Nonlinear DynDOI 10.1007/s11071-015-2137-7

ORIGINAL PAPER

An adaptive chattering-free PID sliding mode control basedon dynamic sliding manifolds for a class of uncertainnonlinear systems

Saleh Mobayen

Received: 3 October 2014 / Accepted: 2 May 2015 Springer Science+Business Media Dordrecht 2015

Abstract This paper proposes a new dynamic PIDsliding mode control technique for a class of uncertainnonlinear systems. The offered controller is formulatedbased on the Lyapunov stability theory and guaran-tees the existence of the sliding mode around the slid-ing surface in a finite time. Furthermore, this approachcan eliminate the chattering phenomenon caused by theswitching control action and can realize high-precisionperformance. Moreover, an adaptive parameter tuningmethod is proposed to estimate the unknown upperbounds of the disturbances. Simulation results for aninverted pendulum system demonstrate the efficiencyand feasibility of the suggested technique.

Keywords Adaptive sliding mode controller Chattering-free performance PID sliding surface Dynamic sliding mode Nonlinear system

1 Introduction

1.1 Background and motivations

Stabilization and tracking control of uncertain nonlin-ear systems have significant applications in electron-

S. Mobayen (B)Department of Electrical Engineering, Faculty ofEngineering, University of Zanjan, P.O. Box 38791-45371,Zanjan, Irane-mail: smobayen@gmail.com; mobayen@znu.ac.irURL: http://www.znu.ac.ir/members/mobayen_saleh

ics, mechanics, and robotic systems [13]. In partic-ular, various dynamical systems such as flexible-linkrobots, oscillators, chaotic systems, and synchronousmachines may be modeled by an uncertain nonlinearstructure. In previous years, investigation of nonlinearcontrol problems has attracted much attention due tothe nonlinear structure of most physical and dynami-cal systems [4]. Nonlinear systems with time-varyingand uncertain terms can model many important phe-nomena and display various behaviors such as differ-ent equilibrium points and multiple periodic orbits fornon-autonomous systems [5].

As an effective and famous robust control scheme,sliding mode control (SMC) has been extensivelyapplied for the improvement of stability, performance,and robustness of linear and nonlinear systems in thepresence of uncertainties [68]. The main features ofSMC are the fast response, robustness against uncer-tainties, insensitivity to the bounded disturbances, goodtransient performance, and computational simplicitywith respect to other control techniques [911]. Theprocedure of SMC can be separated into two phaseswhich are the sliding phase and the reaching phase[12,13]. Because of the influence of sliding surface onthe stability and transient performance of control sys-tems, the major step is to introduce a proper switchingsurface so that tracking errors reduce to a satisfactoryvalue [6,14]. In SMC method, the robust tracking per-formance can be satisfied after the system states reachthe sliding surface, and consequently robustness is notguaranteed during the reaching phase. On the other

123

S. Mobayen

hand, SMC suffers from the undesired high-frequencyoscillations known as chattering phenomena which iscreated by the discontinuous control law and is veryharmful for actuators used in practical systems [1517].To suppress the chattering, a second-order sliding mode(SOSM) control approach has been presented [1820].The main idea of SOSM is that the discontinuous signfunction is forced to act on the time derivative of thecontrol law, and thus the actual control input resultedafter integration is continuous which removes the chat-tering [18]. The other difficult issue in designing SMCis the requirement of the norm of uncertainty properlywhich is used for the switching control gain. Pertur-bation estimation techniques have been investigated in[2128] to overcome the problem of the knowledge ofperturbation upper bounds.

1.2 Literature review

In [22], an observer-based SMC is presented for theexcitation control of synchronous generators whichemploys output feedback and perturbation estimation.In [23], a sliding perturbation observer is offered andintegrated into the proportional integral (PI)-controlledsystem to compensate for the perturbations causedby parameter variations and unknown disturbances. In[24], a variable structure control technique combinedwith the adaptive perturbation estimation method isproposed for trajectory tracking of the piezoelectricactuators. In [25], a high-order SMC observer is sug-gested to guarantee the perturbation estimation of thelinear time-invariant systems affected by unmatchedperturbations. In [26], an adaptive second-order SMCwith a PID sliding surface is designed to overcome thefaults in the heat recovery steam generator boilers andachieve good performance for the boilers. The differ-ence of our offered method in comparison with what isproposed in [26] is the nonlinear structure of the con-trolled system; however, besides good tracking perfor-mance stated in [26], the controlled heat recovery steamgenerator boilers tolerate the faults in input and systemmatrices. In [27], a SMC system with a PID slidingsurface is adopted to control the speed of an electro-mechanical system, and the desired trajectory is trackedin the presence of uncertainties and disturbances. Thenovelties of the suggested method of this paper in com-parison with [27] are the application of adaptive para-meter tuning scheme, using a dynamic sliding surface,

and finite-time tracking controller design for uncertainnonlinear systems; however, using high-speed comput-ers and some application tools in [27], the experimentalresults are presented which confirm the efficiency andsimplicity of the method. In [28], an adaptive fuzzySMC methodology with a PID sliding surface is pro-posed for the control of robot manipulators. Similarto our proposed method, an adaptive SMC approachcombined with a PID sliding surface is presented in[28]; however, fuzzy logic control scheme of [28] isused to generate the hitting control signal, and the out-put gain of the fuzzy SMC is tuned online by a super-visory fuzzy structure, where the chattering problemis avoided. Also, compared with the proposed methodin this paper, an adaptive robust PIDSMC controlleris presented in [29] which is optimized by a multi-objective genetic algorithm to control a mechanicalsystem in the presence of external disturbances andperturbations. Moreover, the application of PID-SMCto a MIMO system which is proposed in [30] is a morechallenging action due to the coupling effects betweenthe control variables. All the methods introduced inthe above-mentioned references motivate researchersto actively develop the proposed method of this paper.

1.3 Contributions

To the best of the authors knowledge, very littleattempts have been made to design a robust adaptiveSMC using a PID sliding surface for the tracking con-trol problem of nonlinear system with time-varyinguncertainties. In the recent years, very little attentionhas been paid to this problem, which is still openin the literature. In this paper, a PID sliding surface-based SMC control approach is proposed for the track-ing problem of nonlinear uncertain systems with time-varying uncertainties and nonlinearities. Using the sug-gested control technique, the tracking errors convergeto zero in a finite time and the existence of the slidingmode around the sliding surface in a finite time is guar-anteed. This approach eliminates the reaching phaseof SMC to improve the global robustness of the sys-tem. In addition, this method overcomes the chatteringproblem caused by the discontinuous sign function andeliminates the need for pre-design information aboutthe upper bound of external disturbances. The stabil-ity and robustness of the proposed control method areproved using Lyapunov stability theory. To justify the

123

An adaptive chattering-free PID sliding mode control

feasibility and efficiency of the introduced method, thetracking problem of an inverted pendulum system issimulated.

1.4 Paper organization

This paper is organized as follows: the problem descrip-tion is given in Sect. 2. In Sect. 3, the main resultsare proposed and the convergence and stability of theclosed-loop systems are analyzed via Lyapunov stabil-ity theory. Simulation results on an inverted pendulumsystem are provided in Sect. 4, and the conclusions arepresented in Sect. 5.

2 Problem description

The nonlinear second-order system with time-varyinguncertainties is described as:

x1(t) = x2(t),x2(t) = f (x, t) + f (x, t)

+ (b(x, t) + b(x, t)) u(t) + d0(x, t), (1)where x(t) = [x1(t), x2(t)]T is the state vector, u(t)is the control input, b(x, t) and f (x, t) are the knownbounded nonlinear functions, and b(x, t), f (x, t)and d0(x, t) are the nonlinear functions which intro-duce the system uncertainties and external distur-bances. Defining d(x, t) = f (x, t)+b(x, t)u(t)+d0(x, t), the nonlinear second-order system can be con-sidered as:

x1(t) = x2(t),x2(t) = f (x, t) + b(x, t)u(t) + d(x, t). (2)

The uncertain nonlinear system (2) is supposed totrack the desired trajectory xd(t) = [x1d(t), x2d(t)]T ,where x2d(t) = x1d(t), and x2d(t) is a differentiablefunction of time. The tracking error is defined as:

E(t) = x(t) xd(t) = [e(t), e(t)]T , (3)where e(t) = x1(t) x1d(t) and e(t) = x2(t) x2d(t).

The dynamic sliding mode equation for system (2)can be defined as:

s(t) = G (E(t) L(t)) , (4)where G = [g1, g2] are the gain coefficients andL(t) = [l(t), l(t)]T are the terminal functions.

Assumption 1 [31]: Consider the function l(t) :R+ R, l(t) Cn[0,), l(t) L, l(t) is finitein interval [0, T ], E(0) = L(0), E(0) = L(0), that is,l(0) = e(0), l(0) = e(0), and l(0) = e(0). Moreover,for every t T , l(t) = 0, l(t) = 0, and l(t) = 0.Cn[0,) is the set of n rank differentiable continuousfunctions defined in [0,).

Remark 1 The sliding surface (4) makes that the statetrajectories arrive at the sliding surface right from thebeginning. Thus, the reaching interval is eliminated,and the global robustness of the whole system is guar-anteed. From (4) and Assumption 1, it follows that:

s(0) = G (E(0) L(0)) = 0, (5)s(0) = G (E(0) L(0)) = 0. (6)

The terminal function l(t) is defined as [31]:

l(t) =

(a20T 5 e0 + a21T 4 e0 + a22T 3 e0

)t5

+(

a10T 4 e0 + a11T 3 e0 + a12T 2 e0

)t4

+(

a00T 3 e0 + a01T 2 e0 + a02T e0

)t3

+ 12 e0t2 + e0t + e0,

if t T

0, if t T(7)

In light of Assumption 1 and the following equa-tions, functions l(t), l(t), and l(t) can all be equal tozero at time t = T by designing ai j (i, j = 0, . . . , 2)as:

a00 + a10 + a20 = 1,3a00 + 4a10 + 5a20 = 0,6a00 + 12a10 + 20a20 = 0,

(8)

a01 + a11 + a21 = 1,3a01 + 4a11 + 5a21 = 1,6a01 + 12a11 + 20a21 = 0,

(9)

a02 + a12 + a22 = 12 ,3a02 + 4a12 + 5a22 = 1,6a02 + 12a12 + 20a22 = 1.

(10)

From (8)(10), the values of the parameters ai j(i, j = 0, . . . , 2) can be obtained as follows:a00 = 10, a01 = 6, a02 = 1.5, a10 = 15,a11 = 8, a12 = 1.5, a20 = 6, a21 = 3,a22 = 0.5.

123

S. Mobayen

3 Main results

The proposed PID sliding manifold can be defined as:

(t) = kps(t) + kit

0

s( )d + kd s(t), (11)

where kp, ki , and kd are the positive constants repre-senting the proportional, integral, and derivative coef-ficients, respectively. The time derivative of the PIDsliding manifold (11) can be obtained as: (t) = kps(t) + ki s(t) + kd s(t). (12)

If the condition (t) = 0 is satisfied and the con-stant gains kp, ki , and kd are designed properly, thenthe dynamic sliding surface s(t) will converge to zeroexponentially, and hence the right-hand side expressionof (12) is strictly Hurwitz.

In the following theorem, the tracking action of thedesired trajectory xd(t) is guaranteed and the trackingerror E(t) converges to zero in finite time T .

Theorem 1 Consider the uncertain nonlinear second-order system (2). Suppose that the control law is definedas:

u(t) = (g2b(x, t))1{kp

(g1

(e(t) l(t))

+ g2( f (x, t) + b(x, t)u(t) x2d l(t)

))

+ kd(g1

( f (x, t) + b(x, t)u(t) x2d l(t))

+ g2( f (x, t) + b(x, t)u(t) x2d ...l (t)

)),

+ ki G (E(t) L(t)) + sgn( ) | |+ + sgn( )} , (13)

where and are arbitrary positive constants and is a scalar value which satisfies: max () , (14)where = (kpg2 + kd g1

)d(x, t)+kd g2d(x, t). Then,

the sliding manifold (t) converges to zero in finitetime, the dynamic sliding surface s(t) converges to zeroexponentially, and the tracking action of the desiredtrajectory xd(t) is guaranteed.Proof Consider the following candidate Lyapunovfunction:

V1(t) = 12(t)T (t). (15)

The first and second time derivatives of the dynamicsliding surface can be obtained from (2)(4) as:

s(t) = G (E(t) L(t))

= g1(e(t) l(t)) + g2 ( f (x, t) + b(x, t)u(t)

+ d(x, t) x2d l(t), (16)s(t) = G (E(t) L(t))

= g1( f (x, t) + b(x, t)u(t) + d(x, t) x2d l(t)

)

+ g2( f (x, t) + b(x, t)u(t) + b(x, t)u(t)

+ d(x, t) x2d ...l (t)). (17)

If (4), (16), and (17) are substituted into (12), one gives:

(t) = k p(g1

(e(t) l(t))

+ g2( f (x, t) + b(x, t)u(t) + d(x, t) x2d l(t)

)

+ ki G (E(t) L(t)) + kd (g1 ( f (x, t)+ b(x, t)u(t) + d(x, t) x2d l(t)

)

+ g2( f (x, t) + b(x, t)u(t) + b(x, t)u(t)

+ d(x, t) x2d ...

l (t)))

. (18)Differentiating V1(t) and using (18) yields:

V1(t) = (t)T (t)= (e)T {kp

(g1

(e(t) l(t)) + g2 ( f (x, t)

+ b(x, t)u(t) + d(x, t) x2d l(t)))

+ ki G (E(t) L(t)) + kd (g1 ( f (x, t)+ b(x, t)u(t) + d(x, t) x2d l(t)

)

+ g2( f (x, t) + b(x, t)u(t) + b(x, t)u(t)

+ d(x, t) x2d ...l (t)))}

, (19)where substituting (13) in (19), one can obtain:

V1(t) = T sgn( ) | | T T sgn( )+ T ((kpg2 + kd g1

)d(x, t) + kd g2d(x, t)

),

(20)where based on the condition (14) follows that:V1(t) | |2 | |+1

= V ( ) V ( ), (21)where = ( + 1) /2 < 1, = 2 > 0, and = 2 > 0. This means that the Lyapunov func-tion (15) decreases gradually and the sliding manifold(t) converges to zero in finite time. This completesthe proof. unionsqRemark 2 From (5), (6), and (11), one can obtain(0) = 0. Based on the Lyapunov stability analysis,the sliding manifold (t) = 0 can be achieved all thetime. This demonstrates that the reaching phase in thesliding mode control can be removed and the globalrobustness can be satisfied.

123

An adaptive chattering-free PID sliding mode control

Remark 3 Using the discontinuous control law u(t) in(13), the sliding manifold (t) is forced to convergeto zero in finite time. Thus, by integrating (13), theactual control law u(t) becomes continuous and elim-inates destructive high-frequency oscillations calledchattering.

In practice, the upper bound of the system distur-bances is often unknown and therefore is difficultto determine. In the following theorem, an adaptiveparameter tuning technique is proposed to estimate theunknown upper bound of the system disturbances.

Theorem 2 Let the PID sliding manifold be in theform of (11) and assume that the external disturbancesd(x, t) and d(x, t) are unknown but bounded, where in (14) is an unknown positive constant. Also, supposethat is the estimation value of which is estimatedby using the following adaptation law: = |(t)| , (22)where is a positive constant. Using the adaptiveparameter tuning control law given by:

u(t) = (g2b(x, t))1{kp

(g1

(e(t) l(t))

+ g2( f (x, t) + b(x, t)u(t) x2d l(t)

))

+ kd(g1

( f (x, t) + b(x, t)u(t) x2d l(t))

+ g2( f (x, t) + b(x, t)u(t) x2d ...l (t)

))

+ ki G (E(t) L(t)) + sgn( ) | |+ + sgn( )} , (23)

then the finite-time convergence of the sliding manifoldto zero is guaranteed and also the zero tracking errorobjective is satisfied.Proof The positive definite Lyapunov function is deter-mined as:

V2(t) = 12T + 1

2 T (t) (t), (24)

where L = L L and is a positive coefficient withthe condition < 1

. Taking the time derivative of (24)

and using (18) and (22) yield:

V2(t) = + T (t) (t)= ( ) + T (t) {kps(t) + ki s(t) + kd s(t)

}

= ( ) |(t)| + T (t) {kp(g1

(e(t) l(t))

+ g2 ( f (x, t) + b(x, t)u(t)+ d(x, t) x2d l(t)

))

+ ki G (E(t) L(t)) + kd (g1 ( f (x, t)+ b(x, t)u(t) + d(x, t) x2d l(t)

)

+ g2( f (x, t) + b(x, t)u(t) + b(x, t)u(t)

+ d(x, t) x2d ...l (t)))}

, (25)where substituting (23) into (25), one can obtain:

V2(t) = ( ) |(t)| + T (t) ( sgn( (t)) |(t)|

(t) sgn( (t)) + )

( ) |(t)| + || |(t)| |(t)|+1 |(t)|2 |(t)|

( ) |(t)| + || |(t)| |(t)|+ |(t)| |(t)|

= ( ||) |(t)| (1 ) ( ) |(t)| .(26)

Since > || and < 1, therefore (26) can beexpressed by:

V2(t)

2 ( ||) |(t)|2

2

(1 ) 2

|(t)|

min{

2 ( ||) ,

2

(1 ) |(t)|}

|(t)|2

+ 2

= V2(t) 12 , (27)where = min

{2 ( ||) ,

2

(1 ) |(t)|}

> 0. In conclusion, using the adaptive tuning controllaw (23), the finite-time convergence to the sliding man-ifold (t) = 0 is guaranteed. unionsq

Since the discontinuous switching function sgn(.)presented in (23) can cause the chattering problem,undesired responses can occur for the uncertain nonlin-ear system. To avoid this problem, the function sgn(.)can be replaced by the following continuous saturationfunction:

sat( ) ={

sgn( ), | | >

, | | (28)

where is the boundary layer thickness. Furthermore,although the existence of the proposed NFTSMC canbe guaranteed outside the, it cannot be satisfied inside

123

S. Mobayen

Fig. 1 Schematic diagramof the proposed controller

the . In the worst situation, the state trajectories of thesystem would just reach . This will obtain a consider-able influence on the steady-state characteristics of thesystem.

4 Simulation results

The inverted pendulum system is a famous test plat-form for evaluating control techniques [32]. The controlgoal is to balance the pendulum in the inverted posi-tion. This system has particular and significant real-lifeapplications such as position control, aerospace vehiclecontrol, and robotics [33]. In this section, the proposedcontrol technique is applied to an inverted pendulumsystem. Dynamical equations of the system are denotedas:

x1 = x2,

x2 =g(mc + m p) sin(x1) m pl px22 sin(x1) cos(x1)

l p(

43 (mc + m p) m p cos2(x1)

)

+ cos(x1)l p

(43 (mc + m p) m p cos2(x1)

)u(t) + d(x, t),

(29)where x = [x1, x2]T , x1 and x2 are the angular positionand angular velocity of the pendulum, l p is the lengthof the pendulum, mc is the cart mass, m p is the pendu-lum mass, and g is the acceleration due to gravity. Thevalues of the system parameters are set as follows:l p = 0.5 m, m p = 0.1 kg, mc = 1 kg,

g = 9.8 m/s2.The external disturbance is assumed as d(x, t) =

0.3 cos(x1) + 0.2 cos( t) + 0.1 sin(1.5t)u. From (2)and (29), the bounded nonlinear functions f (x, t) andb(x, t) are determined as:

f (x, t) = g(mc + m p) sin(x1) m pl px22 sin(x1) cos(x1)

l p(

43 (mc + m p) m p cos2(x1)

) ,

0 5 10 15 20 25 30-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time (s)

an

gula

r pos

ition

(rad)

x1(t)x1d(t)

Fig. 2 Angular position tracking trajectory

0 5 10 15 20 25 30-1.5

-1

-0.5

0

0.5

1

1.5

2

time (s)

an

gula

r vel

ocity

(rad

/s)

x2(t)x2d(t)

Fig. 3 Angular velocity tracking trajectory

b(x, t) = cos(x1)l p

(43 (mc + m p) m p cos2(x1)

) .

The constant parameters are selected as: kp = 4,ki = 2, kd = 1, g1 = 5, g2 = 0.6, = 4.5, = 0.05, = 0.2, = 3, = 15, and = 35 . The initialconditions of the system states are assumed as:x(0) = [0.5 1]T .

123

An adaptive chattering-free PID sliding mode control

0 5 10 15 20 25 30-3

-2

-1

0

1

2

3

time (s)

con

trol i

nput

(volt

)

Fig. 4 Control input

0 5 10 15 20 25 30-2

0

2

4

6

8

10

12

time (s)

ada

ptive

par

amet

er

Fig. 5 Adaptive parameter

The desired trajectory is chosen as:xd(t) = 0.5 sin(t). (30)

The schematic diagram of the control configurationis shown in Fig. 1. The angular position and angu-lar velocity tracking trajectories are shown in Figs. 2and 3. It can be observed from these figures that theposition and velocity states appropriately track thedesired reference signals. The time responses of thecontrol input and adaptive parameter are displayedin Figs. 4 and 5. It is demonstrated that the proposedcontrol law yields the excellent vibration control androbustness features and overcomes the external dis-turbances and nonlinearities. The time responses ofthe dynamic sliding surface s(t) and the PID slidingmanifold (t) are shown in Figs. 6 and 7. Clearly,it can be seen that the designed sliding surfaces con-verge to the origin quickly. Noticeably, the simulations

0 5 10 15 20 25 30-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

time (s)

slid

ing

surfa

ce s

(t)

Fig. 6 Dynamic sliding surface s(t)

0 5 10 15 20 25 30-5

-4

-3

-2

-1

0

1

2

3

4

time (s)

slid

ing

man

ifold

(t)

Fig. 7 PID sliding manifold (t)

confirm the efficiency and feasibility of the proposedmethod.

5 Conclusions

In this paper, a novel dynamic PID sliding mode con-troller is successfully designed for a class of uncertainnonlinear systems. Using the proposed technique, thechattering problem caused by the switching term of theSMC method is eliminated. Also, the proposed con-trol method can guarantee that the system states reachthe sliding manifolds at all times, and then the globalrobustness of the closed-loop system is guaranteed. Inaddition, an adaptive parameter tuning scheme is pre-sented to estimate the unknown upper bound of the dis-turbances. Finally, usefulness and effectiveness of theproposed technique for an inverted pendulum systemhave been verified by simulations.

123

S. Mobayen

References

1. Skruch, P.: Feedback stabilization of a class of nonlinearsecond-order systems. Nonlinear Dyn. 59, 681693 (2010)

2. Aghababa, M.P.: Fractional modeling and control of a com-plex nonlinear energy supply-demand system. Complexity.(2014). doi:10.1002/cplx.21533

3. Yue, M., Wei, X.: Dynamic balance and motion control forwheeled inverted pendulum vehicle via hierarchical slidingmode approach. Proc. Inst. Mech. Eng. Part I J. Syst. ControlEng. 228(6), 351358 (2014)

4. Mobayen, S., Majd, V.J.: Robust tracking control methodbased on composite nonlinear feedback technique for linearsystems with time-varying uncertain parameters and distur-bances. Nonlinear Dyn. 70, 171180 (2012)

5. Rosas, D., Alvarez, J., Alvarez, E.: Robust synchronizationof arrays of uncertain nonlinear second-order dynamical sys-tems. Nonlinear Dyn. 67, 27352746 (2012)

6. Mobayen, S.: Design of LMI-based global sliding mode con-troller for uncertain nonlinear systems with application toGenesios chaotic system. Complexity. (2014). doi:10.1002/cplx.21545

7. Mobayen, S.: Design of a robust tracker and disturbanceattenuator for uncertain systems with time delays. Complex-ity. (2014). doi:10.1002/cplx.21569

8. Mobayen, S.: Finite-time robust-tracking and model-following controller for uncertain dynamical systems. J. Vib.Control (2014). doi:10.1177/1077546314538991

9. Wang, L., Sheng, Y., Liu, X.: A novel adaptive high-ordersliding mode control based on integral sliding mode. Int. J.Control Autom. Syst. 12, 459472 (2014)

10. Mobayen, S., Majd, V.J., Sojoodi, M.: An LMI-based com-posite nonlinear feedback terminal sliding-mode controllerdesign for disturbed MIMO systems. Math. Comput. Simul.85, 110 (2012)

11. Mobayen, S.: An LMI-based robust controller design usingglobal nonlinear sliding surfaces and application to chaoticsystems. Nonlinear Dyn. 79(2), 10751084 (2014)

12. Mobayen, S.: Design of CNF-based nonlinear integral slid-ing surface for matched uncertain linear systems with mul-tiple state-delays. Nonlinear Dyn. 77, 10471054 (2014)

13. Mobayen, S.: Robust tracking controller for multivariabledelayed systems with input saturation via composite non-linear feedback. Nonlinear Dyn. 76, 827838 (2014)

14. Pisano, A., Usai, E.: Sliding mode control: a survey withapplication in math. Math. Comput. Simul. 81, 95479(2011)

15. Liu, L., Han, Z., Li, W.: Global sliding mode control andapplication in chaotic systems. Nonlinear Dyn. 56, 193198(2009)

16. Efimov, D., Fridman, L.: Global sliding-mode observer withadjusted gains for locally Lipschitz systems. Automatica 47,565570 (2011)

17. Jing, Y., Mao, Y., Dimirovski, G.M., Zhang, S.: Adaptiveglobal sliding mode control strategy for the vehicle antilockbraking systems. In: American Control Conference (ACC09), St. Louis, MO (2009)

18. Joe, H., Kim, M., Yu, S.C.: Second-order sliding-mode con-troller for autonomous underwater vehicle in the presence

of unknown disturbances. Nonlinear Dyn. 78(1), 183196(2014)

19. Li, X., Yu, X., Han, Q.L.: Stability analysis of second-order sliding mode control systems with input-delay usingPoincar map. IEEE Trans. Autom. Control 58, 24102415(2013)

20. Chang, J.L.: Dynamic compensator-based second-ordersliding mode controller design for mechanical systems. IETControl Theory Appl. 7, 16751682 (2013)

21. Zeinali, M., Notash, L.: Adaptive sliding mode control withuncertainty estimator for robot manipulators. Mech. Mach.Theory 45(1), 8090 (2010)

22. Jiang, L., Wu, Q.H., Zhang, C., Zhou, X.X.: Observer-basednonlinear control of synchronous generators with perturba-tion estimation. Int. J. Elect. Pow. Energy Syst. 23(5), 359367 (2001)

23. Lu, Y.S., Cheng, C.M., Cheng, C.H.: Non-overshooting PIcontrol of variable-speed motor drives with sliding pertur-bation observers. Mechatronics 15(9), 11431158 (2005)

24. Ghafarirad, H., Rezaei, S.M., Abdullah, A., Zareinejad, M.,Saadat, M.: Observer-based sliding mode control with adap-tive perturbation estimation for micropositioning actuators.Precis. Eng. 35(2), 271281 (2011)

25. Loza, A.F., Punta, E., Fridman, L., Bartolini, G., Delprat, S.:Nested backward compensation of unmatched perturbationsvia HOSM observation. J. Frankl. Inst. 351(5), 23972410(2014)

26. Aliakbari, S., Ayati, M., Osman, J.H.S., Sam, Y.M.: Second-order sliding mode fault-tolerant control of heat recoverysteam generator boiler in combined cycle power plants.Appl. Therm. Eng. 50(1), 13261338 (2013)

27. Eker, I.: Sliding mode control with PID sliding surface andexperimental application to an electromechanical plant. ISATrans. 45(1), 109118 (2006)

28. Amer, A.F., Sallam, E.A., Elawady, W.M.: Adaptive fuzzysliding mode control using supervisory fuzzy control for 3DOF planar robot manipulators. Appl. Soft. Comput. 11(8),49434953 (2011)

29. Taherkhorsandi, M., Mahmoodabadi, M.J., Talebipour, M.,Castillo-Villar, K.K.: Pareto design of an adaptive robusthybrid of PID and sliding control for a biped robot viagenetic algorithm optimization. Nonlinear Dyn. 79(1), 251263 (2014)

30. Cao, Y., Chen, X.B.: An output-tracking-based discretePID-sliding mode control for MIMO systems. IEEE/ASMETrans. Mech. 19(4), 11831194 (2014)

31. Liu, J., Sun, F.: A novel dynamic terminal sliding modecontrol of uncertain nonlinear systems. J. Control TheoryAppl. 5, 189193 (2007)

32. Hassanzadeh, I., Mobayen, S.: Controller design for rotaryinverted pendulum system using evolutionary algorithms.Math. Probl. Eng. 2011, 17 pp (2011)

33. Hassanzadeh, I., Mobayen, S.: PSO-based controller designfor rotary inverted pendulum system. J. Appl. Sci. 8(16),29072912 (2008)

123

An adaptive chattering-free PID sliding mode control based on dynamic sliding manifolds for a class of uncertain nonlinear systemsAbstract1 Introduction1.1 Background and motivations1.2 Literature review1.3 Contributions1.4 Paper organization

2 Problem description3 Main results4 Simulation results5 ConclusionsReferences