Electr Eng (2010) 92:257268DOI 10.1007/s00202-010-0180-4

ORIGINAL PAPER

Time-varying sliding-mode control for finite-time convergence

Yu-Sheng Lu Chien-Wei Chiu Jian-Shiang Chen

Received: 18 September 2008 / Accepted: 1 October 2010 / Published online: 19 October 2010 Springer-Verlag 2010

Abstract This paper proposes a time-varying sliding-modecontrol scheme, in which a piecewise defined function of timethat contains trigonometric functions is incorporated into theswitching function not only to eliminate the reaching phase,but also to guarantee finite-time convergence of the trackingerror. Without using fractional-power functions of certainstate variable, the proposed control is both free from singu-larity around the equilibrium and also convenient for prac-tical implementation. Moreover, the proposed scheme hasthe terminal time as an explicit parameter, which makes thecontroller design simple to meet the terminal-time require-ment. Experiments were conducted on a two-link direct-drivemanipulator to demonstrate the effectiveness of the proposedscheme.

Keywords Sliding-mode control Finite-timeconvergence Continuous equivalent control Control systems

1 Introduction

Sliding-mode control (SMC) [1] has been recognized asan effective approach in controlling uncertain systems with

Y.-S. Lu (B)Department of Mechatronic Technology, National Taiwan NormalUniversity, 162, He-ping East Rd., Sec. 1, Taipei 106, Taiwane-mail: luys@ntnu.edu.tw

C.-W. ChiuDepartment of Mechanical Engineering, National YunlinUniversity of Science and Technology, Yunlin 640, Taiwan

J.-S. ChenDepartment of Power Mechanical Engineering, National TsingHua University, Hsinchu 300, Taiwan

highly coupled and nonlinear dynamics. With SMC, statetrajectories are directed toward and constrained on somepredefined sliding hyperplanes. During sliding motion, theclosed-loop system dynamics are totally governed by param-eters of the sliding hyperplanes, despite the presence of exter-nal disturbances and parametric uncertainties satisfying theso-called matching condition. In conventional SMC, thereusually exists a reaching phase before system state arrivesat the sliding hyperplanes. The existence of such a reach-ing phase; however, reduces the performance robustness ofthe SMC system. Speeding up the reaching phase can alle-viate this problem, but this involves excessive control effortand may excite unmodeled dynamics. In addition, the con-ventional SMC schemes adopt linear sliding manifolds toguarantee the asymptotic stability of the sliding dynamics.This asymptotic stability, however, implies that the systemstate trajectories converge to the equilibrium as time goes toinfinity, leading to sluggish responses.

Previous studies [2,3] reported a moving switching linescheme with a piecewise-constant function, a function withmultiple steps, for second-order systems to pass arbitraryinitial conditions, hence reducing the reaching phase. Royand Olgac [4] extended this scheme to nth order systemsin companion form. However, these schemes [24] cannotensure the existence of a sliding mode throughout an entireresponse. For second-order systems, Chang and Hurmuzlu[5] redefined a new tracking error vector by subtracting atime-dependent vector from the original tracking error vec-tor, such that the redefined tracking error vector is initially azero vector, and the reaching phase is eliminated by replacingthe original tracking error vector with the redefined one inthe conventional switching function. Yilmaz and Hurmuzlu[6] expanded this approach to nth-order systems in compan-ion form. Bartoszewicz [7] proposed time-varying switch-ing functions for constant-acceleration sliding mode and

123

258 Electr Eng (2010) 92:257268

constant-speed sliding mode. Betin et al. [8] shifted theswitching line with a constant speed as proposed by Bart-oszewicz [7] to have a sliding mode from the beginning ofthe position control of a dc motor drive. Fuzzy logic algo-rithms were integrated with the sliding mode controller, inwhich the slope of the switching line was tuned to catchthe system state on the switching line rapidly [9,10]. Globalsliding-mode control (GSMC) [1114] has been proposedto provide a general framework for eliminating the reach-ing phase so that a sliding mode exists throughout an entireresponse, and the system response is then completely invari-ant to matched system perturbations. In GSMC, forcing func-tions that have to be designed to meet certain constraintsare introduced into switching functions to shape slidingdynamics that are equivalent to closed-loop system dynam-ics, for sliding motion exists all the time. In [11], the robusteigenvalue assignment GSMC (REA-GSMC) scheme wasdeveloped to place closed-loop poles invariably at desiredlocations so that the robust system can be characterizedin the conventional way. Although most of the previouslymentioned time-varying SMC schemes ensure global slidingmotion, they do not guarantee finite-time convergence of thetracking error.

To enhance the convergence properties of dynamical sys-tems, the design of terminal sliding mode controllers havingcertain state variable with fractional power was developed in[15,16]. When compared with linear hyperplane-based SMC,the terminal SMC offers the superior property of finite-timeconvergence and speeds up the rate of convergence aroundan equilibrium point. However, the terms with negative frac-tional powers existing in the terminal SMC laws may causethe singularity problem near the equilibrium [1719]; that is,the control takes on infinite values in the vicinity of the steadystate, and the resulting system has an unbounded right-handside. In fact, Wu et al. [20] proposed a two-phase controlscheme to avoid the singularity in their original control. Fur-ther studies [21,22] on solving the singularity problem adopta switching function with the velocity-error variable raisedto the power of greater than one, similar to the switchingfunction in the time-optimal control of a double-integratorplant. However, there exists a reaching phase in the previ-ously mentioned terminal SMCs, and system responses maythus be sensitive to system perturbations [23]. Bartoszewicz[7] proposed a time-varying terminal SMC that integrates thetime-varying sliding mode into the terminal SMC to elimi-nate the reaching phase. Nevertheless, the singularity aroundthe equilibrium is still a problem, and this approach assumeszero velocity error at the initial time that is not always true.In addition, the terminal time is not a controller parameter inthe previous scheme, and it is not obvious how to determinecontroller parameters for a desired terminal time.

This paper proposes a time-varying SMC for finite-timeconvergence of the tracking error. Beside the elimination of

the reaching phase, the proposed scheme offers the advantageof finite-time convergence without using fractional-powerfunctions of certain state variable, which both reduces therequired implementation effort and also makes the controlfree from singularity. Moreover, the initial velocity error doesnot necessarily have to be zero in the proposed scheme. Whenconcerning the controller design, the terminal time explicitlyappears in the formulation of the proposed scheme; that is, noparameter needs to be tuned to achieve the specified terminaltime. In addition to the parameters required in the conven-tional SMC, there is only one parameter to be adjusted inthe proposed scheme. Because the proposed scheme is basedon the GSMC framework [11], this paper first reviews theGSMC framework and the REA-GSMC design that origi-nates from the GSMC framework. Subsequently, the pro-posed design is presented, and experiments on a two-linkrobot manipulator are performed to investigate the effective-ness of the proposed scheme.

2 GSMC scheme for robot manipulators

2.1 Robot manipulators and GSMC

Consider an n-link robot manipulator whose dynamic modelis [24]M(q)q + B(q, q)q + g(q) = u(t) + d(t) (1)in which q n is the joint-displacement vector, u n isthe applied joint-torque vector, M(q) is the n n symmetricinertia matrix, B(q, q) is the nn matrix containing Coriolisand centrifugal terms, g(q) n is the gravitational forcevector, and d(t) n includes bounded unknown distur-bances referred to the actuator inputs.

Let qd denote the reference trajectory, and define the track-ing error as e = q qd . In the GSMC [11], the switchingfunction is defined as

s = e + Ce f(t) (2)in which C = diag [c1, c2, . . . , cn] , ci > 0 for i =1, 2, . . . , n, and f(t) n is referred to as the forcing func-tion in sliding dynamics described by s = 0. The switchingfunction in conventional SMC can be obtained by settingf(t) = 0, giving s = e + Ce. The conventional SMC isto force state trajectories to slide along the fixed switchinglines, s = 0. However, there exists a reaching phase beforesliding motion occurs, and system performance is sensitiveto perturbations during the reaching phase [23]. It is thus dif-ficult to determine the switching function and the gains in thereaching control law according to the performance specifi-cations. Consequently, the design has been much involved inthe conventional SMC scheme. In GSMC, the vector func-tion f(t) has to be designed to meet the following conditions:(C1) f(0) = e0 + Ce0; (C2) f(t) 0 as t ; and (C3)

123

Electr Eng (2010) 92:257268 259

f(t) has a bounded first time derivative. Here, e0 and e0 arevectors of velocity error and position error, respectively, att = 0. With the satisfaction of the first condition, state tra-jectories originate from the sliding regime, i.e. s(0) = 0. Thesecond condition implies asymptotic stability of the GSMCsystem, while the third condition is required for the existenceof a sliding control. If all three conditions are satisfied, andthe control law is designed such that the sliding conditionholds, then asymptotic stability is guaranteed, and the slid-ing mode exists throughout an entire response, i.e. s(t) = 0for t 0. In GSMC, the design of reaching control laws isunnecessary because state trajectories are constrained on thesliding regime from the very beginning of system responses.This both simplifies the controller design and also enhancesperformance robustness.

In this paper, the switching function is modified to

= s + Ct

0

s( )d (3)

Since s(0) = 0 in GSMC, we have (0) = 0. This gives (t) = 0 for t 0 provided that the sliding conditionholds. On the other hand, the solution of (t) = s(t) +C

t0

s( )d = 0 for t 0 is s(t) = 0 for t 0. Therefore,the introduction of the integral compensation to the switchingfunction in this way has no influence on the sliding dynamicsin theory. The conventional SMC that employs the switchingfunction s = e + Ce forces the tracking error e(t) to con-verge to zero in the ideal sliding mode s = e + Ce = 0, inwhich the switching frequency of control effort is assumed tobe infinitely high. However, due to the existence of parasiticunmodeled dynamics and/or digital implementation, infinite-frequency switching control cannot be achieved, and theswitching function s = e+Ce cannot be constrained exactlyto zero, leading to nonzero tracking errors. To eliminate thedirect-current components of nonzero tracking errors, theintegral compensation is required in practical implementa-tions.

2.2 GSMC law for robot manipulators

It is well known that the manipulator dynamics is linear interms of a suitably selected set of equivalent manipulatorparameters [24]. More specifically, for rigid robot dynam-ics (1), there exists a vector of unknown physical parametersp = [p1 p2 . . . pm]T m such thatM(q)v + B(q, q)v + g(q) = H(q, q, v, v)p (4)in which v n, and H(q, q, v, v) is a known n m matrix.Let p = [ p1 p2 pm]T be an estimate of the vector p, anddefine an estimated linearly parameterized model as

M(q)v + B(q, q)v + g(q) = H(q, q, v, v)p (5)in which M, B, and g are the estimates of M, B, and g, respec-tively. Assume that the estimation error for each parame-ter is bounded, i.e.

pi pi Pi for i = 1, 2, . . . , m.

Assume also that the unknown disturbance is bounded, i.e.d j (t) D j for j = 1, 2, . . . , n. Note that the assumptionof the boundedness of d(t) is for deriving a sliding controllaw, rather than for obtaining (5). No assumption concerningd(t) is required to obtain (5). The sliding control law thatsatisfies the sliding condition is

u = M(q)v + B(q, q)v + g(q) K sgn( ) (6)

in which v = qd Ce + f(t) Ct

0s( )d ,

K = diag [k1, k2, . . . , kn] , ki > 0, i = 1, 2, . . . , n,sgn( ) = [ sgn(1) sgn(2) sgn(n) ]T , = diag [1, 2, , n] , i = Di + mj=1

hi j Pj ,i = 1, 2, . . . , n,

and hi j denotes element ij of matrix H(q, q, v, v). Since thesliding control law (6) satisfies the sliding condition, the slid-ing mode is ensured throughout an entire response providedthat the forcing function f(t) fulfills conditions (C1C3). Inother words, if f(t) meets conditions (C1C3), then closed-loop system dynamics are equivalent to the global slidingdynamics described by (t) = 0 for t 0, and the out-put response is completely invariant to system perturbations.Please refer to Appendix A for the stability analysis of theclosed-loop system with the control input (6).

3 Time-varying sliding-mode controlfor finite-time convergence

The sliding dynamics (t) = 0 for each link are decoupledand can thus be independently dealt with in the same fashion.Let us discard the index of each component of a vector forsimplicity and clarity, and use c and z to denote ci and thei th component of a vector z, respectively.

3.1 Revisit of robust eigenvalue assignment GSMC(REA-GSMC) design

The GSMC scheme provides a general framework for ensur-ing the existence of a global sliding mode. However, it is anopen question about how to design the forcing function f (t)to have satisfactory sliding dynamics. Lu and Chen [11] pro-posed a robust eigenvalue assignment GSMC (REA-GSMC)design that assigns

f (t) = (e0 + ce0) exp(ct) (7)

123

260 Electr Eng (2010) 92:257268

in which the initial time is set to zero. It is obvious that thisforcing function satisfies conditions (C1C3), implying thatthe sliding dynamics completely govern closed-loop systemresponses. Solving s = 0 subject to (7) and e(0) = e0, wehave

e(t) = e0 exp(ct) + (e0 + ce0)t exp(ct) (8)It can be shown that the solution of the initial value problem

e + 2ce + c2e = 0 (9)subject to e(0) = e0 and e(0) = e0 is the same as (8). Hence,the forcing function (7) forces the tracking error to convergein the way described by (9), which can be determined accord-ing to the desired eigenvalues, c of multiplicity two. Theerror response governed by (9) is valid for all t 0, and isindependent of the parametric uncertainties and external dis-turbances, implying performance robustness. Therefore, theREA-GSMC robustly places two eigenvalues throughout anentire response. In contrast, the conventional SMC robustlyplaces one eigenvalue only after the reaching phase, and theresponse during the reaching phase can be much influencedby system perturbations, yielding non-robust system behav-ior. Although the REA-GSMC design achieves robust eigen-value assignments, it leads to infinite convergence time. Toincrease the convergence rate, the poles must be far from theorigin on the left-half complex plane, bringing as a conse-quence a high-gain controller. When considering the actu-ator saturation and the excitation of unmodeled dynamicsin practical implementation, this high-gain control may beundesirable.

3.2 Time-varying sliding-mode control for finite-timeconvergence

Based on the GSMC, a kind of sliding dynamics is devisedto reach the desired trajectory in finite time. Considering thefollowing forcing function

f (t) =

e0 + ce0 sin(

t2tn

), 0 t tn

12 (e0 + ce0 )

[1 + cos

(

ttnt f tn

)], tn < t t f

0, t > t f(10)

in which t f is the terminal time assigned according to designspecifications, and tn are constant parameters to be deter-mined, and 0 < tn < t f . Note that the forcing functionis continuous on [0,). Moreover, since limtt+n f (t) =limttn f (t) = 0 and limtt+f f (t) = limttf f (t) = 0,the forcing function f has a continuous first derivative withrespect to time. This continuity ensures no abrupt jumps inthe equivalent control associated with the proposed slidingdynamics provided that the desired trajectory qd has a contin-uous second derivative and the disturbance d is continuous as

well. The continuity of the equivalent control implies smoothoperations even though the design is intended to have finite-time convergence.

It can be verified that the forcing function (10) satisfiesconditions (C1C3), meaning that the closed-loop systemresponses are completely governed by the sliding dynamicsdescribed by s = 0 with the forcing function (10). To deter-mine the constants and tn , the error response in the slidingmode s = 0 needs to be analyzed. From (10), it is seen that fis piecewise defined. Consequently, the error response has tobe solved in three parts corresponding to the three intervalsover which f is defined. For 0 t tn , solving s = 0subject to e(0) = e0 yields

e(t) = A1 + A2 sin(

t

2tn

)+ A3 cos

( t

2tn

)

+A4 exp(ct) for 0 t tn (11)in which A1 = c1e0 + e0, A2 = c A0, A3 =(2tn)1 A0, and A4 = (c1e0 + A3) with the constantA0 =

[c2 +

(

2tn

)2]1. Then for tn < t t f , solving s = 0

with limtt+n e(t) = e(tn) that is obtained from (11) gives

e(t) = A5 + A6 sin(

t tnt f tn

)+ A7 cos

(

t tnt f tn

)

+A8 exp(ct) for tn < t t f (12)in which

A5 = (2c)1(e0 + ce0 ),A6 =

[2(t f tn)

]1(e0 + ce0 )A9,

A7 = 21c(e0 + ce0 )A9,A8 = (A1 A5 + A2 A7) exp(ctn) + A4

with A9 =[

c2 +(

t f tn

)2]1. Setting e(t f ) = 0 in (12)

gives the relation

A5 A7 + A8 exp(ct f ) = 0 (13)After rearranging (13), we have

= L1(e0 + ce0)[1 + exp (c(tn t f ))] + c1e0 exp(ct f )

L1 + L2 exp(c(tn t f )

) + L3 exp(ct f )(14)

in which L1 = (2c)1 + 21cA9, L2 = (2c)1 cA0 +21cA9, and L3 = (2tn)1 A0. From (14), can be deter-mined by assigning the constant parameter tn a positive valuesmaller than the terminal time t f . Generally speaking, thereis no unique choice for tn , and decreasing tn increases therequired control effort during an initial period and vice versa.With c = 10, e0 = 0, e0 = 1, and t f = 0.8, Fig. 1 depictsthe ideal error responses and forcing functions with the pro-posed sliding dynamics for various values of tn . It is found

123

Electr Eng (2010) 92:257268 261

Fig. 1 Theoretical responsesassociated with the proposedsliding dynamics subject tovarious values of tn (the topright and the bottom rightsubplots provide magnifiedviews of the top left and thebottom left subplots,respectively)

0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Time (s)

e(t)

tn=0.2

tn=0.4

tn=0.6

0 0.5 1-10

-8

-6

-4

-2

0

Time (s)

f(t)

IAE=0.170IAE=0.240IAE=0.307

0.76 0.78 0.8 0.82-20

-15

-10

-5

0x 10-6

Time (s)

e(t)

0 0.5 1-1

-0.5

0

Time (s)

f(t)

ITAE=0.020ITAE=0.037ITAE=0.060

that the tracking error converges to zero at the specified ter-minal time, despite different values of tn . The correspondingintegral of absolute error (IAE) and the integral of time abso-lute error (ITAE) values defined as

IAE =1

0

|e(t)|dt, and ITAE =1

0

t |e(t)|dt

are also indicated in Fig. 1. It is seen that decreasing tnincreases the convergence rate during an initial period andthus decreases the IAE and ITAE values.

With the parameter determined according to (14), theproposed sliding dynamics give e(t f ) = 0, which impliese(t f ) = 0 because f (t f ) = 0 and s(t f ) = e(t f )+ce(t f )f (t f ) = 0. Moreover, since f (t) = 0 for t > t f accordingto (10), solving s = 0 subject to limtt+f e(t) = 0 givese(t) = 0 for t > t f . Therefore, the tracking error remainsat zero for all t t f . In contrast to the REA-GSMC, whoseforcing function has a constant algebraic sign throughout anentire response, the proposed forcing function, as shown inFig. 1, changes signs during the transient response so thatthe convergence time is finite. In conclusion, the proposedtime-varying SMC brings the system state to the target statein finite time and is then bumplessly turned into the conven-tional SMC after the terminal time when the tracking errorvector converges to zero. In other words, the sliding con-trol law (6) with the switching function defined by (3) and(10) forces the system state to reach the desired trajectoryin finite time and remain on it thereafter. It is seen that theproposed control law does not involve fractional-power func-

tions of certain state variable, and no singularity occurs in theproposed scheme. Beside its simplicity in implementationcompared with the previous terminal SMCs, the proposedscheme has the terminal time t f as an explicit parameter forthe designers convenience.

4 Application to a two-link manipulator

4.1 System description

Consider the position control of a two-link revolute-jointmanipulator subject to an uncertain payload. For a two-linkrobot manipulator moving in the vertical plane as shown inFig. 2, the dynamic equation is given by[

M11 M12M21 M22

] [q1q2

]+

[B11 B12B21 B22

] [q1q2

]+

[g1g2

]

=[

u1u2

]+

[d1d2

](15)

in which

M11 = m121c + I1 + Im1 +(m2 + m2 + m p

)21

+ m222c + I2 + Im2 + IM2 + m p22+ 2 (m22c + m p2) 1 cos q2,

M12 = M21 = m222c + I2 + Im2 + m p22+ (m22c + m p2) 1 cos q2,

M22 = m222c + I2 + Im2 + m p22,B11 = 2

(m22c + m p2

)1q2 sin q2,

123

262 Electr Eng (2010) 92:257268

2m

g

nm

C1

1

2

1q

2q

C2

Fig. 2 Experimental planar robot manipulator with two directly drivenlinks

B12 = (m22c + m p2

)1q2 sin q2,

B21 =(m22c + m p2

)1q1 sin q2,

B22 = 0, g2 = g(m22c + m p2)cos(q1 + q2),g1 = g

[(m11c + m21 + m21 + m p1) cos q1

+(m22c + m p2)cos(q1 + q2)]

Here, mi is the mass of link i , Ii is its moment of inertiaabout mass centre, i denotes the length of link i , ic is thedistance between the joint i and the mass centre of link i , Imiis the rotor moment of inertia of motor i , IM2 is the statormoment of inertia of motor 2, m2 is the mass of motor 2, gis the gravitation constant, and m p denotes the mass of anuncertain payload.

The robot in the experimental setup is directly driven bytwo permanent-magnet ac servomotors. Figure 3 shows theconfiguration of the robot control system, in which ac motors1 and 2 are with rated output power of 400 and 100 W,respectively, and both are SGMAH servomotors manufac-tured by Yaskawa Electric. Estimated parameter values of therobot are m1 = 0.064, m2 = 0.039, I1 = 1.93 104,I2 = 6.63 105, 1 = 0.117, 2 = 0.105, 1c =0.055, 2c = 0.037, Im1 = 6.16 105, Im2 = 3.64 106, IM2 = 1.8 104, and m2 = 0.54 in SI units. Therange of the uncertain payload mass is between 0.1 and 0.3kg, i.e. m p [0.1, 0.3] . The shaft encoder mounted to eachac servomotor has 2,048 lines, which yields a resolution of8,192 pulses/rev after the A and B signals from the encoderhave been processed by the FPGA (Field-ProgrammableGate Array), model XCV50PQ240-C6 from Xilinx, Inc. Thevelocity information is measured by a digital tachometer thatmeasures the time interval of the encoder pulses to achieve

FPGA

Encoder

A, B, Z signals

TMS320C6711 DSP

AC motor 1

A, B, Z signals

AC motor 2

Encoder

DSP interface

DAC interface

Position counting

Velocity detection

DAC + Analog

processing

DAC interface

Position counting

DAC + Analog

processing

Regulated current

converter

Regulated current

converter

Fig. 3 Hardware configuration of the robot control system

more accurate estimation than the direct differentiation ofthe position signal. The tasks of both position counting andvelocity detection are implemented in the FPGA. The con-troller core is a floating-point TMS320C6711 digital sig-nal processor (DSP) that obtains information on positionand velocity from the FPGA, calculates control algorithms,and sends control efforts to the regulated current converterthrough a 12-bit digital-to-analog converter and some analogsignal processing circuits. A sampling period of 0.08192 mswas chosen. In the experimental system, a personal com-puter was used to develop the control program written in Clanguage, to compile it, to download the resulting code intoDSP for execution, and to acquire experimental data.

4.2 Controller design

The REA-GSMC and the proposed time-varying SMC aredesigned and implemented for performance comparisons. Tohave objective comparisons, both schemes use the same slid-ing control law (6) except with different forcing functions.Since the main system uncertainties are due to the payloadand external disturbances including joint frictions, the onlyparametric uncertainty considered in the controller design isassociated with the payload, and the switching gains in thesliding control law (6) becomei = Di + |hi | Mp, i = 1, 2 (16)in which Mp denotes the bounds on payload uncertainty,and hi represents the term associated with m p in the i throw of H(q, q, v, v). Since m p [0.1, 0.3], we specifyMp = (0.3 0.1)/2 = 0.1 and m p = (0.3 + 0.1)/2 = 0.2.

123

Electr Eng (2010) 92:257268 263

Fig. 4 Experimental responsesby the REA-GSMC and theproposed method with t f = 0.8

0 0.5 1-0.2

-0.1

0

0.1

0.2

Time (s)

e1

(rad)

REA-GSMCProposed Method

0 0.5 1-0.2

-0.1

0

0.1

0.2

Time (s)

e2

(rad)

0 0.05 0.1 0.15 0.2-1

0

1

2

3

4

Time (s)

u1

(Nm)

IAE1=0.205IAE1=0.248

0 0.05 0.1 0.15 0.2-0.2

0

0.2

0.4

0.6

0.8

Time (s)

u2

(Nm)

IAE2=0.218IAE2=0.247

0 0.05 0.1 0.15 0.2

-2

0

2

4

Time (s)

1

ITAE1=0.033ITAE1=0.040

0 0.05 0.1 0.15 0.2

-2

0

2

4

Time (s)

2

ITAE2=0.035ITAE2=0.040

Other parameters in the control law (6) are chosen to bec = 10, K = diag [0.10, 0.02] , D1 = 0.10, and D2 =0.02. In the proposed time-varying SMC, the parameter tn isnominally set to one-half of t f , i.e. tn = t f /2. To diminishchattering, the boundary-layer approach [24] that replacesthe discontinuous sign function by a continuous saturationfunction is applied to the sliding control law (6) with a bound-ary-layer thickness of 1.0.

4.3 Experimental evaluation

The manipulator was initially at rest, i.e. q1(0) = /2 radand q2(0) = 0. Let the tracking errors of the first and the sec-ond joints be denoted as e1 and e2, respectively. Moreover,the IAEi and ITAEi values shown in the following figuresrepresent the IAE and ITAE values, respectively, for the i thjoint, i = 1, 2. With a payload of 0.2 kg, Fig. 4 shows theresponses to qd1 = q1(0) + 1.0 rad and qd2 = 1.0 rad by

the REA-GSMC and the proposed time-varying SMC witht f = 0.8. It is seen that the output response around the equi-librium by the proposed scheme is less sluggish than thatby the REA-GSMC. Moreover, with nearly the same settingtime corresponding to a 5% tolerance band, the IAE andITAE values with the REA-GSMC are smaller than thosewith the proposed method, whereas the maximum controlrequired by the REA-GSMC is much larger than that by theproposed scheme. Figure 5 shows the dynamic responses bythe proposed scheme with different payloads, revealing thatthe system performance is robust to the variation of the pay-load. With a payload of 0.2 kg, Fig. 6 presents the dynamicresponses by the proposed scheme for various values of t f . Itis seen that the smaller the specified terminal time, the largerthe required maximum control. By comparing Figs. 6 with 4,it is found that although the proposed control with t f = 0.6demands less maximum control magnitude than the REA-GSMC, its corresponding output response settles faster than

123

264 Electr Eng (2010) 92:257268

Fig. 5 Experimental responsesby the proposed method subjectto various payloads

Fig. 6 Experimental responsesby the proposed method withdifferent values of t f

0 0.5 1-0.2

-0.1

0

0.1

0.2

Time (s)

e1

(rad)

tf=0.6tf=0.8tf=1.0

0 0.5 1-0.2

-0.1

0

0.1

0.2

Time (s)

e2

(rad)

0 0.05 0.1 0.15 0.2-1

0

1

2

3

4

Time (s)

u1 (N

m)

IAE1=0.205IAE1=0.248IAE1=0.288

0 0.05 0.1 0.15 0.2-0.2

0

0.2

0.4

0.6

0.8

Time (s)

u2

(Nm)

IAE2=0.206IAE2=0.247IAE2=0.287

0 0.05 0.1 0.15 0.2

-0.5

0

0.5

1

1.5

Time (s)

1

ITAE1=0.028ITAE1=0.040ITAE1=0.054

0 0.05 0.1 0.15 0.2

-0.5

0

0.5

1

1.5

Time (s)

2

ITAE2=0.028ITAE2=0.040ITAE2=0.054

123

Electr Eng (2010) 92:257268 265

Fig. 7 Experimental responsesby the proposed method withvarious values of tn

0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Time (s)

e1

(rad)

tn=0.2

tn=0.4

tn=0.6

0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Time (s)

e2

(rad)

0 0.05 0.1 0.15 0.2-1

0

1

2

3

4

Time (s)

u1

(Nm)

IAE1=0.179IAE1=0.248IAE1=0.317

0 0.05 0.1 0.15 0.2-0.2

0

0.2

0.4

0.6

0.8

Time (s)

u2

(Nm)

IAE2=0.178IAE2=0.247IAE2=0.316

0 0.05 0.1 0.15 0.2

-0.5

0

0.5

1

1.5

Time (s)

1

ITAE1=0.022ITAE1=0.040ITAE1=0.064

0 0.05 0.1 0.15 0.2

-0.5

0

0.5

1

1.5

Time (s)

2

ITAE2=0.022ITAE2=0.040ITAE2=0.064

that by the REA-GSMC, and its corresponding ITAE valuesare smaller than those with the REA-GSMC, illustrating thatthe proposed control with t f = 0.6 is superior to the REA-GSMC in terms of minimizing the ITAE performance indexwith less demands on control effort.

With fixed t f = 0.8, Fig. 7 shows the error responses bythe proposed scheme for various values of tn . It demonstratesthat the smaller the parameter tn , the larger the required max-imum control. The plots of the switching functions have beenincluded in Figs. 4, 6 and 7 to demonstrate the elimination ofthe reaching phase. Note that the boundary-layer approach[24] has been applied to the sliding control law (6) to dimin-ish chattering, which however only ensures the boundednessof the switching functions, rather than the existence of slid-ing modes. It can be seen that all switching functions shownin Figs. 4, 6, and 7 begin from zero and remain bounded.Despite the non-existence of ideal sliding modes, the IAE and

ITAE values given in Fig. 7 are close to the correspondingideal values indicated in Fig. 1, and the error responses shownin Fig. 7 are similar to the ideal ones presented in Fig. 1.Moreover, by setting tn = 0.2 while keeping t f = 0.8, theIAE and ITAE values are much smaller than those shown inFig. 4. This suggests that the proposed control outperformsthe REA-GSMC in the sense that it can provide good IAEand ITAE performance without requiring excessive controleffort.

Consider the task of tracking a circular path in the verticalplane. Corresponding to q1(0) = /2 rad and q2(0) =0, the end effector of the manipulator is initially locatedat (0,/L) in the xy plane, and the circular path tobe followed is described by x = 0.1 + r cos(1.28t) andy = 0.1 + r sin(1.28t), in which the radius r = 0.05 m.As shown in Fig. 8 are the trajectories of the end effectorin the vertical plane. It is seen that, when the end effector

123

266 Electr Eng (2010) 92:257268

approaches the desired trajectory, the contour error of theproposed system with t f = 0.8 converges only a little betterthan that of the REA-GSMC system. However, the corre-sponding dynamic responses during an initial time periodshown in Fig. 9 reveal that the proposed scheme requiresmuch less starting torque, but gives less sluggish outputresponses near the target than the REA-GSMC while bothyield almost the same settling time. From the previous exper-imental results, it is observed that the proposed time-varyingSMC scheme is superior to the REA-GSMC scheme in thefollowing sense: With nearly the same settling time, the pro-

Fig. 8 Trajectories of the end effector for tracking a circular path

posed SMC requires much less maximum control effort andyields less sluggish output responses around the referencethan the REA-GSMC. In addition, the proposed scheme hasthe terminal time as an explicit parameter with an ease inobtaining a controller that fulfills the terminal-time require-ment.

5 Conclusions

This paper has proposed a time-varying SMC scheme forfinite-time convergence. The reaching phase is eliminatedto enhance performance robustness, and the resulting outputresponse around the reference is less sluggish than that by theSMC with linear sliding dynamics. The continuity of the firsttime derivative of the proposed forcing function in the slid-ing dynamics ensures smooth operation, while guaranteeingfinite-time convergence. Moreover, the proposed switchingfunction does not involve fractional-power functions of cer-tain state variable, and is thus both free from the singularityproblem and also convenient for practical implementation.When concerning the determination of controller parameters,the terminal time can be explicitly specified in the proposeddesign, and the controller that achieves the desired termi-nal time can be easily obtained using the proposed scheme.Experimental results on a two-link direct-drive robot demon-strate that the proposed time-varying SMC can achieve fastand smooth responses with moderate control magnitude.

Fig. 9 Experimental responsesby the REA-GSMC and theproposed method with t f = 0.8for tracking the circular path

123

Electr Eng (2010) 92:257268 267

Acknowledgments The authors would like to thank the anonymousreferees and the editors for their valuable comments and suggestions,as well as the National Yunlin University of Science and Technologyfor the use of its facilities. This work was partially supported by theNational Science Council of ROC under Grant Number NSC 98-2221-E-003-008-MY2.

Appendix A: Stability Proof of the GSMC System

To demonstrate the stability of the proposed system, considera Laypunov candidate

2V = TM(q) (17)Since M(q) is uniformly positive definite, V is a positivedefinite function. Differentiating V with respect to time andsubstituting (1)(3) into the resulting equation yields

V = TM + 12TM

= TM (q qd + Ce f + Cs) + 12TM

= T [M (qd + Ce f + Cs) Bq g + u + d]

+12TM (18)

Using the relation M(q) = 2B(q, q) + J(q, q), in whichJ(q, q) is skew symmetric [24], one has

TBq + 12TM

= TB(qd + Ce f + C

t0

s( )d)

(19)

Substituting (6) and (19) into (18) gives

V = T[

M(qd + Ce f + Cs)

+ B(qd + Ce f + C

t0

s( )d)

g + u + d]

= T[(

M M)

v +(

B B)

v + (g g)

+ d sgn( ) K]

(20)

Subtracting (4) from (5), one has(M M

)v +

(B B

)v + (g g) = H (p p) (21)

According to (20) and (21), the time derivative of V alongany system trajectoryV = T [H (p p) + d sgn( ) K ]

TK 0 (22)is a negative definite function of , implying that = 0 is aglobally asymptotically stable equilibrium point. Moreover,since (0) = 0, one has (t) = 0 for t 0. Accordingto (3), the solution of (t) = s(t) + C t0 s( )d = 0 for

t 0 is s(t) = 0 for t 0, ensuring the asymptotic stabil-ity of the proposed system provided that the three conditions(C1)(C3) are fulfilled. unionsq

References

1. Utkin VI (1977) Variable structure systems with sliding mode.IEEE Trans Autom Control 22:212222

2. Choi SB, Cheong CC, Park DW (1993) Moving switching surfacesfor robust control of second-order variable structure systems. IntJ Control 58:229245

3. Choi SB, Park DW, Jayasuriya S (1994) A time-varying slidingsurface for fast and robust tracking control of second-order uncer-tain systems. Automatica 30:899904

4. Roy RG, Olgac N (1997) Robust nonlinear control via moving slid-ing surfacesn-th order case. In: Proceedings of IEEE conferenceon decision and control. San Diego, pp 943948

5. Chang TH, Hurmuzlu Y (1993) Sliding control without reachingphase and its application to bipedal locomotion. J Dyn Syst MeasControl Trans ASME 115:447455

6. Yilmaz C, Hurmuzlu Y (2000) Eliminating the reaching phasefrom variable structure control. J Dyn Syst Meas Control TransASME 122(4):753757

7. Bartoszewicz A (1996) Time-varying sliding modes for secondorder system. IEE Proc Contr Theory Appl 143(5):455462

8. Betin F, Pinchon D, Capolino GA (2002) A time-varying slidingsurface for robust position control of a dc motor drive. IEEE TransInd Electron 49(2):462473

9. Yagz N, Hacoglu Y (2005) Fuzzy sliding modes with mov-ing surface for the robust control of a planar robot. J Vib Control11:903922

10. Yorgancoglu F, Kmrcgil H (2008) Single-input fuzzy-likemoving sliding surface approach to the sliding mode control. ElectrEng 90(3):199207

11. Lu YS, Chen JS (1994) Design of a global sliding mode controllerfor robot manipulator with robust tracking capability. Proc Natl SciCouncil Part A Phys Sci Eng 18(5):463476

12. Lu YS, Chen JS (1995) A global sliding mode controller design formotor drives with bounded control. Int J Control 62(5): 10011019

13. Choi HS, Park YH, Cho Y, Lee M (2001) Global sliding-modecontrol: improved design for a brushless dc motor. IEEE ContrSyst Mag 21(3):2735

14. Lin JY, Lu YS, Chen JS (2002) A global sliding-mode controlscheme with adjustable robustness for a linear variable reluctancemotor. JSME Int J Series C 45(1):215225

15. Venkataraman ST, Gulati S (1993) Control of nonlinear systemsusing terminal sliding modes. J Dyn Syst Meas Control TransASME 115:554560

16. Man Z, Paplinski AP, Wu H (1994) A robust MIMO terminal slid-ing mode control scheme for rigid robotic manipulators. IEEETrans Autom Control 39(12):24642469

17. Barambones O, Etxebarria V (2002) Energy-based approach tosliding composite adaptive control for rigid robots with finite errorconvergence time. Int J Control 75(5):352359

18. Parra-Vega V, Hirzinger G (2001) Chattering-free sliding modecontrol for a class of nonlinear mechanical systems. Int J RobustNonlinear Control 11(12):11611178

19. Tang Y (1998) Terminal sliding mode control for rigid robots.Automatica 34(1):5156

20. Wu Y, Yu X, Man Z (1998) Terminal sliding mode control designfor uncertain dynamic systems. Syst Control Lett 34:281288

21. Feng Y, Yu X, Man Z (2002) Non-singular terminal sliding modecontrol of rigid manipulators. Automatica 38(12):21592167

123

268 Electr Eng (2010) 92:257268

22. Yu S, Yu X, Man Z (2005) Continuous finite-time control forrobotic manipulators with terminal sliding mode. Automatica41:19571964

23. Harashima F, Hashimoto H, Kondo S (1985) MOSFET converter-fed position servo system with siding mode control. IEEE TransInd Electron 32:238244

24. Slotine JJE, Li W (1991) Applied nonlinear control. Prentice Hall,New Jersey

123

Time-varying sliding-mode control for finite-time convergenceAbstract1 Introduction2 GSMC scheme for robot manipulators2.1 Robot manipulators and GSMC2.2 GSMC law for robot manipulators

3 Time-varying sliding-mode control for finite-time convergence3.1 Revisit of robust eigenvalue assignment GSMC (REA-GSMC) design3.2 Time-varying sliding-mode control for finite-time convergence

4 Application to a two-link manipulator4.1 System description4.2 Controller design4.3 Experimental evaluation

5 ConclusionsAcknowledgmentsAppendix A: Stability Proof of the GSMC SystemReferences

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 149 /GrayImageMinResolutionPolicy /Warning /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 150 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 599 /MonoImageMinResolutionPolicy /Warning /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/CreateJDFFile false /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles false /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /DocumentCMYK /PreserveEditing true /UntaggedCMYKHandling /LeaveUntagged /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice