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A New Smooth Robust Control Design For Uncertain Nonlinear Systems with

Non-Vanishing Disturbances

Article  in  International Journal of Control · December 2015

DOI: 10.1080/00207179.2015.1128561

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A New Smooth Robust Control Design ForUncertain Nonlinear Systems with Non-VanishingDisturbances

Bin Xian & Yao Zhang

To cite this article: Bin Xian & Yao Zhang (2015): A New Smooth Robust Control Design ForUncertain Nonlinear Systems with Non-Vanishing Disturbances, International Journal ofControl, DOI: 10.1080/00207179.2015.1128561

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December 4, 2015 International Journal of Control TCON˙A˙1128561

To appear in the International Journal of ControlVol. 00, No. 00, Month 20XX, 1–26

Publisher: Taylor & FrancisJournal: International Journal of ControlDOI: http://dx.doi.org/10.1080/00207179.2015.1128561

A New Smooth Robust Control Design For Uncertain Nonlinear Systems with

Non-Vanishing Disturbances

Bin Xiana∗ and Yao Zhanga

a The Institute of Robotics and Autonomous System, the Tianjin Key Laboratory of Process Measurementand Control, School of Electrical Engineering and Automation, Tianjin University;

(Received 00 Month 20XX; accepted 00 Month 20XX)

In this paper, we consider the control problem for a general class of nonlinear system subjected touncertain dynamics and non-varnishing disturbances. A smooth nonlinear control algorithm is presentedto tackle these uncertainties and disturbances. The proposed control design employs the integral of anonlinear sigmoid function to compensate the uncertain dynamics, and achieve a uniformly semiglobalpractical asymptotic stable (USPAS) tracking control of the system outputs. A novel Lyapunov basedstability analysis is employed to prove the convergence of the tracking errors and the stability of the closedloop system. Numerical simulation results on a two-link robot manipulator are presented to illustratethe performance of the proposed control algorithm comparing with the layer boundary sliding modecontroller and the robust of integration of sign of error (RISE) control deisign. Further more, real-timeexperiment results for the attitude control of a quadrotor helicopter are also included to confirm theeffectiveness of the proposed algorithm.

Keywords: Smooth algorithm; non-varnishing disturbance; USPAS; Lyapunov analysis;

1. Introduction

Disturbance rejecting is a fundamental issue in control theory and has been paid great attentionfor several decades. It is naturally that the disturbances would deteriorate the control performanceand even cause the damage of the whole system. Moreover, the existence of the disturbance ishardly able to fit in lots of well-established control frames, such as online adaptive method andsystem states observer, etc. Adaptive control algorithm and internal mode control are often usedto estimate a class of sine-type disturbances or exgenous-generated disturbances (Chen , 2004;Feemster, Fang, & Dawson , 2006; Serrani, Isidori, & Marconi , 2001). But in most situations, thedisturbances render no fixed forms like the sine-type.

Variable structure control is the common method to deal with the disturbances without any priorknowledge but the norm-bounded upper boundedness. Specifically, a discontinuous term sign(∙)would be applied to facilitate the stability analysis in sliding mode control (Efe , 2008; Islam, &Liu , 2011; Su, Leung, & Zhou , 1990). A simple nonlinear PI structure which is discontinuous

∗Corresponding author. Email: xbin@tju.edu.cn

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on the equilibrium was proved to have asymptotically tracking performance in (Ortega, Astolfi, &Barabanov , 2002). Terminal sliding mode control yielding finite-time stability also had been utilizedto deal with uncertain dynamics (Hong, Xu, & Huang , 2002). The control schemes aforementionedsuffered the issue of chattering caused by the discontinuity on the equilibrium. It is well knownthat the infinite control bandwidth and infinitesimal switching time constraint the utilization ofthe kind of discontinuous control algorithms in practical application.

Several approaches to reduce the chattering issue had been developed (i.e. adaptive sliding modecontrol, integral sliding mode control, high-order sliding model control)(Levant , 2007; Xu, Mirmi-rani, & Ioannou , 2004; Zong, Zhao, & Jing , 2010). In (Xu, Mirmirani, & Ioannou , 2004), adaptivelaw for the robust gains was designed to minimize the chattering amplitude of the control inputs.The augmented integration of error in the dynamic sliding surface help to weaken the chattering ef-fect of the system state (Zong, Zhao, & Jing , 2010). Even though the above control designs resultedin asymptotic tracking result, the discontinuous drawback still existed. Continuous methods ap-peared in recent years (Girin, & Plestan , 2009; Laghrouche,Plestan, & Glumineau , 2007; Wilcox,MacKunis, Bhat, Lind, & Dixon , 2010), in which the high order sliding mode technique consistedof the increased order dynamics such that the derivative of the input appeared. Even though theaugmented system was designed to be discontinuous, the integration of the input was continuous.This kind of strategy was utilized for the tracking problem of the hypersonic vehicle’s velocity,altitude and attack angle, and global exponential stability was achieved (Wilcox, MacKunis, Bhat,Lind, & Dixon , 2010). The challenge of this strategy is that the augmented system involve withthe high order derivative of the sliding surface (often equals to the order of the dynamics). For mostmechetronic systems, it is very difficult to obtain third order or higher order time derivatives of thesystem states. Levant, aiming to drop the unmeasured signals, proposed second-order sliding modecontrol (Levant , 2007) where the phase plane was introduced to carry out the stability analysis.A similar method, RISE, was first presented in (Xian, Dawson, De Queiroz,& Chen , 2004), andthen further developed in (Patre, MacKunis, Makkar, & Dixon , 2008; Patre, MacKunis, Kaiser, &Dixon , 2008; Patre, MacKunis, Dupree, & Dixon , 2011). Unlike the second-order SMC technique,the RISE control provided a constructive Lyapunov stability analysis and expanded the solutionto high order MIMO systems. However, these two methods were first order differentiable but notsmooth enough. The first order differentiable property doesn’t provide itself conveniently stackinginto backstepping control design where second-order or higher order derivative of the virtual con-trol input signal might be needed (Qu., 1998), or output control where lipschitz condition has tobe meet (Khail., 2003). A smooth high-gain controller for a nonlinear dynamics was presented in(Dasdemir,& Zergeroglu, 2015), and achieved a practical tracking result. To our best knowledge,there is no such smooth control scheme that achieve asymptotically stable while the dynamics issubjecting to system uncertainties and unknown disturbances.

In this paper, motivated by the RISE structure, we propose a simple smooth nonlinear controlalgorithm for a full-actuated dynamics which is suffering the unknown disturbance and uncertaindynamics. An integration of sigmoid function is employed to compensate the system uncertain-ties (including non-vanishing disturbances and un-model dynamics). Unlike the Layer boundarycontrol method, the proposed control design has the ability to learn the uncertain dynamics. Anovel Lyapunov candidate function which contains the desired uncertainties plus a small constantis presented to prove the USPAS result. Since the existence of the small constant offset, it is a chal-lenge that derive the correlation between the Lyapunov candidate function and its time derivative.By utilizing the class K∞ function, we are able to conclude the ISS property with respect to asmall constant as the input. Thus, the USPAS result can arises eventually from the ISS property.Numerical simulation and experiments have been performed on a two-link robot manipulator and aquadrotor attitude control testbed, respectively, to illustrate the good performance of the presentedcontrol algorithm.

Different from feedforward compensating methods like neural network and fuzzy system, theproposed smooth controller is able to account for the non-vanishing disturbances. And by dividing

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the system uncertainty into an uncertain part associated with desired trajectories and an uncertainpart associated with tracking errors, we can eliminate the assumption that the uncertain dynamicsis global bounded. Therefore the main contributions of the controller include the following: 1) it ishigher order differentiable and is suitable to be combined with a lot of control strategies, i.e., inner-outer loop control, backstepping control, and so on; 2) the controller can deal with the parametricuncertainties and disturbances under very little knowledge about the system dynamics; 3) it isconvenient to blend feedforward design, i.e., the neural network, fuzzy system into the proposedcontroller due to its simple structure.

The paper is organized as follows. Section 2 describes the dynamic model. Section 3-5 presentthe control objective and the design procedure of the proposed controller. Section 6 shows the Lya-punov based stability analysis. Simulation and experiment results that demonstrate the improvedperformance of the proposed controller are included in Section 7. Some conclusion remarks aregiven in Section 8.

2. Dynamic model

Consider a class of m-th order MIMO nonlinear systems of the following formulation:

x(m) = f(x, x, . . . x(m−1)) + G(x, x, . . . x(m−1))u + d(t), (1)

where (∙)(i)(t) denote the i-th order derivative with respect to time, and x(i)(t) ∈ Rn, i = 1, 2, . . .m−

1 are the system states. Let x =[xT , xT , xT , . . . , (x(m−1))T

]T∈ Rmn. In (1), f(x) ∈ Rn and

G(x) ∈ Rn×n are unknown nonlinear C2 functions, u(t) ∈ Rn is the control input, d(t) ∈ Rn

represents some unknown external disturbances. The subsequent control development is basedupon the assumption that the state vector x(t) is measurable in the following control development.Additionally, the following reasonable assumptions are made about the dynamics in (1).

Assumption 1: The input matrix G(x) is symmetric, positive-definite, and satisfies the the fol-lowing inequality

g ‖ξ‖2 ≤ ξT G−1(x)ξ ≤ g(x) ‖ξ‖2 ∀ξ ∈ Rn, (2)

where g ∈ R+ is a positive constant, g(x) ∈ R is a positive function, and ‖∙‖ denotes the standardEuclidean norm.

Assumption 2: If x(i)(t) ∈ L∞, i = 1, 2, . . .m + 1, then f(x), G(x) are bounded, and the first andsecond order partial derivatives of f(x), G(x) with respect to x(i)(t), i = 1, 2, . . .m − 1 exist andare bounded.

Assumption 3: The external disturbances d(t) and their first two order time derivatives arebounded in the sense that

d(i)(t) ∈ L∞ i = 0, 1, 2. (3)

3. Control objective

The main objective of this paper is to develop a smooth robust nonlinear controller that will ensurethat the state x(t) of the dynamic system in (1) tracks the desired trajectory xd(t) despite systemuncertainties and unknown external disturbances. To meet the aim, the following tracking error

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vector, denoted by e1(t) ∈ Rn, is defined:

e1(t) = xd(t) − x(t), (4)

where xd(t) ∈ Rn denotes the time-varying reference trajectory vector which is designed such that

x(i)d (t) ∈ L∞, i = 0, 1, . . . ,m + 2.To provide flexibility in the design of feedback controller, filtered tracking error vectors, denoted

by ei(t) ∈ Rn, i = 1, 2, . . .m, are defined as follows

e2(t) = e1(t) + e1(t)e3(t) = e2(t) + e2(t) + e1(t)e4(t) = e3(t) + e3(t) + e2(t)

...em(t) = em−1(t) + em−1(t) + em−2(t).

(5)

The filtered error signals ei(t), i = 2, 3, . . .m can be expressed in terms of e(j)1 (t), j = 0, 1, . . . i − 1

as

ei =i−1∑

j=0

aije(j)1 , (6)

and the constant coefficient aij , i = 1, 2, . . .m, j = 0, 1, . . . i− 1 can be calculated via the Fibonaccisequence as follows (Xian, Dawson, De Queiroz,& Chen , 2004):

ai0 = B(i) = 1√5

[(1+

√5

2

)i−(

1−√

52

)i]

i = 2, 3, . . .m

aij =∑i−1

k=1 B(i − k − j + 1)ak+j−1,j−1,i = 3, 4, . . .m and j = 1, 2, . . . i − 2

ai,i−1 = 1, i = 1, 2, . . .m.

(7)

4. Open loop error system

An auxiliary filtered error signal r(t) ∈ Rn is defined in the following form

r(t) = em(t) + αem(t), (8)

where α ∈ R is a positive constant. Since the filtered error vector r(t) contains the unmeasurablesignal x(m)(t), it can not be used in the feedback control design directly. After taking the timederivative of r(t) and invoking (4) and (5), the following equation can be obtained

G−1(x)r = G−1(x)em + αG−1(x)em

= G−1(x)(∑m−1

j=0 amje(j+2)1 + αem

)

= G−1(x)x(m+1)d + G−1(x)

(∑m−2j=0 amje

(j+2)1 + αem

)−

d(G−1(x(t))f(x(t)))dt − d(G−1(x(t))d(t))

dt − u + G−1(x)x(m),

(9)

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and then re-arranged into the following form:

G−1(x)r = −12G−1(x)r − em + N − u, (10)

where the auxiliary function vector N(x, x,t) ∈ Rn contains the uncertain terms related with thedynamic system, and is defined as follows

N = G−1(x)x(m+1)d + G−1(x)

(∑m−2j=0 amje

(j+2)1 + αem

)−

d(G−1(x(t))f(x(t)))dt − d(G−1(x(t))d(t))

dt + G−1(x)x(m)+12G−1(x)r + em.

(11)

Before constructing the nonlinear feedback controller, we perform the following manipulation on(15). Let the auxiliary function Nd(t) ∈ Rn be Nd(t) = N(xd, xd, t), we have

Nd = G−1(xd)x(m+1)d − d(G−1(xd(t))f(xd(t)))

dt −d(G−1(xd(t))d(t))

dt + G−1(xd)x(m)d

(12)

Note that by applying Assumption 1-3, it is not difficult to check that Nd(t), Nd(t) are boundedand satisfying the following inequality

‖Nd(t)‖L∞≤ ϑ1,

∥∥∥Nd(t)

∥∥∥L∞

≤ ϑ2, (13)

in which ϑ1 and ϑ2 are known constants, and the L∞-norm is defined as

ζ = [ζ1, . . . , ζn]T

‖ζ‖∞ = max{ζi, i = 1, . . . , n}‖ζ‖L∞

= supt>0 ‖ζ‖∞ .(14)

After adding and subtracting Nd(t) to the right side of the equation (10), the following open looperror for r(t) dynamics can be obtained

G−1(x)r = −12G−1(x)r − em + Nd + N − u, (15)

where

N(x, x,t) = N − Nd. (16)

Remark 1: Since N(x, x,t) is continuously differentiable with respect to x(i), i = 1, 2, . . . ,m, wecan show that N(t) can be upper bounded by the following inequality (Xian, Dawson, De Queiroz,&Chen , 2004)

∥∥∥N∥∥∥ ≤ ρ1(‖z‖) ‖z‖ , (17)

where z(t) ∈ R(m+1)n is defined as

z(t) = [ eT1 eT

2 ∙ ∙ ∙ eTm rT ]T (18)

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and ρ1(∙) is a non-decreasing function. By multiplying a monotone increasing function π(‖z‖) withπ(0) = 1, a new function is developed as ρ(‖z‖) = π(‖z‖) ∙ ρ1(‖z‖), and it is easy to check that

ρ(‖z‖) is a global invertible, non-decreasing function and the inequality∥∥∥N∥∥∥ ≤ ρ(‖z‖) ‖z‖ holds.

5. Control development

Based on the open loop error dynamic system in (15) and subsequent stability analysis, the controlinput u(t) is designed as

u(t) = (ks + 1)em(t) − (ks + 1)em(0)+∫ t0 [(ks + 1)αem(τ) + βTanh(nem(τ)/ε)]dτ.

(19)

where ks, β ∈ R+ are some positive constant control gains. The vector Tanh(nem(τ)/ε) ∈ Rn isdefined as

Tanh(nem(τ)/ε) = [tanh(nem1(τ)/ε), . . . , tanh(nemn(τ)/ε)]T , (20)

where emi(τ), i = 1, 2, . . . , n is the ith element of the error vector em(τ), and ε ∈ R+ denotes asmall positive constant to be adjusted. The term −(ks +1)em(0) in (19) is included for driving theinitial condition of u(t) to u(0) = 0, but it dose not affect the open-loop dynamics in (15).

By taking the time derivative of u(t) in (19) and substituting the resulting equation into (15),we can obtain the close loop error dynamics for r(t) as follows

G−1(x)r = −12G−1(x)r − em − (ks + 1)r + Nd + N

− βTanh(nem(t)/ε).(21)

In (21), the second and third term are the direct feedback of the system error, the dis-match ofthe uncertainty N(t) will be conquered by the direct error feedback. Furthermore, the boundeduncertain vector Nd(t) can be approximately compensated by the robust term βTanh(nem(t)/ε).

6. Stability analysis

Before presenting the main result of this section, we now state the following definitions and lemmaswhich will be invoked later. The definitions about uniform semiglobal practical asymptotic stability(USPAS) come from (Chaillet, & Loria , 2008).

Definition 1: Denote by Rδ the closed ball of radius δ centered at the origin such that Rδ = {z ∈R(m+1)n : ‖z‖ ≤ δ}. And we define ‖z‖δ = infζ∈Rδ

‖z − ζ‖.

Definition 2: For some positive constants δ and Δ satisfying Δ > δ, the ball Rδ is said to beuniformly stable (US) on RΔ if there exists a class K∞ function η(∙) such that the trajectory of zfrom any initial state z0 ∈ RΔ satisfy ‖z(t)‖δ ≤ η(‖z0‖) for any t ≥ t0 .

Definition 3: The ball Rδ is said to be uniformly attractive (UA) on RΔ if there exists a class Lfunction σ(∙) such that the trajectory of z(t) from any initial state z0 ∈ RΔ satisfy ‖z(t)‖δ ≤ σ(t−t0)for all t ≥ t0 .

Definition 4: Let Θ = [α, ks, β, ε] ∈ R4 be the set of control gains, the dynamics is said to beUSPAS on Θ if, given any Δ > δ > 0, there exists θ∗(δ, Δ) ∈ Θ such that Rδ is US and UA on RΔ

for the dynamics.

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The USPAS definition differs from the uniformly ultimate boundedness (UUB) definitions, inthe sense that the size of the ultimate boundedness and the region of attraction can be adjustedarbitrarily via the control gains.

Lemma 1: Define an auxiliary function y(t) ∈ R as follows

y = βn∑

i=1

ε

nln cosh

(nemi

ε

)−

n∑

i=1

Ndiemi + σ (22)

where Ndi denotes the ith element of the vector Nd, and the positive constant σ ∈ R+ is

σ = ε ‖Nd‖L∞ln 2. (23)

Provided that the control gain β is selected according to the following sufficient condition

β > ϑ1, (24)

then

y ≥ 0. (25)

Proof. From (22), the norm definition and (24) can be employed to show that

y ≥ ‖Nd‖L∞

n∑

i=1

ε

nln cosh

(nemi

ε

)−

n∑

i=1

‖Nd‖L∞|emi| + σ. (26)

Let the auxiliary functions yi(t) ∈ R, for i = 1, . . . , n are defined as

yi = ‖Nd‖L∞

ε

nln cosh

(nemi

ε

)− ‖Nd‖L∞

|emi| + σ/n (27)

which includes the ith element of em(t) exclusively. It is not difficult to show that y ≥∑n

i=1 yi

from (26).Next, we will prove the upperbound and lowerbound of yi(t) ∈ R, i = 1, . . . , n.By taking the partial derivative of yi with respect to emi, we have

dyi

demi= ‖Nd‖L∞

tanh(

nemi

ε

)− ‖Nd‖L∞

if emi > 0dyi

demi= ‖Nd‖L∞

tanh(

nemi

ε

)+ ‖Nd‖L∞

if emi < 0not exist if emi = 0.

(28)

Note that∣∣tanh

(nemi

ε

)∣∣ < 1 for any emi ∈ [−∞, +∞], it can be seen form (28) that yi is adecreasing function with respect to emi as emi > 0, and yi is a increasing function with respect toemi as emi < 0. Since yi is continuous w.r.t emi on the whole defined domain, we have

yi max = yi |emi=0= σ/nyi min = min{yi |emi=−∞, yi |emi=+∞}.

(29)

To compute the limit of yi as emi → −∞ and emi → +∞, respectively, we perform the following

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manipulation on yi

yi = ‖Nd‖L∞emi + ‖Nd‖L∞

εn ln

(1+exp(−2nemi/ε)

2

)−

‖Nd‖L∞|emi| + σ/n

yi = −‖Nd‖L∞emi + ‖Nd‖L∞

εn ln

(1+exp(2nemi/ε)

2

)−

‖Nd‖L∞|emi| + σ/n.

(30)

Based on the first equation in (30), we can conclude that

limemi→+∞ yi = ‖Nd‖L∞emi − ‖Nd‖L∞

εn ln 2−

‖Nd‖L∞emi + σ/n,

(31)

and based on the second equation, we have

limemi→−∞ yi = −‖Nd‖L∞emi − ‖Nd‖L∞

εn ln 2+

‖Nd‖L∞emi + σ/n.

(32)

And then by using the condition in (23), it can be obtained that

yi min = limemi→+∞

yi = limemi→−∞

yi = 0. (33)

It is obviously that (25) holds.�

Theorem 1: The proposed controller in (19) achieves USPAS tracking control for the dynamicsystem listed in (1) which is subject to dynamic uncertainties and external unknown disturbances,provided that

α > 12

β > ϑ1 + 1αϑ2,

(34)

and the control gain ks is selected sufficiently large relative to the initial condition of the system.

Proof. Noticing that the selecting condition for β in (34) is sufficient condition of (24), so theinequality in (25) still holds.

We now define the Lyapunov candidate function V (t) as

V =12

m∑

i=1

eTi ei +

12rT G−1(x)r + y, (35)

where y is defined in (22). It follows that

V ≥ 12

∑mi=1 eT

i ei + 12rT G−1(x)r

≥ λ1 ‖z‖2 (36)

where λ1 = 12 min{1, g}. By substituting (22) into (35), we have

V = V1 + σ (37)

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where the auxiliary function V1(t) ∈ R is defined as

V1 = 12

∑mi=1 eT

i ei + 12rT G−1(x)r+

β∑n

i=1εn ln cosh

(nemi

ε

)−∑n

i=1 Ndiemi,(38)

and V1(t) can be upper bounded as

V1 ≤ 12

∑mi=1 eT

i ei + 12rT G−1(x)r+(

β + ‖Nd‖L∞

)∑ni=1 |emi| .

(39)

By examining the resulting inequality in (39), we can check that the right-side of (39) is positivedefinite w.r.t z in (18), and by applying lemma 4.3 in Khail. (2003), we can prove that

V1 ≤ λ1(‖z‖) (40)

where the function λ1(∙) ∈ R ≥ 0 → R ≥ 0 belongs to class K∞.By using (36), (37), and (40), it yields

λ1 ‖z‖2 ≤ V ≤ λ1(‖z‖) + σ. (41)

Remark 2: From (38), the K∞-class function λ1(∙) is correlated with the designed control gainβ. And β is just related with the maximum value of the pre-determined reference trajectory of thesystem and the control gain α. If α satisfies the condition in (34), β is restricted in the domainthat β > ‖ϑ1‖L∞

+ 2 ‖ϑ2‖L∞which is independent with any control gains and time-varying system

states. Thus, both the control gain β and the function λ1(∙) are independent with the other controlgains.

Taking the time derivative of V (t) yields

V =∑m

i=1 eTi ei + rT G−1(x)r + 1

2rT G−1(x)r+β∑n

i=1 tanh(nemi/ε)emi − ddt(N

Td em).

(42)

By substituting (5) and the closed loop error dynamics in (21) into the resulting equation andcancelling the common terms, the following equation can be obtained

V = −∑m−1

i=1 ‖ei‖2 − α ‖em‖2 + eT

m−1em − ‖r‖2 − ks ‖r‖2 +

rT N + αeTm

(Nd − 1

αNd − βTanh(nem/ε))

.(43)

Based on the fact that

eTm−1em ≤

12‖em−1‖

2 +12‖em‖2 , (44)

the expression in (43) can be upper bounded as

V ≤ −∑m−2

i=1 ‖ei‖2 − 1

2 ‖em−1‖2 −

(α − 1

2

)‖em‖2 − ‖r‖2 −

ks ‖r‖2 + rT N + αeT

m

(Nd − 1

αNd − βTanh(nem/ε))

≤ −δ ‖z‖2 − ks ‖r‖2 + ρ(‖z‖) ‖z‖ ‖r‖+

αeTm

(Nd − 1

αNd − βTanh(nem/ε))

(45)

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where δ = min{12 ,(α − 1

2

)}, and α are chosen according to the conditions listed in (34). After

completing the triangle inequality squares of the terms in (45), the following expression can beobtained

V ≤ −12δ ‖z‖2 −

(12δ − 1

4ksρ2(‖z‖)

)‖z‖2 +

αeTm

(Nd − 1

αNd − βTanh(nem/ε))

≤ −12δ ‖z‖2 −

(12δ − 1

4ksρ2(‖z‖)

)‖z‖2 +

α(∑n

i=1 |emi|(|ϑ1| + 1

α |ϑ2|)−∑n

i=1 β tanh(nemi/ε)emi) .

(46)

Since the fact that

0 ≤ |x| − x tanh (x/ε) ≤ 0.2785ε, (47)

it follows:

V ≤ −γ ‖z‖2 −(

12δ − 1

4ksρ2(‖z‖)

)‖z‖2 +

0.2785α(|ϑ1| + 1

α |ϑ2|)ε,

(48)

where γ = 12δ is a positive constant.

Bearing the inequality (41) in mind, the following bounds can be developed:

‖z‖2 ≥

{ [λ−1

1 (V − σ)]2

if V ≥ σ0 if V < σ

≥

{ (λ−1

1 (V2 ))2

−(λ−1

1 (σ))2

if V ≥ σ0 if V < σ

(49)

where the following inequality is utilized

(λ−1

1 (θ1 + θ2))2

≤(λ−1

1 (2θ1))2

+(λ−1

1 (2θ2))2

for any θ1 ≥ 0, θ2 ≥ 0.(50)

The function λ−11 (∙) denotes the inverse of λ1(∙), and due to the lemma 4.2 in Khail. (2003), λ−1

1 (∙)also belongs to class K∞.

Substituting (49) into (48) yields

V ≤ −γ(λ−1

1 (V2 ))2

+ γ(λ−1

1 (σ))2

−(12δ − 1

4ksρ2(‖z‖)

)‖z‖2 +

0.2785α(|ϑ1| + 1

α |ϑ2|)ε.

(51)

The time derivative of V (t) can be further upper bounded by

V ≤ −ω(V ) + c1(ε) + c2(ε)for ‖z‖ ≤ ρ−1(

√2ksδ),

(52)

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where ω(V ) = γ(λ−1

1 (V2 ))2

denotes a class K∞ function, and c1(ε) = γ(λ−1

1 (ε ‖Nd‖L∞ln 2)

)2,

c2(ε) = 0.2785α(|ϑ1| + 1

α |ϑ2|)ε are positive constants. Defining ε0 = ω−1(c1(ε)+ c2(ε)), the upper

inequality for V (t)can be re-written as follows:

d(V −ε0)dt ≤ −ω(V ) + ω(ε0)

for ‖z‖ ≤ ρ−1(√

2ksδ).(53)

Next, by invoking the comparison theorem in (Khail., 2003), we can show that V ≤ V if V satisfiesthat

d(V −ε0)dt = −ω(V ) + ω(ε0)

V (0) = V (0)(54)

Defining the positive definite function W = 12(V − ε0)2 with respect with (V − ε0) and taking the

time derivative of W , we obtain

W = (V − ε0)(−ω(V ) + ω(ε0)). (55)

Notice that the right hand of the above equation is negative definite with respect with ( V − ε0),thus it can be obtained that

V ≤ V ≤ ϕ(∣∣V (0) − ε0

∣∣ , t − t0) + ε0

for ‖z‖ ≤ ρ−1(√

2ksδ).(56)

ϕ(∙) is a class-KL function. It follows that the error vector ‖z‖ is bounded by a class-KL functionplus a small positive constant as follows

‖z‖ ≤√

1λ1

ϕ(|V (0) − ε0| , t − t0) + 1λ1

ε0

for ‖z‖ ≤ ρ−1(√

2ksδ).(57)

From Definition 1, we can transform the norm ‖∙‖δ into the following piecewise mapping

‖z‖√ 1λ1

ε0=

‖z‖ −√

1λ1

ε0 if ‖z‖ ≥√

1λ1

ε0

0 if ‖z‖ <√

1λ1

ε0

. (58)

After combining (57) and (58), it is concluded that

‖z‖√ 1λ1

ε0≤√

1λ1

ϕ(|V (0) − ε0| , t − t0) + 1λ1

ε0 −√

1λ1

ε0

for ‖z‖ ≤ ρ−1(√

2ksδ)(59)

where the UA and US conditions for z(t) are concluded, and for any ultimate boundedness√

1λ1

ε0

we can adjust ε to meet it.Now, we are going to show the semiglobal property. In (57), two inequalities exist. The second

one is the pre-condition of the first one, that is to mean the upperboundedness of the first inequalityis smaller than the second. Thus we have

ρ−1(√

2ksδ) ≥

√1λ1

ϕ(|V (0) − ε0| , 0) +1λ1

ε0, (60)

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which can be transformed as

ks ≥12δ

ρ2

(√1λ1

ϕ(|V (0) − ε0| , 0) +1λ1

ε0

)

. (61)

Note that the region of attraction can be made arbitrarily large to include any initial condition byincreasing the control gain ks (i.e., a semi-global stability result).�

Remark 3: Different from the Layer-boundary SMC, the proposed control design exhibits theability of learning the uncertain dynamics of the system (1). By expanding the equation (8) andmultiplying G−1(x) on both sides, the following equation can be obtained

G−1(x)r = G−1(x)x(m)d + G−1(x)

(∑m−2j=0 amje

(j+1)1 + αem

)−

G−1(x)f(x) − u − G−1(x)d.(62)

Then substituting the control input (19) into the above equation yields

G−1(x)r = G−1(x)x(m)d + G−1(x)

(∑m−2j=0 amje

(j+1)1 + αem

)−

G−1(x)f(x) − G−1(x)d − (ks + 1)em(t) + (ks + 1)em(0)−∫ t0 [(ks + 1)αem(τ) + βTanh(nem(τ)/ε)]dτ

(63)

Let M ∈ Rn be defined as

M = G−1(x)x(m)d + G−1(x)

(∑m−2j=0 amje

(j+1)1 + αem

)−

G−1(x)f(x) − G−1(x)d − (ks + 1)em(t) − G−1(x)r,(64)

the equation in (63) is able to be written as

∫ t

0[(ks + 1)αem(τ) + βTanh(nem(τ)/ε)]dτ − (ks + 1)em(0) = M. (65)

By separating M into the terms Md(t) ∈ Rn and M(t) ∈ Rn, where Md(t) = M(xd, xd, t) andM(t) = M(t) − Md(t). Note that all the bounded uncertain terms of the close-loop dynamics

are included in Md, and M(t) meets the upperbound∥∥∥M

∥∥∥ ≤ ρM (‖z‖) ‖z‖ (Xian, Dawson, De

Queiroz,& Chen , 2004). ρM (∙) is a global invertible, non-decreasing function. After re-arranging(65), we have

∥∥∥∫ t0 [(ks + 1)αem(τ) + βTanh(nem(τ)/ε)]dτ − (ks + 1)em(0) − Md

∥∥∥

≤ ρM (‖z‖) ‖z‖(66)

Since that the error vector z is USPAS, it is clear that the integration terms in u can practicallyasymptotically learn the uncertain dynamics Md.

7. Numerical simulation and real-time experiments

In this section, an numerical simulation performed on a planar two-link manipulator and a rea-timeexperiment performed on a quadrotor are employed to validate the effectiveness of the proposedcontrol design methodology.

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7.1 Numerical simulation

The proposed control algorithm in (19) is simulated for a two-link robot manipulator. The dynamicsof the manipulator is given by

[m1 + 2m2 cos(q2) m2 + m3 cos(q2)m2 + m3 cos(q2) m2

] [q1

q2

]

+[−m3 sin(q2)q2 −m3 sin(q2)(q1 + q2)m3 sin(q2)q1 0

] [q1

q2

]

+[

fd1 00 fd2

] [q1

q2

]

+

[τd1

τd2

]

=

[τ1

τ2

](67)

where m1 = 3.473 kg/m2, m2 = 0.193 kg/m2, m3 = 0.242 kg/m2, fd1 = 5.3 N/m.s, fd2 = 1.1 N/m.sde Queiroz, Dawson, & Agarwal (1999). And the external disturbances are set as τd1 = 4 sin(3t)N.m, τd2 = 2 sin(3t) N.m. The reference trajectories of the positions are selected as follows

qd(t) =

[40.11 sin(t)(1 − exp(−0.3t3))68.75 sin(t)(1 − exp(−0.3t3))

]

deg . (68)

The exponential term in (68) is included to ensure that qd(0) = qd(0) = qd(0) = 0.The control gains are adjusted as

α = 20 ks = 80β = 25 ε = 2 × 10−5 (69)

to obtain the results given in Figure 1. To check the ultimate boundedness of the tracking errorsclearly, we set the initial condition of the errors as 0. From Figure 1, we can see that the trackingerrors are regulated in a arbitrary small region while keeping the torque input signals smooth. Thetwo control torques are with some reasonable values.

To make a comparison, RISE controller (Xian, Dawson, De Queiroz,& Chen , 2004) and a layerboundary sliding model controller are also simulated for the same manipulator. Figure 2 depictsthe tracking performance of the RISE control structure. It can be seen that the tracking errorsare with almost the same amplitude as the ones in Figure 1 while the controller exhibiting a bigdeal of chattering behavior. Figure 3 shows the simulation results that sliding mode controlleris utilized. The discontinuity in the controller can be dealt by replacing sign(e) with continuoussaturation function sat(e/ε). From Figure 3, we can notice that even though the torque inputs arerather smooth, the tracking errors are much bigger than the ones in Figure 1. Figure 4 depictsthe learning ability of the proposed control design, in which we can see that the integration termslearn the system uncertainties with high accuracy.

7.2 Real-time experiment

A quadrotor helicopter is an aircraft with X-shaped frame and equipped with two pairs of counter-rotating rotors at each tail-end. We developed a attitude control system for a small scale quadrotor.(See Figure 5). The helicopter is fixed on a spherical joint. It gives the quadrotor free yaw motionand ±40◦ freedom of pitch and roll rotation. The thrust generated by the four rotors are denoted byf1, f2, f3, and f4. As stated in (Castillo, Dzul, & Lozano , 2004), the quadrotor attitude dynamicscan be modeled as follows

Jη = −C(η, η)η + τ. (70)

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In (70), η = [θ, φ, ψ]T represents the vector of the quadrotor’s Euler angles where θ(t), φ(t),and ψ(t) are the roll, pitch, and yaw angle respectively. The vector J = [J1, J2, J3]T denotes themoments of inertia of the corresponding axes directly dependent on η, the matrix C(η, η) is theCentrifugal-Coriolis matrix. The elements of the control torque vector τ = [τθ, τφ, τψ]T are definedas

τθ = l(−(f1 + f4) + (f2 + f3))τφ = l(−(f1 + f2) + (f3 + f4))τψ = l(−(f1 + f3) + (f2 + f4))

(71)

where l denotes the distance from each motor to the center of mass of the quadrotor. Let u1 =f1+f2+f3+f4 be a constant here to compensate the gravity effect. Thus the term fi for i = 1, 2, 3, 4can be easily solved. We apply the proposed smooth control design and the Layer boundary slidingmode control, with respective, for the attitude control of the quadrotor in three different testingcases, in each of them only one Euler angle is set to track a time-varying trajectory while therest two angles are stabilized to 0 degree. It is worth noting that in the experiment, to ease thecode-programming, we remove the bias of initial value −(ks + 1)em(0) in (19).

Case 1: The tracking performance for roll angle is tested under the proposed controller and theL-Boundary SMC while the pitch angle and the yaw angle are required to stay with 0 degree. Thepre-defined trajectory for roll angle is set as a sinusoidal like trajectory in the following form:

φd(t) = 15 sin(2π

15t) sin(

2π

150t) (72)

Case 2: The tracking performance for pitch angle is tested under the proposed controller andthe L-Boundary SMC while the roll angle and the yaw angle are required to stay with 0 degree.The pre-defined trajectory for pitch angle is set as the same as trajectory in (72).

Case 3: The tracking performance for yaw angle is tested under the proposed controller and theL-Boundary SMC while the roll angle and the pitch angle are required to stay with 0 degree. Thepre-defined trajectory for yaw angle is set as the same as trajectory in (72).

In every case, the experiment last 150 seconds, providing sufficient time to judge the performanceof the proposed controller and a comparison of Layer boundary sliding mode controller. The ex-periment results of the three testing cases are given in Figures 6-17 where the actual trajectoriesand desired trajectories for θ(t), φ(t), and ψ(t), the tracking errors and the control inputs aredepicted. The analysis data of each experiment which contains the maximum steady state (definedas 20 seconds to 150 seconds) error and the root mean square error is displayed in Table 1.

The experiment shows that both the proposed controller and the Layer boundary sliding modecontrol are able to achieve stable tracking result. From Figures 6-17, the two control design sharethe similar convergent speed that the tracking errors get into the specified small region around 0within 20s. The data in Table 1 indicates that the proposed controller result in better trackingperformance comparing to the Layer boundary sliding mode control. Comparing to L-BoundarySMC, the Max SS Error and the RMS Error obtained in the proposed control structure are almosttwo-thirds smaller in Case 1 and 50% smaller in Case 2&3. Videos of the experiments for theproposed control design are available at website as follows:

Case1: http://youtu.be/-XGQGsa2K9kCase2: http://youtu.be/XC IZ3178dMCase3: http://youtu.be/dUSk-pkdkq4

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8. CONCLUSIONS

A nonlinear smooth robust control is developed for a class of nonlinear general uncertain systemsthat achieve the USPAS tracking of the desired time-varying trajectories despite the uncertaindynamics and the unknown non-varnishing disturbances. The main contribution of this paper isto develop a integral of the nonlinear sigmoid function structure to approximately conceal thesystem uncertainties and to prove the USPAS stability of the controller by constructing a novelLyapunov function. From the simulation and experiment results, the proposed controller in thispaper is arbitrary smooth, while maintaining the ultimately tracking bounded errors with specifiedboundaries.

References

Castillo, P., Dzul, A., & Lozano, R. (2004). Real-time stabilization and tracking of a four-rotor mini rotor-craft. IEEE Transactions on Control Systems Technology, 12 , 510–516.

Chen, W. H. (2004). Disturbance Observer based Control for Nonlinear Ssytems. IEEE/ASME Transactionson Mechatronics, 9 (4), 706–710.

Chaillet, A., & Loria, A. (2008). Uniform Semiglobal Practical Asymptotic Stability for Non-autonomousCascaded Systems and Applications. Automatica, 44 , 337–347.

Dasdemir, J., & Zergeroglu, E. (2015). A New Continuous High-Gain Controller Scheme for a Class ofUncertain Nonlinear Systems. International Journal of Robust and Nonlinear Control, 25 , 125–141.

de Queiroz, M. S., Dawson, D. M., & Agarwal, M. (1999). Adaptive Control of Robot Manipulators withController/Update Law Modularity. Automatica, 35 , 1379–1390.

Efe, M. O. (2008). Fractional Fuzzy Adaptive Sliding Model Control of a 2-DOF Direct-Drive Robot Arm.IEEE Transactions on Systems, Man, and Cybernetics-Part B:Cybernetics , 38 (6), 1561–1570.

Feemster, M. G., Fang, Y. C., & Dawson, D. M. (2006). Disturbance Rejection for a Magnetic LevitationSystem. IEEE/ASME Transactions on Mechatronics, 11 (6), 709–717.

Girin, A., & Plestan, F. (2009). A New Experimental Setup for A High Performancedouble ElectropneumaticActuators System. Proc. of the 2009 American Control Conference,Saint-Louis, Missouri, USA.

Hong, Y., Xu, Y. & Huang, J. (2002). Finite-time Control for Robot Manipulators. Systems and ControlLetter, 46 (4), 243–253.

Islam, S., & Liu, X. P. (2011). Robust Sliding Mode Control for Robot Manipulators. IEEE Transactionson Industrial Electronics, 58 (6), 2444–2453.

Khail, H. K. (2003). Nonlinear Systems(3rd ed.). Prentice-Hall. Inc New JerseyLaghrouche, S., Plestan, F., & Glumineau, A. (2007). Higher Order Sliding Mode Control Based on Integral

Sliding Surface. Automatica, 43 , 531–537.Levant, A. (2007). Principles of 2-sliding Mode Design. Automatica, 43 , 576–586.Ortega, R., Astolfi, A. & Barabanov, N. (2002). Nonlinear PI Control of Uncertain Systems: an Alternative

to Parameter Adaptation. Systems and Control Letter, 47 , 259–278.Patre, P. M., MacKunis, W., Makkar, C., & Dixon, W. E. (2008). Asymptotic Tracking for Systems with

Structured and Unstructured Uncertainties. IEEE Transactions on Control System Technology, 16 (2),373–379.

Patre, P. M., MacKunis, W., Kaiser, K., & Dixon, W. E. (2008). Asymptotic Tracking for Uncertain DynamicSystems Via a Multilayer Neural Network Feedforward and RISE Feedback Control Structure. IEEETransactions on Automatic Control, 53 (9), 2180–2185.

Patre, P. M., MacKunis, W., Dupree, K., & Dixon, W. E. (2011). Modular Adaptive Control of UncertainEuler–Lagrange Systems With Additive Disturbances. IEEE Transactions on Automatic Control, 56 (1),155–160.

Qu, Z. H. (1998). Robust Control of Nonlinear Uncertain Systems. John Wiley and Sons, Interscience Divi-sion, New York, NY.

Serrani, A., Isidori, A.,& Marconi, L. (2001). Semiglobal Nonlinear Output Regulation With AdaptiveInternal ModeL. IEEE Transactions on Automatic Control, 46 (8), 1178–1194.

Su, C. Y., Leung, T. P., & Zhou, Q. J. (1990). A Novel Variable Structure Control Scheme for Robot

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Trajectory Control. Proc. of the 1990 IFAC Triennial World Congress 1990 (pp. 117–120).Wilcox, Z., MacKunis, W., Bhat, S., Lind, R., & Dixon, W. E. (2010). Lyapunov-Based Exponential Tracking

Control of a Hypersonic Aircraft with Aerothermoelastic Effects. AIAA Journal of Guidance, Control,and Dynamics, 33 (4), 1213–1224.

Xian, B., Dawson, D. M., de Queiroz, M. S., & Chen, J. (2004). A Continuous Asymptotic Tracking ControlStrategy for Uncertain Nonlinear Systems. IEEE Transactions on Automatic Control, 49 (7), 1206–1211.

Xu, H. J., Mirmirani, M. D., & Ioannou, P. A. (2004). Adaptive Sliding Mode Control Design for A Hyper-sonic Flight Vehicle. AIAA Journal of Guidance, Control, and Dynamics, 27 (5), 829–838.

Zong, Q., Zhao, Z. S., & Jing, Z. (2010). Higher Order Sliding Mode Control with Self-tuning Law based onintegral sliding mode. IET Control Theory and Applications, 4 (7), 1282–1289.

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0 10 20 30 40 50-5

0

5x 10

-5

Time (sec)

e q1(t)

(deg

)0 10 20 30 40 50

-4

-2

0

2x 10

-5

Time (sec)

e q2(t)

(deg

)

0 10 20 30 40 50-10

-5

0

5

10

Time (sec)

τ 1(t)(

N.m

)

control input

0 10 20 30 40 50-2

-1

0

1

2

Time (sec)

τ 2(t)(

N.m

)

Figure 1. The tracking errors and control inputs of the proposed controller

0 10 20 30 40 50-2

-1

0

1

2x 10

-4

Time (sec)

e q1(t)

(deg

)

0 10 20 30 40 50-4

-2

0

2

4x 10

-5

Time (sec)

e q2(t)

(deg

)

0 10 20 30 40 50-10

-5

0

5

10

Time (sec)

τ 1(t)(

N.m

)

control input

0 10 20 30 40 50-2

-1

0

1

2

Time (sec)

τ 2(t)(

N.m

)

23.8 24 24.24.55

4.64.65

4.7

control input

Figure 2. The tracking errors and control inputs of the RISE controller

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0 20 40-0.2

-0.1

0

0.1

0.2

Time (sec)

e q1(t)

(deg

)

0 20 40-0.04

-0.02

0

0.02

0.04

Time (sec)

e q2(t)

(deg

)0 20 40

-10

-5

0

5

10

Time (sec)

τ 1(t)(

N.m

)

control input

0 20 40-2

-1

0

1

2

Time (sec)τ 2(t

)(N

.m)

Figure 3. The tracking errors and control inputs of the layer boundary sliding mode

control

0 5 10 15 20 25-10

-5

0

5

10

15

20

Uncertainty1Uncertainty2Learning term1Learning term2

Figure 4. The learning ability for the system uncertainties

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Figure 5. The testbed of the quadrotor helicopter

0 50 100 150-20

-10

0

10

20

Time (sec)

φ &

φd (

deg)

0 50 100 150-1

0

1

2

Time (sec)

θ &

θd (

deg)

0 50 100 150-0.5

0

0.5

Time (sec)

ψ &

ψd (

deg)

Figure 6. Case1: Actual trajectories (blue line) and desired trajectories (red line) for

roll, pitch and yaw angles under the proposed controller.

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0 50 100 150-2

-1

0

1

Time (sec)e φ (

deg)

0 50 100 150-1

0

1

2

Time (sec)

e θ (de

g)

0 50 100 150-0.5

0

0.5

Time (sec)

e ψ (

deg)

Figure 7. Case1: Tracking error signals of roll, pitch and yaw angles under the pro-

posed controller.

0 50 100 150-2

-1

0

1

Time (sec)

τ φ (N

)

0 50 100 150-2

-1

0

1

2

Time (sec)

τ θ (N

)

0 50 100 150-5

0

5

Time (sec)

τ ψ (

N)

Figure 8. Case1: Control torques τφ, τθ and τψ under the proposed controller.

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0 50 100 150-20

-10

0

10

20

Time (sec)φ

& φ

d (de

g)

0 50 100 150-2

0

2

4

Time (sec)

θ &

θd (

deg)

0 50 100 150-1.5

-1

-0.5

0

0.5

Time (sec)

ψ &

ψd (

deg)

Figure 9. Case1: Actual trajectories (blue line) and desired trajectories (red line) for

roll, pitch and yaw angles under the Layer boundary sliding mode controller.

0 50 100 150-1

0

1

2

3

Time (sec)

φ &

φd (

deg)

0 50 100 150-20

-10

0

10

20

Time (sec)

θ &

θd (

deg)

0 50 100 150-2

-1

0

1

Time (sec)

ψ &

ψd (

deg)

Figure 10. Case2: Actual trajectories (blue line) and desired trajectories (red line)

for roll, pitch and yaw angles under the proposed controller.

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0 50 100 150-1

0

1

2

3

Time (sec)

e φ (de

g)

0 50 100 150-1

0

1

2

Time (sec)

e θ (de

g)

0 50 100 150-2

-1

0

1

Time (sec)

e ψ (

deg)

Figure 11. Case2: Tracking error signals of roll, pitch and yaw angles under the

proposed controller.

0 50 100 150-2

-1

0

1

Time (sec)

τ φ (N

)

0 50 100 150-2

-1

0

1

2

Time (sec)

τ θ (N

)

0 50 100 150-4

-2

0

2

4

Time (sec)

τ ψ (

N)

Figure 12. Case2: Control torques τφ, τθ and τψ under the proposed controller.

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December 4, 2015 International Journal of Control TCON˙A˙1128561

0 50 100 150-2

-1

0

1

2

Time (sec)φ

& φ

d (de

g)

0 50 100 150-20

-10

0

10

20

Time (sec)

θ &

θd (

deg)

0 50 100 150-1

-0.5

0

0.5

1

Time (sec)

ψ &

ψd (

deg)

Figure 13. Case2: Actual trajectories (blue line) and desired trajectories (red line)

for roll, pitch and yaw angles under the Layer boundary sliding mode controller.

0 50 100 150-1

-0.5

0

0.5

1

Time (sec)

φ &

φd (

deg)

0 50 100 150-1

0

1

2

3

Time (sec)

θ &

θd (

deg)

0 50 100 150-20

-10

0

10

20

Time (sec)

ψ &

ψd (

deg)

Figure 14. Case3: Actual trajectories (blue line) and desired trajectories (red line)

for roll, pitch and yaw angles under the proposed controller.

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December 4, 2015 International Journal of Control TCON˙A˙1128561

0 50 100 150-1

-0.5

0

0.5

1

Time (sec)

e φ (de

g)

0 50 100 150-1

0

1

2

3

Time (sec)

e θ (de

g)

0 50 100 150-1

-0.5

0

0.5

1

Time (sec)

e ψ (

deg)

Figure 15. Case3: Tracking error signals of roll, pitch and yaw angles under the

proposed controller.

0 50 100 150-1

-0.5

0

0.5

1

Time (sec)

τ φ (N

)

0 50 100 150-2

-1

0

1

2

Time (sec)

τ θ (N

)

0 50 100 150-2

-1

0

1

2

Time (sec)

τ ψ (

N)

Figure 16. Case3: Control torques τφ, τθ and τψ under the proposed controller.

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December 4, 2015 International Journal of Control TCON˙A˙1128561

0 50 100 150-1

0

1

2

Time (sec)

φ &

φd (

deg)

0 50 100 150-2

0

2

4

Time (sec)

θ &

θd (

deg)

0 50 100 150-20

-10

0

10

20

Time (sec)

ψ &

ψd (

deg)

Figure 17. Case3: Actual trajectories (blue line) and desired trajectories (red line)

for roll, pitch and yaw angles under the Layer boundary sliding mode controller.

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December 4, 2015 International Journal of Control TCON˙A˙1128561

Table 1. Analysis data of the tracking errors for the proposed controller

and the layer boundary sliding mode controller.

Controller The proposed Layer-boundary SMC

Max SS Error(degree)-Case1 0.9838 2.8510RMS Error(degree)-Case1 0.3338 1.1761Max SS Error(degree)-Case2 0.7765 1.7046RMS Error(degree)-Case2 0.3158 0.5147Max SS Error(degree)-Case3 0.7988 1.7382RMS Error(degree)-Case3 0.4262 0.9903

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