In addition, both sufficiently small observation error and stabilisation of the
closed-loop system are proved in finite time. Finally, experimental results show the effectiveness of the proposed method.
1 Introduction
In recent years, pneumatic servo systems have been investigated
widely because of their various advantages such as low cost,
simple structure and convenient maintenance [1]. Meanwhile,
pneumatic servo systems are used extensively in mechanical
operations, medical areas, automation systems and many other
fields [2, 3]. However, it is difficult to achieve satisfactory
performances for pneumatic servo systems due to some negative
effects caused by non-linearity of servo valves, various friction
forces, air compressibility and so on [4]. At present, many
researchers have put forward a lot of control strategies to improve
these negative problems of pneumatic servo systems, please see
[5]. A neural network control scheme based on proportional–
integral-derivative has been introduced to ensure good tracking
performance of a pneumatic X–Y table [6]. In [7], a non-linear
model-based control methodology of pulse width has been
presented to modulate pneumatic servo actuators. A function
approximation technique-based adaptive controller has been
proposed for a pneumatic servo system with variable payload [8].
A backstepping controller has also been used to achieve
satisfactory control effects for a pneumatic servo system [9].
However, above control schemes require highly complete
mathematical models which are hardly obtained because of strong
non-linearity, unmodelled dynamics and external disturbances for
pneumatic servo systems. Therefore, it is a key idea to estimate
these uncertain factors effectively by using an extended state
observer for pneumatic servo systems.
Extended state observer, which was introduced by Han Jingqing
in 1990s, is a core part of active disturbance rejection control [10].
Strong non-linearity, unmodelled dynamics and external
disturbances of uncertain systems are estimated by the special
feedback mechanism of extended state observer [11]. Since the
extended state observer is not dependent on accurate mathematical
models, it has been used in various complex plants such as
autonomous underwater vehicle [12] and permanent-magnet
synchronous motors [13]. In addition, various controllers have also
been designed to improve performances of complicated systems
via extended state observers. In [14], a feedback linearisation
controller based on an extended state observer has been designed
for trajectory tracking control of a flexible-joint robotic system. A
missile guidance law has been proposed by using an extended state
observer in [15]. Both a predictive functional controller and an
extended state observer have been applied to speed control for a
permanent-magnet synchronous motor servo system [16]. Adaptive
IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816
© The Institution of Engineering and Technology 2017
control has been investigated by combining with an extended state
observer for attitude synchronisation control of an spacecraft
formation [17]. Therefore, in order to obtain good performances for
pneumatic servo systems, a sliding mode control (SMC) method is
designed by combining with an extended state observer.
The SMC is an effective control approach which has been
widely applied to various complicated systems such as non-linear
active suspension vehicle systems [18] and trajectory tracking of
non-holonomic wheeled mobile robots [19]. For a pneumatic servo
system, it is expected that stabilisation of the closed-loop system is
achieved in finite time. However, traditional linear SMC only
guarantees asymptotic tracking performance, which means the
tracking error converges to zero only when time goes to infinity
[20]. Therefore, to obtain finite-time stabilisation of a system,
terminal SMC has been presented in [21]. The terminal SMC not
only has strong robustness with respect to various uncertainties, but
also makes the tracking error converge to zero in finite time [22].
However, there exists a singularity problem for the terminal SMC,
please see [23]. To solve the singularity problem, a non-singular
terminal SMC method has been proposed in [24]. Then, a nonsingular fast terminal SM (NFTSM) control method has been
presented to avoid the singularity problem, and it has a faster
convergence rate than a non-singular terminal SMC method [25].
To the best of our knowledge, very few results are available on the
extended state observer-based NFTSM finite-time tracking control
for pneumatic servo systems. This problem is important and
challenging in both theory and practise, which motivates us to
carry on this research work.
In this paper, the finite-time tracking control for a pneumatic
servo system is investigated by combining an extended state
observer with an NFTSM controller. The extended state observer is
used to estimate strong non-linearity, unknown modelling and
external disturbances of the pneumatic servo system. Moreover, the
NFTSM controller is applied to ensure satisfactory performances
of the closed-loop system. Then, both sufficiently small
observation error and stabilisation of the closed-loop system are
analysed in finite time. Finally, experimental results are given to
illustrate the effectiveness of the proposed method. Furthermore,
the main contributions of this paper are summarised as follows:
i.
An extended state observer-based NFTSM finite-time tracking
control scheme is proposed to ensure response rapidity and
control precision for a pneumatic servo system.
2808
2 Problem formulations
2.1 System structure
Fig. 1 Platform of pneumatic servo system
The experiment platform of pneumatic servo system is shown in
Fig. 1. It contains six main parts: a pneumatic rod cylinder (SMC,
MBBQ80-200B), a five-way proportional valve (Festo,
MPYE-5-3/8-010-B),
a
displacement
sensor
(LVDTPB41MS150X), an acquisition card (Advantech, PCL-812), an
output card (Advantech, PCL-1727) and an industrial control
computer (Advantech, IPC-610H).
The schematic representation of pneumatic servo system is
introduced in Fig. 2. Running process of the pneumatic servo
system is described as follows: a voltage signal, which is given by
the industrial control computer, is transmitted to the five-way
proportional valve at the beginning. Then, valve ports of the fiveway proportional valve are opened accordingly, and supplied gas
flows into chamber A or chamber B to drive the piston rod of
pneumatic rod cylinder. Meanwhile, the displacement sensor is
applied to measure the position of piston rod in real time and the
collected data is processed by the industrial control computer.
2.2 System model
According to Gulati and Barth [27], the flow dynamics in chamber
A and chamber B of the pneumatic servo system are given as
follows:
Ṗ(a, b)(t) =
Fig. 2 Schematic representation of pneumatic servo system
ii. An extended state observer is proposed to deal with incomplete
mathematical models via estimating strong non-linearity,
unmodelled dynamics and external disturbances.
iii. Both sufficiently small observation error and stabilisation of
the closed-loop system are proved in finite time by using
appropriate Lyapunov functions.
The remainder of this paper is organised as follows. Section 2
introduces the system structure and the system model. Design of
control strategy, sufficiently small observation error in finite time
and stabilisation analysis of the closed-loop system in finite time
are presented in Section 3. Finally, experimental results are shown
in Section 4 and conclusions are given in Section 5.
Notations: In this paper, the superscript T stands for the
transpose of a matrix, the superscript H stands for the conjugate
transpose of a matrix, ∥ v ∥2 denotes the Euclidian 2-norm of a
vector v. For a matrix M of order m × n, σmin{M} denotes the
minimum singular value of matrix M. For a square matrix N of
order n, λmin{N} denotes the minimum eigenvalue of matrix N,
λmax{N} denotes the maximum eigenvalue of matrix N. An
important definition is shown as follows:
sigδ(x) = sgn(x) | x|δ
where δ ∈ (0, 1), x ∈ R, sgn( ⋅ ) is the standard symbolic function.
Remark 1: According to Zhang [26], the singular values for a
matrix M of order m × n are defined to be the square roots of the
eigenvalues for matrix M H M. Note that M H is the conjugate
transpose matrix of M. For a matrix M of order m × n, there exist
singular values and the minimum singular value σmin{M} is nonnegative. However, the calculation of eigenvalues is only applied
to a square matrix N of order n and the minimum eigenvalue
λmin{N} can be negative.
IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816
© The Institution of Engineering and Technology 2017
KV̇ (a, b)(t)
KRT
q (t) +
P (t)
V (a, b)(t) (a, b)
V (a, b)(t) (a, b)
(1)
where subscripts a and b represent the properties of chambers A
and B, respectively. P(a, b)(t) is the pressure, q(a, b)(t) is the mass flow
rates, V (a, b)(t) is the volume, K is the ratio of specific heat of air, R
is the universal gas constant, and T is the system temperature.
Considering [28, 29] and related mechanics theories, the following
mathematical model of pneumatic servo system is given as
ÿ(t) = −
Fs(t)
Pa(t)Aa − Pb(t)Ab
+ w(t) +
m
m
(2)
where y(t) is the displacement, ẏ(t) is the velocity, ÿ(t) is the
acceleration, Fs(t) are various friction forces, m is the mass of
piston rod, w(t) is the external disturbances and the uncertain parts
of pneumatic servo system, Aa and Ab are the piston areas of
chambers A and B, respectively. According to (1) and (2), the
pneumatic servo system will become complicated if the
relationship between mass flow rates and pressures is taken into
account. For simplicity, the pneumatic servo system is taken as a
second-order system [29, 30]. Equation (2) is processed as follows:
Pa(t)Aa − Pb(t)Ab
= b0u(t) + Δu(t)
m
Fs(t)
Δu(t) −
= f (t, X(t))
m
(3)
where b0 is the control gain, u(t) is the control input signal, Δu(t) is
the non-linear input signal, X(t) = [x1(t)x2(t)]T is the state vector of
pneumatic servo system, and f (t, X(t)) are various uncertain factors
of the pneumatic servo system. Then, (2) is rewritten as follows:
ẋ1(t) = x2(t)
ẋ2(t) = f (t, X(t)) + w(t) + b0u(t)
y(t) = x1(t)
(4)
where x1(t) and x2(t) are set as y(t) and ẏ(t), respectively.
Assumption 1: In the pneumatic servo system (4), f (t, X(t)) and
w(t) are regarded as a class of generalised disturbance L(t), i.e.
L(t) = f (t, X(t)) + w(t). In addition, L(t) is continuous
2809
^
^
s1(t) = ε(t) + k1sigς1(ε(t)) + k2sigς2(ε̇(t))
^
(10)
^
where k1 > 0, k2 > 0, 1 < ς2 < 2 and ς1 > ς2. If s1 = 0, the
convergence time T 3 of ε(t) is given as follows:
T3 =
Fig. 3 Block diagram of the proposed design
differentiable and bounded, i.e. |L̇(t) | ≤ Ld, where Ld is a positive
constant.
3 Design of control strategy
Considering strong non-linearity, unknown modelling and external
disturbances of the pneumatic servo system (4), an extended state
observer-based NFTSM finite-time tracking control scheme is
designed to study the positioning problem of the system. The block
diagram of the proposed design is shown in Fig. 3. In addition, in
order to make this paper express clearly, three lemmas are
introduced as follows.
Lemma 1 [31]: Consider the system of differential equations
ẋa(t) = f (xa(t))
(5)
where f : D → ℝn is continuous on an open neighbourhood
D ⊆ ℝn of the origin and f (0) = 0. Suppose there exists a
continuous function V : D → ℝ such that the following conditions
hold:
i. V is positive definite.
ii. There exist c > 0, ϱ ∈ (0, 1) and an open neighbourhood
ϖ ⊆ D of the origin such that
V̇ a(xa(t)) + c(V a(xa(t)))ϱ ≤ 0,
xa(t) ∈ ϖ∖{0}
(6)
Then, the origin is a finite-time-stable equilibrium of (5).
Furthermore, the setting-time function T 1 is shown as follows:
T1 ≤
1
(V (x (t )))1 − ϱ
c(1 − ϱ) a a 0
(7)
where V a(xa(t0)) is the initial value of V a(xa(t)) and T 1 is
continuous. In addition, if D = ℝn, V a(xa(t)) is proper, and
V̇ a(xa(t)) takes negative values on ℝn∖{0}, then the origin is a
globally finite-time-stable equilibrium of (5).
Lemma 2 (reaching time) [32]: Consider the non-linear system
ẋb(t) = f (xb(t)), f (0) = 0 and xb(t) ∈ Rn. Suppose there exists a
positive definite scalar function V b(xb(t)) such that
V̇ b(xb(t)) ≤ − τ1V b(xb(t)) − τ2V b(xb(t))θ
(8)
where τ1 > 0, τ2 > 0 and 0 < θ < 1, then the system is finite-time
stable. Furthermore, the setting time T 2 is obtained as follows:
τ1V b1 − θ(xb(t0)) + τ2
1
T2 ≤
ln
τ1(1 − θ)
τ2
(9)
where V b(xb(t0)) is the initial value of V b(xb(t)).
Lemma 3 (sliding time) [33]: Suppose an NFTSM surface is
chosen as follows:
2810
^1/ ς2
ς | ε(0)|1 − 1/ ς2
k2
dx = ^2−1/ ς
^ ς
1/ ς2
2
1
(ε(t) + k1ε (t))
0
k2 (ς2 − 1)
^
ς −1
1 ς −1
⋅F , 2
;1 + 2
; − k1 | ε(0)|ς1 − 1
ς2 (ς1 − 1)ς2
(ς1 − 1)ς2
∫
|ε(0)|
(11)
where ε(0) is the initial value of ε(t) and F( ⋅ ) is the Gaussian
hypergeometric function.
Remark 2: According to Beukers [34] and Abramowitz and Stegun
[35], the Gaussian hypergeometric function is given as follows:
F(a, b; c; z) =
∞
(a)n(b)n n
z
(c)nn!
n=0
∑
where z is the variable, real numbers a, b, c are the function
parameters and c ≠ 0, − 1, − 2, …. The Pochhammer symbol (x)n
is defined by (x)0 = 1 and (x)n = x(x + 1)⋯(x + n − 1). Since
1 < ς2 < 2 and ς1 > ς2, function F( ⋅ ) in (11) is the Gaussian
hypergeometric function. According to Yang and Yang [33], the
^
conditions of ς1, ς2 and k1 induce that function F( ⋅ ) will keep
convergent. Moreover, the exact form of function F( ⋅ ) changes
with the involved parameters. For example
F(a, b; b; z) = (1 − z)− a,
F
1
3
, 1; ; − z2 = z−1arctan z
2
2
3.1 Design of extended state observer
In this section, various negative factors, which contain strong nonlinearity, unknown modelling and external disturbances of the
pneumatic servo system (4), are estimated by using an extended
state observer. In pneumatic servo system (4), the generalised
disturbance L(t) = f (t, X(t)) + w(t) is regarded as an extended state
x3(t), i.e. L(t) = x3(t). Then (4) is rewritten as follows:
ẋ1(t) = x2(t)
ẋ2(t) = x3(t) + b0u(t)
(12)
ẋ3(t) = L̇(t)
On the basis of (12), an extended state observer is presented as
follows:
ż1(t) = z2(t) − β1 sig(α + 1)/2(z1(t) − x1(t))
ż2(t) = z3(t) − β2 sig(α + 1/2)(z1(t) − x1(t)) + b0u(t)
(13)
ż3(t) = − β3 sigα(z1(t) − x1(t))
where α is the given parameter with α ∈ (0, 1), zi(t) is the
observation value of xi(t), and βi is the observation gain with
i = 1, 2, 3. Let ei(t) = zi(t) − xi(t), where ei(t) is the observation
error. Considering (12) and (13), the following observer error
system is given as:
ė1(t) = e2(t) − β1 sig(α + 1)/2(e1(t))
ė2(t) = e3(t) − β2 sig(α + 1)/2(e1(t))
(14)
α
ė3(t) = − L̇(t) − β3 sig (e1(t))
Subsequently, finite-time sufficiently small observation error is
analysed by Lyapunov theory as follows.
IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816
© The Institution of Engineering and Technology 2017
G(s~) = | s~E − A|
s~ + rμβ1
Theorem 1: Considering the extended state observer (13) and
Assumption 1, there exist gains β1, β2, β3 and α
(β1 > 0, β2 > 0, β3 > 0, 1 > α > 0) such that the following
inequation is obtained in finite time ts (ts > 0) as
∥ η(t) ∥2 ≤
ILd
σmin{A2}σmin{P}
−β2
2
0
−β3
0
2
.
V(η(t)) = η(t)TPη(t)
(16)
2β1
| e (t) |α + 1 + e22(t) + e32(t)
α+1 1
+ e2(t) − β2 sig
(e1(t))
2
ATP + PA = − Q
λmin{P} ∥ η(t) ∥22 ≤ V(η(t)) ≤ λmax{P} ∥ η(t) ∥22
∥ η(t) ∥2 ≥ | e1(t)|(α + 1)/2
=
−μβ3
0
Considering (22), there exists
(27)
Note that A and P are non-singular matrices when e1 ≠ 0, it follows
that:
σmin{Q} = 2σmin{ − AP} ≥ 2σmin{ − A}σmin{P}
(19)
1 η − 0 L̇(t)
1
0
The
(28)
The non-singular matrix A is rewritten as follows:
rμ
−A = 0
0
= Aη(t) − BL̇(t)
1 > r = (α + 1)/2 > 1/2,
where
μ = | e1(t) |(α − 1)/2 > 0.
characteristic equation of the matrix A is written as follows:
(25)
(26)
Q = ( − A)TP + P( − A)
e3(t) − β2 sig(α + 1/2)(e1(t))
0
(24)
For the symmetrical positive definite matrix Q, the following
equation is gotten as:
σmin{Q} = λmin{Q}
(18)
α+1
| e1(t) |(α − 1)/2 e2(t) − β1 sig(α + 1/2)(e1(t))
2
−β2
the
^
V̇(η(t)) = − η(t)TQη(t) + 2L̇(t)Bη(t)
≤ − λmin{Q} ∥ η(t) ∥22 + 2ILd ∥ η(t) ∥2
= − (λmin{Q} ∥ η(t) ∥2 − 2ILd) ∥ η(t) ∥2
Hence, the following equation is obtained as:
−rμβ1
Meanwhile,
(23)
According to (21) and (22), the inequality is obtained as follows:
Remark 3: Obviously, in addition to e1 = 0, V(η(t)) is continuous
and differentiable everywhere. Moreover, before reaching the
origin, the observer error system (14) is not possible to stay in
e1(t) = 0. Therefore, when e1(t) ≠ 0, V̇(η(t)) is calculated in
accordance with the conventional method.
The equation is calculated as follows:
−β3 sigα(e1(t)) − L̇(t)
rμ 0
0
(22)
According to (16), the following inequality is obtained as:
(17)
2
d sig(α + 1)/2(e1(t))
(α + 1) | e1(t) |(α − 1)/2 ė1(t)
=
dt
2
^
where B = − BTP = [β3 0 − 2], then I = ∥ B ∥2 = β32 + 4.
Since the matrix A is Hurwitz, there exists a corresponding
symmetrical positive definite matrix Q > 0 such that P is the
solution of Lyapunov equation. One has that
where ∥ η(t) ∥22 = | e1(t) |α + 1 + e22(t) + e32(t).
following inequality is acquired as
+ e3(t) − β3 sig(α + 1/2)(e1(t)) ≥ 0
η̇(t) =
^T
^T
The following formula is confirmed as:
(α + 1/2)
(21)
= η(t)T(ATP + PA)η(t) + 2L̇(t)B η(t)
−β2 −β3
Moreover, select the appropriate parameters β1, β2 and β3 to meet
ILd ≪ σmin{A2}σmin{P}, then ∥ η(t) ∥2 is limited to be small enough
in finite time.
Proof: A appropriate Lyapunov function is constructed as follows:
V(η(t)) =
(20)
s~
+η(t)TP(Aη(t) − BL̇(t))
2β1 /(α + 1) + β22 + β32
P=
0
= (Aη(t) − BL̇(t))TPη(t)
−1 ,
0
0
μβ3
0
−1
V̇(η(t)) = η̇(t)TPη(t) + η(t)TPη̇(t)
0
0
s~
where E is the identity matrix. If there exists βi > 0 with i = 1, 2, 3,
then all coefficients of G(s~) is positive. Therefore, the matrix A is
Hurwitz which means that A is stable. The time differential
equation of V(η(t)) is expressed as follows:
η(t) = [sig(α + 1)/2(e1(t)), e2(t), e3(t)]T,
A2 = β2
β3
−rμ
β2
= s~3 + rμβ1s~2 + rμβ2s~ + rμ2 β3
(15)
where
I = β32 + 4,
β1 −1
=
0
1
0
0 β1 −1
0 β2 0
μ β3 0
0
−1 = A1 A2
0
(29)
One has that
σmin{ − A} = σmin{A1 A2} ≥ σmin{A1}σmin{A2}
(30)
Since A1 is a diagonal matrix and rμ < μ, there exists
IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816
© The Institution of Engineering and Technology 2017
2811
σmin{A1} =
If |e1(t) | ≥ 2/(α + 1)
One has that
1,
|e1(t) | <
2
α+1
2/(α − 1)
rμ,
|e1(t) | ≥
2
α+1
2/(α − 1)
2/(α − 1)
, then ∥ η(t) ∥2 ≥ 2/(α + 1)
λmin{Q} = σmin{Q} ≥ 2rμσmin{A2}σmin{P}
If the following relationship:
(31)
(α + 1)/(α − 1)
.
(32)
Considering formulas (24), (25) and (32), the following
relationship is obtained as:
(α + 1)/2
− 2ILd
α
= (α + 1)σmin{A2}σmin{P} | e1(t) | − 2ILd
≥ (α + 1)
2
α+1
(33)
2α /(α − 1)
σmin{A2}σmin{P} − 2ILd
2
α+1
2α /(α − 1)
≥ℏ
(34)
Therefore, inequality (33) is rewritten as follows:
q1 ≥ ℏσmin{A2}σmin{P} − 2ILd
= q1, min
(35)
(36)
Considering formulas (23), (25) and (35), one has that
V̇(η(t)) ≤ − q1 ∥ η(t) ∥2
q1, min
≤ −
V 1/2(η(t))
λmax{P}
= − C1V 1/2(η(t))
1 1/2
1 1/2
V (η(t)) ≤
V (η(t0))
C1 /2
C1 /2
(37)
(38)
(39)
According to formulas (24), (25) and (39), the following
relationship is acquired as:
q2 = λmin{Q} ∥ η(t) ∥2 − 2ILd
≥ 2σmin{A2}σmin{P} ∥ η(t) ∥2 − 2ILd
2812
V̇(η(t)) ≤ − q2 ∥ η(t) ∥2
q2
≤ −
V 1/2(η(t)) = − C2V 1/2(η(t))
λmax{P}
1 1/2
1 1/2
V (η(t)) ≤
V (η(t1))
C2 /2
C2 /2
(42)
(43)
Therefore, ∥ η ∥2 is a decreasing function of time. Meanwhile, the
following in equation is obtained in finite time ts = ts1 + ts2 as:
ILd
σmin{A2}σmin{P}
(44)
Obviously, if the appropriate parameters β1, β2 and β3 are selected
to make σmin{A2}σmin{P} large enough, then the observation error
∥ η(t) ∥2 is limited to be sufficiently small in finite time. This
completes the proof. □
In this section, an NFTSM controller is designed to ensure good
performances of the pneumatic servo system (4). The tracking error
is presented as follows:
ε1(t) = y(t) − yd(t) = x1(t) − yd(t)
ε2(t) = ẋ1(t) − ẏd(t) = x2(t) − ẏd(t)
(40)
(45)
where yd(t) is the reference signal. According to (4) and (45) as
well as Assumption 1, the following tracking error system is given
as:
ε̇1(t) = ε2(t)
ε̇2(t) = L(t) + b0u(t) − ÿd(t)
(46)
Considering (45), the sliding mode surface s(t) is designed as
follows:
γ
If ∥ η(t) ∥2 < (2/(α + 1))(α + 1)/(α − 1), then |e1(t) | < (2/(α + 1))2/(α − 1).
Therefore, there exists
λmin{Q} = σmin{Q} ≥ 2σmin{A2}σmin{P}
(41)
is satisfied, then there exist q2 > 2σmin{A2}σmin{P}Md − 2ILd = 0
and V̇(η(t)) < 0. Similar to (37) and (38), it follows that:
γ
s(t) = ε1(t) + k1sig 1ε1(t) + k2sig 2ε2(t)
According to Lemma 1, the observer error system (14) converges
into the region ∥ η(t) ∥2 < (2/(α + 1))(α + 1)/(α − 1) in finite-time time
ts1. Meanwhile, the finite-time time ts1 is expressed as follows:
ts1 ≤
ILd
= Md
σmin{A2}σmin{P}
3.2 Design of non-singular fast terminal sliding mode
controller
Note that β1, β2 and β3 are the adjustable parameters. Before β1, β2
and β3 are given, it is necessary to take the adjustable parameters
into q1, min = ℏσmin{A2}σmin{P} − 2ILd. When q1, min ≤ 0, the
adjustable parameters β1, β2 and β3 are reassigned such that
q1, min > 0. When q1, min > 0, corresponding experiments are carried
out using the adjustable parameters β1, β2 and β3. That is, β1, β2 and
β3
are
determined
when
both
the
conditions
ℏσmin{A2}σmin{P} > 2ILd and experimental effects are satisfied.
Therefore, inequality q1, min > 0 is established by selecting the
appropriate β1, β2 and β3. Then it follows that:
V̇(η(t)) < 0
> ∥ η(t) ∥2 >
∥ η(t) ∥2 ≤
Since α is the given parameter with α ∈ (0, 1), there exists a
minimum value ℏ such that
(α + 1)
(α + 1)/(α − 1)
ts2 ≤
q1 = λmin{Q} ∥ η(t) ∥2 − 2ILd
≥ 2rμσmin{A2}σmin{P} | e1(t) |
2
α+1
(47)
where k1 > 0, k2 > 0, 1 < γ2 < 2 and γ1 > γ2. The sliding mode
reaching condition is expressed as follows:
s(t)ṡ(t) ≤ 0
(48)
Considering in (48), the following reaching law is selected as:
ṡ(t) = − k3 sgn(s(t)) − k4s(t)
(49)
where k3 > 0, k4 > 0, k4 = kl + kd, kl > 0 and kd > 0. To obtain
satisfactory tracking position, the controller u(t) is designed as
follows:
u(t) = −
1
1
2−γ
z (t) +
sig 2ε2(t)
b0 3
k2γ2
γ
× (1 + k1γ1 | ε1(t) | 1
−1
) + k3 sgn(s(t))
(50)
+k4s(t) − ÿd(t)
The controller u(t) is a voltage signal, which is used to determine
the opening of proportional valve [36].
IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816
© The Institution of Engineering and Technology 2017
γ
V̇ 1(s(t)) ≤ − k2γ2 | ε2(t) | 2
−k2γ2 | ε2(t) |
−1
γ2 − 1
k3 | s(t)|
kls2(t)
γ
≤ − 2k2γ2 | ε2(t) | 2
−2k2γ2 | ε2(t) |
γ2 − 1
−1
k3V 1(t)1/2
(55)
klV 1(t)
≤ − ψ 1(t)V 1(t)1/2 − ψ 2(t)V 1(t)
γ
Fig. 4 Phase-trajectory plot of the pneumatic servo system
Remark 4: Total disturbances are estimated and system states are
tracked in real time by using the extended state observer.
Moreover, the estimation and tracking processes are synchronised
with controlling the system. According to Theorem 1, the
observation error ∥ η(t) ∥2 is limited to be sufficiently small in
finite time. Meanwhile, total disturbances of the pneumatic servo
system are bounded in practise. The disturbance estimation z3 is
also bounded. Therefore, system states do not diverge, please see
[10–17, 37]. That is, states are not divergent so seriously that the
closed-loop system cannot be controlled using the extended state
observer.
Subsequently, finite-time stabilisation of the closed-loop system
(46) is analysed by Lyapunov theory as follows:
Theorem 2: Consider the closed-loop system (46) with the
NFTSM controller (50). When the sliding mode surface is selected
as (47), and by choosing appropriate parameters k4 = kl + kd such
that (L(t) − z3(t))s(t) ≤ kds2(t), |L(t) − z3(t) | ≤ k4 | s(t)|, then tracking
error of the closed-loop system (46) converges to zero in finite
time.
Proof: Considering the closed-loop system (46), the following
Lyapunov function is designed as:
1 2
s (t)
2
V 1(s(t)) =
γ
= s(t)(ε2(t) + k1γ1 | ε1(t) | 1
+k2γ2 | ε2(t) |
γ2 − 1
−1
ε2(t)
V̇ 1(s(t)) = s(t)[ε2(t) + k1γ1 | ε1(t) |
+k2γ2 | ε2(t) |
γ
−1
−k2γ2 | ε2(t) |
= k2γ2 | ε2(t) |
γ2 − 1
(53)
(L(t) + b0u(t) − ÿd(t))]
−k2γ2 | ε2(t) |
−k2γ2 | ε2(t) |
γ
k3 | s(t) | − k2γ2 | ε2(t) | 2
−1
k4s2(t)
(L(t) − z3(t))s(t)
γ2 − 1
γ2 − 1
1
2−γ
sig 2ε2(t)
k2γ2
ε̇2(t) = L(t) − z3(t) −
γ
× (1 + k1γ1 | ε1(t) | 1
−1
(57)
) − k3 sgn(s(t)) − k4s(t)
When ε2(t) = 0, the (57) is rewritten as follows:
ε̇2(t) = L(t) − z3(t) − k3 sgn(s(t)) − k4s(t)
(54)
γ2 − 1
k3 | s(t) | − k2γ2 | ε2(t) |
2
kds (t)
2
kls (t)
By choosing appropriate parameter k4, the following relationship is
established as:
(59)
When (L(t) − z3(t))s(t) ≤ kds (t), the following relationship is
obtained as:
(60)
Then, there still holds ε̇2(t) ≤ − k3 or ε̇2(t) ≥ k3 under the
conditions of s(t) > 0 or s(t) < 0, respectively [33]. Moreover, the
trajectories cross, which include from ε2 ≤ ξ to ε2(t) ≥ − ξ for
s(t) > 0 and from ε2(t) ≥ − ξ to ε2(t) ≤ ξ for s(t) < 0, are realised
in finite time. In addition, the trajectories in the region |ε2(t) | > ξ
also converge to the boundaries ±ξ in finite time [38].
To sum up, wherever trajectories of the pneumatic servo system
(4) are, the sliding mode surface s(t) = 0 is reached in finite time.
Moreover, according to Lemma 3 and the sliding mode surface
(47), trajectories of the pneumatic servo system (4) converge to
zero along the sliding mode surface s(t) = 0 in finite time and the
sliding time T s is expressed as follows:
Ts =
2
IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816
© The Institution of Engineering and Technology 2017
(58)
(52)
(L(t) − z3(t))s(t)
γ2 − 1
where V 1(t0) is the initial value of V 1(t).
In another case, ε2(t) = 0 is considered in the pneumatic servo
system (4), please see path II of Fig. 4. Substituting (50) into
differential formula of the tracking velocity error ε2(t) in (46), it
follows that:
|ε2(t) | ≤ ξ
ε2(t)
(56)
where e3(t) = L(t) − z3(t) is the observer error. According to
formulas (58) and (59), when s(t) > 0 or s(t) < 0, ε̇2(t) ≤ − k3 or
ε̇2(t) ≥ k3 is set up, respectively. Therefore, it is known that
ε2(t) = 0 is not an attractor. It also means that there exists a small
enough positive constant ξ such that
Substituting (50) into (53), it follows that:
V̇ 1(s(t)) = k2γ2 | ε2(t) | 2
ψ (t)V 1(s(t0))1/2 + ψ 1(t)
1
ln 2
ψ 2(t)/2
ψ 1(t)
|L(t) − z3(t) | ≤ k4 | s(t)|
According to (46), the (52) is rewritten as follows:
γ2 − 1
Tr ≤
ε̇2(t))
γ1 − 1
−1
(51)
Differentiating (51), there exists
V̇ 1(s(t)) = s(t)ṡ(t)
γ
−1
where ψ 1(t) = 2k2γ2 | ε2(t) | 2 k3, ψ 2(t) = 2k2γ2 | ε2(t) | 2 kl. In
addition, the phase-trajectory plot of the pneumatic servo system
(4) is shown in Fig. 4. Subsequently, two cases are analysed as
follows.
In the first case, the reachability of NFTSM control is not
affected under the condition of ε2(t) ≠ 0, please see path I of Fig. 4.
When ε2(t) ≠ 0, ψ 1(t) > 0 and ψ 2(t) > 0 are established. According
to Lemma 2 and inequality (55), the tracking error (45) moves to
the sliding model surface s(t) = 0 in finite time and the reaching
time T r is expressed as follows:
=
∫
|ε1(t0)|
0
1/ γ2
k2
1 − 1/ γ2
γ2 | ε1(t0)|
−1/ γ2
2
k
γ
1/ γ2
(ε1(t) + k1ε1 1(t))
F
dx
1 γ2 − 1
,
;
γ2 (γ1 − 1)γ2
(γ2 − 1)
γ −1
γ −1
1+ 2
; − k1 | ε1(t0)| 1
(γ1 − 1)γ2
(61)
2813
Fig. 5 Experimental platform with variable load
where ε1(t0) is the initial value of ε1(t). That is, by choosing
k4 = kl + kd
appropriate
parameters
such
that
(L(t) − z3(t))s(t) ≤ kds2(t) and |L(t) − z3(t) | ≤ k4 | s(t)|, tracking error
of the closed-loop system (46) converges to zero in finite time.
This completes the proof. □
4 Experiments and results
In this section, two experiments with variable load, which include a
step signal at 100 mm and a sinusoidal signal at 0.4 Hz, 40 mm
amplitude, are used to verify the effectiveness of extended state
observer-based NFTSM controller for the pneumatic servo system
(4). The experimental platform with variable load is shown in Fig.
5.
Note that maximum stroke of the pneumatic cylinder is 200 mm, the spring coefficient is 0.4 N/mm and the supply absolute
pressure is 0.6 MPa. Meanwhile, adjustable parameters of the
proposed method are listed in Table 1. In addition, the
experimental results are shown in Figs. 6 and 7. Moreover, r is the
given tracking signal, z1 is the observation value of x1, z3 is the
estimate of non-linearity, x1 and L are the tracking trajectories of
pneumatic servo system (4) based on the proposed method and the
linear active disturbance rejection control (LADRC) method,
respectively. u1 and u2 are the control inputs of pneumatic servo
system (4) based on the proposed method and the LADRC method,
respectively. e1 and e2 are the tracking errors of pneumatic servo
system (4) based on the proposed method and the LADRC method,
respectively.
In the first experiment, a step signal at 100mm is tracked as
shown in Fig. 6. Moreover, parameters of the NFTSM controller
(50) are set as k1 = 4.29, k2 = 7.8, k3 = 9.7 and k4 = 0.86. Fig. 6a
shows the displacement for tracking the given step signal via the
extended state observer-based (13) NFTSM method. Meanwhile,
the observation value z1 is almost coincident with x1. Fig. 6b shows
that the piston moves to the given tracking position at about 0.69
and 0.85 s based on the proposed method and the LADRC method,
respectively. As shown in Fig. 6c, due to the control inputs u1 and
u2 eventually tend to be stable, the pneumatic servo system (4) is
able to stay at the given tracking position based on the proposed
method and the LADRC method, respectively. Fig. 6d shows that
the tracking error of the proposed method is <0.2 mm and the
tracking error of the LADRC method is <0.42 mm. In Fig. 6e, the
estimate of non-linearity z3 reaches a steady state finally under the
condition of variable load.
In the second experiment, a given sinusoidal signal at 0.4 Hz,
40 mm amplitude is tracked as shown in Fig. 7. Moreover,
parameters of the NFTSM controller (50) are set as k1 = 2.5,
k2 = 4.5, k3 = 2.7 and k4 = 3.36. In Fig. 7a, tracking trajectory of
the pneumatic servo system (4) based on the proposed method is
better than that based on the LADRC method. In addition, the
observation value z1 is almost coincident with x1. The control inputs
2814
Fig. 6 Experimental results of the proposed method and the LADRC
method for tracking a step signal (a) Tracking trajectory of the proposed
method, (b) Comparison of tracking trajectories, (c) Comparison of control
inputs, (d) Comparison of tracking errors, (e) Estimate of non-linearity
IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816
© The Institution of Engineering and Technology 2017
Table 2 Comparison of experimental results
Controller
Step signal
Sinusoidal signal
Convergence Tracking Delay Amplitude
time, s
error, mm time, s error, mm
LADRC
in this paper
0.85
0.69
0.42
0.2
0.42
0.11
3.1
1.2
arranged in Table 2. From Table 2, it is known that the pneumatic
servo system (4) based on the proposed method in this paper has
better response rapidity and control precision compared with the
LADRC method. Therefore, the effectiveness of the extended state
observer-based NFTSM method is confirmed in the experimental
platform with variable load.
5 Conclusion
A NFTSM finite-time tracking control strategy has been designed
in this paper to guarantee the response rapidity and control
precision of pneumatic servo system via an extended state
observer. The extended state observer has been introduced to
estimate the total disturbances which include strong non-linearity,
unmodelled dynamics and external disturbances. The NFTSM
controller has been designed by combining with the estimation
value of extended state observer. Moreover, the corresponding
theoretical analyses on sufficiently small observation error and
stabilisation of the closed-loop system have also been proved in
finite time. Finally, by comparing the experimental results, the
proposed method has obvious improvement for performances of
the pneumatic servo system.
6 Acknowledgment
The authors would like to thank the anonymous reviewers for their
detailed comments which helped to improve the quality of the
paper. The work was supported by the National Natural Science
Foundation of China (51505413, 61573301).
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IET Control Theory Appl., 2017, Vol. 11 Iss. 16, pp. 2808-2816
© The Institution of Engineering and Technology 2017
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