1. Multiple Sliding Surface Control for Systems with
Mismatched Uncertainties
Mayuresh B Patil
∗Divyesh Ginoya
College of Engineering, Pune
Email: mayur.mayuresh@gmail.com
∗Email: dlginoya007@gmail.com
S B Phadke
∗P D Shendge
College of Engineering, Pune
Email: sbp.instru@coep.ac.in
∗Email: pds.instru@coep.ac.in
Abstract—This paper proposes a new method for sliding
mode control of systems with mismatched uncertainties. The
proposed method employs a multiple sliding surface (MSS)
approach combined with inertial delay control (IDC) for
estimating unmatched disturbances. It has the advantage of
tackling constant as well as time-variant and state dependent
disturbances/uncertainties. The proposed method is validated by
simulation and compared with traditional sliding mode control
(SMC) and integral sliding mode control (I-SMC) method.
Index Terms—mismatched disturbance, sliding mode control
(SMC), integral sliding mode control (I-SMC), multiple sliding
surface control (MSSC), inertial delay control (IDC)
I. INTRODUCTION
Sliding Mode Control (SMC) has been widely studied
and extensively employed in industrial applications as it’s
conceptually simple, and in particular its powerful ability to
reject disturbances and plant uncertainties [1]-[3]. Most of the
SMC results are satisfactory when systems satisfy matching
conditions i.e. the disturbance is present in the same channel as
that of the input. However, the matching condition assumption
is fairly restrictive and is not satisfied by many practical
systems such as under-actuated mechanical system, electro-
mechanical systems connected in series, flight control systems
[4] and others.
As the demand for attenuation of mismatched uncertain-
ties/disturbances has increased, the research is going on to
solve this problem and to develop different control strategies
[5]-[10]. These mismatched uncertainties affect system sta-
bility and cause deterioration in the output. Hence, one of
the approaches to handle this issue is to focus on system
stability using some classical control design tools such as
Riccati approach [10],[11], adaptive approach [12] etc. The
mismatched uncertainties considered by the authors stated
above must be bounded or vanishing. Again, this assumption
may not be satisfactory for many engineering systems may
suffer from these types of disturbances.
We can state another approach as the integral sliding mode
control (I-SMC). It introduces an integral term into sliding
surface, which makes the system initial states start from
the sliding mode and eliminates the reaching phase. Thus,
I-SMC enhances robustness against matched uncertainty of
the conventional SMC. Compared with SMC for mismatched
uncertainties, the I-SMC method is more useful due to its
simplicity and robustness and has been applied to many
practical systems [13]-[15]. Still, the drawbacks of integral
action come along with the results such as large overshoot
and long settling time.
Recently, many authors introduced a disturbance observer
(DOB) for SMC to eliminate the the effect of uncertain-
ties. Also Non-linear Disturbance Observer (NDOB) along
with SMC has been proved as a good strategy for systems
with mismatched uncertainties [16]. But the major limita-
tion of this strategy is that it can only handle bounded
slow varying/constant disturbances. Multiple sliding surface
control (MSSC) can be used for systems with mismatched
disturbances/uncertainties. For the estimation of uncertainty,
several techniques have been used by several authors. Function
approximation technique is one of them. It is used to transform
the uncertainties into a finite combination of orthonormal basis
function [17].
In this paper, multiple sliding surface control (MSSC)
method is proposed to handle such mismatched uncertainties
using inertial delay control (IDC) [18] - [20]. The key features
of this proposed method are :
1) It allows a sliding controller to be designed as if the
system has a reduced relative degree by defining one of
the states as a synthetic input to the reduced order plant.
2) The synthetic states are then controlled by a second
controller to make sure that the synthetic control will
track the profile defined by the second controller.
3) For estimation of these mismatched distur-
bances/uncertainties inertial delay control (IDC) is
proposed.
This paper is organised as follows : In section II control
using traditional SMC and I-SMC is derived. In Section III
multiple sliding surface control (MSSC) with inertial delay
control (IDC) is proposed. Section IV shows the simulation
results to support the proposed method.
978-1-4673-6190-3/13/ $31.00 c 2013 IEEE

2. II. EXISTING SMC METHODS AND THEIR DRAWBACKS
Consider the following third-order system with mismatched
disturbance, stated as
˙x1 = x2 + d1(t)
˙x2 = x3 + d2(t)
˙x3 = a(x) + b(x)u + d3(t)
y = x1



(1)
where x1, x2 and x3 are states, u is the control input, d1(t),
d2(t) are the disturbances, d3(t) is the lumped uncertainty
and y is the output. It is assumed that these disturbances are
bounded.
A. Traditional SMC
The sliding mode surface and the control input of traditional
SMC are designed as follows:
σ = x3 + c1x2 + c2x1 (2)
u = −b−1
(x)[a(x) + c1x3 + c2x2 + ksgn(σ)] (3)
Combining (1), (2) and (3) we’ll get,
˙σ = −ksgn(σ) + (c2d1(t) + c1d2(t)) (4)
(4) shows that the states of the system (1) will reach
the sliding surface σ = 0 in finite time as long as the
switching gain in control input (3) is designed such that
k > c2||d1(t)|| + c1||d2(t)||. When we analyse the sliding
mode condition, as σ = 0 we’ll get
¨x1 = −c1 ˙x1 − c2x1 + ˙d1 + d2 + c1d1 (5)
Remark 1: From (5) it can be stated that the states can’t
not be drawn to the desired point even if the control law in
(3) can force the states to reach the sliding surface in finite
time. This is the major drawback of traditional SMC and it
can’t be used alone for an unmatched system.
B. Integral SMC
Considering traditional SMC, the very less modified solu-
tion for these unmatched systems is integral SMC (I-SMC).
The sliding surface is designed as
σ = x3 + c1x2 + c2x1 + c3 x1 (6)
The control input can be designed as
u = −b−1
(x)[a(x) + c1x3 + c2x2 + c3x1 + ksgn(σ)] (7)
Considering (1), (6) and (7) we get,
˙σ = −ksgn(σ) + (c2d1(t) + c1d2(t)) (8)
Similar to the above method, the states of the system (1)
will reach the sliding surface in finite time but the condition
is k > c2||d1(t)|| + c1||d2(t)||. Considering the condition σ =
0, we get,
...
x1 = −c1 ¨x1 − c2 ˙x1 − c3 ˙x1 + ¨d1 + ˙d2 + c1
˙d1 (9)
Remark 2: From (9) it can be stated that the state can slide
to the desired point if the system has reached the sliding
surface in finite time and the disturbance has a constant
steady state value. If the disturbances are time-varying or state
dependable this method fails to achieve the desired output.
Also, as stated earlier in this paper, due to integral action
some adverse effects are seen in the control action such as
large overshoot and long settling time.
III. MULTIPLE SLIDING SURFACE CONTROL (MSSC)
DESIGN WITH INERTIAL DELAY CONTROL (IDC)
The goal of the controller is to make x1 track the desired
trajectory x1d. Applying MSS control to the system in (1), the
first sliding surface is defined as
s1 = x1 − x1d (10)
To minimize the initial control we modify our sliding
surface as-
s∗
1 = s1 − s1(0)e−α1t
(11)
where s1(0) is the initial value for s1.
˙s∗
1 = x2 + d1 − ˙x1d + α1s1(0)e−α1t
d1 = ˙s∗
1 − x2 + ˙x1d − α1s1(0)e−α1t
(12)
Using IDC, we can estimate d1. Let the estimation of d1 be
ˆe1 and is defined as-
ˆe1 =
1
τ1s + 1
d1 (13)
A modified second sliding surface is defined as-
s∗
2 = s2 − s2(0)e−α2t
(14)
s∗
2 = x2 − x2d − s2(0)e−α2t
(15)
x2d is the synthetic input and is to design in such a way
that it’ll make s∗
1 ˙s∗
1 < 0 and it will negate the effect of the
uncertainty d1. Solving (12), (13) and (15) together we’ll get
x2d and ˆe1 as-
x2d = ˙x1d − ˆe1 − α1s1(0)e−α1t
− k1s∗
1 (16)
ˆe1 =
1
τ1
[s∗
1 + (k1s∗
1 − s∗
2 − s2(0)e−α2t
] (17)
Moving to next step,
˙s∗
2 = x3 + d2 − ˙x2d + α2s2(0)e−α2t
˙s∗
2 = x3 + e2 + α2s2(0)e−α2t
e2 = ˙s∗
2 − x3 − α2s2(0)e−α2t
(18)
Using IDC, we can estimate e2. Let the estimation of e2 be
ˆe2 and is defined as-
ˆe2 =
1
τ2s + 1
e2 (19)
A modified third sliding surface is defined as-
s∗
3 = s3 − s3(0)e−α3t
(20)
s∗
3 = x3 − x3d − s3(0)e−α3t
(21)

3. x3d is the synthetic input and is to design such that it’ll make
s∗
2 ˙s∗
2 < 0 and it’ll negate the effect of lumped uncertanity e2.
Solving (18), (19) and (21) together we’ll get x3d and ˆe2 as
x3d = − ˆe2 − α2s2(0)e−α2t
− k2s∗
2 (22)
ˆe2 =
1
τ2
[s∗
2 + (k2s∗
2 − s∗
3 − s3(0)e−α3t
)] (23)
Now solving to design control input u,
˙s∗
3 = a(x) + b(x)u + d3 − ˙x3d + α3s3(0)e−α3t
˙s∗
3 = a(x) + b(x)u + e3 + α3s3(0)e−α3t
e3 = ˙s∗
3 − a(x) − b(x)u − α3s3(0)e−α3t
(24)
Again using IDC, we can estimate e3. Let the estimation of
e3 be ˆe3 and is defined as-
ˆe3 =
1
τ3s + 1
e3 (25)
Now, designing control input u such that it will drive s∗
3 to
0 and will negate the effect of e3. Hence the desired control
input u is-
u = −b−1
(x)[a(x) + α3s3(0)e−α3t
+ ˆe3 + k3s∗
3] (26)
and ˆe3 is
ˆe3 =
1
τ3
[s∗
3 + k3s∗
3] (27)
The ultimate boundedness can be proved using Lyapunov
stability criterion.
IV. NUMERICAL SIMULATION
Consider the following system for simulation studies,
˙x1 = x2 + d1(t)
˙x2 = x3 + d2(t)
˙x3 = −2x1 − x2 + ex1
+ u + d3(t)
y = x1



(28)
Both traditional SMC and I-SMC methods are applied
and compared along with the proposed strategy on above
mentioned system. It is assumed that all above disturbances
are acting on the system from initial. These are considered
as uncertainties in the system. These methods are applied for
various types of disturbances stated below. At the end of the
discussion of each case, advantages of MSSC are stated.
A. Case 1 : Constant disturbances
Consider the initial states of system (28) as x(0) =[1 0 0]T
.
Initially, the disturbances are of different values but are consid-
ered as constants. They are taken as- d1(t) = 0.25, d2(t) = 0.1
and d3(t) = 0.2.
If we consider the output, it can be observed from Fig.1(a)
that the traditional SMC fails to drive the state to desired
position. In other words, steady state error is present. This
shows that traditional SMC method is sensitive to mismatched
disturbances.
TABLE I: control parameters for case 1 and case 2
CONTROLLERS PARAMETERS
SMC c1 = 2, c2 = 2, k = 3
I-SMC c1 = 2, c2 = 2, c3 = 3, k = 3
MSSC with IDC τ1 = τ2 = τ3 =0.01, k1 = k2 = k3 =1,
α1 = α2 = α3 =0.1
It is also vary much visible from Fig.1(a) that I-SMC tries to
drive the state to desired equilibrium point but sacrifices more
time. This is relevant to the fact about integral action that it
increases the settling time. Also, it brought the overshoot in
the output and is another drawback of it.
From Fig.2(b) and 2(c), it can be seen that the chattering
in the control input is present due to signum function used in
the design procedure. It is another drawback of the traditional
SMC as well as I-SMC method.
In Fig. 1(a), both MSSC and I-SMC methods can finally
suppress the mismatched uncertainties, but I-SMC method
brings the state around 65 sec whereas MSSC method brings it
around 7 sec to desired equilibrium point. It shows that MSSC
has a much quicker convergence rate than that of the I-SMC
method. Also, Fig.2(a) shows that the control input designed
using MSSC method has no chattering. It provides smooth
control input.
B. Case 2 : Time varying disturbances
Here, we consider initial states of system (28) as well as the
control parameters same as Case 1. As stated earlier, our major
concern is about the fact that the disturbances/uncertainties
can’t be always constant. The state variable in one channel
can be a disturbance in the other channel as that of the
flexible joint or flexible link system. They can be time-variant
or state dependant. Our aim is to reduce the effect of such
disturbances/uncertainties present in the system. Here, the
disturbance in first and second channel is considered time
dependent which are d1(t) = 0.1sin(t) and d2(t) = 0.5sin(t)
respectively. And the third channel has state dependent uncer-
tainty and let’s assume it as d3(t) = x1x2.
Fig.3(a) shows the comparison of output for the three
methods. It can be clearly seen that the traditional SMC as
well as I-SMC method has failed to drive the output to desired
point. Hence these two methods can’t give the desired results
for mismatched systems with disturbances apart from constant
or slow varying ones. Also, as seen in Fig.4(b) and 4(c), in
the control input chattering is present which is not desirable.
The MSSC method drives the output to the desired point
though the disturbances are not constant. Also, Fig.4(a) shows
that the control input designed using MSSC method has no
chattering. This is another advantage of this method.

4. 0 2 4 6 8 10 12 14 16 18 20
−1.5
−1
−0.5
0
0.5
1
1.5
2
time (sec)
statex1
SMC
I−SMC
MSSC
(a)
0 2 4 6 8 10 12 14 16 18 20
−2
−1.5
−1
−0.5
0
0.5
1
1.5
time (sec)
statex2
SMC
I−SMC
MSSC
(b)
0 2 4 6 8 10 12 14 16 18 20
−2
−1
0
1
2
3
time (sec)
statex3
SMC
I−SMC
MSSC
(c)
Fig. 1: state variables in simulation studies of Case 1
0 2 4 6 8 10 12 14 16 18 20
−6
−4
−2
0
2
4
time (sec)
controlu
MSSC
(a)
0 2 4 6 8 10 12 14 16 18 20
−4
−2
0
2
4
6
time (sec)
controlu
SMC
(b)
0 2 4 6 8 10 12 14 16 18 20
−8
−6
−4
−2
0
2
4
6
time (sec)
controlu
I−SMC
(c)
Fig. 2: control input showing chattering reduction for constant
disturbances

5. 0 2 4 6 8 10 12 14 16 18 20
−1
−0.5
0
0.5
1
1.5
time (sec)
statex1
SMC
I-SMC
MSSC
(a)
0 2 4 6 8 10 12 14 16 18 20
−2
−1.5
−1
−0.5
0
0.5
1
time (sec)
statex2
SMC
I-SMC
MSSC
(b)
0 2 4 6 8 10 12 14 16 18 20
−3
−2
−1
0
1
2
3
4
time (sec)
statex3
SMC
I-SMC
MSSC
(c)
Fig. 3: state variables in simulation studies of Case 2
0 2 4 6 8 10 12 14 16 18 20
−8
−6
−4
−2
0
2
4
6
time (sec)
controlu
MSSC
(a)
0 2 4 6 8 10 12 14 16 18 20
−6
−4
−2
0
2
4
6
time (sec)
controlu
SMC
(b)
0 2 4 6 8 10 12 14 16 18 20
−8
−6
−4
−2
0
2
4
6
time (sec)
controlu
I-SMC
(c)
Fig. 4: control input showing chattering reduction for time
varying disturbances

6. C. Effect of Change in linear Gain
We can see the chattering in the control input when we
design it using traditional SMC and I-SMC. This is due to
the discontinuous switching gain present in the control input.
We can get less chattering if we reduce the switching gain,
but at the same time these two methods fail to reject these
disturbances effectively. While using MSSC with IDC, the
output reaches the desired point after some time (6 sec in
the Case 1 and around 9 sec in the Case 2). We can increase
the rate by increasing the value of linear gains (k1, k2 and k3)
or by decreasing time constants (τ1, τ2 and τ3).
V. CONCLUSION
In this paper, a new approach for controlling systems
affected by mismatched disturbances/uncertainties is proposed.
The proposed approach successfully counters the effect of
mismatched uncertainties and outperforms the traditional SMC
and I-SMC approaches. In case of traditional SMC and I-SMC
methods ,while handling these mismatched uncertainties, the
error in the output is distinctly large whereas MSSC with IDC
almost eliminates the error. This conclusion is validated by
simulation for the cases of constant and time varying as well
as state dependent disturbances. Unlike the traditional SMC
and I-SMC approaches, the proposed control is chatter free.
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