This document describes the design methodology for first, second, and third order sliding mode control. It presents the main concepts of sliding mode control and discusses how to design controllers for single-input single-output systems of different orders. Simulation results in MATLAB and Simulink demonstrate the robustness of sliding mode control in the presence of uncertainties and its ability to drive system states and outputs to zero.

1. Pole Placement in Higher Order Sliding -
Mode Control
June 13, 2015
1

2. Abstract
This paper presents the design methodology for simple sliding mode (first order) control to
higher order (third order) sliding mode control for different control laws. The design methods
for first, second and third order sliding mode control is shown on MATLAB. To stabilize the
system the sliding mode controllers for different orders cited above for single input single out-
put system are built with end results obtained on Simulink. Finally we conclude this paper
depicting the robustness of sliding mode control in the presence of uncertainties.
2

3. Contents
1 Introduction 4
2 Main Concepts of Sliding Mode Control 5
3 Main Result 6
3.1 First order sliding mode control . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Second order sliding mode control . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Third order sliding mode control . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Results 11
4.1 First order sliding mode control . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Second order sliding mode control . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.3 Third order sliding mode control . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.4 Advantages of Sliding mode control . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Conclusion 16
3

4. 1 Introduction
We all know that there will be lack of similarity in actual plant parameters and its mathe-
matical model in order to design a controller. This dissimilarity arises because of unknown
external disturbances and perturbations. The aim of any control engineer will be to design a
controller which offers overall performance to the control systems with feedback even in the
presence of uncertainties, disturbances. The control law which offers desired performance and
is well known for its robustness is Sliding Mode Control.
The sliding mode control is also known as Variable Structure Control Method, is a non-linear
control method which allows the system to slide not only on one control structure but also on
multiple control structure, these motion which makes the system to slide is known as sliding
mode [1]. There are four types of sliding mode control which is depicted below:
• Reduced Order Sliding Mode Control.
• Integral Sliding Mode Control
• Continuous Time Sliding Mode Control.
• Discrete Time Sliding Mode Control.
The Sliding Mode Control was first applied to continuous time systems and then after with the
advancement in the digital systems this approach extended to discrete time systems. Both in
continuous time and discrete time systems, the sliding mode control consists of set of integra-
tors and states moreover the system states and integrators are well defined using sliding sub
space. The primary and main concern of the sliding mode control is all its system states along
with output must converge asymptotically at zero even in the presence of perturbations and
uncertainties. In the succeeding sections we will present the design methodology for sliding
mode control along with its advantages and disadvantages.
4

5. 2 Main Concepts of Sliding Mode Control
In order to explain the whole methodology of sliding mode control for higher order systems,
first we have to consider the linear time invariant system which is depicted below:
x = Ax + B (u + w)
x → System State
u → Control Law
w → Unknown perturbation at time
In addition (A, B) must be a controllable pair. In the above equation the control law (u) is
supposed to drive state variables to zero. In order to make these states variables converge at
zero we are going to introduce a new variable in the above state space representation which is
shown below
σ = Cx, σ R
Our aim is to drive the above variable which has been introduced to zero in finite time by means
of control law (u). The variable which is introduced above can also be expressed as shown below
σ = ε1
In other words we can express the output (Cx) as
ε = Cx
We can convert the above state space system to transfer function by the expression which is
shown below
g(S) = C (sI − A)−1
B
g(S) → Transfer function which consists of poles and zeros
The zeros of the above transfer function must have a real part less than zero which completely
ensures that the system in minimum phase. The transfer function g(s) representations for first,
second and third order sliding mode controls are generated using MATLAB which is shown in
the appendix section of the report. Hence we can conclude the main concept of sliding mode
control. In the succeeding sections we will briefly describe the main results for first, second and
third order sliding mode control.
5

6. 3 Main Result
The main difference between the conventional and higher order sliding mode control is that the
conventional sliding mode control is restricted to the outputs whose relative degree is equal to
one on the contrary the higher order sliding mode control allows the sliding variables whose
relative degree greater than one. The whole methodology of sliding mode control is based upon
the proof which is shown below which in turn makes sure that the system is in minimum phase.
The Matrix C is obtained using the Ackermann and Utkin
e1 = [0 0...0 1]
C = e1P−1
γ(A)
P → Systems Controllability Matrix
The controllability matrix can be determined by the following
P = [B AB A2
B ... ... An−1
B]
Also, the roots of γ(A) are the eigenvalues of sliding mode dynamics. The original system ma-
trices and the matrices after similarity transformation will have same eigenvalues. The results
obtained with original matrices will exactly replicate the results obtained with new system ma-
trices under similarity transformation. Now we will further expand this sliding mode control
topic depicting the whole system and controller model for first, second and third order systems
respectively.
3.1 First order sliding mode control
Figure 1:
6

7. The Simulink model for first order sliding mode control with its controller is depicted above,
the Simulink results along with design methodology generated on MATLAB is shown in results
and appendix section of this paper.
u = −
CAx + k0 sign(ε1)
CB
3.2 Second order sliding mode control
Figure 2:
We apply the control law which is shown below in order to enforce sliding motion for second
order sliding mode control.
u = −
1
CAB
CA2
x + 10
σ + |σ|
1
2 sign(σ)
|σ | + |σ|
1
2
The controller which is designed using above control for the second order sliding mode control
is depicted below which is included as a subsystem block in the above Simulink model which
has two incoming ports and one outgoing port.
7

8. Figure 3:
The design of methodology for second order sliding mode control is generated on MATLAB
with Simulink results show in appendix and results section respectively.
3.3 Third order sliding mode control
The Simulink model for the third order sliding mode control is shown below. The control law
which is enforced for the third order system is shown below:
8

9. u = −
1
CA2B
CA3
x + 10
σ + 2(|σ | + |σ |
2
3 )
−1
2
(σ + |σ|
2
3 sign(σ)
|σ | + 2(|σ | + |σ|
2
3 )
1
2


Figure 4:
The controller designed for third order using above control law is depicted below
9

10. Figure 5:
10

11. 4 Results
4.1 First order sliding mode control
The Simulink results obtained for the first order sliding mode control is shown below:
Figure 6:
Figure 7:
11

12. Figure 8:
Hence the above are the results for first order sliding mode control. We can clearly see that all
the system states along with the output are asymptotically converging at zero as per theory of
sliding mode control. Here the system is perturbed by w=0.5sin (10t), moreover the control
law is sampled and held every τ = 0.001 seconds.
4.2 Second order sliding mode control
Figure 9:
12

13. Figure 10:
Figure 11:
Hence above are the Simulink results obtained for second order sliding mode control. We can
clearly see that all the system states along with output are asymptotically converging at zero.
Here system is as usual perturbed by w=0.5 sin(10t), the control law is sampled and held
every τ = 0.001 seconds.
13

14. 4.3 Third order sliding mode control
Figure 12:
Figure 13:
14

15. Figure 14:
Hence above are the Simulink results obtained for third order sliding mode control. Here
the system is perturbed by w=0.5sin(10t) with control law sampled and held every τ =
0.001 seconds.
4.4 Advantages of Sliding mode control
• Sliding mode control is known for its robustness because it makes sure that all the states
along with output converge asymptotically at zero despite uncertainties/disturbances.
• It also makes sure that all the system states along with output converge at zero in finite
time.
• Moreover they exhibit reduced-order compensated dynamics in Simulink results.
The only disadvantage of sliding mode control is chattering, there are many ways to avoid and
is very important to eliminate this chattering by providing continuous/smooth control signal.
This chattering is common in many practical control system problems like DC motor control,
aircraft control. One way to avoid this chattering is using quasi-sliding mode.
15

16. 5 Conclusion
Hence we can conclude this paper on Pole placement in higher order sliding mode control
stating that sliding mode control is known for its robustness even in the presence of uncertain-
ties/disturbances and perturbations which was clearly depicted in the preceding sections with
Simulink results. The only repercussion in sliding mode control is chattering which was dis-
cussed earlier and can be attenuated using quasi sliding mode. We have presented a complete
design methodology for first, second and third order sliding mode control with results obtained
on MATLAB/SIMULINK.
16

17. References
[1] www.google.com
[2] Intuitive Theory of Sliding Mode Control.
[3] Pole Placement in Higher Order Sliding - Mode Control.
17

18. Appendix
% Kedar Bhalkiker
% First order sliding mode control design
clear all
clc
format shortG
System Matrices given
A=[0 1 0 0;0 0 -1.56 0;0 0 0 1;0 0 46.87 0]
I=eye(4)
g=(A+5*I)^3
B=[0;0.97;0;-3.98]
e1=[0 0 0 1]
A =
0 1 0 0
0 0 -1.56 0
0 0 0 1
0 0 46.87 0
I =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
g =
125 75 -23.4 -1.56
0 125 -190.12 -23.4
0 0 828.05 121.87
0 0 5712 828.05
B =
0
0.97
0
-3.98
e1 =
0 0 0 1

19. Controllability Matrix
P=[B A*B A^2*B A^3*B]
P1=inv(P)
p=P(4,:)
t=[p;p*A;p*A^2;p*A^3]
Pcap_inv = P1*t
t1=inv(t)
g1 = t1*g*t
P =
0 0.97 0 6.2088
0.97 0 6.2088 0
0 -3.98 0 -186.54
-3.98 0 -186.54 0
P1 =
0 1.194 0 0.03974
1.194 0 0.03974 0
0 -0.025474 0 -0.0062086
-0.025474 0 -0.0062086 0
p =
-3.98 0 -186.54 0
t =
-3.98 0 -186.54 0
0 -3.98 0 -186.54
0 0 -8737 0
0 0 0 -8737
Pcap_inv =
0 -4.7521 0 -569.94
-4.7521 0 -569.94 0
0 0.10139 0 58.997
0.10139 0 58.997 0
t1 =
-0.25126 0 0.0053645 0
0 -0.25126 0 0.0053645
0 0 -0.00011446 0
0 0 0 -0.00011446
g1 =

20. 125 75 -84320 -5621.4
0 125 -6.8508e+05 -84320
0 0 828.05 121.87
0 0 5712 828.05
Matrix C obtained
C = e1*Pcap_inv*g1*t1
C =
-3.1843 -1.9106 -4.5449 -0.7169
Transfer function obtained
[num,den]=ss2tf(A,B,C,0)
gs=tf(num,den)
num =
0 1 15 75 125
den =
1 -8.8818e-16 -46.87 0 0
gs =
s^3 + 15 s^2 + 75 s + 125
-------------------------------
s^4 - 8.882e-16 s^3 - 46.87 s^2
Continuous-time transfer function.
New System Matrices after similarity transformation
Acap=t1*A*t
Bcap=t1*B
Ccap = C*t
Acap =
0 1 0 -4.4157e-16
0 0 -5621.4 0
0 0 0 1
0 0 46.87 0

21. Bcap =
0
-0.26507
0
0.00045553
Ccap =
12.674 7.6041 40303 6620
% Kedar Bhalkiker
% Second order sliding mode control design
clear
clc
format shortG
System Matrices Given
A=[0 1 0 0;0 0 -1.56 0;0 0 0 1;0 0 46.87 0]
I=eye(4)
g=(A+5*I)^2
B=[0;0.97;0;-3.98]
e1=[0 0 0 1]
A =
0 1 0 0
0 0 -1.56 0
0 0 0 1
0 0 46.87 0
I =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
g =
25 10 -1.56 0
0 25 -15.6 -1.56
0 0 71.87 10
0 0 468.7 71.87
B =
0
0.97
0

22. -3.98
e1 =
0 0 0 1
Controllability Matrix
P=[B A*B A^2*B A^3*B]
P1=inv(P)
p=P(4,:)
t=[p;p*A;p*A^2;p*A^3]
Pcap_inv = P1*t
t1=inv(t)
g1 = t1*g*t
P =
0 0.97 0 6.2088
0.97 0 6.2088 0
0 -3.98 0 -186.54
-3.98 0 -186.54 0
P1 =
0 1.194 0 0.03974
1.194 0 0.03974 0
0 -0.025474 0 -0.0062086
-0.025474 0 -0.0062086 0
p =
-3.98 0 -186.54 0
t =
-3.98 0 -186.54 0
0 -3.98 0 -186.54
0 0 -8737 0
0 0 0 -8737
Pcap_inv =
0 -4.7521 0 -569.94
-4.7521 0 -569.94 0
0 0.10139 0 58.997
0.10139 0 58.997 0
t1 =

23. -0.25126 0 0.0053645 0
0 -0.25126 0 0.0053645
0 0 -0.00011446 0
0 0 0 -0.00011446
g1 =
25 10 -5621.4 -3.1892e-14
0 25 -56214 -5621.4
0 0 71.87 10
0 0 468.7 71.87
Matrix C obtained
C = e1*Pcap_inv*g1*t1
C =
-0.63686 -0.25474 -0.40647 -0.062086
Transfer function obtained
[num,den]=ss2tf(A,B,C,0)
gs=tf(num,den)
num =
0 0 1 10 25
den =
1 -8.8818e-16 -46.87 0 0
gs =
s^2 + 10 s + 25
-------------------------------
s^4 - 8.882e-16 s^3 - 46.87 s^2
Continuous-time transfer function.
New Matrices Obtained after similarity transformation
Acap=t1*A*t
Bcap=t1*B
g1 = t1*g*t
C = e1*Pcap_inv*g1*t1
Ccap = C*t

24. Acap =
0 1 0 -4.4157e-16
0 0 -5621.4 0
0 0 0 1
0 0 46.87 0
Bcap =
0
-0.26507
0
0.00045553
g1 =
25 10 -5621.4 -3.1892e-14
0 25 -56214 -5621.4
0 0 71.87 10
0 0 468.7 71.87
C =
-0.63686 -0.25474 -0.40647 -0.062086
Ccap =
2.5347 1.0139 3670.2 589.97
% Kedar Bhalkiker
Third Order Sliding Mode Control Design
clear all
close all
clc
format shortG
System Matrices Given
A=[0 1 0 0;0 0 -1.56 0;0 0 0 1;0 0 46.87 0]
B=[0;0.97;0;-3.98]
I=eye(4)
g=(A+5*I)
A =
0 1 0 0
0 0 -1.56 0

25. 0 0 0 1
0 0 46.87 0
B =
0
0.97
0
-3.98
I =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
g =
5 1 0 0
0 5 -1.56 0
0 0 5 1
0 0 46.87 5
Controllability Matrix
P=[B A*B A^2*B A^3*B]
P1=inv(P)
p=P(4,:)
t=[p;p*A;p*A^2;p*A^3]
Pcap_inv=P1*t
t1=inv(t)
g1=t1*g*t
e1=[0 0 0 1]
P =
0 0.97 0 6.2088
0.97 0 6.2088 0
0 -3.98 0 -186.54
-3.98 0 -186.54 0
P1 =
0 1.194 0 0.03974
1.194 0 0.03974 0
0 -0.025474 0 -0.0062086
-0.025474 0 -0.0062086 0
p =

26. -3.98 0 -186.54 0
t =
-3.98 0 -186.54 0
0 -3.98 0 -186.54
0 0 -8737 0
0 0 0 -8737
Pcap_inv =
0 -4.7521 0 -569.94
-4.7521 0 -569.94 0
0 0.10139 0 58.997
0.10139 0 58.997 0
t1 =
-0.25126 0 0.0053645 0
0 -0.25126 0 0.0053645
0 0 -0.00011446 0
0 0 0 -0.00011446
g1 =
5 1 -1.5946e-14 -4.4157e-16
0 5 -5621.4 -1.5946e-14
0 0 5 1
0 0 46.87 5
e1 =
0 0 0 1
Matrix C Obtained
C=e1*Pcap_inv*g1*t1
C =
-0.12737 -0.025474 -0.031043 -0.0062086
Transfer Function Obtained
[num,den]=ss2tf(A,B,C,0)
gs=tf(num,den)
num =

27. 0 0 0 1 5
den =
1 -8.8818e-16 -46.87 0 0
gs =
s + 5
-------------------------------
s^4 - 8.882e-16 s^3 - 46.87 s^2
Continuous-time transfer function.
Matrices Obtained after similarity transformation
Acap=t1*A*t
Bcap=t1*B
Ccap=C*t
Acap =
0 1 0 -4.4157e-16
0 0 -5621.4 0
0 0 0 1
0 0 46.87 0
Bcap =
0
-0.26507
0
0.00045553
Ccap =
0.50694 0.10139 294.98 58.997
The above are the codes for the design of first, second and third order sliding mode control
which clearly shows the system matrices given, Matrix C obtained, transfer function and new
system matrices under similarity transformation.