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Accepted Manuscript
Adaptive Sliding Mode Control of a Class of Nonlinear Systems with
Artificial Delay
Spandan Roy, Indra Narayan Kar
PII:
DOI:
Reference:
S0016-0032(17)30528-8
10.1016/j.jfranklin.2017.10.010
FI 3180
To appear in:
Journal of the Franklin Institute
Received date:
Revised date:
Accepted date:
9 February 2016
19 January 2017
15 October 2017
Please cite this article as: Spandan Roy, Indra Narayan Kar, Adaptive Sliding Mode Control of
a Class of Nonlinear Systems with Artificial Delay, Journal of the Franklin Institute (2017), doi:
10.1016/j.jfranklin.2017.10.010
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ACCEPTED MANUSCRIPT
Adaptive Sliding Mode Control of a Class of Nonlinear
Systems with Articial Delay
Spandan Roy ∗, Indra Narayan Kar
a,
a
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a Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, India
Abstract
In this paper, an adaptive-robust control (ARC) strategy, christened as Adap-
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tive Time-delayed Sliding Mode Control (ATSMC) is presented for trajectory
tracking control of a class of uncertain Euler-Lagrange systems. The proposed
control framework brings together the best features of the switching control
logic and time-delayed logic. ATSMC uses articial time delay to approximate
the unknown dynamics through time-delayed logic, and the switching logic provides robustness against the approximation error. The adaptation law for the
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switching gain of the conventional ARC methodologies suer from over- and
under-estimation problem. The novel adaptive law of ATSMC alleviates the
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over- and under-estimation problem of switching gain. Moreover, a new design methodology and stability criterion for time-delayed control is proposed
which provides an upper bound on the allowable delay time. Experimental re-
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sults of the proposed methodology using a nonholonomic wheeled mobile robot
(WMR) is presented and improved tracking accuracy of the proposed control
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law is noted compared to time-delayed control and conventional adaptive sliding
mode control.
Keywords: Adaptive sliding mode control, time-delayed control, Razumikhin
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theorem, wheeled mobile robot
∗ Corresponding
author
Email addresses:
Narayan Kar)
sroy002@gmail.com (Spandan Roy), ink@ee.iitd.ac.in (Indra
A
Preprint submitted to Journal of L TEX Templates
January 19, 2017
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1. Introduction
Adaptive control and Robust control are the two popular control strategies to deal with uncertain nonlinear systems. In general, adaptive control
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uses predened parameter adaptation laws which adjusts the parameters of the
controller online according to the pertaining uncertainties [1]. However, online computation of the unknown system parameters and controller gains for
complex systems is intensive [2]. Whereas, robust control reduces computa-
tion complexity for complex systems compared to adaptive control as exclusive
online estimation of uncertain parameters is not required [3]. Robust control
tackles the uncertainties of the system within a predened uncertainty bound.
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However, dening a prior uncertainty bound is not always possible in practice.
Further, to increase the operating region of the controller, often higher uncertainty bounds are assumed. This in turn leads to problems like higher controller
gain and consequent possibility of chattering for the switching law based robust
controller like Sliding Mode Control (SMC). This in eect reduces controller
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accuracy [4]. Higher order sliding mode [5] can alleviate the chattering problem
but prerequisite of uncertainty bound still exists.
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Time-Delayed Control (TDC) is utilized in [6], [7], [15], [31], [37] to provide
robustness against uncertainties. In this process, all the uncertain terms are
20
represented by a single function which is then approximated using control in-
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put and state information of the immediate past time instant. The advantage of
this robust control approach is that it is easy to implement and reduces the bur-
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den of tedious modelling of complex system to a great extent. In spite of this,
the unattended approximation error, commonly termed as time-delayed error
25
(TDE) causes detrimental eect to the performance of the closed loop system
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and its stability. In this front, a few works have been carried out to tackle TDE
which include internal model [8], gradient estimator [9], ideal velocity feedback
[11], nonlinear damping [12] and sliding mode based approach [13]-[14]. The
stability of the closed loop system, as proposed in [8]-[11], [31], [37], depends
30
on the boundedness of TDE as shown in [6]. An auto-tuning algorithm is pro2
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posed in [10] so that the nominal mass/inertia matrix can follow the perturbed
mass/inertia matrix to maintain the system stability as outlined in [6]. Stability of the system in [13] is established in frequency domain, which makes the
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approach dicult to analyse the stability of complex nonlinear systems. More-
over, the controllers designed in [12] and [14] require nominal modelling and
upper bound of the TDE respectively which is not always possible in practical
circumstances. In the eld of TDC based controllers, to the best knowledge
of the authors, controller design issues such as selection of controller gains and
sampling interval to achieve ecient performance is still an open problem for
Euler-Lagrange (EL) systems. In contrast to TDC, works reported in [16]-[18]
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use low pass lter to approximate the unknown uncertainties and disturbances.
However, frequency range of system dynamics and external disturbances are
required to determine the time constant of the lter. Furthermore, the order of
the low pass lter needs to be adjusted according to the order of disturbance to
45
maintain stability.
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Considering the constraints of adaptive and robust control, recently global
research is reoriented towards adaptive-robust control (ARC). The series of pub-
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lications [2], [19]-[25] estimate the uncertain terms online based on projection
function which requires predened bound of individual uncertain system pa50
rameters. The works reported in [26]-[30] attempt to estimate the maximum
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uncertainty bound but the integral adaptive law makes the controller susceptible to very high switching gain and consequent chattering [31]. Recently,
researchers have applied ARC in quantization process for industrial networked
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control systems [32]. The adaptive sliding mode control (ASMC), as presented
in [33]-[34], evaluates the switching gain depending on a predened threshold
value. However, until the threshold value is achieved, the switching gain will be
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increasing (resp. decreasing) even if tracking error decreases (resp. increases)
and this creates overestimation (resp. underestimation) problem of switching
gain [35].
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1.2. Organization
The article is organized as follows: a new stability analysis of TDC along
with its design issues are rst discussed in Section 2. This is followed by the
proposed ATSMC methodology and its detailed analysis. Section 3 presents the
experimental results of the proposed controller and its comparison with TDC
([6], [31], [37]) and ASMC ([33]-[34]). Section 4 concludes the paper.
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1.1. Contribution
The paper has two major contributions. First, a new stability analysis for
TDC, based on the Razumikhin-type approach [38], is provided. The proposed
stability analysis is carried out in continuous time domain unlike the ones in [6],
[31], [37] which approximates the closed loop system in discrete time. Proper
choice of sampling interval and controller gain is key to the performance of TDC
and this design issue is still to be addressed in the literature for EL systems.
Through the proposed stability analysis, an upper bound of the sampling interval and its relation with the controller gain is established, which is necessary
to stabilize the system. Further, it is shown that the discrete nature of the
sampling interval does not violate the stability of the overall system which is in
continuous time domain.
As the second contribution of this article, Adaptive Time-delayed Sliding
Mode Control (ATSMC) has been devised for EL systems. ATSMC approximates the overall (or lumped) uncertainties by past data using time-delayed logic
and provides robustness against the approximation error by switching control.
The proposed adaptive law of ATSMC alleviates the over- and under-estimation
problem of switching gain without any prior knowledge of the bound of uncertainties. For a proof of concept, experimental results of the proposed control
methodology in comparison to TDC ([6], [31], [37]) and ASMC ([33]-[34]) are
provided using the "PIONEER-3" mobile robot.
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1.3. Notations
The following notations are used in this paper: any variable (•)(t) delayed
by an amount h as (•)(t − h) would be denoted as (•)h unless stated otherwise;
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λmin (•) and || • || represent minimum eigen value and Euclidean norm of the
argument respectively; P > 0 denotes a positive denite matrix; I denotes
2. Controller Design
2.1. Time-delayed Control (TDC): Revisited
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identity matrix of appropriate dimension.
In general, an Euler-Lagrange system with second order nonlinear dynamics
can be written as
(1)
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M (q(t))q̈(t) + H(q(t), q̇(t)) = τ (t),
where q(t) ∈ Rn denotes the system state, τ (t) ∈ Rn denotes the control input,
M (q) ∈ Rn×n represents the mass/inertia matrix and H(q, q̇) ∈ Rn denotes
combination of other system dynamics terms based on system properties such
as Coriolis, friction force, slip, damping etc. In practice, it can be assumed that
is dened as [6]
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unmodelled dynamics and disturbances are subsumed by H . The control input
τ = M̂ u + Ĥ,
(2)
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where u is the auxiliary control input; M̂ and Ĥ are the nominal values of M
and H respectively. To reduce the modelling eort of the complex systems, Ĥ
is approximated from the input and feedback data of the previous instant using
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the time-delayed logic ([6], [31], [37]) and the system denition (1) as
Ĥ(q, q̇) ∼
= H(qh , q̇h ) = τh − M̂h q̈h ,
(3)
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where h is a xed small delay time and τh = τ (t − h), M̂h = M̂ (q(t − h)), qh =
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q(t − h). Putting (2) and (3) into (1), the system dynamics is converted as
M̂ (q)q̈ + H̄(q, q̇, q̈, q̈h ) = τh ,
(4)
where H̄ = (M − M̂ )q̈ + M̂h q̈h − M̂ u + H . Let q d (t) be the desired trajectory to
be tracked and e1 (t) = q(t) − q d (t) is the tracking error. The auxiliary control
input u is dened in the following way:
u(t) = q̈ d (t) − K2 ė1 (t) − K1 e1 (t),
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(5)
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where K1 and K2 are two positive denite matrices with appropriate dimensions.
Due to the presence of τh in (4), substituting the time-delayed versions of (5)
M̂ q̈ + H̄ = M̂h uh + Ĥh
⇒ q̈ = M̂ −1 M̂h uh + M̂ −1 (Ĥh − H̄)
= uh + (M̂ −1 M̂h − I)uh + M̂ −1 (Ĥh − H̄)
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and (2) into (4) yields
= (q̈hd − K2 ė1h − K1 e1h ) + (M̂ −1 M̂h − I)uh + M̂ −1 (Ĥh − H̄),
(6)
where e1h = e1 (t − h), q̈hd = q̈ d (t − h). Adding and subtracting q̈ d to the right
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hand side of (6), the following error dynamics is obtained:
ë1 = −K2 ė1h − K1 e1h + σ1 ,
(7)
where σ1 = (M̂ −1 M̂h − I)uh + M̂ −1 (Ĥh − H̄) + q̈hd − q̈ d and it is treated as the
overall uncertainty or time-delayed error (TDE). Further, (7) can be formulated
in state space as
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ė = A1 e + B1 eh + Bσ1 ,
(8)
 




 
e1
0 I
0
0
0
 , B1 = 
, B =  . As e(t − h) =
where e =  , A1 = 
ė1
0 0
−K1 −K2
I
R0
e(t) − −h ė(t + θ)dθ, the error dynamics (8) is rewritten as
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ė(t) = Ae(t) − B1
Z
0
ė(t + θ)dθ + Bσ1 ,
−h
(9)
where A = A1 + B1 and the derivative inside the integral is with respect to θ.
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The time-delayed error (TDE) σ1 remains bounded if M̂ is selected
such that [6], [31]:
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Lemma 1.
||M −1 M̂ − I|| < 1.
(10)
In practice, perturbation in the mass matrix M is generally caused due to
the payload variation. It is always possible to have an prior idea about the
payload variation. So, it is always plausible to nd a M̂ so that (10) is satised
[6], [31].
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It is to be noted that the original system (1) does not possess any
time delay inherently. However, an articial delay h gets invoked into the system
while using the time-delayed logic (3).
Remark 1.
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2.1.1. Stability Analysis
The design procedure of TDC, as provided in [6], [31], [37], does not provide
105
any selection criterion of delay h and its relation with controller gains K1 and
K2 . However, the choice of h, K1 and K2 are important for the performance
of TDC. This design issue is an open problem for EL systems in the domain
of TDC based controllers.
In this paper, a new stability criterion, based
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on the Razumikhin-type theorem [38], is presented through Theorem 1 which
addresses the aforementioned issues. Before stating the results, the following
standard assumptions are made:
The controller gains K1 and K2 are selected in a way such
that A is a Hurwitz matrix, which is always possible from the structure of A.
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Assumption 1.
Let V (e) be a Lyapunov function candidate. Then following
the Razumikhin-type theorem, the following inequality holds [38]:
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Assumption 2.
V (e(ν)) < rV (e(t)),
t − 2h ≤ ν ≤ t,
(11)
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where r > 1 is a constant.
Assumption 2 from the Razumikhin theorem [38] is a well established tool for
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stability analysis for time-delayed systems using Lyapunov stability convention.
With V (e), there are two possible situations i.e. (i) V (e(ν)) ≥ rV (e(t)) and (ii)
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V (e(ν)) < rV (e(t)). These two conditions are depicted in Fig. 1.
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First let us consider condition (i) i.e., V (e(t)) ≤ (1/r)V (e(ν)) ≤ V (e(ν)) (as
r > 1). This condition implies that the error trajectories decrease during the
time interval t − 2h ≤ ν ≤ t where h > 0. This would mean that the error
trajectories are not growing and moving from the ball V (e(t)) to V (e(ν)). This
implies that the time delayed system is already stable. Hence, the condition
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V (e(t ))
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V (e(t ))
V (e( ))
V (e( ))
V (e( ))  rV (e(t ))
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V (e( ))  rV (e(t ))
Figure 1: Razumikhin condition.
(i) V (e(ν)) ≥ rV (e(t)) (i.e. the reverse condition of (11)) does not require any
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analysis.
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Now consider the situation (ii) V (e(ν)) < rV (e(t)) (i.e. Assumption 2). This
condition implies that the error trajectories starting within the ball V (e(ν))
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approach towards the ball V (e(t)). Now, if one can prove that the ball V (e(t))
does not grow (i.e. V̇ (e(t)) is not increasing which implies V (e(t)) is bounded)
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then the error trajectories will be contained within the ball V (e(t)) and the
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system would remain stable. The detailed physical interpretation of Razumikhin
theorem and its relation with conventional Lyapunov stability is provided in [38].
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Further, it is to be noted that Assumption 2 is only used for analysis and it does
not put any restriction on system itself.
The system (4) employing the control input (2), having auxiliary
input (5) is UUB. The delay time is upper bounded as
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Theorem 1.
h<
λmin (Q)
:= h̄max ,
||E||
(12)
∀h,
(13)
if controller gains K1 and K2 are selected such that
λmin (Q) > h||E||
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where E = βP B1 (A1 P −1 AT1 + B1 P −1 B1T + P −1 )B1T P + 2(r/β)P , β > 0 is a
scalar and P > 0 is the solution of the Lyapunov equation AT P + P A = −Q
for some Q > 0.
Proof. Let V (e) be a Lyapunov function dened as:
1 T
e P e.
2
V (e) =
Using (9), the time derivative of V (e) yields
1
V̇ = − eT Qe − eT P B1
2
Z
0
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ė(t + θ)dθ + sT1 σ1 ,
−h
(14)
(15)
where s1 = B T P e. Expanding the second term of (15) and using (8) we have
Z
0
−h
Z
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−eT P B1
ė(t + θ)dθ = −
0
eT P B1 [A1 e(t + θ)+
−h
B1 e(t − h + θ) + Bσ1 (t + θ)]dθ.
(16)
Applying (11) to (14) the following relation is achieved:
(17)
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eT (ν)P e(ν) < reT (t)P e(t).
For any two non-zero vectors z1 and z2 , there exists a constant β > 0 and matrix
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D > 0 such that the following relation holds:
±2z1T z2 ≤ βz1T D−1 z1 + (1/β)z2T Dz2 .
(18)
Applying (18) and (17) to (16) and taking D = P the following inequalities are
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obtained:
−2
Z
0
−h
eT P B1 A1 [e(t + θ)] dθ ≤
1 T
e (t + θ)P e(t + θ)]dθ
β
−h
≤ heT βP B1 A1 P −1 AT1 B1T P + (r/β)P e.
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0
Z
−2
Z
0
Z
[βeT P B1 A1 P −1 AT1 B1T P e +
(19)
0
−h
eT P B1 B1 e(t − h + θ)dθ ≤
1 T
e (t − h + θ)P e(t − h + θ)]dθ
β
−h
≤ heT βP B1 B1 P −1 B1T B1T P + (r/β)P e.
[βeT P B1 B1 P −1 B1T B1T P e +
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(20)
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−2
Z
0
−h
eT P B1 [Bσ1 (t + θ)] dθ ≤
Z
0
[βeT P B1 P −1
−h
(21)
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× B1T P e + (1/β)(Bσ1 (t + θ))T P Bσ1 (t + θ)]dθ
Z 0
T
−1 T
(1/β)(Bσ1 (t + θ))T P Bσ1 (t + θ)dθ.
≤ he βP B1 P B1 P e +
−h
Further, assuming the system to be locally Lipschitz within the delay, the following inequality always holds:
Z 0
1 T
(Bσ1 (t + θ)) P Bσ1 (t + θ) dθ
≤ Γ1 .
2β −h
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Substituting (19)-(22) into (15) we have
1
V̇ (e) ≤ − eT [Q − hE] e + Γ1 + ||B T P ||||σ1 ||||e||.
2
(22)
(23)
It is to be noticed from (23) that for stability, the condition [Q − hE] > 0 ⇒
λmin (Q) > h||E|| is required to be satised. So, if Q, K1 and K2 are selected in
a way such that λmin (Q) > h||E|| ∀h, then the maximum allowable delay can
λmin (Q)
:= h̄max .
||E||
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be found from (23) as
h<
(24)
Let Ψ = Q − hE > 0 and the uncertainties are bounded. So, V̇ (e) < 0 would be
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established if λmin (Ψ)||e||2 > 2Γ1 + 2||B T P ||||σ1 ||||e||. Thus (4) would be UUB
with the error bound
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||e|| = µ0 +
||B T P ||||σ1 ||
λmin (Ψ)
2Γ1
+ µ20 := $0 ,
λmin (Ψ)
(25)
. Let Ξ denote the smallest level surface of V containing
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where µ0 =
s
the ball B$0 with radius $0 centred at e = 0. For initial time t0 , if e(t0 ) ∈ Ξ
then the solution remains in Ξ. If e(t0 ) ∈/ Ξ then V decreases as long as e(t) ∈/ Ξ.
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The time required to reach $0 is zero when e(t0 ) ∈ Ξ, otherwise, while e(t0 ) ∈/ Ξ
the nite time tr to reach $0 , for some scalar c0 > 0, is given by [36]
tr − t0 ≤ (V (k e(t0 ) k) − V ($0 ))/c0
where V̇ (t) ≤ −c0 .
It is to be noted that Γ1 is only used for the purpose of analysis and not used
for control law design.
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It can be noted from (25) that smaller h would increase the value
λmin (Ψ) and reduce the error bound. Thus, tracking accuracy can be improved
by selecting h as small as possible. However in practice, information of past
instances are available only at sampling intervals. Thus, h cannot be selected
smaller than sampling interval. Further, (12) provides an upper bound to the
choice of delay (i.e. sampling interval) h for given K1 , K2 , Q, r and β to
maintain system stability. This design issue was previously unaddressed in the
literature for TDC based controllers of EL systems. However, depending on applications, choice of sampling interval is governed by the corresponding hardware
response time, computation time etc.
Remark 2.
As mentioned earlier, h is selected as the sampling interval in
practice, which is discrete in nature. The stability analysis in Theorem 1 is
independent of ḣ (i.e. independent of variation of delay). Hence, the system
remain stable ∀h < h̄max . Therefore, the discrete time nature of h does not
violate stability of the overall closed loop system.
Remark 3.
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2.2. Adaptive Time-delayed Sliding Mode Controller
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It can be observed from (7) and (25) that due to the absence of any inherent robustness term, the tracking performance of TDC gets degraded in the
face of TDE. The works reported in [8]-[9] can negotiate only slowly varying
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or constant TDE. Due to the eect of sensor noise and other unmodelled dynamics this prerequisite may not hold in real-life environment. The controller
reported in [10] can only provide robustness against the perturbation pertain-
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ing to the mass matrix, while other unmodelled disturbances in TDE degrades
its performance. On the other hand, the controllers designed in [2], [11]-[15],
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[19]-[25] require predened bound of uncertainties which is not always possible
165
in practical circumstances. Further, the adaptive laws of [33]-[34] suer from
overestimation-underestimation problem.
In this endeavour, a novel adaptive-robust control law, Adaptive Time-
delayed Sliding Mode Controller (ATSMC) is proposed. The proposed ATSMC
approximates the uncertainties using the time-delayed logic and thus removes
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170
the requirement of complete nominal model of the system. ATSMC provides
robustness against the TDE without any prior knowledge of the bound of the uncertainties as well as it does not depend on the variation in TDE. Furthermore,
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the adaptive law of ATSMC addresses the overestimation-underestimation issue of the switching gain. Before stating the control structure of ATSMC, the
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following assumption is made:
Let there exist constant matrices P̄ > 0, Q̄ > 0 and K > 0
such that the following conditions hold:
Assumption 3.
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

P̄1 P̄2T
, where P̄1 > 0, P̄2 > 0, P̄3 > 0,
(i) P̄ = 
P̄2 P̄3


0
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 is Hurwitz, where Ω = P̄3−1 P̄2 ,
(ii) Ā = 
−K −2Ω
(iii) ĀT P̄ + P̄ Ā = −Q̄.
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To facilitate the ATSMC controller design, the system dynamics (1) is rewritten as
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q̈ = f + b̂τ,
(26)
where f = −M̂ −1 ((M − M̂ )q̈ + H), b̂ = M̂ −1 . The control input τ is designed
τ = b̂−1 (û + ∆u),
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as
(27)
where û is the equivalent control input and ∆u is the switching control input
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which negotiates the uncertainties. These terms would be dened later individ-
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ually. The sliding surface s is designed as
s = B T P̄ e = P̄3 ė1 + P̄2 e1
⇒ P̄3−1 s = ė1 + Ωe1 .
(28)
Time derivative of (28) yields
P̄3−1 ṡ = ë1 + Ωė1
= f + b̂τ − q̈ d + Ωė1 .
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(29)
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The equivalent control û in (27) is designed as
(30)
û = q̈ d − Ωė1 − fˆ,
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where fˆ is the nominal value of f . However, knowledge of f is not always avail-
able due to the eect of unmodelled dynamics and disturbances in practice. So,
to avoid any prior information of the nominal model of f , ATSMC approximates
fˆ through the time-delayed logic as
fˆ(t) ∼
= fh = q̈h − b̂h τh ,
(31)
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where fh = f (t − h), b̂h = b̂(t − h). Putting (27), (30) and (31) into (29) the
following is obtained:
P̄3−1 ṡ = f − fˆ + ∆u
= f − q̈h + b̂h τh + ∆u
= σ2 + ∆u + ∆uh − Ωė1h ,
(32)
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where σ2 = f −q̈h +q̈hd −fˆh , ∆uh = ∆u(t−h), fˆh = fˆ(t−h). Further, substituting
(32) in the time derivative of (28) yields the following error dynamics:
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ë1 = −Ke1 − Ωė1 + P̄3−1 ṡ + Ke1
= −Ke1 − Ωė1 − Ωė1h + ∆u + ∆uh + σ2 + Ke1 .
(33)
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The error dynamics (33) can be rewritten in the sate-space form as
AC
CE
ė = Ā1 e + B̄1 eh + B(∆u + ∆uh + σ3 ),
(34)



0
I
0 0
 , B̄1 = 
, σ3 = σ2 + Ke1 . The switching
where Ā1 = 
−K −Ω
0 −Ω
control input ∆u is evaluated as


−αc̄(e, t) s
if k s k≥ ,
ksk
(35)
∆u =

−αc̄(e, t) s
if
k
s
k<
,

where c̄ = c̆ + ĉ is the overall switching gain; c̆ = ||Ke1 − ξΩė1h − (1 + ξ)ë1h ||; ĉ
185
tackles the uncertainties; α > 0 is a scalar adaptive gain; ξ > 0 is a user dened
13
ACCEPTED MANUSCRIPT
scalar and > 0 is a small scalar used to avoid chattering. Utility of ξ will be
explained later in Section 3. In this paper, the following novel adaptive control
law for evaluation of ĉ is proposed:
CR
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


||s|| if ĉ > γ and G(e) > 0,



ĉ˙ = −||s|| if ĉ > γ and G(e) ≤ 0,




 γ
if ĉ ≤ γ,
(36)
where γ > 0 is a small scalar to keep ĉ always positive and G(e) is any suitable
190
function of tracking error selected by the designer. It can be observed form
AN
US
(36) that evaluation ĉ eliminates the requirement of any prior knowledge of the
bound of uncertainties. Here, G(e) is selected as G(e) = ||s|| − ||sh ||. According
to the adaptive law (36) and the present selection of G(e), ĉ increases (resp.
decreases) when the error trajectories move away (resp. do not move away)
195
from ||s|| = 0. As h is to be selected as sampling interval, it is to be noted that
M
G(e) > 0 i.e. ||s|| > ||sh || (resp. G(e) ≤ 0 i.e. ||s|| ≤ ||sh ||) denote the instances
when error trajectories move away (resp. do not move away) from ||s|| = 0 for
the proposed ATSMC. The stability analysis of ATSMC is carried out in the
200
ED
sense of UUB and stated in Theorem 2.
2.2.1. Stability Analysis of ATSMC
Let f is locally Lipschitz for every delay time θ within the
interval θ ∈ [−2h 0].
PT
Assumption 4.
CE
Assumption 4 is very common for practical systems [39] as well as time-
delayed systems [41]. Let x = [q T q̇ T q̈ T ]T and σ = (1 + ξ)(f (x(t)) − fˆ(x(t −
h))) = (1 + ξ)(f (x(t)) − f (x(t − 2h))) (as fˆ(x(t)) = f (x(t − h)) from (31)).
AC
Since f is locally Lipschitz ∀θ ∈ [−2h 0], then ∃l1 ∈ R+ such that ||f (x(t)) −
f (x(t − 2h))|| ≤ l1 ||x(t) − x(t − 2h)||. Further, the EL system (26) is at least
C 2 continuous i.e., q, q̇ are contiuous and dierentable. Then Locally Lipschtiz
condition on f and the system being C 2 continuous implies that (26) x(t) is
also locally Lipschitz ∀θ ∈ [−2h 0]. This implies that ∃l2 ∈ R+ such that
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||x(t) − x(t − 2h)|| ≤ 2l2 h ∀t ∈ [t − 2h t]. Then ∃c ∈ R+ such that ||σ|| ≤ c.
Here, l1 , l2 , c are only used for analytical purpose and they are not used to
compute control law. It is to be noted that locally Lipschitz condition on x(t)
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∀t ∈ [t − 2h t] does not guarantee system stability. For example, a ramp or
an exponentially increasing function are locally Lipschitz, but these functions
are unbounded. So, to verify stability of the system, the following Lyapunov
function candidate is selected:
(37)
V̄ = V1 + V2 ,
AN
US
where V1 = 21 eT P̄ e and V2 = 12 (ξsT P̄3−1 s + (ĉ − c)2 ).
The maximum allowable delay for system (26) employing (27), (30)
and (35) is obtained as
Lemma 2.
h<
λmin (Q̄)
:= hmax ,
||F ||
if K and P̄ are selected such that
M
λmin (Q̄) > h||F ||
(38)
(39)
∀h,
r
β
ED
where F = β P̄ B̄1 (Ā1 D̄−1 ĀT1 + B̄1 D̄−1 B1T + D̄−1 )B1T P̄ + 2 D̄,
D̄ = P̄ + ξ P̄ B P̄3−1 B T P̄ .
Proof. Applying (11) to (37) we have
PT
eT (ν)D̄e(ν) < reT (t)D̄e(t) + ϕ(ν),
(40)
where ϕ(ν) = r(ĉ(t) − c)2 − (ĉ(ν) − c)2 .
CE
Following the similar procedure, as detailed while proving Theorem 1 and se-
AC
lecting D = D̄ we have
V̇1 (e) ≤ −(1/2)eT Υe + Γ + sT σ4 ,
(41)
where Υ = Q̄ − hF,
Γ≥
1
||
2β
Z
0
−h
[ϕ(t + θ) + ϕ(t − h + θ) + (Bσ4 (t + θ))T P̄ Bσ4 (t + θ)]dθ||,
σ4 = ∆u + ∆uh + σ3 .
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So, if P̄ , K and Q̄ are selected in a manner such that λmin (Q̄) > h||F || ∀h, then
the maximum allowable delay is found to be
λmin (Q̄)
.
||F ||
(42)
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hmax =
Inclusion of ξ enables the designer to lengthen or shorten hmax while keeping
205
other design parameters unchanged and this is shown later in Section 3.
The system (26) employing (27), (30) and (35) and having the
adaptation law (36) is UUB with the following error bounds
Theorem 2.
$i =
s
2(Γ + ι)
+ µ2i
λmin (Υ)
i = 1, 2, 3
for ||s|| ≥ ,
AN
US
$i = µi +
s
8(1 + ξ)α(Γ + ι)c̄ + µ2i
4(1 + ξ)αc̄λmin (Υ)
i = 4, 5, 6
for ||s|| < ,
||Θ||||B T P̄ ||
, µ4 = c̄ + ||Θ||, µ5 = 2c + c̆ − ĉ + ||Θ||,
λmin (Υ)
(2c − (2(1 + ξ)α + 1)ĉ + ||Θ||)||B T P̄ ||
µ2 =
,
λmin (Υ)
(c − 2(1 + ξ)αĉ + ||Θ||)||B T P̄ ||
, µ6 = c + c̆ + ||Θ||,
µ3 =
λmin (Υ)
ED
M
where µ1 =
ι = γ2
for ĉ ≤ γ and ι = 0 for ĉ > γ.
CE
PT
Proof. Investigating various combinations of ∆u and ĉ the following possible
cases have been identied:
Case (i): ĉ > γ, ||s|| ≥ and G(e) > 0
Case (ii): ĉ > γ, ||s|| < and G(e) > 0
Case (iii): ĉ > γ, ||s|| ≥ and G(e) ≤ 0
Case (iv): ĉ > γ, ||s|| < and G(e) ≤ 0
Case (v): ĉ ≤ γ, ||s|| ≥ and any G(e)
Case (vi): ĉ ≤ γ, ||s|| > and any G(e)
Now, utilizing (32), (35), (36) and (37), the stability of (26) employing ATSMC
is analysed for various cases as follows:
AC
210
215
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Case (i):
ĉ > γ,
Using (32) we have
||s|| ≥ and G(e) > 0
(43)
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˙
V̇2 = ξsT (∆u + ∆uh + σ2 − Ωė1h ) + (ĉ − c)ĉ.
Utilizing (36), (41) and (43), the time derivative of (37) yields
AN
US
1
V̄˙ ≤ − eT Υe + Γ + sT (2(1 + ξ)∆u + σ) + sT Θ + sT (Ke1 − ξΩė1h ) + (ĉ − c)||s||
2
s
1
+ σ) + sT Θ
= − eT Υe + Γ + sT (−2(1 + ξ)αĉ
2
||s||
s
+ sT (−2(1 + ξ)αc̆
+ Ke1 − ξΩė1h − (1 + ξ)ë1h ) + (ĉ − c)||s||, (44)
||s||
where Θ = (1 + ξ)(∆uh − ∆u). Since ξ > 0 and c̆ = ||Ke1 − ξΩė1h − (1 + ξ)ë1h ||,
we have the following relation for a choice of α > 0.5
sT (−2(1 + ξ)αc̆
s
+ Ke1 − ξΩė1h − (1 + ξ)ë1h )
||s||
≤ (−2(1 + ξ)αc̆ + ||Ke1 − ξΩė1h − (1 + ξ)ë1h ||)||s|| < 0.
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Again
sT (−2(1 + ξ)αĉ
s
+ σ) ≤ (−2(1 + ξ)αĉ + c)||s||.
||s||
(45)
(46)
ED
As system is locally Lipschitz, using (45) and (46), from (44) we have
PT
1
V̄˙ ≤ − λmin (Υ)||e||2 − (2(1 + ξ)α − 1)ĉ||s|| + ||Θ||||s|| + Γ.
2
CE
Further, ξ > 0 and α > 0.5 yields
1
V̄˙ ≤ − λmin (Υ)||e||2 + ||Θ||||B T P̄ ||||e|| + Γ.
2
(47)
AC
Thus, (26) would be UUB with the error bound
Case (ii):
||e|| = µ1 +
ĉ > γ,
s
2Γ
+ µ21 = $1 .
λmin (Υ)
(48)
||s|| < and G(e) > 0
The term sT σ turns out to be:
sT (σ + Ke1 − ξΩė1h − (1 + ξ)ë1h ) ≤ (c + c̆)
17
sT s
.
ksk
(49)
ACCEPTED MANUSCRIPT
Using (36), (41), (43) and (49), the time derivative of (37) gives
(50)
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1
V̄˙ ≤ − λmin (Υ)||e||2 + Γ + ||s||||Θ|| + (ĉ − c)||s||
2
s
s
T
+s
−2(1 + ξ)αc̄ + (c + c̆)
.
||s||
The combination of third, fourth and fth term of (50) takes the maximum
value of (µ24 )/(8(1 + ξ)αc̄) for ||s|| = (µ4 )/(4(1 + ξ)αc̄). Thus V̄˙ < 0 would
be achieved if λmin (Υ)||e||2 > 2Γ + ((c̄ + ||Θ||)2 )/(4(1 + ξ)αc̄). So, the system
(26) is UUB with the error bound is calculated as
8(1 + ξ)αΓc̄ + µ24
= $4
4(1 + ξ)αc̄λmin (Υ)
AN
US
||e|| =
s
(51)
Following the similar procedure as outlined above, Case (iii) and Case (iv)
yields
Case (iii):
ĉ > γ,
||s|| ≥ and G(e) ≤ 0
Case (iv):
ĉ > γ,
M
1
V̄˙ ≤ − λmin (Υ)||e||2 + (2c − (2(1 + ξ)α + 1)ĉ + ||Θ||)||s|| + Γ.
2
||s|| < and G(e) ≤ 0
ED
1
V̄˙ ≤ − λmin (Υ)||e||2 + Γ + ||s||||Θ|| − (ĉ − c)||s||
2
s
s
T
+s
−2(1 + ξ)αc̄ + (c + c̆)
.
||s||
CE
PT
Since ĉ ≤ γ for Cases (v) and (vi) and c > 0, the following holds:
(ĉ − c)ĉ˙ = (ĉ − c)γ ≤ γ 2 − cγ ≤ γ 2 .
Then the following stability conditions are derived for Case (v) and Case (vi):
AC
Case (v):
ĉ ≤ γ,
||s|| ≥ and any G(e)
1
V̄˙ ≤ − λmin (Υ)||e||2 + Γ + (c − 2(1 + ξ)αĉ)||s|| + ||s||||Θ|| + γ 2 .
2
Case (vi):
ĉ ≤ γ,
||s|| > and any
G(e)
1
s
s
2
2
T
˙
V̄ ≤ − λmin (Υ)||e|| + Γ + ||s||||Θ|| + γ + s
−2(1 + ξ)αc̄ + (c + c̆)
.
2
||s||
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Cases (iii) to (vi) can also be shown to be UUB in similar way like Cases (i)
and (ii). The error bounds for Cases (iii), (iv), (v) and (vi) are given by $2 ,
some scalar c1 > 0, is computed to be [36]
∀i = 1, · · · , 6 where V̄˙ (t) ≤ −c1 . (52)
tri ≤ t0 + (V̄ (||e(t0 )||) − V̄ ($i ))/c1 ,
AN
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225
The performance of ATSMC can be characterized by the various
error bounds $i ( i = 1, · · · , 6), which are also functions of α and h. It can be
noticed that high value of α and smaller h (which renders higher λmin (Υ)) would
result in better accuracy. However, too large α may result in high control input
requirement. Further, choice of sampling interval depends hardware response
time. Also, one may choose dierent values of α for G(e) > 0 and G(e) ≤ 0.
Moreover, the proposed ATSMC methodology has the exibility that user can
select any suitable error function G(e).
Remark 4.
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220
The boundary layer is inserted in (35) to avoid chattering [40].
However, it sacrices tracking accuracy as switching does not occur inside ||s|| <
[40]. So, it can be noticed from the expressions of $4,5,6 in Theorem 2 that
smaller values of would reduce the error bound and improve tracking accuracy.
However, one cannot select arbitrarily small as selection of must not invite
chattering. Moreover, unlike µ4,5,6 (when ||s|| < ), the expressions of µ1,2,3
(when ||s|| ≥ , from Theorem 2) involve the term ||B T P̄ ||. Since P̄ can be
designed by the user, one can reduce µ1,2,3 by reducing ||B T P̄ ||. However, this
exibility is lost when ||s|| < .
CE
PT
ED
Remark 5.
230
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$5 , $3 and $6 respectively. The reaching time tri to each error bound $i , for
AC
235
Comparison with Existing Adaptive Sliding Mode Control:
The
Adaptive Sliding Mode Control (ASMC), reported in [33]-[34], does not require
any predened uncertainty bound. The control input of ASMC is given by
τ = Σ−1
n (−κn + ∆us ),
19
(53)
ACCEPTED MANUSCRIPT
where Σn and κn are the nominal values of Σ and κ respectively; ∆us is the
switching control input. For a choice of sliding surface s̄, the vectors Σ and κ
are dened as follow [33]-[34]:
(54)
CR
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s̄˙ = κ + Στ.
Further, it is assumed in [33]-[34] that
(55)
||Σn || ≥ ||Σ − Σn ||,
(56)
||κn || ≥ ||κ − κn ||.
AN
US
The switching control ∆us is evaluated as:
∆us = −%̂
s̄
,
||s̄||
(57)
where %̂ is the switching gain and it is evaluated as


γ
M
%̂˙ =


%̄||s̄||sgn(||s̄|| − ς) if %̂ > γ
if %̂ ≤ γ
, ς = 4%̂ts .
(58)
Here %̄ > 0 is a scalar adaptive gain and ts is the sampling interval. It can be
ED
noted from (58) that switching gain %̂ does not decrease unless (resp. increase)
||s̄|| < ς (resp. ||s̄|| ≥ ς ). So, even if the the error trajectories move close
240
to (resp. move away from) ||s̄|| = 0, %̂ will be increasing (resp. decreasing)
PT
unless ||s̄|| < ς (resp. ||s̄|| ≥ ς(t)). This situation creates overestimation (resp.
underestimation) problem of switching gain. Whereas the proposed adaptive
law (36) does not involve any threshold value like ς . In fact the gain ĉ increases
CE
(resp. decreases) when error trajectories move away (resp. do not move away)
245
from ||s|| = 0 and this alleviates the underestimation (resp. overestimation)
AC
problem.
Moreover, for fair comparison, let us consider s̄ = s. Then comparing (29)
and (54) one can nd that Σn for ASMC is equivalent to b̂ of ATSMC. For
EL systems, b̂ corresponds to nominal mass matrix M̂ (b̂ = M̂ −1 ). Hence, the
250
condition (55) is similar to (10) which is not restrictive for practical systems.
However, ASMC also needs to satisfy ||κn || ≥ ||κ − κn |||. Again comparing
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ACCEPTED MANUSCRIPT
(29) and (54) one can nd that for this class of system, the term κ would
involve f which in turn subsumes the unmodelled dynamics H . So, to satisfy
||κn || ≥ ||κ − κn || ASMC would require fˆ (nominal value of f ) which is not
always possible due to unmodelled dynamics. Whereas, ATSMC avoids the
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255
requirement of knowledge of fˆ by approximating fˆ using past data with the
time-delayed logic (31).
Thus, the advantages of ATSMC can be summarized as follows:
• ATSMC only requires the knowledge of b̂ to design the control law since fˆ
is approximated from the input-output data using the time-delayed logic
260
AN
US
(31). This in turn reduces the tedious modelling eort of complex nonlinear systems. To illustrate the fact with an example, friction, slip, skid etc.
for WMR can be included in f as unmodelled dynamics and consequently
can be approximated by time-delayed logic.
• Evaluation of switching gain is independent of any predened thresh-
265
M
old value like [33]-[34]. Therefore, ATSMC alleviates the overestimationunderestimation problem of the switching gain.
ED
Since ASMC ([33]-[34]) also provides robustness against the uncertain system
without having any prior knowledge of the uncertainty bounds, it would be
prudent to compare the performance of ASMC with the proposed ATSMC.
PT
However, the switching input of ASMC in the form (57) induces chattering.
CE
Hence, it is modied as below
∆us =


−%̂ s̄
||s̄||

−%̂ s̄
if ||s̄|| ≥ ,
if ||s̄|| < .
(59)
The control structures of TDC ([6], [31], [37]), ASMC ([33]-[34]) and the pro-
AC
posed ATSMC are provided in Table 1.
270
3. Application: Nonholonomic WMR
The performance of the proposed ATSMC can be suitably judged in practical circumstances where real life uncertainties creep in. Nonholonomic WMR
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Table 1: Control Structure of Various Controllers
Corresponding Structure.
TDC ([6], [31], [37])
(2), (3), (5)
ASMC ([33]-[34])
(53), (54), (58), (59)
ATSMC (proposed)
(30), (31), (35), (36)
M
AN
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Controller
ED
PT
Figure 2: Schematic of a WMR.
provides a unique platform to test the proposed control law since under practi-
CE
cal circumstances a WMR is always subjected to uncertainties like friction, slip,
skid etc. These terms are dicult to model and in many cases they are not
considered while modelling. The dynamic equation of a WMR (Fig. 2) after
AC
solving the Lagrange multipliers as in [14] can be written as follows
M̄ (q)q̈ + C̄(q, q̇) = N τ,
22
(60)
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ACCEPTED MANUSCRIPT

m̄
0
AN
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Figure 3: Control architecture for ATSMC while employing on PIONEER-3.
m̄dsinφ
k1
k2

PT
ED
M




 
 0
m̄
−m̄dcosφ k3 k4 


τr

where M̄ = 
m̄dsinφ −m̄dcosφ
I¯
−k5 k5  , τ =  


τl


 k1
k3
−k5
Iw
0


k2
k4
k5
0
Iw




2
0 0
m̄dφ̇2 cosφ + m̄rw
sinφ(θ̇r2 − θ̇l2 )/2b̄








2
0 0
m̄dφ̇2 cosφ − m̄rw
sinφ(θ̇r2 − θ̇l2 )/2b̄








2
N = 0 0 , C̄ = 

(θ̇r2 − θ̇l2 )/2b̄
m̄drw








1 0


−m̄drw φ̇2 /2




2
0 1
−m̄drw φ̇ /2
CE
k1 = k2 = − m̄rw cosφ/2, k3 = k4 = −m̄rw sinφ/2, k5 = rw (I¯ − m̄d2 )/b̄.
Here q ∈ R5 = {xc , yc , φ, θr , θl } is the generalized coordinate vector of the sys-
tem. The position of the WMR can be specied by three generalized coordinates
AC
(xc , yc , φ) where (xc , yc ) are the coordinates of the center of mass of the system
and φ is the heading angle; (θr , θl ) and (τr , τl ) denote rotation and torque inputs
275
of the right and left wheels respectively; m̄ and I¯ represent the mass and inertia
of the overall system respectively; rw denotes the wheel radius.
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3.1. Experimental Results and Comparison
The control architecture of the proposed ATSMC with application to PIONEER3 WMR is depicted in Fig. 3. The performance of ATSMC is compared with
CR
IP
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TDC ([6], [31], [37]) as well as ASMC [33]-[34] while the robot is directed to
track the following circular path:
xdc = 1.5sin(0.35t) + .1, ycd = 1.5cos(0.35t) + 1.5, φd = 0.35t, θrd = 4t, θld = 3t.
For a choice of K1 = K2 = Q = I, β = 1, and r = 1.1, the maximum allowable
delay for TDC is found to be h̄max = 125ms. While selecting similar β and r like
TDC, and K = I , P̄ = P , hmax for ATSMC with various values of ξ is provided
AN
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280
in Table 2. It can be seen from Table 2 that stricter upper bound of h can be
Table 2: hmax (ms) for ATSMC with Various ξ
0.3
0.5
hmax
195.4
179.7
2
3
8
10
15
147.1
83.3
39
32.2
22.3
M
ξ
ED
achieved for ATSMC as ξ increases while other parameters are kept unchanged.
As mentioned earlier, h is to be selected as small as possible for better accuracy
and simultaneously it cannot be selected smaller than the sampling interval.
Considering the hardware response time, the sampling interval and thus the
PT
285
delay time h is selected as h = 30ms for all the controllers. Consequently
ξ = 10 is selected for ATSMC from Table 2. Other necessary design parameters
CE
are dened as α = %̄ = 2, s̄ = s, = 0.1, γ = 0.001. To dene b̂, M̂ , κn and Σn ,
the following parametric values are selected for the WMR as supplied by the
manufacturer: m̄ = 18kg, rw = 0.097m, d = 0.02m, b̄ = 0.381m, I¯ = 5.13kg −
AC
290
m2 , Iw = 0.05kg − m2 . Again, to create a dynamic payload variation, a further
3.5kg payload is added and kept for 5sec and then removed. This process is
carried out for the entire duration of experimentation. A time gap of 5sec is
maintained between two successive instances of addition of the payload. Further,
295
the payload was added randomly at dierent places on the robotic platform every
24
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3.5
3
2.5
Desired path
Path tracked with ATSMC
1.5
1
0.5
0
−2
−1
0
xc (m)
1
2
AN
US
−0.5
CR
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yc (m)
2
Figure 4: Trajectory tracking performance of ATSMC.
time to create dynamic variation in center of mass and inertia. Since nominal
system mass is m̄ = 18kg and payload variation is 3.5kg , one can verify that,
with the selection of aforementioned system parameters, the condition (10) is
M
satised.
The trajectory tracking performance of ATSMC is depicted in Fig. 4 while
300
following the desired circular path. Tracking performance comparison of ATSMC
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against TDC and ASMC is illustrated in Fig. 5 in terms of path error (dened
as the Euclidean distance in xc and yc position error) and control input require305
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ment (dened as ||τ ||). TDC does not possess any measure to negotiate the
approximation error TDE that arises from the time-delayed approximation. On
the other hand, both ASMC and ATSMC have robustness properties against
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the uncertainties and provide better tracking compared to TDC which is clearly
evident from the error plots. ASMC requires the condition (56) to be fullled
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which requires nominal knowledge of all components in f (q, q̇, q̈). The WMR
310
dynamics (60) is built upon the assumption that rolling without slipping/ pure
rolling of wheels hold. However, in practice, one cannot neglect the eects of
friction, slip, skid etc. which are subsumed under f as unmodelled dynamics.
On the contrary, ATSMC does not require any knowledge of f or its nominal
value. ATSMC approximates f (which includes unmodelled dynamics and dis25
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TDC
200
ASMC
ATSMC
150
100
50
0
0
5
10
0
5
10
15
20
25
30
0.4
0.2
0.1
0
35
AN
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||τ|| (Nm)
0.3
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IP
T
path error (mm)
250
15
20
time (sec)
25
30
35
Figure 5: Performance comparison of ATSMC with TDC and ASMC.
turbances) with its embedded time-delayed logic and provides robustness against
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315
the approximation error using the switching law. As a matter of fact, ATSMC
provides the best tracking accuracy amongst the three controllers in contention.
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Another important attribute for any ARC law is the evaluation of switching
gain which tackles the uncertainties. The switching gain (%̂) evaluation of ASMC
320
with respect to sliding surface for both the wheels are provided in Fig. 6 and
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Fig. 7. It can be noticed from Fig. 6 that ||s̄|| decreases during t = 0.1−19.2 sec
and then again increases for the rest of the time. However, the switching gain %̂
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increases monotonically during t = 0.1 − 7.8 sec (when ||s̄|| ≥ ς ) even when ||s̄||
decreases and move towards ||s̄|| = 0. This situation gives rise to overestimation
problem of switching gain. This phenomenon arises as according to (58), %̂ does
not decrease when ||s̄|| ≥ ς . Further, %̂ decreases monotonically when ||s̄|| < ς
during t = 19.2−25 sec even though ||s̄|| increases during t = 19.2−25 sec. This
phenomenon gives rise to underestimation problem of switching gain. Similar
situations can be noticed for left wheel also in Fig. 7.
The switching gain (ĉ) evaluation of ATSMC is depicted through Fig. 8 for
AC
325
330
26
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0.25
0.15
0.1
0
5
10
15
20
time (sec)
25
30
0.06
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T
̺
ˆ
0.2
35
ς
||s̄||
0.04
0
0
5
10
AN
US
0.02
15
20
time (sec)
25
30
35
Figure 6: Switching gain evaluation of ASMC for right wheel.
M
0.16
̺
ˆ
0.14
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0.12
0
5
10
15
20
time (sec)
25
30
PT
0.1
35
0.02
ς
||s̄||
0.015
AC
CE
0.01
0.005
0
0
5
10
15
20
time (sec)
25
30
35
Figure 7: Switching gain evaluation of ASMC for left wheel.
both the wheels. To demonstrate how ATSMC has been able to avoid the over-
and under-estimation problem of switching gain, two magnied versions of ĉ are
27
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0.11
right wheel
left wheel
ĉ
0.105
0.095
0
5
10
15
20
time (sec)
25
30
0.08
0.04
0.02
0
5
10
AN
US
||s||
35
right wheel
left wheel
0.06
0
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0.1
15
20
time (sec)
25
30
35
Figure 8: Switching gain properties for ATSMC.
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shown in Fig. 9 and Fig. 10 for right and left wheel respectively. It is to be noted
that ĉ, according to (36), increases whenever ||s|| > ||sh || (i.e. ||s|| increases)
335
and decreases whenever ||s|| ≤ ||sh || (i.e. ||s|| decrease or does not change). It
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can be observed from Fig. 9 that ĉ increases during t = 0 − 0.1 sec when ||s||
increases and ĉ decreases during t = 0.1 − 0.8 sec when ||s|| decreases. Similarly,
ĉ replicates the wavy nature of ||s|| during t = 22 − 24 sec. Further, ĉ decreases
340
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when ||s|| does not change during t = 22.5 − 22.6 sec. This happens because ĉ
decreases when ||s|| = ||sh || (i.e. ||s|| does not change) according to (36). Thus,
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ATSMC avoids any overestimation (resp. underestimation ) phenomenon as ĉ
does not increase (resp. decrease) when ||s|| decreases (resp. increases). Similar
AC
arguments can also be followed from Fig. 10.
345
To verify the eect of h on the controller performance, similar experiments
were carried out for various sampling intervals. The tracking performances
are tabulated in Table 3 for ATSMC and TDC in terms of average path error
(APE). The percentage path error is calculated with respect to the diameter of
the circular path. In this scenario, ξ = 1.5 is selected for ATSMC which provides
28
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0.105
0.099
ĉ
ĉ
0.097
0.095
0
1
time (sec)
0.096
22
2
0.02
0.04
0.019
0.02
0
24
AN
US
||s||
||s||
0.06
23
time (sec)
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0.098
0.1
0.018
0
1
time (sec)
0.017
22
2
23
time (sec)
24
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Figure 9: Switching gain evaluation of ATSMC for right wheel (magnied).
0.104
0.1
0.103
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AC
0
1
time (sec)
2
0.102
22
−3
10
0.02
9.5
23
time (sec)
24
23
time (sec)
24
x 10
||s||
0.03
||s||
CE
0.099
ĉ
ĉ
ED
0.101
0.01
0
9
0
1
time (sec)
8.5
22
2
Figure 10: Switching gain evaluation of ATSMC for left wheel (magnied).
29
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Table 3: Path Error (mm) Comparison of TDC and ATSMC for Various Sampling Interval
h (ms)
ATSMC
TDC
% APE
APE
% APE
30
41.23
1.37
165.11
5.50
60
49.63
1.65
173.54
5.78
90
60.17
2.01
189.39
6.31
120
71.56
2.38
203.76
6.79
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APE
350
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the upper bound of the delay of hmax = 124.4ms. It is evident from Table 3
that higher choice of sampling intervals result in poorer tracking accuracy for
both the controllers. These phenomena can be substantiated from the error
bounds of both TDC and ATSMC where higher values of delay have pervasive
4. Conclusions
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eect on the error bounds.
The choice of sampling interval and controller gains are crucial for the per-
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355
formance of TDC and this previously unsolved design issue is addressed in this
paper. An upper bound of the sampling interval and its relation with the con-
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troller gains are formulated through a new stability analysis based on the Razumikhin approach. The proposed ATSMC control law approximates unknown
360
dynamics and perturbations through time-delayed logic and compensates the
CE
time-delayed approximation error by switching logic without any prior knowledge of the bound of uncertainties. Moreover, the proposed adaptive switching
AC
law of ATSMC alleviates the overestimation-underestimation problem of switching gain. Experimental results with a WMR show improved path tracking per-
365
formance of ATSMC compared to conventional ASMC and TDC.
However, due to the inclusion of boundary layer, accuracy of ATSMC gets
sacriced. So, the future work would be to extend the proposed adaptive-robust
law to higher order sliding mode control.
30
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370
References
Control Design, Wiley, New York, 1995.
CR
IP
T
[1] M. Kristic, I. Kanellakopoulos, P. V. Kokotovic, Nonlinear and Adaptive
[2] X. Liu, H. Su, B. Yao, J.Chu, Adaptive robust control of a class of uncer-
tain nonlinear systems with unknown sinusoidal disturbances, 47th IEEE
Conference on Decision and Control (2008) 2594-2599.
375
AN
US
[3] M. J. Corless, G. Leitmann, Continuous state feedback guaranteeing uni-
form ultimate boundness for uncertain dynamic system, IEEE Tran. Autom. Control. 26(10) (1981) 1139-1144.
[4] H. Lee, V. I. Utkin, Chattering suppression methods in sliding mode control
systems, Annu. Reviews. Control. 31 (2007) 179-188.
380
M
[5] A. Levant, Higher-order sliding modes, dierentiation and output-feedback
control, Int. J. Control. 76(9/10) (2003) pp. 924-941.
ED
[6] T. C. Hsia and L. S. Gao, Robot manipulator control using decentralized linear time-invariant time-delayed joint controllers, IEEE International
Conference on Robotics and Automation (1990) 2070-2075.
PT
385
[7] K. Youcef-Toumi, O. Ito, A time delay controller for systems with unknown
CE
dynamics, ASME J. Dyn. Sys., Meas., Control. 112(1) (1990) 133-142.
[8] G. R. Cho, P. H. Chang, S. H. Park, M. Jin, Robust tracking under non-
AC
linear friction using time delay control with internal model, IEEE Tran.
390
Control. Syst. Technol. 17(6) (2009) 1406-1414.
[9] D. K.Han, P. H. Chang, Robust tracking of robot manipulator with nonlinear friction using time delay control with gradient estimator, J. Mechanical
Sci. Technol. 24(8) (2010) 1743-1752.
31
ACCEPTED MANUSCRIPT
[10] S-J Cho, M. Jin, T-Y Kuc, J. S. Lee, Stability guaranteed auto-tuning
algorithm of a time-delay controller using a modied Nussbaum function,
395
Int. J. Control. 87(9) (2014) 1926-1935.
CR
IP
T
[11] M. Jin, S. H. Kang, P. H. Chang, Robust compliant motion control of
robot with nonlinear friction using time-delay estimation, IEEE Tran. Indus. Electron. 55(1) (2008) 258-269.
400
[12] Y. Jin, P. H. Chang, M. Jin, D. G. Gweon, Stability guaranteed time-delay
control of manipulators using nonlinear damping and terminal sliding mode,
AN
US
IEEE Tran. Indus. Electron. 60(8) (2013) 3304-3318.
[13] P. H. Chang, S. H. Park, On improving time-delay control under certain
hard nonlinearities, Mechatron. 13(4) (2003) 393-412.
405
[14] S. Roy, S. Nandy, R. Ray, S. N. Shome, Time delay sliding mode control nonholonomic wheeled mobile robot: experimental validation, IEEE
M
International Conference on Robotics and Automation (2014) 2886-2892.
[15] J. Kim, H. Joe, S-C Yu, J. S. Lee, M. Kim, Time delay controller design
for position control of autonomous underwater vehicle under disturbances,
ED
IEEE Tran. Indus. Electron. DOI 10.1109/TIE.2015.2477270, 2015.
410
[16] A. Kuperman, Q. C. Zhong, Robust control of uncertain nonlinear sys-
PT
tems based on an uncertainty and disturbance estimator, Int. J. Robust
Nonlinear Control. 21 (2011) 79-92.
CE
[17] S. E. Talole, S. B. Phadke, Robust input-output linearization using uncertainty and disturbance estimator, Int. J. Control. 82 (2009) 1794-1803.
415
AC
[18] P. V. Suryawanshi, P. D. Shengde, S. B. Phadke, Robust sliding mode control for a class of nonlinear systems using inertial delay control, Nonlinear
Dynamics, 78(3) (2014) 1921-1932.
[19] X. Zhu, G. Tao, B. Yao, J. Cao, Adaptive robust posture control of parallel
420
manipulator driven by pneumatic muscles with redundancy, IEEE/ASME
Tran. Mechatron., 13(4) (2008) 441-450.
32
ACCEPTED MANUSCRIPT
[20] X. Zhu, G. Tao, B. Yao, J. Cao, Integrated direct/indirect adaptive robust
posture control of parallel manipulator driven by pneumatic muscles, IEEE
Tran. Control. Syst. Technol. 17(3) (2009) 576-588.
[21] G. Zhang, J. Chen, Z. Lee, Adaptive robust control of servo mechanisms
CR
IP
T
425
with partially known states via dynamic surface control approach, IEEE
Tran. Control. Syst. Technol. 18(3) (2010) 723-731.
[22] W. Sun, Z. Zhao, H. Gao, Saturated adaptive robust control for active
suspension systems, IEEE Tran. Indus. Electron. 60(9) (2013) 3889-3896.
[23] S. Islam, P. X. Liu, A. E. Saddik, Robust control of four rotor unmanned
AN
US
430
aerial vehicle with disturbance uncertainty, IEEE Tran. Indus. Electron.
63(3) (2015) 1563-1571.
[24] Z. Chen, B. Yao, Q. Wang, µ-synthesis based adaptive robust posture control of linear motor driven stages with high frequency dynamics: a case
study, IEEE/ASME Tran. Mechatron. 20(3) (2015) 1482-1490, 2015.
M
435
[25] Z. Liu, H. Su, S. Pan, A new adaptive sliding mode control of uncertain
ED
nonlinear dynamics, Asian J. Control. 16(1) (2014) 198-208.
[26] A. Nasiri, S. K. Nguang, A. Swain, Adaptive sliding mode control for a
PT
class of MIMO nonlinear systems with uncertainty, J. Franklin Inst. 351
(2014) 2048-2061.
440
CE
[27] A. Nasiri, A. Swain, S. K. Nguang, Passive actuator fault tolerant control
for a class of MIMO non-linear systems with uncertainties, IEEE Conference on Control Applications (2014) 1184-1189.
AC
[28] A. K. Khalaji, S. A. A. Moosavian, Robust-adaptive controller for a tractor-
445
trailor mobile robot, IEEE/ASME Tran. Mechatron. 19(3) (2014) 943-953.
[29] Q. Meng, T. Zhang, X. Gao, J. Y. Song, Adaptive sliding mode faulttolerant control of the uncertain stewart platform based on oine multibody Dynamics, IEEE/ASME Tran. Mechatron. 19(3) (2014) 882-894.
33
ACCEPTED MANUSCRIPT
[30] S. Liu, H. Zhou, X. Luo, J. Xiao, Adaptive sliding fault tolerant control for
nonlinear uncertain active suspension systems, J. Franklin Inst. 353 (2016)
450
180-199.
CR
IP
T
[31] S. Roy, S. Nandy, R. Ray, S. N. Shome, Robust path tracking control of
nonholonomic wheeled mobile robot, Int. J. Control., Autom. Syst. 13(4)
(2015) 897-905.
455
[32] D. Almakhles, A. K. Swain, A. Nasiri, N. Patel, An adaptive two-level
quantizer for networked control systems, IEEE Tran. Control. Syst. Tech-
AN
US
nol. (2016) DOI: 10.1109/TCST.2016.2574768.
[33] F. Plestan, Y. Shtessel, V. Bregeault, A. Poznyak, New methodologies for
adaptive sliding mode control, Int. J. Control. 83(9) (2010) 1907-1919.
460
[34] F. Plestan, Y. Shtessel, V. Bregeault, A. Poznyak, Sliding mode control
with gain adaptation - application to an electropneumatic actuator state,
M
Control. Eng. Pract. 21 (2013) 679-688.
[35] B. Bandyopadhayay, S. Janardhanan and S. K. Spurgeon, Advances in
465
ED
Sliding Mode Control, Springer-Verlag, New York, 2013.
[36] G. Leitmann, On the eciency of nonlinear control in uncertain linear
systems, ASME J. Dyn. Sys., Meas., Control. 103 (1981) 95-102.
PT
[37] Y. H. Shin, K. J. Kim, Performance enhancement of pneumatic vibration
isolation tables in low frequency range by time delay control, J. Sound.
CE
Vib. 321 (2009) 537-553.
470
[38] J. K. Hale, Theory of functional dierential equation, Springer-Verlag,
AC
1977.
[39] H. Khalil, Nonlinear systems, 3rd ed. Prentice Hall, 2002.
[40] J.-J. E. Slotine, W. Li, Applied Nonlinear Control, Prentice-hall, 1991.
[41] E. Fridman, Tutorial on Lyapunov-based methods for time-delay systems,
475
Eur. J. Control 20 (2014) 271-283.
34
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