A revised Terminal Sliding Mode Controller Designfor Servo Implementation
Khalid Abidi
∗
, and JianXin Xu
∗
∗
National University of SingaporeFaculty of Engineering, Department of Electrical and Computer Engineering10 Kent Ridge Crescent, Singapore, 119260
Abstract
—Terminal Sliding Mode (TSM) control is known forits high gain property nearby the vicinity of the equilibriumwhile retaining reasonably low gain elsewhere. This is desirablein digital implementation where the limited sampling frequencymay incur chattering if the controller gain is overly high. In thiswork we integrate a linear switching surface with a terminalswitching surface. The mixed switching surface can be designedaccording to the precision requirement. The analysis, simulationsand experimental investigation show that the mixed SMC designoutperforms the linear SMC as well as the pure TSMC.
I. I
NTRODUCTION
SLIDING MODE control is a powerful technique thathas been successfully used for the control of the linear andnonlinear systems. In order to design sliding mode controlsystems, a switching surface or a sliding mode is deﬁned ﬁrst,and a sliding mode controller is then designed to drive thesystem state variables to the sliding mode so that the desiredconvergence property can be obtained in the sliding mode,which is not affected by any modeling uncertainties and/ordisturbances [1]. Reviewing the history of the developmentof the sliding mode control systems [2][4], it can be foundthat linear sliding mode has been widely used to describe thedesired performance of closed loop systems, that is, the systemstate variables reach the system srcin asymptotically in thelinear sliding mode. Although the parameters of the linear sliding mode can be adjusted such that the convergence rate maybe arbitrarily fast, the system states in the sliding mode cannotconverge to zero in ﬁnite time. Recently, a new techniquecalled terminal sliding mode control has been developed in [1]to achieve ﬁnite time convergence of the system dynamics inthe terminal sliding mode. In [5][7], the ﬁrstorder terminalsliding mode control technique is developed for the controlof a simple secondorder nonlinear system and an 4thordernonlinear rigid robotic manipulator system with the result thatthe output tracking error can converge to zero in ﬁnite time.In this paper, a revised terminal sliding mode control lawis developed. It is shown that the new method can achievebetter performance than with the linear SM or pure TSM. Tovalidate the proposed method simulation and experiments areconducted on a piezomotor system.The paper is organized as follows. The problem formulationis presented in
§
2. Appropriate terminal sliding surface andSMC are designed in
§
3. In
§
4, numerical and experimentalresults will be presented. Conclusions are given in
§
5.II. P
ROBLEM
F
ORMULATION
A. System Properties
Consider the following continuoustime model of piezomotor driven linear stage
˙
x
1
(
t
) =
x
2
(
t
)˙
x
2
(
t
) =
−
k
fv
m x
2
(
t
) +
k
f
m u
(
t
)
−
1
mf
(
x
,t
)
y
(
t
) =
x
1
(
t
)
(1)where
x
1
is the position,
x
2
is the velocity,
u
is the voltageinput, and
f
(
x
,t
)
is the friction disturbance and is assumedbounded such that

f
(
x
,t
)
 ≤
f
max
. The constants
m
,
k
fv
,and
k
f
are the nominal mass, damping, and force constantsrespectively.The objective is to design a TSM such that the output,
y
(
t
)
,of system (1) will converge to a desired reference trajectory
r
(
t
)
in ﬁnite time.III. TSM C
ONTROLLER
D
ESIGN
In this section we will discuss the TSM controller design.The controller will be designed based upon an appropriateselection of a Lyapunov function. Further, the closedloopsystem will be analyzed to derive the stability conditions. Thissection will conclude with a discsussion on the tracking errorbound.
A. Controller Design
Consider the terminal sliding surface deﬁned below,
σ
=
c
1
e
+
c
2
˙
e
+
c
3
e
p
(2)where
e
=
r
−
y
is the tracking error,
σ
is the sliding function,and
c
1
,
c
2
,
c
3
,
p
are design constants.Before we proceed with the controller design, we need torewrite the system (1) in terms of the tracking error
e
. Considerthe new state
z
= ˙
e
, it can be shown that the system can berewritten as
˙
e
(
t
) = ˙
r
−
x
2
(
t
) =
z
(
t
)˙
z
(
t
) =
k
fv
m z
(
t
)
−
k
f
m u
(
t
) + ¨
r
−
k
fv
m
˙
r
+ 1
mf
(
x
,t
)
(3)
1599781424422005/08/$25.00 ©2008 IEEE
Now, select the Lyapunov function
V
(
t
) =
12
σ
2
. The derivative of the Lyapunov function is given as
˙
V
=
σ
˙
σ
=
σ
(
c
1
˙
e
+
c
2
˙
z
+
pc
3
e
p
−
1
z
)
.
(4)To achieve
˙
V
=
−
c
4
σ
2
<
0
, where
c
4
is a constant, we set
c
1
˙
e
+
c
2
˙
z
+
pc
3
e
p
−
1
z
=
−
c
4
σ
(5)and substitute the expression of
˙
z
from (3) into (5) to obtain
c
1
z
+
c
2
k
fv
m z
−
k
f
m u
+ ¨
r
−
k
fv
m
˙
r
+ 1
mf
+
pc
3
e
p
−
1
z
(6)
=
−
c
4
σ.
From (7) the control can be derived as
u
=
mk
f
¨
r
−
k
fv
m
˙
r
+
c
1
c
2
+
k
f
vm
+
pc
3
c
2
e
p
−
1
z
+
c
4
c
2
σ
+ 1
k
f
f.
(7)Note that the disturbance
f
is unknown so the controller willbe revised to the form
u
=
mk
f
¨
r
−
k
fv
m
˙
r
+
c
1
c
2
+
k
f
vm
+
pc
3
c
2
e
p
−
1
z
+
c
4
c
2
σ
.
(8)Ideally the term
c
4
c
2
σ
in (8) is actually
c
4
c
2
sign
(
σ
)
, however, since the controller will be implemented in a digitalenvironment the switching term will not be used as it cancause chattering. Once the controller is designed, we need toexamine the stability.
B. Convergence Analysis
In order to check the stability we will revisit the Lyapunovfunction
V
=
12
σ
2
. Substitution of (8) in (4) will lead to
˙
V
=
σ
(
−
c
4
σ
+
c
2
f
)
.
(9)From (9) it can be concluded that

σ
≥
c
2
c
4
f
max
(10)which gives a bound on the sliding function
σ
and the properselection of the constants
c
2
and
c
4
would minimize thisbound. However, our main concern is the tracking error
e
.Consider the function
V
e
=
12
e
2
and (2). The derivative
˙
V
e
is given by
˙
V
e
=
e
˙
e
=
e
−
c
1
c
2
e
−
c
3
c
2
e
p
+ 1
c
2
σ
(11)which provides a minimum bound on the error once the systemhas entered sliding mode. This bound is the solution of
c
1

e

+
c
2

e
p

=
c
2
c
4
f
max
(12)which can be rewritten as

e

=
c
2
f
max
c
4
(
c
1
+
c
2

e
p
−
1

)
(13)if the exponent is set between
0
< p <
1
then it can be bepossible to plot the relationship between
e
,
c
1
and
p
whileselecting
c
2
= 1
and
c
2
c
4
f
max
in order to understand betterthe effects of having a linear SM incorporated with a TSM.From Fig.1 we see that increasing
c
1
and decreasing
p
lead toa smaller error bound. Note that if the linear term was absent,
Fig. 1. Error bound as a function of
c
1
and
p
in other words
c
1
= 0
, the error would converge to a minimumbound of
1
c
4
f
max
1
p
which in this example is 1 as seen fromFig.1. This shows the beneﬁts of having a combined linear andnonlinear term in the TSM.IV. N
UMERICAL AND
E
XPERIMENTAL
I
NVESTIGATION
Conisder the system with the following nominal parameters:
m
= 1
kg
,
k
fv
= 144
N
and
k
f
= 6
N/V olt
. This simplelinear model does not contain any nonlinear and uncertaineffects such as the frictional force in the mechanical part, highorder electrical dynamics of the driver, loading condition, etc.,which are hard to model in practice. In general, producing ahigh precision model will require more efforts than performinga control task with the same level of precision.
A. Numerical Investigation
The system is simulated at a sampling time of
T
= 1
ms
and the disturbance force
f
acting on the system is modeledsimply as
f
=
10
if x
2
<
00
if x
2
= 0
−
10
if x
2
>
0
.
(14)For the new TSM the control parameters are selected as
c
1
=50
,
c
2
= 1
,
c
3
= 10
, and
c
4
= 1000
while the exponent
p
=
2131
. These values are found after a few trials to get the mostoptimum performance. The new TSM is compared against anSM controller where
c
3
= 0
and a TSM controller where
c
1
= 0
. In Fig.2 we see the desired trajectory and the trackingperformance of all three controllers. A better idea about thetracking can be found by looking at Fig.3 where the trackingerror of all three controllers is plotted. We see that the new
160
TSM outperforms the other controllers. From Fig.4 we seethat the control signal is almost identical for all three cases.
Fig. 2. Position trajectory and comparison of the new TSM, SM and TSMcontrollers’ performanceFig. 3. Tracking error comparison of the new TSM, SM and TSM controllers’performance
B. Experimental Investigation
Using the experimental system with the block diagram Fig.5that was modelled in the previous section, the controller isimplemented. For the new TSM the control parameters areinitially selected as in the previous section and are retuned asit is expected not to exactly match the experimental case. Thetuned parameters are
c
1
= 70
,
c
2
= 3
,
c
3
= 25
, and
c
4
= 1200
while the exponent
p
=
2131
. The new TSM is compared againstan SM controller where
c
3
= 0
and a TSM controller where
c
1
= 0
. In Fig.6 we see the desired trajectory and the trackingperformance of all three controllers. Similar to the simulationin Fig.7 we can see that the new TSM outperforms the othercontrollers, however, the difference between the TSM and thenew TSM is around 7 microns with the new TSM going aslow as 17 microns for the given 40mm displacement. FromFig.8 we see that the control signal for all three cases.
Fig. 4. Control input comparison of the new TSM, SM and TSM controllers’performanceFig. 5. System Block DiagramFig. 6. Position trajectory and comparison of the new TSM, SM and TSMcontrollers’ performance
V. CONCLUSIONThis work presents a revised TSM controller based on a linear SM combined with a TSM. Theoretical investigation showsthat the revised controller can provide better performance thanwith either the pure SM or the pure TSM. Numerical andExperimental comparison with the SM and TSM controllers
161
Fig. 7. Tracking error comparison of the new TSM, SM and TSM controllers’performanceFig. 8. Control input comparison of the new TSM, SM and TSM controllers’performance
prove the effectiveness of the proposed method.R
EFERENCES[1] Z. Man, and X. H. Yu, ”Terminal sliding mode control of mimo linearsystems,”
IEEE Transaction on Circuits and Systems I: FundamentalTheory and Applications
, vol. 41, pp. 10651070, 1997.[2] R. A. Decarlo, S. H. Zak, and G. P. Matthews, ”Variable structure controlof nonlinear multivariable systems: A tutorial,”
Proc. IEEE
, vol. 76, pp.212231, 1988.[3] V. Utkin, ”Sliding mode control in discretetime and difference systems.In A.S.I. Zinober (Eds.)”,
Variable Structure and Lyapunov Control
,pp.87107, 1994.[4] K. Abidi, and X.J. Xu, ”On the discretetime integral sliding modecontrol,”
IEEE Transactions on Automatic Control
, vol. 52, pp. 709715, 2007.[5] G. G. Morgan, and U. Ozguner, ”A decentralised variable structurecontrol algorithm for robotic manipulators,”
IEEE Journal of Roboticsand Automation
, vol. 1, pp. 5765, 1985.[6] C. M. Dorling, and A. S. I. Zinober, ”Two approaches to hyperplanedesign in multivariable variable structure control systems,”
International Journal of Control
, vol. 44, pp. 6582, 1986.[7] H. Khurana, S. I. Ahson, and S. S. Lamba, ”On the stabilization of largescale control systems using variable structure system theory,”
IEEE Transactions on Automatic Control
, vol. 31, pp. 176178, 1986.
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