nonsingular fast terminal sliding-mode stabilizer for...

$: Nonsingular fast terminal sliding-mode stabilizer for …scientiairanica.sharif.edu/article_4123_4f7cf0ef6ee7d45c...Scientia Iranica D (2017) 24(3), 1410{1418 Sharif University of$
Scientia Iranica D (2017) 24(3), 1410{1418

Sharif University of TechnologyScientia Iranica

Transactions D: Computer Science & Engineering and Electrical Engineeringwww.scientiairanica.com

Nonsingular fast terminal sliding-mode stabilizer for aclass of uncertain nonlinear systems based ondisturbance observer

S. Mobayena;� and F. Tchierb

a. Department of Electrical Engineering, Faculty of Engineering, University of Zanjan, Zanjan, Iran.b. Department of Mathematics, King Saud University, P.O. Box 22452, Riyadh 11495, Saudi Arabia.

Received 7 November 2015; received in revised form 17 April 2016; accepted 24 May 2016

KEYWORDSFinite-time stabilizer;Nonsingular fastterminal sliding mode;Third-order system;Disturbance observer;Robustness.

Abstract. This paper investigates a novel nonsingular fast terminal sliding-modecontrol method for the stabilization of the uncertain time-varying and nonlinear third-order systems. The designed disturbance observer satis�es the �nite-time convergenceof the disturbance approximation error and the suggested �nite-time stabilizer assuresthe presence of the switching behavior around the switching curve in the �nite time.Furthermore, this approach can overcome the singularity problem of the fast terminalsliding-mode control technique. Moreover, knowledge about the upper bounds of thedisturbances is not required and the chattering problem is eliminated. Usefulness ande�ectiveness of the o�ered procedure are con�rmed by numerical simulation results.

© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

1.1. Background and motivationsThe rigorous foundation for the theory of �nite-timestabilization was �rst presented by Bhat and Bern-stein [1]. Sliding-Mode Control (SMC) is e�cientlyapplied for the stabilization and controller design ofvarious linear and nonlinear dynamical systems suchas singular systems [2], robotic manipulators [3], non-holonomic mobile robots [4,5], fractional order sys-tems [6], chaotic systems [7,8], under-actuated sys-tems [9], etc. SMC is an in uential control technique,which has access to the favorite responses in spite of theuncertainties and disturbances [10,11]. The importantfeatures of SMC technique are robustness against para-metric uncertainties, superior transient performance,quick responses, insensitivity to the bounded externaldisturbances, and computational simplicity compared

*. Corresponding author. Tel.: +98 24 33054208E-mail address: [email protected] (S. Mobayen)

to other robust control techniques [12]. The processof SMC design is divided into two phases: (a) thesliding phase, and (b) the reaching phase. Firstly,in sliding phase, a switching curve is speci�ed suchthat the controlled system displays favorable dynamicperformance during the sliding mode [13]. Secondly,in reaching phase, a sliding control law is employedfor the state trajectories of the controlled system toconverge on the sliding curve. Due to the e�ect ofthe sliding curve on the stabilization and transientresponse of the controlled system, the design procedureof the switching curve is the most important subjectin the sliding method [14]. SMC utilizes a discon-tinuous controller to drive the state trajectories toa predesigned switching surface on which the desiredperformance besides stability of the system can beobtained [15]. Young et al. [13] proposed SMC observerscheme with only the discontinuous term being fed backthrough a suitable gain. Generally, traditional SMChas some important problems, such as discontinuouscontrol, which often yields chattering phenomenon [16].

S. Mobayen and F. Tchier/Scientia Iranica, Transactions D: Computer Science & ... 24 (2017) 1410{1418 1411

Also, it is required in [14] that the external disturbanceshould be matched, that is, act in the channels of theinputs. To cope with these problems and attain higheraccuracy, some new forms of SMC have been proposed.Moreover, SMC cannot guarantee the converging per-formance of the state trajectories to the origin inthe �nite time. To tackle the mentioned problem,the Terminal Sliding-Mode Control (TSMC) procedurehas been suggested and performed in several controlsystems. TSMC technique proposes some excellentspeci�cations such as rapid response and �nite-timestability in comparison with linear SMC [17]. TSMCis principally suitable for high-precision stabilizationand control as it precipitates the convergence rate nearthe origin [18]. However, when the state trajectoriesof the system are far away from the origin, TSMCcannot present appropriate convergence e�ciency likeSMC [19]. The Fast Terminal Sliding-Mode Control(FTSMC) idea guarantees fast transient convergenceand strong robustness [20]. In the last decade, there hasbeen more attention to the utilization of the mentionedtechnique for various control problems [21]. Neverthe-less, it should be mentioned that the FTSMC techniquestill requires to be further considered in robustnessperformances to tackle the system disturbances.

1.2. Literature reviewIn most of the considered research, the boundariesof the perturbations are directly employed in thedesign of the TSMC law [22,23]. To estimate thedisturbances, various design procedures based on thedisturbance observer have been planned in the recentyears [24]. In [25], SMC has been established for thecontrol and stabilization of uncertain and nonlineardynamical systems using the disturbance observers.In [26], a novel multiple-surface SMC is recommendedfor the uncertain nonlinear systems and a disturbance-observer-based approach is de�ned to estimate themismatched uncertainties of the system. In [27], thedesign procedure of the adaptive �nite reaching timecontrollers for the �rst- and second-order dynamicalsystems with perturbations is investigated, where thesuggested controllers are continuous and retain robust-ness to the disturbances. The combination of FTSMCand Global Sliding-Mode (GSM) surface for the robusttracking control of nonlinear second-order systems withtime-varying uncertainties is investigated in [28]. Arecursive FTSMC technique for tracking control of non-holonomic systems in the chained form is proposedin [29], where the tracking errors are allowed to decayto the origin in the �nite time with an exponentialdecay rate. A disturbance-observer-based recursiveTSMC tracker is presented in [30] for the �nite-timetracking control of third-order non-holonomic systemswith unknown external disturbances. An adaptiveFTSMC technique combined with GSM scheme is

suggested in [31] for the tracking problem of uncer-tain nonlinear third-order systems. A Linear MatrixInequalities (LMI)-based second-order FTSMC methodis investigated in [32] for the tracking control of non-linear uncertain systems with matched and mismatcheduncertainties. However, the singularity problem isnot fully resolved in [26-30]. In [33], the robustsynchronization problem of disturbed chaotic systemsis investigated, where, using an LMI-based disturbanceobserver, the boundedness conditions of disturbanceerrors are satis�ed. In [22,34], the disturbance ob-servers are applied to estimate the disturbances andsome robust control approaches are considered usingthe outputs of the disturbance observers. To satisfythe approach of the approximation errors to the originin the �nite time, TSMC disturbance observer has beenestablished in [20,35,36]. In [33,34], composite controldesign procedures of the disturbance-observer-basedcontroller and TSMC are o�ered for the uncertainstructural and nonlinear systems where the proposeddisturbance observers are based on the regional poleplacement and D-stability theories. In [37], by com-bining TSMC and second-order SMC approaches, anonlinear robust control technique and a disturbanceobserver are designed for the longitudinal dynamicsof hypersonic vehicles with uncertainties and distur-bances, which can provide high-precision and fastconvergence.

1.3. ContributionsTo the best of our knowledge, there are two disadvan-tages of TSMC and FTSMC, which are the singularityproblem and the requirement of the bounds of thedisturbances. In the recent years, very little attentionhas been paid to both these problems, which is stillopen in the literature. In this study, based on thedisturbance observer presented in [20], we apply anew Nonsingular Fast Terminal Sliding-Mode Control(NFTSMC) approach for the �nite-time control of un-certain and nonlinear third-order systems with externaldisturbances. The singularity problem of FTSMCmethod is solved by the designed NFTSMC and thedisturbance observer is developed to quickly force thedisturbance approximation errors to converge to theorigin in a �nite time.

1.4. Paper organizationThe structure of the paper is as follows: The for-mulation of the problem is described in Section 2.In Section 3, the proposed control mechanism andstability analysis are introduced. Simulation resultsof the application of the o�ered disturbance-observer-based NFTSMC method on an uncertain nonlinearthird-order system and an uncertain chaotic system areobtained in Section 4. Finally, Section 5 draws theconcluding remarks of this research.

1412 S. Mobayen and F. Tchier/Scientia Iranica, Transactions D: Computer Science & ... 24 (2017) 1410{1418

2. Problem description

Consider the uncertain nonlinear system as follows:_x1 = x2;

_x2 = x3;

_x3 =f(x; t) + �f(x; t) + (b(x; t) + �b(x; t))u

+ d0(x; t);

y = x1; (1)

where x = [x1; x2; x3]T is the state vector; u is thecontrol signal; y is the system output; b(x; t) and f(x; t)are the bounded nonlinear functions; and �b(x; t),�f(x; t), and d0(x; t) are the nonlinear functions rep-resenting the uncertainties and external disturbances.Furthermore, it is supposed that a positive constantvalue �, which is the lower bound of b(x; t), is de�ned,that is, � = inf(jb(x; t)j). De�ning d(x; t) = �f(x; t) +�b(x; t)u+ d0(x; t), one can obtain System (1) as:

_x1 = x2;

_x2 = x3;

_x3 = f(x; t) + b(x; t)u+ d(x; t);

y = x1: (2)

In order to examine the stabilization problem of theuncertain system, we can rewrite Dynamics (2) in thefollowing form:

_x = Ax+B ff(x; t) + b(x; t)u+ d(x; t)g ; (3)

where:

A =

240 1 00 0 10 0 0

35 ; B

24001

35 :Now, based on the pole-placement procedure, a

term �kx with �k = b�k0; �k1; �k2c is considered, where �ki'sare selected such that the deterministic equation S3 +�k2S2 + �k1S + �k0 = 0 is stable system. Hence, theexponentially stable dynamics are achieved as:

...x + �k2�x+ �k1 _x+ �k0x = 0; (4)

which designates that x converges to zero. Conse-quently, System (3) is modi�ed as:

_x=(A�B�k)x+B�

�kx+f(x; t)+b(x; t)u+d(x; t)�;(5)

where the control input can be considered by thefollowing transformation:

u = b(x; t)�1 f�v � f(x; t)g ; (6)

where �v is a new control input. The control sig-nal (Eq. (6)) contains two portions: one portion is

�b(x; t)�1f(x; t), which is employed to remove the sys-tem nonlinearities, and the other portion is b(x; t)�1�v,which is used to weaken the impacts of the perturba-tions. Substituting Eq. (6) into Eq. (5) gives:

_x = (A�B�k)x+B(�v + �H); (7)

where �H = �kx + d(x; t). The function d(x; t) is acontinuous function and, hence, �H is also continuous.Then, System (7) is completely controllable and can becontrolled by several robust control procedures.

Remark 1. In the case that b(x; t) is not invert-ible, similar to the methods planned in control pa-pers such as [38], one can use the term b(x; t)+ =�b(x; t)T b(x; t)

��1 b(x; t)T as the pseudo-inverse ofb(x; t).

Lemma 1. Consider the candidate positive-de�nitefunctional V (t), which ful�lls a di�erential inequality,as [19]:

_V (t) ��V (t)��V(t)� 8t� t0; V (t0) � 0; (8)

where � and � are two positive coe�cients, and � is afraction of two odd positive numbers with 1 > � > 0.As a result, for certain time, t0, the above-mentionedfunction, V (t), approaches the origin in the �nite timeas [29]:

ts = t0 +1

�(1� �)ln��V (t0)1�� + �

�

�: (9)

Proof. If two sides of Inequality (8) are divided toV �(t), we obtain:

V (t)�� _V (t) � ��V (t)1�� ; (10)

and thus:

dt �� V (t)��V (t)1�� + �

�dV (t): (11)

Integrating two sides of Relation (11) from t0 to tsyields:

ts � t0 �Z 0

V (t0)

�V (t)��

�V (t)1�� + �

�dV (t)

= � 1�(1� �)

�ln(�)� ln

��V (t0)1�� + �

��=

1�(1� �)

ln��V (t0)1�� + �

�

�; (12)

which completes the proof of the lemma.


3. Main results

The auxiliary variable of the sliding disturbance ob-server can be described as:

s = z � x3; (13)

with z, which is speci�ed by:

_z =� ks� �sgn(s)� "sp0=q0 � jf(x; t)jsgn(s)

+ b(x; t)u; (14)

where q0 and p0 are odd positive integers with p0 < q0.The design coe�cients k, �, and " are some positiveconstants and the condition � � jd(x; t)j is obtained.The TSMC disturbance estimator d̂ is speci�ed as:

d̂=�ks� �sgn(s)�"sp0=q0�jf(x; t)jsgn(s)�f(x; t):(15)

Theorem 1. Consider the disturbed third-order sys-tem (Eq. (1)) and TSMC disturbance observer givenby Eqs. (13)-(15). Then, the disturbance approxima-tion error of the suggested TSM disturbance observerconverges to the origin in the �nite time.

Proof. Considering Eqs. (1) and (14) and di�erenti-ating Eq. (13), one achieves:

_s = _z � _x3 = �ks� �sgn(s)� "sp0=q0

� jf(x; t)jsgn(s)� f(x; t)� d(x; t): (16)

Construct the candidate Lyapunov function as:

V (s) =12s2: (17)

The time derivative of V (s) is given by:

_V (s)=s _s=s��ks��sgn(s)�"sp0=q0�jf(x; t)jsgn(s)

� f(x; t)� d(x; t)�� ks2 � "s(p0+q0)=q0

� �jsj+ jsjjd(x; t)j � jf(x; t)jjsj � sf(x; t)

� �ks2 � "s(p0+q0)=q0

��2kV(s)�2(p0+q0)=2q0

"V(s)(p0+q0)=2q0 ; (18)

where based on Lemma 1 and Eq. (18), one obtainsthat the auxiliary variable s can converge to the originin the �nite time. Consequently, the error ~d is obtainedusing Eqs. (1), (13), and (15) as:

~d =d̂� d(x; t) = �ks� �sgn(s)� "sp0=q0

� jf(x; t)jsgn(s)� f(x; t)� d(x; t) = �ks� �sgn(s)� "sp0=q0 � jf(x)jsgn(s)� _x3

+ b(x; t)u = _z � _x3 = _s: (19)

It clari�es that according to the �nite-time convergenceof the switching surface, s, to the origin, the approxi-mation error, ~d, reaches zero in the �nite time.�

In order to develop FTSMC stabilizer using thedisturbance observer, the following switching surface isproposed:

� = _s+ �s+ �s�; (20)

where � and � are two positive coe�cients and � isa fraction of two odd positive integer numbers with1 > � > 1

2 .In order to assure that FTSMC curve approaches

zero in the �nite time and system states quicklyconverge to zero, the following theorem is proposed.

Theorem 2. The uncertain third-order system(Eq. (2)) is considered. If the control law is employedas:

_u =b(x; t)�1��sgn(�)j�j� + � + (�+ ��s��1) _s

+ �z � _f(x; t)� _b(x; t)u+ �s� _̂d�; (21)

where and � are two arbitrary positive coe�cients,then the trajectories of Eq. (2) are enforced to move tothe switching curve (Eq. (13)) in the �nite time and tostay on it.

Proof. The candidate Lyapunov function is given as:

V (�) =12�T�: (22)

From Eqs. (2), (13), and (20), the derivative of � canbe obtained as:

_� =�s+ � _s+ ��s��1 _s = (�+ ��s��1) _s+ �z � _f(x; t)

� _b(x; t)u� b(x; t) _u� _d(x; t): (23)

Di�erentiating V (�) and using Eq. (23), one can �nd:

_V (�) =�T _� = �T�

(�+ ��s��1) _s+ �z � _f(x; t)

� _b(x; t)u� b(x; t) _u+ �s� _̂d�; (24)

where, substituting Eq. (21) in Eq. (24), we obtain:


_V (�) = ��T�sgn(�)j�j� � �T �= � j�j2 � �j�j�+1

= ��V (�)� �V ��(�); (25)

where �� = (� + 1)=2 < 1, � = 2 > 0 and � = 2�� >0. Then, the Lyapunov function (Eq. (22)) convergesgradually to zero and the switching curve approacheszero in the �nite time.�

Now, considering the FTSMC law demonstratedin Eq. (21), the term ��s��1 _s may lead to a singularityproblem if _s 6= 0 when s = 0 due to the negative powerof s. Thus, the FTSMC cannot satisfy a boundedcontrol action if _s 6= 0 when s = 0. Then, the NFTSMCis suggested to overcome the singularity phenomenonof the FTSMC.

De�ne the NFTSMC stabilizer based on the dis-turbance observer as follows:

� = s+1�

( _s+ �s)1� : (26)

In order to dominate the singularity phenomenonof the FTSMC and guarantee the �nite-time conver-gence of the state trajectories to zero, the followingtheorem is proposed.

Theorem 3. The uncertain and nonlinear third-order system (Eq. (2)) is considered. Applying theNFTSMC law:

_u =b(x; t)�1��( _s+ �s)1� 1

� (�sgn(�)j�j� + �)

+ �z � _f(x; t)� _b(x; t)u+ � _s+ �s� _̂d�; (27)

with some positive coe�cients and �, the states ofEq. (2) are forced to move from the initial conditionsto the switching curve (Eq. (13)) in the �nite time andto stay on it.

Proof. The Lyapunov function is considered as:

V (�) =12�T�: (28)

From Eqs. (2), (13), and (26), the derivative of � canbe found as:

_� = _s+1��

(�s+ � _s)( _s+ �s)1��1 = _s

+1��

��z � _f(x; t)� _b(x; t)u� b(x; t) _u

� _d(x; t) + � _s�

( _s+ �s)1��1: (29)

Di�erentiating V (�) and using Eq. (29), we obtain:

_V (�) = �T _� = �T�

_s+1��

��z � _f(x; t)u

� _b(x; t) _u+ � _s+ �s� _̂d�

( _s+ �s)1��1

�; (30)

where, substituting Eq. (27) in Eq. (30), one canachieve:

_V (�) = ��T�sgn(�)j�j� � �T � = � j�j2 � �j�j�+1

= ��V (�)� �V ��(�); (31)

where �� = (�+1)=2 < 1, � = 2 > 0, and � = 2�� > 0.Thus, the NFTSMC surface converges to the origin inthe �nite time and the states of the system quicklyconverge to zero.�

Since the discontinuous switching function sgn(:)shown in Eqs. (21) and (27) can result in chatter-ing problem, undesired responses can occur in thenonlinear third-order system. To avoid this problem,the function sgn(:) can be replaced by the followingcontinuous saturation function:

sat(�) =

(sgn(�); j�j > �� ; j�j � �

(32)

where � is the boundary-layer thickness. Furthermore,although the existence of the proposed NFTSMC canbe guaranteed outside �, it cannot be satis�ed inside�. In the worst situation, the state trajectories of thesystem would only reach �. This will considerablyin uence the steady-state characteristics of the system.

Remark 2. From Eq. (31), the derivative ofthe Lyapunov function is negative semi-de�niteand guarantees that V (�) and � are bounded. Itis deduced from Eq. (20) that s and _s are twobounded functions. Since V (0) is a bounded scalarand V (�) is non-increasing, it can be concludedthat limt!1

R t0 jj�jjdt and limt!1

R t0 jjsjjdt are also

bounded. Thus, according to Barbalat's lemma andbased on the boundedness of limt!1

R t0 jjsjjdt and _s,

the auxiliary variable s converges asymptotically tozero, that is, limt!1

R t0 sdt = 0.

Remark 3. In the situations that some of the vari-ables such as:

�z; �s; _̂d; and _f(x; t)

are not measurable, one can employ the delayed-feedback control method to model these variables.Specially, the delay term in the form of Euler approx-imation of the derivative function can be applied forthe derivatives of the variables [31]. The variable �z isreplaced by a delay function, �z = 1

h [ _z(t)� _z(t� h)], ifthe delay h > 0 is su�ciently small.


4. Simulation results

Example 1. Consider the following uncertain non-linear third-order system [39]:

_x1 = x2;

_x2 = x3;

_x3 =�3x3�4x2�2x1+x1x2+�f(x; t)+u+d(x; t);(33)

where �f(x; t) = 0:39 sin(x1x2 + x3pt) and d(x; t) =

0:6 sin(10t). The initial conditions are chosen as:x(0) = [�1 1:5 1]T . The constant parameters arechosen as: k = 2, � = 1, � = 1, � = 10, � = 2, = 3," = 1, p0 = 3, q0 = 5, and � = 3

5 . The trajectories ofthe states x1, x2, and x3 are demonstrated in Figure 1.The time response of the control signal is demonstratedin Figure 2. It is shown that the suggested methodcan obtain superior performance and high robustnessand is capable to control the parametric uncertaintiesand system nonlinearities. The time responses ofthe auxiliary variable s and the NFTSMC surface �are demonstrated in Figure 3. Clearly, in can beseen that these sliding curves approach zero quickly.These numerical simulations approve the proposedtechnique.

Figure 1. State trajectories

Figure 2. Control input u.

Figure 3. Sliding surfaces s and �.

Example 2. Consider the Lur'e-like chaotic systemwith an additive control input de�ned by [40]:

_x1 = x2;

_x2 = x3;

_x3 =�6:8x1�3:9x2�x3+12X (x1)�0:8 cos(5t) + u;(34)

where:

X (x1) =

(Px1 if jx1j < 1

Psgn(x1) if jx1j � 1

P(35)

System (34) represents a chaotic behavior for P = 1:5.This chaotic system is numerically simulated using theo�ered control law with the following initial parametersand initial conditions:

k = 2; � = 1:5; � = 2; � = 8; � = 2; " = 1;

p0 = 3; q0 = 5; � =35; x(0) = [0:5 1 1]T :

The tracking trajectories of states x1, x2, and x3 areshown in Figure 4. It is demonstrated in these �guresthat all of the states are stabilized. The dynamic

Figure 4. State trajectories.


Figure 5. Control input u.

Figure 6. Sliding surfaces s and �.

control input, u, is demonstrated in Figure 5. Thetime responses of s and � are shown in Figure 6.Obviously, it can be found that the switching surfaceand NFTSMC surface converge to the origin quickly.Therefore, the uncertain chaotic system is stabilizedby applying the planned controller.

5. Conclusions

A new disturbance-observer-based �nite-time stabilizerfor the uncertain and nonlinear third-order systemsis considered in this work. A novel reaching lawis suggested to assure the presence of the switchingbehavior around the designed NFTSMC curve in the�nite time. Moreover, in order to dominate the singu-larity phenomenon of FTSMC, a nonsingular controlapproach is proposed. Intensive numerical simulationresults are displayed to con�rm the e�ectiveness ofthe suggested technique and acceptable results areobtained. It is pointed out that the suggested approachcan be employed for the tracking control and stabiliza-tion of higher-order uncertain and nonlinear dynamicalsystems.

Acknowledgement

This research project was supported by a grant fromthe \Research Center of the Center for Female Sci-enti�c and Medical Colleges", Deanship of Scienti�cResearch, King Saud University.

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Biographies

Saleh Mobayen received his doctorate in ControlEngineering from Tarbiat Modares University, Iran, in2012, and is currently Assistant Professor of Control


Engineering at University of Zanjan. He has pub-lished several papers in the national and internationaljournals. His research interests include sliding-modecontrol, robust tracking, non-holonomic robots, andchaotic systems.

Fairouz Tchier received her PhD degree in Math-

ematics with the specialty of Theoretical ComputerScience from Universit�e Laval, Quebec City, Qu�ebec,Canada, in 1996, and is currently an Associate Profes-sor in Mathematics Department, King Saud University.Her research interests include theoretical computerscience, software engineering, �xed point theory, formalmethods, and fuzzy set theory.

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