Practically Oriented Finite-Time Control Design and

4166 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 65, NO. 5, MAY 2018

Practically Oriented Finite-Time Control Designand Implementation: Application to a Series

Elastic ActuatorChuanlin Zhang , Member, IEEE, Yunda Yan , Student Member, IEEE, Ashwin Narayan ,

and Haoyong Yu , Member, IEEE

Abstract—This paper proposes a practically orientedfinite-time control design for nonlinear systems under a lessambitious but more practical semiglobal control objective.As one main contribution, the designed finite-time controllercan be expressed as a very simple form while the controlgain tuning mechanism follows a conventional pole place-ment manner, which enhances its facility in practical imple-mentations. Moreover, by utilizing a modified nonrecursivehomogeneous domination design without preverifying anynonlinearity growth constraints, the control scheme can bedirectly obtained by totally neglecting the recursive calcu-lations of series virtual controllers. Rigourous semiglobalattractivity and local finite-time convergence analysis arepresented to ensure the theoretical justification. A controlapplication and experimental verification to a series elasticactuator demonstrates the control effectiveness and signif-icant performance improvements compared with asymptot-ical state feedback controllers.

Index Terms—Finite-time control design, robust control,semiglobal stability, series elastic actuator (SEA).

Manuscript received July 5, 2017; revised September 14, 2017; ac-cepted October 8, 2017. Date of publication October 19, 2017; date ofcurrent version January 16, 2018. This work was supported in part by theNational Natural Science Foundation of China under Grant 61503236, inpart by the Shanghai Science and Technology Commission Key Programunder Grant 15160500800, in part by the Engineering Research Cen-ter of Shanghai Science and Technology Commission Program underGrant 14DZ2251100, in part by the FRC Tier 1 grant with WBS underGrant R397000218112 from the Faculty of Engineering, National Uni-versity of Singapore, and in part by the BMRC Grant 15/12124019 fromthe Agency for Science, Technology and Research (A*STAR), Singaporeand Fundamental Research Funds for the Central Universities, NationalMedical Research Council Grant NMRC/BnB/0019b/2015, Ministry ofHealth, Singapore. (Corresponding author: Haoyong Yu.)

C. Zhang is with the College of Automation Engineering, Shang-hai University of Electric Power, Shanghai 200090, China, and alsowith the Department of Biomedical Engineering, National University ofSingapore, 117576 Singapore (e-mail: [email protected]).

Y. Yan is with the School of Automation, Southeast University, Nanjing210096 China, and also with the Department of Biomedical Engineering,National University of Singapore, 117576 Singapore (e-mail: [email protected]).

A. Narayan and H. Yu are with the Department of Biomedical Engi-neering, National University of Singapore, 117576 Singapore (e-mail:[email protected]; [email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2017.2764843

I. INTRODUCTION

F INITE-TIME control issue for nonlinear systems has beenextensively studied in the last decades owing to its well-

known fast convergence rate and stronger robustness againstsystem uncertainties/disturbances. A pioneer significant theo-retical contribution for continuous finite-time control can bereferred to the research of double integrators in [1] where ahomogeneous second-order controller is proposed to achieve aglobal finite-time stability. Later research interests mainly focuson the nonsmooth extension of recursive backstepping designto realize finite-time stability of nth order control systems withthe presence of nonlinear perturbations. Notably, an effectivenonsmooth design approach, namely, adding a power integra-tor, is first proposed in [2] and [3]. Meanwhile, an extendedbackstepping design strategy is also proposed in [4] to solvethe finite-time stabilization problem for a class of perturbedchain of power integrators. With a few years’ development, thenonsmooth control design method is greatly enriched by takingfully advantage of the weighted homogeneity [5], [6]. It is laterreported in [7] and [8] that a homogeneous domination designmethodology can be systematically addressed to solve the finite-time output feedback stabilization problem for a class inherentnonlinear systems. From a practical point of view, finite-timecontrol design for practical systems have also been extensivelystudied, see, e.g., [9]–[11], etc. On the other hand, finite-timecontrol via an alternative nonsingular terminal sliding mode canalso be widely found in the literature, for examples, [12] and[13], only mention a few.

However, it is well known that a common nonlinear growthconstraint of the considered system is essentially required forglobal continuous finite-time stabilization design, which is de-scribed detailedly in [8]. For general real-life plants, it is clearlyof suspicion that most of the nonlinear hypothesis could bewidely satisfied. Hence, a key question arises that for a generalclass of nonlinear systems of the form (1), how to essentiallyrelax the prerequirement of the system nonlinearity constraints,such that a practically oriented finite-time controller can bewidely applied. On the other hand, due to massive utilizationof recursive domination approaches, the obtained guideline ofthe control gains are usually very conservative and sometimeseven the control gains are required to be smooth functions ofthe system state, see, for instances, [3] and [14], etc. Typically,

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ZHANG et al.: PRACTICALLY ORIENTED FINITE-TIME CONTROL DESIGN AND IMPLEMENTATION 4167

the conservative feature will be much severe when handlinghigh-order systems. In practices, this feature will add muchcomplexity for implementations and might cause an apparentperformance deterioration when the system is suffering frommeasurement noises.

In this paper, inspired by a recent advance on nonsmoothstabilization design [15], we propose a practically orientedfinite-time control design strategy for system (1) under a less am-bitious but more practical control objective, namely, semiglobalrather than restrictive global control. First, a delicate coordi-nates transform is presented by exactly calculating the steady-state generators. Second, we show by a nonrecursive designthat the controller scheme design can be essentially separatedwith its stability analysis, which is inevitable in existing recur-sive design approaches. Third, a rigorous semiglobal attractivityand local finite-time convergence provide the theoretical justi-fications of the proposed method. Compared with the existingrelated finite-time control design results for lower triangularnonlinear systems, the main distinguishable improvements arestated as follows. First, under a semiglobal control infrastruc-ture, we show that existing essentially required nonlinearitygrowth constraints, which are restrictive to be verified, can nowbe fully removed. Second, the involved tuning homogeneous de-gree, which could significantly affect the control performance,has been endowed with much flexibility within a tunable re-gion. Third, the proposed novel nonrecursive synthesis approachcould render a finite-time control scheme to be designed in avery simple expression, whereas its control gain selection fol-lows a conventional pole placement manner. More specifically,the obtained controller can reduce to its linear state feedbackcontroller counterpart simply by assigning the homogeneousdegree to zero.

At the end, an application and experimental verification re-sults to a series elastic actuator (SEA) are included in orderto illustrate the simplicity and effectiveness of the proposedcontrol design strategy. Moreover, to provide better understand-ing of the gain tuning mechanism, a detailed parameter figu-ration procedure is presented. Then, the detailed experimentalcomparison results with a conventional proportional-derivative(PD) controller, a linear state feedback controller, and an opti-mal controller are provided as well to demonstrate the controlperformance improvements.

Notations: For integers j and i satisfying 0 ≤ j ≤ i, denoteNj :i := {j, j + 1, . . . , i}. The symbol Ci denotes the set ofall differentiable functions whose first ith time derivatives arecontinuous. A continuous function �·�a is defined by �·�a =sign(·)| · |a where a ∈ R+ is a constant.

II. PRELIMINARIES AND PROBLEM STATEMENT

A. Preliminaries

The following notations are provided for briefness ofexpressions.

1) (Weighted Homogeneity) [6] For a fixed choice of co-ordinates x = (x1 , . . . , xn ) ∈ Rn , and positive real numbers(r1 , r2 , . . . , rn ) � r, a one-parameter family of dilation is a mapΔr : R+ × Rn → Rn , defined by Δr

ε x = (εr1 x1 , . . . , εrn xn ).

For a given dilation Δr and a real number τ , a continuousfunction V : Rn → R is called Δr−homogeneous of degreeτ , denoted by V ∈ Hτ

Δ r if V ◦ Δrε = ετ V . A continuous vector

field f(x) =∑

fj (x)( ∂∂xj

) is Δr− homogeneous of degree τ ,

if fj ∈ Hτ +rj

Δ r , j ∈ N1:n . Through out this paper, r is givenby r1 = 1, ri = ri−1 + τ = 1 + (i − 1)τ, i ∈ N2:n with a de-gree τ ∈ (− 1

n , 0). A homogeneous vector xτiΔ r is denoted by

xτiΔ r � (x

τr 11 , . . . , x

τr ii )T , xτ

Δ r = xτnΔ r and �x�τ

Δ r = (�x1�τr 1 ,

. . . , �xn� τr n )T . ‖x‖Δ r = (

∑ni=1 |xi |p/ri )1/p denotes a homo-

geneous p−norm and ‖x‖ = (∑n

i=1 |xi |p)1/p denotes a conven-tional Lp norm. In this paper, we choose p = 2 for simplicity.

2) [16] Consider a system x = f(x) + g(x)u, x ∈ Rn , withx = 0 as its equilibrium. This system is said to be semigloballystabilizable if, for each (arbitrarily large) compact subset P ⊂Rn , there exists a feedback law u = u(x), which in generaldepends on P , such that the equilibrium is locally asymptoticallystable and x(t0) ∈ P ⇒ lim

t→∞x(t) = 0.

B. Problem Statement

In this paper, we revisit the finite-time control problem for aclass of lower triangular nonlinear systems of the form [3], [17]

⎧⎨

⎩

xi(t) = xi+1(t) + φi(xi(t)), i ∈ N1:n−1xn (t) = u(t) + φn (x(t))y(t) = x1(t)

(1)

where xi = col(x1 , . . . , xi), i ∈ N1:n−1 and x = xn are the sys-tem partial and full state vector, respectively, y is the system out-put, and φi(·), i ∈ N1:n is a known smooth nonlinear function.The output reference signal, denoted by yr and its nth orderderivative are assumed to be piecewise continuous, known andbounded.

To achieve a realizable control scheme, first, we definex∗

i = col(x∗1 , . . . , x

∗i ), i ∈ N1:n+1 as an auxiliary variable vec-

tor where x∗i is determined by the following steady-state

generators:{

x∗1 = yr

x∗i = dx∗

i−1dt − φi−1(x∗

i−1), i ∈ N2:n+1 .(2)

Second, denote z = col(z1 , . . . , zn ), where zi = (xi − x∗i )/

�i−1 , i ∈ N1:n , v = (u − x∗n+1)/�n , and � ≥ 1 is a bandwidth

factor to be determined later in the stability analysis. In thispaper, we show that, without going through the procedure ofrecursive stability analysis, a simple finite-time controller canbe explicitly prebuilt of the following form:

v = −k1�z1�1 + n τ

1 − k2�z2�1 + n τ1 + τ − · · · − kn�zn�

1 + n τ1 + (n −1 ) τ

� −K�z�1+nτΔ r

u = �nv + x∗n+1 (3)

where K = [k1 , . . . , kn ] is the coefficient vector of a Hurwitzpolynomial p(s) = sn + knsn−1 + · · · + k2s + k1 and the di-lation weight is defined by r = (1, 1 + τ, . . . , 1 + (n − 1)τ)with a homogeneous degree τ ∈ (− 1

n , 0).Remark 2.1: Nonlinear synthesis approaches with simple

controller expression, less calculation tasks, and easier gain



Fig. 1. Block diagram of the proposed finite-time control method.

tuning mechanism are always imperatively demanded inpractices. In this paper, a finite-time control method can nowbe easily implemented without much additional complexitiescomparing to its linear state feedback control counterpart, assketchily depicted by Fig. 1. Comparing with the existing relatednonlinear control results, such as [18]–[20], etc., the trackingcontrol law can now be designed in a very simple manner of theform (3). It is worthy pointing out that by setting the homoge-neous τ = 0, the proposed controller reduces to a conventionallinear state feedback control law, i.e., u = −�nKz + x∗

n+1 .Remark 2.2: The proposed regulation methodology presents

us an alternative, but more practical control design procedureowing to the semiglobal control infrastructure. More distin-guishably, the common requirements of various nonlinearitygrowth conditions on the system nonlinearities that are alwaysemployed in the existing finite-time-control-related literaturessuch as [3] and [21]–[23], are essentially relaxed. Hence, anynonlinear systems of the form (1) can be finite-time controlledby the proposed controller (3) where the main difference relieson the determine guideline of the bandwidth factor �.

III. MAIN RESULT

The main result of this paper is presented by the followingtheorem.

Theorem 3.1: For any given constant ρ ∈ R+ that could bearbitrary large, consider the closed-loop system consisting of(1) satisfying x(0) ∈ �x � [−ρ, ρ]n and the finite-time controllaw (3) with a sufficiently large bandwidth factor �, which isdependent on ρ. Then, the following statements hold.

1) Any trajectory starting from the compact set �x willconverge to its equilibrium x = x∗.

2) There exists a finite-time instant T > 0 such that ∀t ≥T, y(t) = yr .

Proof: Based on the relation (2), the z-system can be directlyexpressed as

{zi = �zi+1 + (φi(xi) − φi(x∗

i ))/�i−1 , i ∈ N1:n−1zn = �v + (φn (x) − φn (x∗

n ))/�n−1 .(4)

Motivated by [6] and [24], construct a simplified Lyapunovfunction V (z) of the following form:

V (z) =(�z�1− τ

2Δ r

)T

P �z�1− τ2

Δ r (5)

where P is a positive definite and symmetrical matrix satisfyingΛT P + PΛ = −I with Λ being a companion matrix of K. Fromthe definition of K, it is obvious that Λ is Hurwitz.

To simplify the proof, the following proposition whose proofis collected in the Appendix is given first.

Proposition 3.1: The following statements hold.1) V (z) ∈ C1 ∩ H2−τ

Δ r .2) There exist constants ε ∈ (0, 1/n) and α ∈ R+ such that

∂V (z)∂zT

(z2 , . . . , zn ,−K�z�1+nτ

Δ r

)T

≤ −αV2

2−τ (z), τ ∈ (−ε, 0).

From the definition of x∗i and yr , there exists a constant ρ >

0 such that maxi∈N1 :n {supt≥0{x∗i (t)}} ≤ ρ. Then, for given

compact set �x , define a level set

Ω =

{

z ∈ Rn |V (z) ≤ supz∈�z �[−(ρ+ ρ),(ρ+ ρ)]n

V (z)

}

.

Let N = supz∈Ω

‖z‖∞, where ‖ · ‖∞ stands for L∞ norm of

vectors, �N = [−N,N ]n . It is not difficult to conclude that�x ⊂ �z ⊂ Ω ⊂ �N .

With Proposition 3.1 in mind, the time derivative of V (z)along the closed-loop system (3)–(4) gives

V (z) =∂V (z)∂zT

�(z2 , . . . , zn ,−K�z�1+nτ

Δ r

)T

+n∑

i=1

∂V (z)∂zi

φi(xi) − φi(x∗i )

�i−1

≤ −α�V2

2−τ (z) +n∑

i=1

∂V (z)∂zi

φi(xi) − φi(x∗i )

�i−1 . (6)

To proceed, the following proposition, whose proof is laboriousand included in the Appendix, is required.

Proposition 3.2: There exists a constant α ∈ R+ , which isdependent on N but independent of �, such that the followingrelation holds:

n∑

i=1

∂V (z)∂zi

(φi(xi) − φi(x∗i ))/�i−1

∣∣∣�N

≤ αV2

2−τ (z).

Substituting the relation in Proposition 3.2 into (6) yields

V (z)∣∣Ω ≤ −(α� − α)V

22−τ (z). (7)

Now one can select a sufficiently large scaling gain � ≥ 1 tosatisfy the following guideline:

α� − α ≥ 1 (8)

which leads to

V (z)∣∣Ω ≤ −V

22−τ (z). (9)

In what follows, we will use a contradiction argument toprove that under the guideline (8), for any nonzero initial statessatisfying x(0) ∈ �x , all the trajectories of z(t) will stay in Ωforever.

If the aforementioned statement is not true, the trajectory ofz(t) will escape the set Ω within a finite time. Due to the fact



that z(0) ∈ �N , it yields

V (z(0)) ≤ −(α� − α)V2

2−τ (z(0)) ≤ −V2

2−τ (z(0)) < 0.(10)

Hence, there must exist two time instants t2 > t1 > 0, such that

1) V (z(t1)) < 0, 2) V (z(t2)) = V (z(t1)), and 3) V (z(t2))> 0.(11)

It is clear that (9) still holds for t ∈ [t1 , t2 ]. The followingrelations hold:

V (z(t2)) − V (z(t1)) =∫ t2

t1

V (z(s))ds ≤ −∫ t2

t1

V2

2−τ (z(s))ds.

(12)

By the relation 2) of (11) and the fact that V (z(s)) > 0, s ∈[t1 , t2 ], (12) leads to an obvious contradiction, expressed as0 ≥ ∫ t2

t1V

22−τ (z(s))ds > 0. Hence, we can arrive at the follow-

ing conclusion ∀x(0) ∈ �x ⇒ z(0) ∈ �z ⇒ z(t) ∈ Ω, ∀t ≥0, which implies that the set Ω is an invariance set.

Further, by Lemma A.2 in mind and owing to the fact that τ ∈(−ε, 0), hence, 0 < 2

2−τ < 1, the relation V (z)∣∣Ω ≤ −V

22−τ (z)

leads to a straightforward conclusion that there exists a finitetime instant T > 0, such that y(t) − yr = 0, t ∈ [T,∞). Thiscompletes the proof of Theorem 3.1. �

Remark 3.1: In practices, different with local control strate-gies, the compact set �x of the initial states can be definedunder an extreme case study. By following the guideline (8),one can determine a proper bandwidth factor � to meet the con-trol performance specifications. On the other hand, the formula(8) could be somehow conservative due to the extensively usedmathematical estimations. A practical “trial and error” way ofthe parameter configuration can be stated as follows. The con-trol gain K can be simply selected following the pole placementmanner first. By setting a large � to guarantee the stability, thenone can tune � to be smaller and smaller while testing the gapbetween current control performance and pregiven performanceindexes until satisfactory response curves appear.

IV. APPLICATION TO ROBUST MOTION CONTROL

FOR AN SEA

A. System Description

SEAs are widely applied in advanced robot applications dueto their advantages over conventional stiff and nonback derivableactuators in force control, e.g., high fidelity, low cost, low stic-tion, etc. [25]–[27]. As a sketch review of the related literature,linear controllers constitute a main choice, the performancescan also be improved by using nonlinear control strategy andadding feed-forward control loops, see for examples [28]–[30],etc. However, it is worthy of pointing out that almost all thoseexisting control applications to SEAs are based on an asymptot-ical control result. Due to the fact that both a fast convergencerate and a stronger robustness are highly demanded for the SEAsystem in practices, finite-time regulation strategy will be ofsignificance compared with the existing related asymptoticalcontrol results. To the best of the authors’ knowledge, only one

Fig. 2. Illustration of the experimental SEA.

TABLE IINVOLVED PARAMETERS OF THE SEA

Parameters Description

mm inertia/mass of the motorml inertia/mass of the linkbm viscous friction coefficient of the motorbl viscous friction coefficient of the linkqm angle/position of the motorql angle/position of the linkk stiffness of the SEAFm motor torque/force

recent work [31] in the literature addresses the finite-time con-trol problem for the SEA by using a terminal sliding-mode con-trol scheme, where the bothersome chattering issue cannot beavoided. In this section, we will show that the proposed theoret-ical result will provided a much easier control implementation,and meanwhile significant control performance improvementscan be achieved compared with the conventional PD and lin-ear state feedback controllers, while there are not much addedcomplexities of the gain tuning mechanism.

In this paper, we use an SEA with a novel design (as de-picted by Fig. 2) that gives the actuator different impedancesat different force ranges. The actuator has two series elasticelements: a linear spring with a low stiffness and a torsionalspring with a high stiffness. In this paper, we verify the pro-posed controller using only the torsional spring. Fig. 2(a) is across section showing the structure of the studied actuator. Themotor (Maxon EC-4pole brushless dc motor operating at 200 W)shown is coupled to a ball screw through a torsional spring. Twoincremental encoders (Renishaw RM22IC) with resolutions of2 048 and 1 024 pulses per revolution are used to measure theangular displacement of the motor shaft and lead screw, re-spectively. Using the analogy of two-mass–spring–damper sys-tem, by neglecting the inevitable unmodeled disturbances, onecan obtain the nominal mathematical model of the followingform [30]:

{mm qm + bm qm = Fm − k(qm − ql)mlql + bl ql = k(qm − ql)

(13)

where the description of all involved parameters are listed inTable I.



Let x = [x1 , x2 , x3 , x4 ]T = [ql , ql , qm , qm ]T . System (13)can be expressed as the following state-space form:

⎧⎪⎪⎨

⎪⎪⎩

x1 = x2

x2 = kml

x3 − kml

x1 − bl

m lx2

x3 = x4

x4 = 1mm

Fm − kmm

(x3 − x1) − bm

mmx4 .

(14)

B. Finite-Time Robust Controller Design

In what follows, we consider a more practical controlobjective, namely, semiglobal control, rather than the restrictiveglobal control target. Hence, without any preverificationsof certain nonlinearity growth conditions, it is straightfor-ward to utilize the proposed tracking control approach todesign a robust finite-time control law (the tracking refer-ence is denoted by qlref ) to realize the accurate positioncontrol. With a series of precalculations as x∗

1 = qlref , x∗2 =

q(1)lref , x∗

3 = ml

k (q(2)lref + k

mlqlref + bl

m lq

(1)lref ), x∗

4 = x∗(1)3 , F ∗

m =

mm (x∗(1)4 + k

mm(x∗

3 − x∗1) + bm

mmx∗

4), z1 = x1 − x∗1 , z2 =

(x2 − x∗2)/�, z3 = (x3 − x∗

3)/�2 , z4 = (x4 − x∗4)/�3 , then we

are able to construct the following implementable finite-timecontrol law:

{v = −K

[�z1�1+4τ , �z2�

1 + 4 τ1 + τ , �z3�

1 + 4 τ1 + 2 τ , �z4�

1 + 4 τ1 + 3 τ

]T

Fm = �4v + F ∗m .

(15)

C. Experimental Setup

In the experimental setup, the control algorithm is imple-mented in real-time at 1 kHz on a dSPACE DS1007 processorboard with the DS3002 incremental encoder board for readingthe encoders. The motor is controlled using the Elmo Gold Whis-tle Servo Drive, which accepts motor torque commands as ananalog signal. The dSPACE DS2102 DAC board is used to gen-erate the motor command. The nominal values of the experimen-tal SEA are identified as follows: mm = 2.2 × 10−6(kg · m2),ml = 4 × 10−6(kg · m2), k = 0.14(N · m/rad), and bm = bl =1 × 10−3(N · m · s/rad).

D. Parameter Configuration and PerformanceVerification

In what follows, starting from PD control, we will showhow to establish the proposed practically oriented finite-timecontroller and elaborate the control performance improvementby choosing appropriate parameters of the proposed finite-timecontroller.

Step 1: From PD control to state feedback control. In the start-ing session, we first implement a conventional PD controllerto the SEA. Figs. 3 and 4 show the set-point and trajectorytracking performances under different proportional gains (kp =0.3, 0.5, 0.7, 0.9, while kd = 0.01), where qlref = 0.5 (rad) inFig. 3 and qlref = 0.5 sin(5t + ϕ) (rad) in Fig. 4, respectively.Generally speaking, choosing a larger proportional gain will re-sult in faster convergence rate, higher precision, but meanwhile,larger overshoot and control energy consumption. Without loss

Fig. 3. Set-point tracking performances under the PD controller withdifferent proportional gains kp .

Fig. 4. Trajectory tracking performances under the PD controller withdifferent proportional gain kp .



Fig. 5. Set-point tracking performances under finite-time controller (15)with different homogeneous degree τ , while K = [0.5, 0.01, 0.1, 0.01]and � = 1.

Fig. 6. Set-point tracking performances under finite-time controller(15) with different bandwidth factor �, while τ = −0.1 and K =[0.5, 0.01, 0.1, 0.01].

of generality, we extend the PD coefficients of the red dash linesin Figs. 3 and 4 as the coefficients of linear state feedback con-troller, i.e., K = [0.5, 0.01, 0.1, 0.01]. Compared with the blackdot line in Figs. 5 and 8, it is obvious to see the progressivetracking performances of state feedback controller.

Step 2: From state feedback control to finite-time control. Firstnotice that the proposed finite-time control law (15) reduces toa linear state feedback controller if we set the homogeneousdegree τ = 0. In this step, by simply modifying the homoge-neous degree τ from 0 to several negative values gradually, wecan implement the proposed finite-time controller to obtain abetter control performance while the control gains can be set asfix values. By understanding that in real-life systems, there are

Fig. 7. Set-point tracking performances under finite-time controller (15)with different control gain k1 , while [k2 , k3 , k4 ] = [0.01, 0.1, 0.01], τ =−0.1, and � = 1.1.

Fig. 8. Trajectory tracking performances under finite-time con-troller (15) with different homogeneous degree τ , while K =[0.5, 0.01, 0.1, 0.01] and � = 1.



Fig. 9. Trajectory tracking performances under finite-time controller(15) with different bandwidth factor �, while τ = −0.1 and K =[0.5, 0.01, 0.1, 0.01].

various disturbances/uncertainties, hence, it is of significancethat the proposed finite-time controller could reduce the settlingtime and improve the system robustness against the inevitabledisturbances/uncertainties. As depicted by Figs. 5 and 8, thecontrol performance is significantly improved if τ is a negativevalue and moreover, a smaller τ will clearly lead to a fasterconvergence speed and lower steady error. Under the proposedfinite-time controller (15), it can be observed from Figs. 6 and 9that the bandwidth factor � has also played a key role as a larger �will lead to an obvious performance variation as well. However,it should be pointed out here that a larger � will cause a cleardeterioration of the system robustness against the measurementnoises, which is a common problem of existing high gain controlmethods. In Figs. 7 and 10, the control performance variationalong with the control gain selection k1 , while the other con-trol parameters are fixed is depicted. To make the comparisons

Fig. 10. Trajectory tracking performances under finite-time controller(15) with different control gain k1 , while [k2 , k3 , k4 ] = [0.01, 0.1, 0.01],τ = −0.1, and � = 1.1.

clearer and more precise, the performance indexes (overshoot,offset) of set-point tracking and integral square error (ISE) in-dex (

∫ t2

t1e2(t)dt, where [t1 , t2 ] is a period of the reference signal

in the steady state and e(t) is the tracking error) for trajectorytracking case are included in Table II.

As a direct conclusion from the previously illustrated figures,the proposed finite-time control strategy will clearly lead to asignificant control performance improvement, while the con-trol gain selection guideline is as simple as the conventionallinear state feedback controllers. Moreover, the added negativehomogeneous degree will endow the control engineers a muchflexibility of tuning the control performances in practical imple-mentations.

E. Performance Comparison With an Optimal Controller

In order to better demonstrate the control performancesuperiorities of the proposed nonsmooth controller with



TABLE IIPERFORMANCE INDEXES OF THE PD CONTROLLER, LINEAR STATE

FEEDBACK CONTROLLER, AND FINITE-TIME CONTROLLER

Methods Parameters Overshoot Offset ISE

PD kp = 0.3 0.00 −20.21 751.14kp = 0.5 8.68 19.87 425.32kp = 0.7 4.90 −7.56 176.71kp = 0.9 68.08 4.40 114.31

FTC τ = 0 25.72 −4.04 133.68τ = −0.05 26.39 −3.34 141.67τ = −0.1 33.74 −2.63 46.29τ = −0.15 44.97 −0.52 10.45

� = 1 33.74 −2.63 46.29� = 1.05 73.30 −2.28 23.66� = 1.1 57.70 −1.23 10.77� = 1.15 88.33 −1.58 17.45k1 = 0.3 42.05 2.29 56.88k1 = 0.4 52.45 0.18 23.54k1 = 0.5 57.70 −1.23 10.77k1 = 0.6 66.28 0.53 9.26

Fig. 11. Set-point tracking performance comparisons under optimalcontrollers and finite-time controllers (FTC 1: K = [0.4, 0.01, 0.1, 0.01],� = 1.1, and τ = −0.1; FTC 2: K = [0.5, 0.01, 0.1, 0.01], � = 1, and τ =−0.15).

existing asymptotical controllers, we present an experimen-tal performance comparison with an optimal controller bythe predictive approach [32]–[34]. The controller is de-rived based on optimizing a performance index J(t) =12

∫ T

0 (x1(t + τ) − x∗1(t + τ))2 dτ , where T is the predictive

period. Utilizing a predictive approach associated with the Tay-lor expansion, the optimal controller is derived in the form of

FOptm = −

4∑

i=1

kOpti (xi − x∗

i ). (16)

The detail process of derivation can be found in [32]. It is worthnoting that the optimal gains kOpt

i , i ∈ N1:4 are only related tothe predictive period T and the control order r. For simplicity,

Fig. 12. Trajectory tracking performance comparisons under optimalcontrollers and finite-time controllers (FTC 1: K = [0.4, 0.01, 0.1, 0.01],� = 1.1, and τ = −0.1; FTC 2: K = [0.5, 0.01, 0.1, 0.01], � = 1, and τ =−0.15).

the control order r is set as 0 and the predictive period T is setas the only tunable parameter.

Under a set-point tracking control objective, the control per-formance comparisons of the proposed finite-time controller(15) and the optimal controller (16) are presented in Fig. 11.By noting that the initial torque amplitude of the candidate con-trollers are placed in a similar level in order to make a faircomparison, the robustness exhibited by the proposed finite-time controllers are much stronger than the optimal controllers.A lower steady-state error can be achieved with the import of anegative homogeneous degree. Similar conclusions can also beobtained from the case of trajectory tracking as shown in Fig. 12.To make the comparison more clearer, the detailed performanceindexes of both set-point tracking and trajectory tracking casesare also included in Table III.



TABLE IIIPERFORMANCE INDEXES OF THE OPTIMAL CONTROLLER AND

FINITE-TIME CONTROLLER

Methods Parameters Overshoot Offset ISE

OC OC1 9.86 −3.69 27.18OC2 7.90 −1.93 18.53

FTC FTC1 52.45 0.18 23.54FTC2 44.97 −0.52 10.45

V. CONCLUSION

In this paper, we investigate a novel nonrecursive trackingcontrol design framework under a semiglobal control objectivefor a class of lower triangular nonlinear systems. Compared withall existing related results, several improvements are achieved.First, the proposed control scheme is much simpler and thecontrol gain can be easily selected following a pole placementpattern. Second, it is shown that a finite-time trajectory trackingresult can be realized for smooth nonlinear systems withoutany additional nonlinearity growth conditions. Moreover, theproposed one-step control design and stability analysis under anew nonrecursive synthesis manner will facilitate the parameterfiguration and practical implementations. An application to anSEA system and experimental performance comparison resultsare provided to illustrate the simplicity and effectiveness ofthe proposed controller. The proposed control strategy couldprovide a more efficient implementation choice for those real-life systems with high demand of fast control speed and strongerrobustness. Future works will focus on the finite-time activedisturbance/uncertainty attenuation problem for systems withlarge modeling error and external disturbances.

APPENDIX

A. Useful Lemmas

Some useful lemmas are stated as follows for the convenienceof readers.

Lemma A.1: [6] Let V1(x) ∈ Hτ1Δ r and V2(x) ∈ Hτ2

Δ r , re-spectively, then the following statements hold.

1) V1(x)V2(x) ∈ Hτ1 +τ2Δ r .

2) ∂V1 (x)∂xi

∈ Hτ1 −ri

Δ r , i ∈ N1:n .3) If V1(x) is positive definite, then the following re-

lation holds(min{x:V1 (x)=1} V2(x)

)V

τ 2τ 1

1 (x) ≤ V2(x) ≤(max{x:V1 (x)=1} V2(x)

)V

τ 2τ 1

1 (x).Lemma A.2: [6] Consider a dynamical system x =

f(x, t), f(0, t) = 0. Suppose there exists a C1 positive-definiteand proper function V : Rn → Rn and real numbers c > 0 andr ∈ (0, 1), such that V + cV r is seminegative definite, then theorigin x = 0 is a globally finite-time stable equilibrium witha settling time T ≤ V 1−r (x0 ,t0 )

c(1−r) for any given initial conditionx0 = x(t0).

Lemma A.3: [6], [23] Consider the following chain ofintegrators:

ηi(t) = ηi+1(t), i ∈ N1:n−1 , ηn (t) = u(t) (17)

under a homogeneous control law of the following form:

u(t) = −K�η�rn +τΔ r (18)

where K is the coefficient vector of a Hurwitz polyno-mial sn + knsn−1 + · · · + k2s + k1 , 0 > τ > − 1

n is a homo-geneous degree. There exists a constant ε ∈ (0, 1

n ), such thatthe closed-loop system (17)–(18) is globally finite-time stablefor τ ∈ (−ε, 0).

B. Proofs of Propositions

This subsection collects the proofs of propositions used inthis paper.

Proof of Proposition 3.1: By using Lemma A.3, one can ob-

tain the conclusion that that ∂V (z )∂zT

(z2 , . . . , zn ,−K�z�rn +τ

Δ r

)T

is negative definite for a homogeneous degree τ ∈ (−ε, 0). With

V (z) ∈ H2−τΔ r and

(z2 , . . . , zn ,−K�z�rn +τ

Δ r

)T ∈ HτΔ r in mind,

using Lemma A.1, the following relation can be achieved for aconstant α ∈ R+ :

∂V (z)∂zT

�(z2 , . . . , zn ,−K�z�rn +τ

Δ r

)T ≤ −α�‖z‖2Δ r . (19)

Proof of Proposition 3.2: Recalling fi, i ∈ N1:n is a smoothfunction, by utilizing the mean-value theorem, we have

φi(xi) − φi(x∗i ) ≤ γi(xi , x

∗i )

×(|x1 − x∗

1 | + |x2 − x∗2 | + · · · + |xi − x∗

i |)

where γi(xi , x∗i ) is a C0 nonnegative function. For all x ∈ �N ,

it is clearly that there exists a constant γi dependent on N , suchthat γi(xi , x

∗i ) ≤ γi .

In what follows, two cases can be discussed as follows.In the case when |xj − x∗

j | ≥ 1 ∀j ∈ N1:i , by noting that|xj | ≤ N and |x∗

j | ≤ ρ, we know that |xj − x∗j | ≤ N + ρ ≤

(N + ρ)|xj − x∗j |

1 + i τ1 + ( j −1 ) τ . In the case when |xj − x∗

j | < 1 ∀j ∈N1:i , by noting 1+iτ

1+(j−1)τ ≤ 1, we know |xj − x∗j | ≤ |xj −

x∗j |

1 + i τ1 + ( j −1 ) τ .

By summarizing the aforementioned two cases, the followingrelation holds with a constant γi = γi max{N + ρ, 1} :

(φi(xi) − φi(x∗i ))/�i−1

≤ γi

(|�0z1 | 1 + i τ

1 + |�1z2 |1 + i τ1 + τ + · · · + |�i−1zi |

1 + i τ1 + ( i−1 ) τ

)

�i−1 ,

x ∈ �N . (20)

Noting that � ≥ 1, and the relation that (j − 1) 1+iτ1+(j−1)τ

− (i − 1) ∈ (−∞, 0) ∀j ∈ N1:i , it implies l(j−1) 1 + i τ1 + ( j −1 ) τ

−(i−1)

< 1. Hence, we can obtain

(φi(xi) − φi(x∗i ))/�i−1 ≤ γi‖z‖1+iτ

Δ r . (21)

It is clear that V (z) ∈ H2−τΔ r and ‖z‖1+iτ

Δ r ∈ H1+iτΔ r . With

Lemma A.1 in mind, there exists a constant α ∈ R+ , which isdependent on N but independent of �, such that the following



relations hold:

n∑

i=1

∂V (z)∂zi

(φi(xi) − φi(x∗i ))/�i−1

∣∣∣�N

≤n∑

i=1

γi

∣∣∣∣∂V (z)∂zi

∣∣∣∣ ‖z‖1+iτ

Δ r ≤ αV2

2−τ (z). (22)

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Chuanlin Zhang (M’14) received the B.S. de-gree in mathematics and Ph.D. degree in controltheory and control engineering from the Schoolof Automation, Southeast University, Nanjing,China, in 2008 and 2014, respectively.

He was a Visiting Ph.D. Student with the De-partment of Electrical and Computer Engineer-ing, University of Texas at San Antonio, San An-tonio, TX, USA, from 2011 to 2012, and a Visit-ing Scholar with the Energy Research Institute,Nanyang Technological University, Singapore,

from 2016 to 2017. He is currently with the Advanced Robotics Cen-ter, National University of Singapore, as a Visiting Scholar. Since 2014,he has been with the College of Automation Engineering, ShanghaiUniversity of Electric Power, Shanghai, where he is currently an Asso-ciate Professor. His research interests include nonlinear system controltheory and applications for power systems, and he has authored andcoauthored more than 25 Journal papers.

Dr. Zhang is the Principal Investigator of several research projects,including Leading Talent Program of Shanghai Science and TechnologyCommission, “Chenguang Program” by Shanghai Municipal EducationCommission, etc. He was the recipient of the Best Poster Paper Awardin the 3rd IFAC International Conference on Intelligent Control and Au-tomation Science (2013).



Yunda Yan (S’15) was born in Yixing, China,in 1990. He received the B.Sc. degree fromthe School of Automation, Southeast University,Nanjing, China, in 2013, where he is currentlyworking toward the Ph.D. degree in control the-ory and control engineering.

He is currently a Visiting Student with the De-partment of Biomedical Engineering, NationalUniversity of Singapore, Singapore, under theguidance of Dr. Y. Haoyong. His research inter-ests include the development of predictive con-

trol methods and disturbance modeling and estimation approaches andtheir applications in motion control systems.

Ashwin Narayan was born in Kerala, India, in1994. He received the B.Tech. degree in elec-tronics and communication engineering from theNational Institute of Technology, Tiruchirappalli,India, in 2016. He is currently working toward thePh.D. degree at the Department of BiomedicalEngineering, National University of Singapore,Singapore.

His current research interests include sens-ing and control for poststroke gait rehabilitationrobots.

Haoyong Yu (M’10) received the Ph.D. de-gree in mechanical engineering from the Mas-sachusetts Institute of Technology, Cambridge,MA, USA, in 2002.

He was a Principal Member of Technical Staffwith the DSO National Laboratories, Singapore,until 2002. He is currently an Associate Profes-sor with the Department of Biomedical Engineer-ing and a Principal Investigator of the SingaporeInstitute of Neurotechnology, National Universityof Singapore, Singapore. His research interests

include medical robotics, rehabilitation engineering and assistive tech-nologies, and system dynamics and control.

Dr. Yu was the recipient of the Outstanding Poster Award at the IEEELife Sciences Grand Challenges Conference 2013. He has served on anumber of IEEE conference organizing committees.


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