design of a sliding mode control with a nonlinear continuous timevaruing switching surface

8/12/2019 Design of a Sliding Mode Control With a Nonlinear Continuous Timevaruing Switching Surface

http://slidepdf.com/reader/full/design-of-a-sliding-mode-control-with-a-nonlinear-continuous-timevaruing-switching 1/7

Design of a sliding mode control with a nonlinear continuous

time-varying switching surface

Li Guang1，a , Liu Ling-hua1，b

1School of Mechanical Engineering , Hunan University of Technology, Zhuzhou 412007a [email protected] , b [email protected]

Key words: sliding mode control; nonlinear time-varying switching surface; second-order system

Abstract: A new sliding mode controller design approach for second-order systems is proposed that

varies the switching surface in a nonlinear continuous time-varying fashion. The sliding mode

control system, which the slope and intercept of switching surface changes continuously, can

improve the system performance in terms of a eliminating in the reaching time process, robustness

to parameters uncertainties and disturbance. The computer simulation results show the validity of

the method.

Introduction

A salient property of the sliding mode control is the sliding motion of the state on the switching

surface. During this sliding motion, the system has invariance properties yielding motion which is

independent of certain parameter uncertainties and disturbances. Therefore the design of the

switching surface completely determines the performance of the system. Most of the switching

surfaces proposed so far have been designed without consideration of the initial conditions, whether

given or arbitrary. Using these sliding surfaces, the sliding mode occurs only after the system

reaches the surface. Therefore, the tracking behaviors can be hindered by the uncertainties

especially during the reaching phase. One easy way to minimize the reaching phase, hence to get

fast and robust tracking is to employ the larger discontinuous control gain to speed up the reaching

phase [1-2]. However, this may cause extreme sensitivity to un-modeled dynamics, and also higher

chattering which is undesirable in a physical system. In this paper, a continuously time-varying

switching surface scheme substituting that conversional constant linear switching surface is

presented. The surface is initially designed to pass the initial errors and subsequently moved

towards a predetermined surface with a continuously rotating and shifting. Comparing with the

conversional switching surface and discretely moving switching surface method[3-4]

, the proposedapproach has improved the system performance in terms of a decrease in the reaching time,

robustness to disturbances and smoother phase plane trajectory.

1 Continuously time-varying switching surface

Consider a second-order nonlinear uncertain dynamic system described by

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

1 2

2 1 2 1 2 1 2

1 1

, , , , , , p q

i j j

i j

x t x t

t f x x t a t g x x t b x x t u t d t

δ = =

=

= + + +∑ ∑ (1)

with given initial conditions ( )1 0 x t and ( )2 0 x t , where ( ) ( )1 2, , j ja t g x x t δ and ( )d t

represent plant uncertainty and external disturbance respectively, are bounded

Applied Mechanics and Materials Vols. 128-129 (2012) pp 1399-1404Online available since 2011/Oct/24 at www.scientific.net © (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/AMM.128-129.1399

All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of TTP,www.ttp.net. (ID: 59.163.196.6, Indian Inst. of Technology, Roorkee, India-04/03/14,18:07:25)

http://www.scientific.net/

http://www.ttp.net/

http://www.ttp.net/

http://www.scientific.net/



( ) ( ) ( )1 2min max j j ja a a d t δ γ γ ≤ ≤ ≤ ≤， (2)

The tracking error ( )t e is defined as

( ) ( ) ( ) ( ) ( ) ( ) ( )T T

1 2 1 1d 2 2dt e t e t x t x t x t x t = = − − e (3)

Defining the following time-varying switching surface

( )( ) ( ) ( ) ( ) ( )1 2, s t t c t e t e t t α = + −e (4)

where, ( )c t and ( )t α are switching surface slop and intercept, respectively, and are selected

to be continuously differentiable with respect to time. (Fig. 1)

1 x

2

( )10 20

x x

0

final s

( )t α

Fig 1 Construction of the time-varying switching surface

Taking into account the function

( ) ( )

21

2

,V s s t = e (5)

which is positive definite with respect to s .

The control law

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1

2 1 2 d

1 1

ˆ, , s ig n ,q q

j j

j j

u t b t k g t s g t c t e t c t e t t x α −

= =

= − + − − − + +

∑ ∑ x x x

(6)

where:

ˆ j = ˆ ( , ),

j ja g x t ( ) ( )1

2 min max ̂j j ja a a = +

( ) ( ) ( ) ( )1 2maxˆ, , , , max ,

j j j j j j g x t a g x t a a a k γ γ = = − > (7)

makes

( ) ( ) ( )

( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )

2 1 2

1 1 1

ˆ, , , sign sign 0q q q

j j j j

j j j

V ss s c t e c t e e t

s a t g t t g t t g t t s d t k s

α

δ = = =

= = + + −

= − − + − ≤

∑ ∑ ∑

x x x

(8)

negative definite. Thus V is a Lyapunov function for the system (1) with control (6), the sliding

mode on the switching surface (4) actually occurs, and the system is truly insensitive to

disturbances and parameter variations. Furthermore, the continuous functions ( )c t and ( )t α are

designed such that ( )c t converges to a predetermined positive c , and ( )t α converges to zero,respectively, for any

0t t ≥ . Thus, the sliding motion ( ), s t x converges to zero, and tracking error

( ) 0t →e[5].

1400 Measuring Technology and Mechatronics Automation IV



2 Illustrative example

To manifest some features of the proposed continuously time-varying switching surface, and

compare with other fashions of sliding mode controllers, simulations are performed for a nonlinear

mass-damper-spring system shown in Fig. 2. The dynamics equation for this system becomes

m

( ), x t µ

( ), x t ν

( )u t

( )t

Fig. 2 A nonlinear spring-damper-mass system

( ) ( ) ( ) ( ), ,mx x t x t u t d t ν µ + + = + (9)

where ( )d t is the extraneous disturbance, ( ), x t ν and ( ), x t µ are taken by

( ) ( ) ( ) ( ) ( ) ( )3 3

0 1 0 1 0 1 0 1, , x t x x x x x x x t x x x xν ν ν δν δν µ µ µ δµ δµ = + + + = + + + (10)

Thus, by letting ( ) ( )1 t x t = and ( ) ( )2 t x t = , the state space formulation for the system (9) is

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

4 4

1 2 2

1 1

1 0 10 2 0 20

, , ,

,

i j j

i j

x t x t x t f t a t g t b t u t d t

x t x x t x

δ = =

= = + + +

= =

∑ ∑ x x

(11)

where

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

31 0 1 2 1 1 3 0 2 4 1 2 2

1 0 2 1 3 0 4 1

3

1 1 2 1 3 2 4 2 2

, , , , , , , ,

, , , , 1 ,

, , , , , , ,

f t x m f t x m f t x m f t x x m

a t a t a t a t b h m

g t x m g t x m g t x m g t x x m

µ µ ν ν

δ δµ δ δµ δ δν δ δν

= − = − = − = −

= − = − = − = − = =

= = = =

x x x x

x x x x

(12)

For the simulation, the certain parameters in the dynamics are chosen as

0 1 0 11, 0.5, 0.3m µ µ ν ν = = = = = , and the uncertainty terms ( ) ja t δ and ( )d t are taken by

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

1 2 3 40.05 0.25sin 5 , 0.05 0.15sin 7

0.05 0.25 cos 3

a t a t t a t a t t

d t t

δ δ π δ δ π

π

= = − + = = − +

= + (13)

The desired state trajectory is chosen as

( ) ( ) ( ) ( )1d 2d0.5cos 0.2 , 0.1sin 0.2 x t t x t t π π = − = (14)

Then, the simulations for the dynamic system are presented by using following three controllers.

a) conversional sliding mode control with the constant switching surface

( )( ) ( ) ( ) p 1 2, s t t c e t e t = +e (15)

and the control law is

( ) ( )4 4 4

p 2 2d

1 1 1

ˆsign j i j

j i j

u t k g s f g c e x m= = =

= − + − − − + ×

∑ ∑ ∑ (16)

b) sliding mode control with the discrete time-varying switching surface

( )( ) ( ) ( )r 1 2 s, s t t c e t e t α = + −e (17)

Applied Mechanics and Materials Vols. 128-129 1401



wherer

c ands

α are stepwise time-varying slop and intercept of the switching surface (17), and

the control law is the same as (16).

c) proposed control methodology with the following continuously function ( )c t and ( )t α

( ) ( ) p 0

0

c1 exp

c c

c t ck t

−

= ++ − , ( ) ( )0

0

0

1 expt k t α

α

α α

−= −+ − (18)

wherec

k and k α

are the convergent rates for ( )c t and ( )t α , and0

c ,0

α are the initial values

which allow the switching surface to pass the initial condition.

Assuming the initial conditions are ( ) ( )1 20 0, 0 1 x x= = , i.e. ( ) ( )1 20 0.5, 0 1e e= = , and the

predetermined p

7c = , control gain k in (6) and (16) is 0.5.

(a) (b)

Fig. 3 Position and velocity tracking errors for various switching surface: (a) conventional fixed

with 0.5k = ; (b) conventional fixed with 3k = ; (c) stepwise shifted and rotated; (d) continuously

shifted and rotated




(a) (b)

(c) (d)

Fig. 4 Control signal of the algorithm for various switching surface: (a) conventional fixed with

0.5k = ; (b) conventional fixed with 3k = ; (c) stepwise shifted and rotated; (d) continuously shifted

and rotated

Compared the simulating results in figure 3 and 4, it is clearly observed that the presented

scheme guarantees the fastest errors convergence and truly robust behavior of the system.

Considering the conversional switching surface, by using a larger discontinuous control gain of

control law (16), the reaching time is shorten to a certain extent, however, this algorithm causes a

higher chattering in the control input during the sliding phase (Fig. 4 (b)).

3 Conclusions

The design of a sliding mode controller by means of a continuously time-varying switching

surface is presented to improve the tracking behaviors of the second-order dynamic system

subjected to parameter variations and disturbances. The switching surface is designed first to pass

initial errors and subsequently moves towards the predetermined surface via continuously rotating

and shifting. Employing the proposed algorithm, it is possible to reduce remarkably the system

sensitivity to extraneous disturbances and uncertainties parameters by means of shortening the

reaching phase without increasing the undesired chattering of the control input signals. Furthermore,

compared to the discretely time-varying switching surface approach, the existence of a sliding modeis guaranteed, and the representative point of the considered system is forced to stay always on the

switching surface, hence, the presented control methodology allows faster tracking and really

possesses robust behaviors.

Applied Mechanics and Materials Vols. 128-129 1403



Reference

[1] R. F. Fung, R. T. Yang，Application of VSC in position control of a nonlinear electrohydraulic

servo system, Computer & Structures, Vol. 66-4，(1998), p. 365

[2] A. J. Koshkouei, K. J. Burnham, A. S. I. Zinober ，

Dynamic sliding mode control. IEE Proc.Control Theory and Application, Vol.152-4，(2005), p. 392

[3] S. B. Chio, C. C. Chenong, D. W. Park, Moving switching surfaces for robust control of

second-order variable structure systems. International Journal of Control, Vol.58-1, (1993), p. 229

[4] S. B. Chio, D. W. Park, S. Jayasuriya, A time-varying sliding surface for fast and robust tracking

control of second-order uncertain systems. Automatica, Vol.30-5, (1994), p. 899

[5] M. Krstic, I. Kanellakopoulos, P. Kokotovic, in: Nonlinear and Adaptive Control Design, New

York, John. Wiley & Sons, Inc. Publishers ( 1995).




Measuring Technology and Mechatronics Automation IV 10.4028/www.scientific.net/AMM.128-129

Design of a Sliding Mode Control with a Nonlinear Continuous Time-Varying Switching Surface 10.4028/www.scientific.net/AMM.128-129.1399

http://dx.doi.org/www.scientific.net/AMM.128-129

http://dx.doi.org/www.scientific.net/AMM.128-129.1399

http://dx.doi.org/www.scientific.net/AMM.128-129.1399

http://dx.doi.org/www.scientific.net/AMM.128-129

design of a sliding mode control with a nonlinear continuous timevaruing switching surface

Documents