1. Research Article
A fast converging robust controller using adaptive second order sliding mode
Sanjoy Mondal n
, Chitralekha Mahanta
Department of Electronics and Electrical Engineering, Indian Institute of Technology Guwahati, Guwahati 781039, India
a r t i c l e i n f o
Article history:
Received 10 May 2012
Received in revised form
5 July 2012
Accepted 24 July 2012
Available online 14 August 2012
This paper was recommended for
publication by Dr. Jeff Pieper
Keywords:
Second order sliding mode control
Composite nonlinear feedback
Adaptive tuning
Uncertain system
a b s t r a c t
This paper proposes an adaptive second order sliding mode (SOSM) controller with a nonlinear sliding
surface. The nonlinear sliding surface consists of a gain matrix having a variable damping ratio. Initially
the sliding surface uses a low value of damping ratio to get a quick system response. As the closed loop
system approaches the desired reference, the value of the damping ratio gets increased with an aim to
reducing the overshoot and the settling time. The time derivative of the control signal is used to design
the controller. The actual control input obtained by integrating the derivative control signal is smooth
and chattering free. The adaptive tuning law used by the proposed controller eliminates the need of
prior knowledge about the upper bound of system uncertainties. Simulation results demonstrate the
effectiveness of the proposed control strategy.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
Good transient performance is one important issue in both
tracking and regulatory control problems. In general, fast settling
time and small overshoot are the two important requirements in
most of the controller design problems. However, it is well known
that quick response produces large overshoot which is not at all
desirable in many practical electromechanical applications [1].
A low overshoot can be achieved only at the cost of high settling
time. Thus, most of the design problems attempt to strike a trade-
off between response time and overshoot keeping the damping
ratio fixed. Hence the user is compelled to choose either fast
response or low overshoot. This particular problem can be solved
by using the composite nonlinear feedback (CNF) technique [2–4].
The CNF method uses a variable damping ratio for achieving
improvement of both these parameters in the transient perfor-
mance. Initially, a low value of damping ratio is chosen to ensure
quick response and as the output approaches the reference, the
damping ratio is increased to reduce overshoot. In [5–8], the CNF
method is used to design a nonlinear sliding surface which,
starting from an initial low value, gradually increases the damp-
ing ratio as the output approaches the set point. Initially, the low
value of damping ratio results in a quick response and as the
output approaches the set point, the damping ratio increases and
reduces the overshoot.
Another important requirement of closed loop control system
is robustness against uncertainty and modeling error. Most of the
physical systems are approximated by lower order mathematical
models for ease in analysis. However, there is always a model
mismatch between the actual plant and the mathematical model
used for the controller design. These model variations along with
external unknown disturbances affect the performance of the
controller. In minimizing the effect of modeling error and external
disturbance, sliding mode control [9] has established itself as an
effective robust control technique as robustness is inherent in this
control scheme, if the uncertainty satisfies the matching condition,
i.e. the uncertainty is spanned over the input matrix or mathema-
tically, uncertainty A span (input matrix) [9–15]. In spite of claimed
robustness, the sliding mode controller (SMC) suffers from a major
drawback known as the chattering phenomenon [16]. Chattering is
the high frequency finite amplitude oscillations occurring because of
the discontinuous control signal used in the SMC. This chattering
phenomenon is extremely dangerous for actuators used in practical
electromechanical systems. Active research is continuing to devise
ways for chattering elimination. One approach is to approximate the
signum function in the discontinuous control by a high gain satura-
tion function. However, in this method the SMC has to compromise
with its insensitivity feature. Another technique is based on the
observer design [16]. A third approach is to design a higher order
sliding mode control which retains the property of robustness and
also eliminates chattering [17,18].
The main idea of higher order sliding mode control is to keep
not only the sliding surface but also its higher order derivatives to
zero. In the case of the r-th order sliding mode control, the (rÀ1)
time derivatives of the control are continuous everywhere except
Contents lists available at SciVerse ScienceDirect
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ISA Transactions
0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.07.006
n
Corresponding author. Tel.: þ91 361 258 2508; fax: þ91 361 258 2542.
E-mail addresses: m.sanjoy@iitg.ernet.in, mondal.sanjoy@gmail.com,
m.sanjoy@iitg.ac.in (S. Mondal), chitra@iitg.ernet.in (C. Mahanta).
ISA Transactions 51 (2012) 713–721

2. on the sliding manifold. In this scheme, instead of influencing the
first time derivative of the sliding variable, the signum function
acts on its higher order time derivatives. The r-th derivative is
mostly supposed to be discontinuous [18–20]. In [21] quasi-
continuous controller based on higher order sliding mode is used.
The second order sliding mode (SOSM) controller has been
extensively used in the last two decades. Different SOSM algorithms
such as twisting and super twisting [17], suboptimal [22,23] and drift
algorithm are reported in the literature. In [24] a nonlinear fluid flow
model is used to analyze and control DiffServ Network using second
order sliding mode. A proportional integral derivative (PID) sliding
surface is used in [25] to design a SOSM controller for controlling a
linear uncertain plant. In both these control approaches [24,25], prior
knowledge about the upper bound of system uncertainty is a design
prerequisite. However, in practice it is very difficult to obtain advance
knowledge about the upper bound of the uncertainty. As such, it is
necessary to devise methods to get rid of this stringent requirement
while designing a SOSM controller. The major contributions of this
paper are the following:
(i) A nonlinear sliding surface based second order sliding mode
controller is proposed to improve the transient performance
of the closed loop system.
(ii) The basic idea of the proposed second order sliding mode
controller is that the discontinuous sign function is made to
act on the time derivative of the control input and actual
control signal obtained after integration is continuous and
hence chattering is removed.
(iii) An adaptive gain tuning law [13,26–30] is adopted in the
proposed SOSM controller to estimate the unknown upper
bound of the system uncertainty.
(iv) The proposed control law results in lesser total variation in
the control input and requires lower control energy as com-
pared to some existing control methods.
This paper is organized as follows:
The proposed nonlinear sliding surface and its stability are
described in Section 2. The design procedure of the proposed
second order sliding mode controller is discussed in Section 3. The
adaptive tuning law used in the proposed SOSM controller is
explained in Section 4. Simulation results obtained by applying
the proposed adaptive SOSM controller are presented in Section 5.
Conclusions are drawn in Section 6.
2. Problem description
A class of dynamic system is considered as follows:
_x ¼ f½xðtÞŠþg½xðtÞŠuðtÞ
y ¼ s½xðtÞŠ ð1Þ
where xðtÞARn
is the state variable, uAR is the control input and
s½xðtÞŠAR is the measured output function known as sliding vari-
able. Moreover, f½xðtÞŠ and g½xðtÞŠ are smooth functions. The sliding
function s½xðtÞŠ is considered as completely differentiable with its r
successive time derivatives completely known. The aim of the first
order sliding mode control is to force the state trajectories to move
along the sliding manifold sðxÞ ¼ 0. In higher order sliding mode
control, for example in the r-th order sliding mode control, the
purpose is to move the state along the switching surface sðxÞ ¼ 0
and to keep its (rÀ1) successive time derivatives to zero by using a
suitable discontinuous control action [17]. The r-th order sliding
mode sr
ðxÞ satisfies the following equation:
sr
ðxÞ ¼ aðxÞþbðxÞu ð2Þ
where aðxÞ ¼ Lr
f sðxÞ and bðxÞ ¼ LgLrÀ1
f sðxÞ. Here Lf and Lg are the lie
derivatives of the smooth functions in (1) [18].
With the relative degree n, the above nonlinear system (1),
using appropriate transformation, can be transformed into the
following Brunowsky canonical form [5]:
_xiðtÞ ¼ xiþ 1ðtÞ, i ¼ 1, . . . ,ðnÀ1Þ
_xnðtÞ ¼ an1x1ðtÞþan2x2ðtÞþ Á Á Á þannxnðtÞþBnuðtÞþBnf mðx,tÞ ð3Þ
The above system can be expressed as
_xðtÞ ¼ AxðtÞþBðuðtÞþf mðx,tÞÞ
yðtÞ ¼ C1xðtÞ ð4Þ
where A, B and C1 matrices are given by
A ¼
0 1 . . . 0 0
0 0 1 . . . 0
:
:
0 0 0 . . . 1
an1 an2 an3 . . . ann
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
, B ¼
0
0
:
:
0
Bn
2
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
5
ð5Þ
C1 is the output matrix, it depends on the no of output if
yðtÞ ¼ x1ðtÞ, then C1 ¼ ½1 0 0 . . . 0Š. In general xðtÞARn
is the state
vector, uðtÞARm
is the control input and fmðx,tÞARm
represents
parametric perturbation and external disturbances and is assu-
med to be matched. Moreover, AARnÂn
, BARnÂm
and C1 ARpÂn
are
known matrices with proper dimension and n, m, p are also
known. Let us consider
Pair ðA; BÞ is controllable.
Pair ðA; C1Þ is observable.
Matrix B is of full rank, i.e. rankðBÞ ¼ m.
The uncertainty f mðx,tÞ is unknown but bounded and its time
derivative exists.
The system (4) can be transformed into regular form by using a
transformation matrix Tr [5], such that zðtÞ ¼ TrxðtÞ ¼ ½z1ðtÞ z2ðtÞŠT
.
Then the following transformed system is obtained:
_z1ðtÞ ¼ a11z1ðtÞþa12z2ðtÞ
_z2ðtÞ ¼ a21z1ðtÞþa22z2ðtÞþB2uðtÞþB2f mðz,tÞ
yðtÞ ¼ Cz1ðtÞ ð6Þ
The matrix sub-blocks in (6) can be obtained as
TrATT
r ¼
a11 a12
a21 a22
#
, TrB ¼
0
B2
#
ð7Þ
Here, z1ðtÞARnÀm
, z2ðtÞARm
and C ¼ C1ðTrÞÀ1
. For design purpose
m¼1, i.e. a SISO system is considered.
The linear sliding surface designed for the above system can be
expressed as
s½zðtÞŠ ¼ ½ci 1ŠeðtÞ ¼ ½ci 1Š½zðtÞÀzdðtÞŠ
cT
¼ ½ci 1Š ð8Þ
where zd is the desired trajectory. Moreover, ci,i ¼ 1, . . . ,nÀ1 are
real positive constants such that the characteristic equation
fnÀ1
þ
PnÀ1
i ¼ 1 cifiÀ1
¼ 0 has all the roots with negative real parts.
For designing the nonlinear sliding surface, the sliding surface
is selected as [5]
cT
ðtÞ ¼ ½FÀCðr,yÞaT
12P 1Š ð9Þ
where Cðr,yÞ is a nonlinear function. Then
s½zðtÞŠ ¼ cT
ðtÞeðtÞ ¼ ½FÀCðr,yÞaT
12P 1Š
z1ðtÞÀz1dðtÞ
z2ðtÞÀz2dðtÞ
#
ð10Þ
where eðtÞ ¼ ½e1ðtÞ e2ðtÞŠT
, e1ðtÞ ¼ z1ðtÞÀz1dðtÞ and e2ðtÞ ¼ z2 ðtÞÀz2d
ðtÞ. Here F is chosen such that ða11Àa12FÞ has stable poles and the
S. Mondal, C. Mahanta / ISA Transactions 51 (2012) 713–721714

3. dominant poles have low damping ratio. Furthermore, Cðr,yÞ is a
negative nonlinear function which decreases from zero to a high
negative value and thus the damping ratio for the overall system
changes from an initial low value to a higher one [5,8]. Moreover, P is
a positive definite matrix which can be obtained by solving the
Lyapunov criterion given by
ða11Àa12FÞT
PþPða11Àa12FÞ ¼ ÀW ð11Þ
where W is a positive definite matrix.
2.1. Stability of the sliding surface
During the sliding mode, s½zðtÞŠ ¼ 0 and hence from (10)
e2ðtÞ ¼ ÀFe1ðtÞþCðr,yÞaT
12Pe1ðtÞ ð12Þ
Using (6) and (12) yields
_e1ðtÞ ¼ ða11Àa12F þa12Cðr,yÞaT
12PÞe1ðtÞþgðtÞ ð13Þ
where gðtÞ ¼ a11z1dðtÞþa12z2dðtÞÀ_z1dðtÞ. Suppose that the desired
trajectory zd(t) is consistently generated by using the system
model. Then there exists some control ud(t) such that
_z1dðtÞ ¼ a11z1dðtÞþa12z2dðtÞ
_z2dðtÞ ¼ a21z1dðtÞþa22z2dðtÞþB2udðtÞ
yðtÞ ¼ Cz1ðtÞ ð14Þ
Using (13) and (14) yields,
_e1ðtÞ ¼ ða11Àa12F þa12Cðr,yÞaT
12PÞe1ðtÞ ð15Þ
To prove the stability of the sliding surface, let us consider the
following Lyapunov function:
V1ðtÞ ¼ eT
1ðtÞPe1ðtÞ
Taking its derivative and using ð15Þ yields
_V 1ðtÞ ¼ _eT
1ðtÞPe1ðtÞþeT
1ðtÞP_e1ðtÞ ¼ eT
1ðtÞða11Àa12FÞT
Pe1ðtÞ
þeT
1ðtÞPða11Àa12FÞe1ðtÞþ2eT
1ðtÞPa12Cðr,yÞaT
12Pe1ðtÞ
¼ eT
1ðtÞ½ða11Àa12FÞT
PþPða11Àa12FÞŠe1ðtÞ
þ2eT
1ðtÞPa12Cðr,yÞaT
12Pe1ðtÞ
¼ eT
1ðtÞ½ða11Àa12FÞT
PþPða11Àa12FÞþ2Pa12Cðr,yÞaT
12PŠe1ðtÞ
¼ eT
1ðtÞ½ÀW þ2Pa12Cðr,yÞaT
12PŠe1ðtÞ
Let there exist a matrix Q ¼ eT
1ðtÞPa12:
Thus _V 1ðtÞ can be simplified as
_V 1ðtÞ ¼ ÀeT
1ðtÞWe1ðtÞþ2QCðr,yÞQT
Since W 40 and Cðr,yÞo0
_V 1ðtÞo0 ð16Þ
Thus the existence of the sliding mode is proved [6].
3. Design of second order sliding mode controller
Let us consider a nonlinear sliding surface
s½zðtÞŠ ¼ cT
ðtÞeðtÞ ð17Þ
where the coefficient matrix of the sliding surface is given by
cT
ðtÞ ¼ ½FÀCðr,yÞaT
12P1Š. So,
_s½zðtÞŠ ¼ _cT
ðtÞeðtÞþcT
ðtÞ_eðtÞ ð18Þ
Using (6) the following can be written as
_eðtÞ ¼ _zðtÞÀ_zdðtÞ ¼ AregzðtÞþBuðtÞþBf mðz,tÞÀ_zdðtÞ ð19Þ
where f mðz,tÞ contains all the uncertain terms along with
disturbances and Areg can be expressed as
Areg ¼
a11 a12
a21 a22
#
ð20Þ
Using (18) and (19), the following is obtained:
_s½zðtÞŠ ¼ _cT
ðtÞzðtÞÀ_cT
ðtÞzdðtÞÀcT
ðtÞ_zdðtÞ
þcT
ðtÞðAregzðtÞþBuðtÞþBf mðz,tÞÞ ð21Þ
Taking the second derivative of s½zðtÞŠ yields
€s½zðtÞŠ ¼ _cT
ðtÞ_zðtÞþ €c
T
ðtÞzðtÞþcT
ðtÞAreg _zðtÞ
þ _cT
ðtÞAregzðtÞþcT
ðtÞB _uðtÞþ _cT
ðtÞBuðtÞ
þcT
ðtÞB_f mðz,tÞþ _cT
ðtÞBf mðz,tÞÀ
d
dt
ð_cT
ðtÞzdðtÞþcT
ðtÞ_zdðtÞÞ
¼ cT
ðtÞA2
regzðtÞþ2_cT
ðtÞAregzðtÞþ €c
T
ðtÞzðtÞþcT
ðtÞAregBuðtÞ
þ2_cT
ðtÞBuðtÞþcT
ðtÞB _uðtÞÀ€c
T
ðtÞzdðtÞÀ2_cT
ðtÞ_zdðtÞÀcT
ðtÞ€zdðtÞ
þcT
ðtÞAregBf mðz,tÞþ2_cT
ðtÞBf mðz,tÞþcT
ðtÞB_f mðz,tÞ ð22Þ
Let all the uncertainties be norm bounded and cT
ðtÞAregBf mðz,tÞþ
2_cT
ðtÞBf mðz,tÞþcT
ðtÞB_f mðz,tÞ ¼ DFðt,zðtÞÞ.
If it is possible to bring s½zðtÞŠ and _s½zðtÞŠ to zero in finite time
by using a discontinuous control signal _uðtÞ, then the actual input
to the system, u(t), can be obtained by integrating the discontin-
uous signal and thus u(t) becomes continuous. Hence the unde-
sired high frequency oscillations in the control signal u(t) present
in the first order sliding mode control are eliminated.
Assuming y1½zðtÞŠ ¼ s½zðtÞŠ and y2½zðtÞŠ ¼ _s½zðtÞŠ, the system
dynamics can be written as
_y1½zðtÞŠ ¼ y2½zðtÞŠ
_y2½zðtÞŠ ¼ F½zðtÞ,uðtÞŠþG½zðtÞŠvðtÞ ð23Þ
where vðtÞ ¼ _uðtÞ and F½zðtÞ,uðtÞŠ collects all the uncertain terms
not involving _uðtÞ. Thus the systems (21) and (22) are now
controlled by the input v(t). A sliding mode controller can be
designed for the above system using the control input v(t) to keep
the system trajectories in the sliding manifold.
Let the sliding manifold be considered as
S½zðtÞŠ ¼ y2½zðtÞŠþEy1½zðtÞŠ ð24Þ
where EZ0 is a positive constant. Taking the derivative of (24)
yields
_S½zðtÞŠ ¼ _y2½zðtÞŠþE_y1½zðtÞŠ ð25Þ
Using (21) and (22), the following is obtained:
_S½zðtÞŠ ¼ cT
ðtÞA2
regzðtÞþ2_cT
ðtÞAregzðtÞþ €c
T
ðtÞzðtÞþcT
ðtÞAregBuðtÞ
þ2_cT
ðtÞBuðtÞþcT
ðtÞB _uðtÞÀ€c
T
ðtÞzdðtÞÀcT
ðtÞ€zdðtÞ
À2_cT
ðtÞ_zdðtÞþE _s½zðtÞŠþDFðt,zðtÞÞ ð26Þ
Using the constant plus proportional reaching law yields
_S½zðtÞŠ ¼ Àk1S½zðtÞŠÀk2 signðS½zðtÞŠÞ ð27Þ
Using (26) and (27), the control law is obtained as
_uðtÞ ¼ ÀðcT
ðtÞBÞÀ1
½cT
ðtÞA2
regzðtÞþ2_cT
ðtÞAregzðtÞþ €c
T
ðtÞzðtÞ
þcT
ðtÞAregBuðtÞþ2_cT
ðtÞBuðtÞÀ€c
T
ðtÞzdðtÞÀcT
ðtÞ€zdðtÞ
À2_cT
ðtÞ_zdðtÞþk1S½zðtÞŠþk2 signðS½zðtÞŠÞþE _s½zðtÞŠŠ ð28Þ
The above control law (28) is realizable if and only if cT
ðtÞB is a
nonsingular square matrix so that its inverse, ðcT
ðtÞBÞÀ1
, exists.
Moreover, k1 Z0 and k2 4f9ðcT
ðtÞAregBf mðz, tÞþ2_cT
ðtÞBf mðz,tÞ
þcT
ðtÞB_f mðz,tÞÞ9 ¼ 9DFðt,zðtÞÞ9g to satisfy the reaching law condi-
tion S½zðtÞŠ_S½zðtÞŠo0.
Proof. A Lyapunov function is defined as V2ðtÞ ¼ 1
2S½zðtÞŠ2
. Now
taking the derivative of V2ðtÞ yields
_V 2ðtÞ ¼ S½zðtÞŠ_S½zðtÞŠ ¼ S½zðtÞŠ½cT
ðtÞA2
regzðtÞþ2_cT
ðtÞAregzðtÞ
þ €c
T
ðtÞzðtÞþcT
ðtÞAregBuðtÞþ2_cT
ðtÞBuðtÞþcT
ðtÞB _uðtÞ
À€c
T
ðtÞzdðtÞÀcT
ðtÞ€zdðtÞÀ2_cT
ðtÞ_zdðtÞþE _s½zðtÞŠþDFðt,zðtÞÞŠ
S. Mondal, C. Mahanta / ISA Transactions 51 (2012) 713–721 715

4. Using the control law ð28Þ yields
_V 2ðtÞ ¼ S½zðtÞŠ½DFðt,zðtÞÞÀk1S½zðtÞŠÀk2 signðS½zðtÞŠÞŠ
o9S½zðtÞŠ99DFðt,zðtÞÞ9Àk29S½zðtÞŠ9
o9S½zðtÞŠ9ð9DFðt,zðtÞÞ9Àk2Þo0 ð29Þ
Clearly, (29) implies that if k1 Z0 and k2 49DFðt,zðtÞÞ9, the control
law (28) forces the sliding manifold S½zðtÞŠ to zero in finite
time.
Remark 1. For exactly measuring or estimating the derivative of
variables an exact robust differentiator [20] is available.
4. Design of adaptive tuning law for the SOSM controller
In practice, the upper bound of the system uncertainty is often
unknown in advance and hence 9DFðt,zðtÞÞ9 is difficult to find. It is
assumed that the uncertainty is unknown but bounded, i.e.
9DFðt,zðtÞÞ9oT, where T is an unknown positive constant. An
adaptive tuning law [13,26–28] is adopted to estimate T using
which the control law (28) can be written as
_uðtÞ ¼ ÀðcT
ðtÞBÞÀ1
½cT
ðtÞA2
regzðtÞþ2_cT
ðtÞAregBzðtÞþ €c
T
ðtÞzðtÞ
þcT
ðtÞAregBuðtÞþ2_cT
ðtÞBuðtÞÀ€c
T
ðtÞzdðtÞÀcT
ðtÞ€zdðtÞ
À2_cT
ðtÞ_zdðtÞþk1S½zðtÞŠþ ^T signðS½zðtÞŠÞþE _s½zðtÞŠŠ ð30Þ
where ^T estimates the value of T. Defining the adaptation error as
~T ¼ ^TÀT, the parameter ^T is estimated by using the adaptation law
given as follows:
_^T ¼ n9S½zðtÞŠ9 ð31Þ
where n is a positive constant. A Lyapunov function V3ðtÞ is selected
as V3ðtÞ ¼ 1
2 S2
½zðtÞŠþ 1
2 g~T
2
whose time derivative is as follows:
_V 3ðtÞ ¼ S½zðtÞŠ_S½zðtÞŠþg~T
_~T
Using ð26Þ yields
_V 3ðtÞ ¼ S½zðtÞŠ½cT
ðtÞA2
regzðtÞþ2_cT
ðtÞAregzðtÞ
þ €c
T
ðtÞzðtÞþcT
ðtÞAregBuðtÞþ2_cT
ðtÞBuðtÞþcT
ðtÞBuðtÞÀ€c
T
ðtÞzdðtÞ
ÀcT
ðtÞ€zdðtÞÀ2_cT
ðtÞ_zdðtÞþE _s½zðtÞŠþDFðt,zðtÞÞŠþgð ^TÀTÞ
_^T
Using ð30Þ and ð31Þ yields
_V 3ðtÞ ¼ S½zðtÞŠ½DFðt,zðtÞÞÀ ^T signðSÞÀk1S½zðtÞŠŠþgð ^TÀTÞn9S½zðtÞŠ9
Since k1 Z0, the above equation can be written as
_V 3ðtÞr9DFðt,zðtÞÞ99S½zðtÞŠ9À ^T9S½zðtÞŠ9þT9S½zðtÞŠ9
ÀT9S½zðtÞŠ9þgð ^TÀTÞn9S½zðtÞŠ9
rð9DFðt,zðtÞÞ9ÀTÞ9S½zðtÞŠ9Àð ^TÀTÞ9S½zðtÞŠ9þgð ^TÀTÞn9S½zðtÞŠ9
rÀðÀ9DFðt,zðtÞÞ9þTÞ9S½zðtÞŠ9Àð ^TÀTÞðÀgn9S½zðtÞŠ9þ9S½zðtÞŠ9Þ
rÀbS½zðtÞŠ
ffiffiffi
2
p
9S½zðtÞŠ=
ffiffiffi
2
p
9Àbn
ffiffiffiffiffiffiffiffi
g=2
p
ð ^TÀTÞ=
ffiffiffiffiffiffiffiffi
g=2
p
where bS½zðtÞŠ ¼ ðTÀ9DFðt,zðtÞÞ9Þ and bn ¼ ð9S½zðtÞŠ9Àgn9S½zðtÞŠ9Þ
So, _V 3ðtÞrÀminfbS½zðtÞŠ
ffiffiffi
2
p
,bn
ffiffiffiffiffiffiffiffi
2=g
p
gð9S½zðtÞŠ=
ffiffiffi
2
p
9þ ~T
ffiffiffiffiffiffiffiffi
g=2
p
Þ
rÀbxV3ðtÞ1=2
ð32Þ
where bx ¼ minfbS½zðtÞŠ
ffiffiffi
2
p
,bn
ffiffiffiffiffiffiffiffi
2=g
p
g with bx 40. The above inequal-
ity holds if
_^T ¼ n9S½zðtÞŠ9, bS½zðtÞŠ 40,bn 40, T Z9DFðt,zðtÞÞ9 and
go1=n. Therefore, finite time convergence to a domain S½zðtÞŠ ¼ 0
is guaranteed from any initial condition [14,26].
Remark 2. Practically, 9S½zðtÞŠ9 cannot become exactly zero in
finite time and thus the adaptive parameter ^T may increase
boundlessly [26]. A simple way of overcoming this disadvantage
is to use the dead zone technique [9,26] to modify the adaptive
tuning law (31) as
_^T ¼
n9S½zðtÞŠ9,9S½zðtÞŠ9Zd
0,9S½zðtÞŠ9od
(
ð33Þ
where d is a small positive constant.
It is evident from (30) that _u is discontinuous but integration
of _u yields a continuous control law u. Hence the undesired high
frequency chattering of the control signal is eliminated. Thus the
above adaptive SOSM control with nonlinear sliding surface offers
following advantages. Firstly, an improved transient performance
can be obtained without the knowledge about the upper bound of
the system uncertainties. Secondly, the chattering in the control
input is eliminated.
4.1. Choice of nonlinear function Cðr,yÞ
The nonlinear function Cðr,yÞ is used to change the system’s
damping ratio as the output y approaches the reference position r
[3,8]. It possesses the properties mentioned as follows:
When the output is far from the reference value, Cðr,yÞ equals
to zero (or a very small value).
As the output approaches its final desired value, Cðr,yÞ value
gradually becomes highly negative.
The nonlinear function ðCðr,yÞÞ should be continuous, differ-
entiable and its higher derivatives must exist.
The choice of Cðr,yÞ is not unique. Some examples of Cðr,yÞ used
in literature are discussed as follows [1]:
Cðr,yÞ ¼ ÀbeÀaðyÀrÞ2
ð34Þ
where r and y are the reference input and the system output
respectively. Here b40,a40 are the tuning parameters. Higher
the value of the tuning parameter a, faster the convergence of
Cðr,yÞ to its final steady state value. Tuning parameter b con-
tributes to decide the final damping ratio along with the matrix P.
It is observed that the nonlinear function decreases its value from
0 to Àb as the value of error ðyÀrÞ decreases from infinity (very
high value) to zero. Another type of nonlinear function Cðr,yÞ
used is given by
Cðr,yÞ ¼ ÀbeÀa9yÀr9
ð35Þ
where a40 and b40 are the tuning parameters. The value of
Cðr,yÞ changes from 0 to Àb as the error ðyÀrÞ approaches zero
[31] from a high value.
Cðr,yÞ can assume the following form also
Cðr,yÞ ¼ À
b
1ÀeÀ1
ðeÀð1ÀðyÀy0Þ=ðrÀy0ÞÞ2
ÀeÀ1
Þ ð36Þ
where r is the reference input, y0 ¼ yð0Þ is the initial state and b is
the tuning parameter. The function Cðr,yÞ changes its value from
0 to Àb as the error ðyÀrÞ approaches zero from a high value.
5. Simulation results
The proposed adaptive SOSM controller is simulated in
MATLAB-Simulink by using ODE 8 solver with a fixed step size
of 0.005 s.
5.1. Time response of second order process with time delay
The proposed adaptive SOSM controller is applied to a second
order uncertain system with time delay. The performance of the
proposed adaptive SOSM controller using nonlinear sliding surface is
S. Mondal, C. Mahanta / ISA Transactions 51 (2012) 713–721716

5. compared with that obtained by using adaptive SOSM controller
with different linear sliding surfaces. The second order process with
time delay is given by [32]
GðsÞ ¼
0:05eÀ0:5s
ð1þ0:1sÞð1þsÞ
ð37Þ
which can be simplified to the following form using Pade´ approx-
imation [32]:
GðsÞ ¼
0:05
ð1þ0:5sÞð1þ0:1sÞð1þsÞ
ð38Þ
For the above system (38), the state space model given by (4) is
obtained as
A ¼
0 1 0
0 0 1
À20 À32 À13
2
6
4
3
7
5, B ¼
0
0
1
2
6
4
3
7
5, C1 ¼ ½1 0 0Š ð39Þ
The uncertainty considered here is fmðx,tÞ ¼ 0:5 sinð10tÞ.
Since the system is already in the regular form, the transfor-
mation matrix is chosen as Tr ¼ I. Using (6), (19) and (20), the
submatrices can be written as z1ðtÞ ¼ ½x1ðtÞ x2ðtÞŠT
, z2ðtÞ ¼ x3ðtÞ and
coefficient matrices
a11 ¼
0 1
0 0
, a12 ¼
0
1
a21 ¼ ½À20 À32Š, a22 ¼ ½À13Š, B2 ¼ ½1Š ð40Þ
Design of nonlinear sliding surface: The nonlinear sliding surface
consists of a linear and a nonlinear term. Initially, the non-
linear term has a very low value and as the system gradually
approaches its desired position, the nonlinear term decreases
its value to a high negative value and thereby changes the
value of the damping ratio. As a result, the system response
has low overshoot and small settling time. Let us recall,
cT
ðtÞ ¼ ½FÀCðr,yÞaT
12P 1Š ð41Þ
Here the gain matrix F is designed for low damping ratio and
high setting time. Moreover, F can be designed by using pole
placement technique. Let initially the system have damping
ratio x1 and settling time ts1. The closed loop poles are located
at ðÀx1on þ
ffiffi
ð
p
x2
1À1ÞonÞ and ðÀx1onÀ
ffiffi
ð
p
x2
1À1ÞonÞ. The nat-
ural frequency of oscillations on is given by
on ¼
4
x1ts1
ð42Þ
Let us choose the initial damping ratio x1 ¼ 0:4 and settling
time ts1 ¼ 3:5. Using (42), the poles can be found to be located
at À1:1429þ2:6186i,À1:1429À2:6186i. Thus F can be calcu-
lated using pole placement technique as F ¼ ½8:1633 2:2857Š.
Solution of the Lyapunov equation (11) yields
P ¼
0:2144 0:0061
0:0061 0:0246
using W ¼
0:1 0
0 0:1
ð43Þ
The nonlinear function Cðr,yÞ is designed as
Cðr,yÞ ¼ À215:04eÀaðyÀrÞ2
where a ¼ 20, r ¼ 1 ð44Þ
Design of control law: Using the nonlinear function (44), the
nonlinear sliding surface cT
ðtÞ is designed as explained earlier.
The overall SOSM control law is obtained from (28) choosing
E ¼ 15 and k1 ¼ 10. The adaptive tuning law is designed as
_^T ¼ 0:0159S½zðtÞŠ9 with T0 ¼ 0. Here Areg,B are obtained by using
Eqs. (20) and (40).
5.2. Comparison with adaptive SOSM controller using linear sliding
surface
Transient performance of the nonlinear sliding surface based
adaptive SOSM controller is compared against the adaptive SOSM
controller using linear sliding surface. Following three different
linear sliding surfaces are chosen for our comparison:
1. Linear sliding surface 1 with x ¼ 0:4, ts ¼ 3:5 s.
2. Linear sliding surface 2 with x ¼ 0:5, ts ¼ 3:0 s.
3. Linear sliding surface 3 with x ¼ 0:7, ts ¼ 2:5 s.
The adaptive SOSM control law with linear sliding surface is
obtained from (28), with _cðtÞT
¼ €cðtÞT
¼ 0, given by
_uðtÞ ¼ ÀðcT
ðtÞBÞÀ1
½cT
ðtÞA2
regzðtÞþcT
ðtÞBuþE _s½zðtÞŠ
þk1S½zðtÞŠþ ^T signðS½zðtÞŠÞÀcT
ðtÞ€zdðtÞŠ ð45Þ
The output y(t) which is the angular position x1ðtÞ is plotted in
Fig. 1 for the nonlinear sliding surface based as well as linear
sliding surface based adaptive SOSM controllers. From Fig. 1 it is
clearly observed that the peak overshoot and settling time both
decrease significantly in the case of the nonlinear sliding surface
based adaptive SOSM controller. A detailed comparison of tran-
sient performances between these two types of adaptive SOSM
controllers is presented in Table 1.
For the above example, the nonlinear function Cðr,yÞ is plotted
against time in Fig. 2. It is clear from Fig. 2 that the magnitude of
the nonlinear function decreases from an initial low value to a
high negative value, as the output approaches the reference.
The convergence of adaptive gains ^T for different sliding surfaces
are confirmed in Fig. 3. Notably, the knowledge of the upper bounds
on the uncertainties is not a required prerequisite for designing the
SOSM controller.
0 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time (sec)
Systemoutput
with proposed controller
with ξ=0.4, t =3.5sec
with ξ=0.5, t =3.0sec
with ξ=0.7, t =2.5sec
reference
Fig. 1. Output response of adaptive SOSM controller with different sliding
surfaces.
Table 1
Transient response indices for different values of x and ts.
Type of sliding
surface
Peak
overshoot (%)
Settling
time (s)
Linear sliding surface with x ¼ 0:4, ts ¼3.5 s 24 3.5
Linear sliding surface with x ¼ 0:5, ts ¼3.0 s 8 3.0
Linear sliding surface with x ¼ 0:7, ts ¼2.5 s 5 2.5
Proposed controller with nonlinear sliding surface 0 1.1
S. Mondal, C. Mahanta / ISA Transactions 51 (2012) 713–721 717

6. 5.3. Stabilization of an uncertain system
The problem of ship roll stabilization is taken up now and the
performance of the proposed controller is compared with that of
Fulwani et al. [7]. The state space representation of the ship roll
model [7] is given by (4) with,
A ¼
0 1 0
0 0 1
À2 À2:7 À1:7
2
6
4
3
7
5, B ¼
0
0
1
2
6
4
3
7
5 ð46Þ
C1 ¼ ½1 0 0Š ð47Þ
The uncertainty is considered as fmðx,tÞ ¼ sinð10tÞ.
The objective is to stabilize the output quickly to the equili-
brium. The parameters for designing the controller are chosen as
F ¼ ½25 4Š, E ¼ 30, k1 ¼ 20 and the initial condition is assumed as
xð0Þ ¼ ½0:1 0 0ŠT
[7]. By solving the Lyapunov equation (11), P is
obtained as
P ¼
0:0071 À0:0170
À0:0170 0:1105
where W ¼ 0:34
1 0
0 1
ð48Þ
The nonlinear function Cðr,yÞ is selected as [7]
CðyÞ ¼ À50eÀ100y2
ð49Þ
The adaptive tuning law is designed as
_^T ¼ 49S½zðtÞŠ9 with T0 ¼ 0.
Fig. 4 shows the system output and the control input using the
control law proposed by Fulwani et al. [7]. From Fig. 4 it is
observed that the output converges fast to the reference without
any overshoot. However, it is clear that the control input is not
smooth and contains excessive chattering. Moreover, the start-up
control input is also very high.
The simulation results obtained by using the proposed adaptive
SOSM controller are plotted in Fig. 5. It is noticed that the system
output reaches the reference trajectory in the same time as in the
method of Fulwani et al. [7]. Furthermore, the proposed adaptive
SOSM controller produces a chattering free control input. Also, the
proposed method reduces the start-up control input considerably.
Convergence of the adaptive gain, sliding manifold and the sliding
surface by using the proposed method is shown in Fig. 6.
Input performance: Although good transient performance is the
primary aim of the controller, large variation in the control input
to achieve the same is undesirable. For rating the controller
performance, an index to measure the total variation (TV) in the
control input u(t) has been proposed in literature [33] as
TV ¼
Xn
i ¼ 1
9ui þ1Àui9 ð50Þ
which is desired to have a small value. The total variation is a
good measure of the ‘‘smoothness’’ of a signal. A large value of TV
0 2 4 6 8 10
−250
−200
−150
−100
−50
0
Time (sec)
Nonlinearfunction(Ψ(r,y))
Fig. 2. Evolution of nonlinear function Cðr,yÞ.
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
with proposed controller
with ξ=0.4, t =3.5 sec
with ξ=0.5, t =3.0 sec
with ξ=0.7, t =2.5sec
Adaptivegain(T)
Fig. 3. Adaptive gains of SOSM controller with different sliding surfaces.
Fig. 4. System output and control input using the control law [7].
Fig. 5. System output and control input using the proposed control law.
S. Mondal, C. Mahanta / ISA Transactions 51 (2012) 713–721718

7. means more input usage or a more complicated controller [33].
Also, input energy is another important index for the controller
and is calculated by using the 2-norm method. The control energy
is expected to be as small as possible. The input performance of
the proposed adaptive SOSM controller is computed in terms of
TV and 2-norm energy for the period from 0 to 5 s with a
sampling time of 0.01 s. The input performance of the proposed
adaptive SOSM controller and that of Fulwani et al. [7] are
tabulated in Table 2.
It is noted from Table 2 that the proposed adaptive SOSM
controller offers comparable output performance by applying a
smoother control input having minimal total variation as com-
pared to the controller designed by Fulwani et al. [7].
5.4. Stabilization of a triple integrator system
Stabilization of a triple integrator system with uncertainty is
considered now and performance of the proposed adaptive SOSM
controller is compared with the third order sliding mode con-
troller which uses twisting algorithm and was proposed by
Defoort et al. [19]. The triple integrator system is described as
follows [14,8]:
_x1 ¼ x2
_x2 ¼ x3
_x3 ¼ uþf mðx,tÞ
y ¼ x1 ð51Þ
where fmðx,tÞ ¼ sinð10x1Þ is the bounded uncertainty, y is the
output and the initial condition of the system is assumed as
xð0Þ ¼ ½1 0 À1ŠT
[14].
As explained in [14,15], the control for the system given by
(51) is obtained as [19]
u ¼ unom þudisc ð52Þ
where unom and udisc are defined as [14]
unom ¼ Àsignðx1Þ9x19
1=2
À1:5 signðx2Þ9x29
3=5
À1:5 signðx3Þ9x39
3=4
udisc ¼ udisc,1 þudisc,2 ð53Þ
Here udisc,1 and udisc,2 are given by [19]
udisc,1 ¼ Àad
Z t
0
signðsdÞ dt
udisc,2 ¼ Àld9sd9
1=2
signðsdÞ ð54Þ
where the sliding surface sd ¼ x3À
R t
0 unom dt and ad, ld are chosen
as 100, 5 respectively.
The system states and the control input obtained by using
Defoort et al.’s method are shown in Fig. 7. It is observed from
Fig. 7 that the third order sliding mode controller using twisting
algorithm is not able to reduce the chattering in the control input
completely.
The proposed adaptive SOSM controller using a nonlinear
sliding surface is now applied to the triple integrator system.
The parameters in the control law (28) are chosen as E ¼ 3, k1 ¼ 5
and the nonlinear function is selected as Cðr,yÞ ¼ À60:07eÀ10y2
.
The adaptive tuning law is designed as
_^T ¼ 0:19S½zðtÞŠ9 with T0 ¼ 0.
The nonlinear sliding surface is designed by choosing the damp-
ing ratio x ¼ 0:3 and settling time ts¼5 s [14,15]. The positive
definite matrix P obtained by solving the Lyapunov equation (11)
is given as
P ¼
0:2674 0:0070
0:0070 0:0356
using W ¼
0:1000 0:0000
0:0000 0:1000
ð55Þ
The system states and the control input of the triple integrator
system using the proposed adaptive SOSM controller are plotted
in Fig. 8. It is evident from Fig. 8 that the proposed adaptive SOSM
controller produces faster convergence of the system states to
equilibrium and smoother chattering free control input as com-
pared to the method proposed by Defoort et al. [19]. Convergence
of the adaptive gain, the sliding manifold and the sliding surface
by using the proposed adaptive SOSM controller are shown in Fig.
9. System output obtained by using the proposed adaptive SOSM
controller employing both nonlinear and linear sliding surfaces
are compared with the output obtained by using Defoort et al.’s
method [19] are shown in Fig. 10. From Fig. 10 it is clearly observed
that the best transient performance is demonstrated by the proposed
Table 2
Transient response indices and input performance comparison.
Type of
controller
Over-
shoot (%)
Settling
time (s)
Total
variance (TV)
Control
energy
Fulwani et al.’s
method [7]
0 1.1 137.31 86.72
Proposed adaptive
SOSM controller
0 1.1 8.35 52.18
0 2 4 6 8 10 12 14 16 18 20
−1
−0.5
0
0.5
1
Time (sec)
Systemstates
x1 x2 x3
0 2 4 6 8 10 12 14 16 18 20
−4
−2
0
2
4
6
8
Time (sec)
Controlinput
u
Fig. 7. System states and control input using the control law [19].
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
2
4
6
8
Time (sec)
Adaptivegain
T
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
50
100
Time (sec)
Slidingmanifold
S[z(t)]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
1
2
3
Time (sec)
Slidingsurface
σ[z(t)]
Fig. 6. Adaptive gain, sliding manifold and sliding surface using the proposed
control law.
S. Mondal, C. Mahanta / ISA Transactions 51 (2012) 713–721 719

8. adaptive SOSM controller using nonlinear sliding surface. A detailed
comparison of the transient as well as input performance between
the proposed adaptive SOSM controller and Defoort et al.’s method
[19] are tabulated in Table 3. From Table 3 it is evident that the
proposed controller produces zero overshoot, faster settling time,
lesser input variation and lower control action than those obtained
by using the controller proposed by Defoort et al. [19]. The input
performance is calculated for the period from 0 to 20 s with a
sampling time of 0.01 s.
6. Conclusion
In this paper, an adaptive second order sliding mode (SOSM)
controller using nonlinear sliding surface is proposed to enhance
the transient performance of uncertain systems. The nonlinear
sliding surface introduces a variable damping ratio which adapts
its value in accordance with the system response to ensure
quicker settling time and lesser overshoot. An adaptive tuning
rule is proposed in the SOSM controller to estimate the upper
bound of the system uncertainty. This is a remarkable advantage
of the proposed controller as in all conventional first order sliding
mode controllers, the upper bound of the system uncertainty is
required to be known a priori. Another major benefit offered by
the proposed SOSM controller is the elimination of chattering in
the control input. The proposed adaptive SOSM controller is
applied to some typical examples of uncertain systems and its
transient performance is evaluated. Simulation results show that
compared to certain already existing sliding mode controllers, the
proposed adaptive SOSM controller demonstrates faster stabiliza-
tion at lesser control effort. Moreover, the control input produced
by the proposed adaptive SOSM controller is chattering free and
smooth which makes it suitable for practical control applications.
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Time (sec)
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u
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