Automatica 76 (2017) 222–229

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Automatica

journal homepage: www.elsevier.com/locate/automatica

Brief paper

Unknown time-varying input delay compensation for uncertainnonlinear systems✩

Serhat Obuz, Justin R. Klotz, Rushikesh Kamalapurkar, Warren DixonDepartment of Mechanical and Aerospace Engineering, University of Florida, Gainesville, USA

a r t i c l e i n f o

Article history:Received 20 January 2015Received in revised form8 April 2016Accepted 26 August 2016Available online 8 December 2016

Keywords:Input delayRobust controlNonlinear control

a b s t r a c t

A tracking controller is developed for a class of uncertain nonlinear systems subject to unknowntime-varying input delay and additive disturbances. A novel filtered error signal is designed usingthe past states in a finite integral over a constant estimated delay interval. The maximum tolerableerror between unknown time-varying delay and a constant estimate of the delay is determined toestablish uniformly ultimately bounded convergence of the tracking error to the origin. The controllerdevelopment is based on an approach which uses Lyapunov–Krasovskii functionals to analyze theeffects of unknown sufficiently slowly time-varying input delays. A stability analysis is provided toprove ultimate boundedness of the tracking error signals. Numerical simulation results illustrate theperformance of the developed robust controller.

© 2016 Published by Elsevier Ltd.

1. Introduction

Time delay commonly exists in many engineering applicationssuch as master–slave robots, haptic systems, chemical systemsand biological systems. The system dynamics, communicationover a network, and sensing with associated sensor processing(e.g., image-based feedback) can induce time delays that canresult in decreased performance and loss of stability. Time delaysin physical systems are often time-varying. For example, theinput delay in neuromuscular electrical stimulation applicationsoften changes with muscle fatigue (Downey, Kamalapurkar,Fischer, & Dixon, 2015; Merad, Downey, Obuz, & Dixon, 2016),communication delays in wireless networks change with thedistance between the communicating agents, etc. Motivated bysuch practical engineering challenges, numerous research effortshave focused on designing controllers to compensate time delaydisturbances effects.

Research in recent years has focused on developing controllersthat provide stability for systems with delays in the closed-loop dynamics. Smith’s pioneering work Smith (1959), Arstein’smodel reduction (Artstein, 1982), and the finite spectrumapproach

✩ The material in this paper was not presented at any conference. This paper wasrecommended for publication in revised form by Associate Editor Hiroshi Ito underthe direction of Editor Andrew R. Teel.

E-mail addresses: serhat.obuz@ufl.edu (S. Obuz), jklotz@ufl.edu (J.R. Klotz),rkamalapurkar@ufl.edu (R. Kamalapurkar), wdixon@ufl.edu (W. Dixon).

http://dx.doi.org/10.1016/j.automatica.2016.09.0300005-1098/© 2016 Published by Elsevier Ltd.

(Manitius & Olbrot, 1979) have heavily influenced the methods ofdesigning controllers that compensate the effects of delays.

In recent years, research has focused on systems that experi-ence a known delay in the control input. The works in Lozano,Castillo, Garcia, and Dzul (2004), Normey-Rico, Guzman, Dormido,Berenguel, and Camacho (2009) andRoh andOh (1999) develop ro-bust controllers which compensate for known input time delay forsystems with linear plant dynamics. Compensation of input delaydisturbances for nonlinear plant dynamics is addressed in promi-nentworks such as Dinh, Fischer, Kamalapurkar, and Dixon (2013),Fischer (2012), Fischer, Dani, Sharma, and Dixon (2011), Fischer,Dani, Sharma, and Dixon (2013), Fischer, Kamalapurkar, Fitz-Coy,and Dixon (2012), Huang and Lewis (2003), Obuz, Tatlicioglu, Ce-kic, and Dawson (2012), Sharma, Bhasin, Wang, and Dixon (2011)and Teel (1998) for nonlinear plant dynamics affected by exter-nal disturbances and in Henson and Seborg (1994), Jankovic (2006)and Mazenc and Bliman (2006) for plant dynamics without exter-nal disturbances. However, the controllers in Dinh et al. (2013),Fischer (2012), Fischer et al. (2011), Fischer et al. (2013), Fischer,Kamalapurkar et al. (2012), Henson and Seborg (1994), Huang andLewis (2003), Jankovic (2006), Mazenc and Bliman (2006), Obuzet al. (2012), Sharma et al. (2011) and Teel (1998), require exactknowledge of the time delay duration. In practice, the duration ofan input time delay can be challenging to determine for some ap-plications, therefore, it is necessary to develop new controllers thatdo not require exact knowledge of the time delay.

Since uncertainty in the delay can lead to unpredictable closed-loop performance (potentially even instabilities), several recent

S. Obuz et al. / Automatica 76 (2017) 222–229 223

results have been developed which do not assume that the delayis exactly known. Compensation for unknown input delay isinvestigated in Bresch-Pietri, Chauvin, and Petit (2010, 2011),Bresch-Pietri, Chauvin, and Petit (2012), Bresch-Pietri and Krstic(2009), Choi and Lim (2010), Herrera, Ibeas, Alcantara, Vilanova,and Balaguer (2008), Li, Gu, Zhou, and Xu (2014), Li, Zhou, and Lin(2014), Polyakov, Efimov, Perruquetti, and Richard (2013), Wang,Wu, and Gao (2005) and Wang, Saberi, and Stoorvogel (2013) forsystemswith exactly knowndynamics and Chen and Zheng (2006),Yue (2004), Yue and Han (2005) and Zhang and Li (2006a,b) forsystems with uncertain dynamics. However, all of the controllersin Bekiaris-Liberis and Krstic (2013), Bresch-Pietri et al. (2010,2011), Bresch-Pietri et al. (2012), Bresch-Pietri and Krstic (2009),Chen and Zheng (2006), Choi and Lim (2010), Herrera et al. (2008),Li, Gu et al. (2014), Li, Zhou et al. (2014), Polyakov et al. (2013),Wang et al. (2013), Wang et al. (2005), Yue (2004), Yue and Han(2005) and Zhang and Li (2006a,b) are developed for linear plantdynamics. The works in Balas and Nelson (2011), Bresch-Pietri andKrstic (2014), Mazenc and Niculescu (2011) and Nelson and Balas(2012) develop controllers for plants with nonlinear dynamicsand an unknown input delay, but require exact model knowledgeof the nonlinear dynamics. The controller designed in Chiu andChiang (2009) compensates for Takagi–Sugeno fuzzy systemsand unknown actuation delay duration by using a memorylessobserver and a fuzzy parallel distributed integral compensatorfor nonlinear, uncertain dynamics. However, the controller inChiu and Chiang (2009) is designed for output regulation anddoes not address the output tracking problem. There remains aneed for a tracking controller that can compensate for the effectsof unknown time-varying input delays for a class of uncertainnonlinear systems.

When uncertain nonlinear dynamics are present, the controldesign is significantly more challenging than when linear orexactly known nonlinear dynamics are present. For example, ingeneral, if the system states evolve according to linear dynamics,the linear behavior can be exploited to predict the system responseover the delay interval. Exact knowledge of the dynamics facilitatesthe ability to predict the state transition for nonlinear systems. Foruncertain nonlinear systems, the state transition is significantlymore difficult to predict, especially if the delay interval is alsounknown and/or time-varying. Given the difficulty in predictingthe state transition, the contribution in this paper (and in Fischer,Kamalapurkar et al., 2012 and Kamalapurkar, Fischer, Obuz, &Dixon, 2016) is to treat the input delay and dynamic uncertainty asa disturbance and develop a robust controller that can compensatefor these effects.

Recently, Fischer et al. presented a robust controller foruncertain nonlinear systems with additive disturbances subject toslowly varying input delay in Fischer, Kamalapurkar et al. (2012),where it is assumed that the input delay duration is measurableand the absolute value of the second derivative of the delay isbounded by a known constant. The approach in this study extendsour previous work in Fischer, Kamalapurkar et al. (2012) by usinga novel filtered error signal to compensate for an unknown slowlyvarying input delay for uncertain nonlinear systems affected byadditive disturbances. In Fischer, Kamalapurkar et al. (2012), afiltered error signal defined as the finite integral of the actuatorsignals over the delay interval is used to obtain a delay-freeexpression for the control input in the closed-loop error system.However, the computation of the finite integral requires exactknowledge of the input delay. In this study, a novel filtered errorsignal is designed using the past states in a finite integral overa constant estimated delay interval to cope with the lack ofdelay knowledge, which requires a significantly different stabilityanalysis that takes advantage of Lyapunov–Krasovskii functionals.Techniques used in this study provide relaxed requirements of

the delay measurement and obviate the need for a bound of theabsolute value of the second derivative of the delay. It is assumedthat the estimated input delay is selected sufficiently close tothe actual time-varying input delay. That is, there are robustnesslimitations, which can be relaxed with more knowledge aboutthe time-delay. Because it is feasible to obtain lower and upperbounds for the input delay in many applications (Richard, 2003), itis feasible to select a delay estimate in an appropriate range. Newsufficient conditions for stability are based on the length of theestimated delay as well as the maximum tolerable error betweenthe actual and estimated input delay. A Lyapunov-based stabilityanalysis is used to prove ultimate boundedness of the error signals.Numerical simulation results demonstrate the performance of therobust controller.

2. Dynamic system

Consider a class of nth-order nonlinear systems

xi = xi+1, i = 1, . . . , n − 1,xn = f (X, t) + d + u (t − τ) , (1)

where xi ∈ Rm, i = 1, . . . , n are the measurable system states,X =

xT1, xT2, . . . , xTn

T∈ Rmn, u ∈ Rm is the control input,

f : Rmn× [t0, ∞) → Rm is an uncertain nonlinear function,

d : [t0, ∞) → Rm denotes sufficiently smooth unknown additivedisturbance (e.g., unmodeled effects), and τ : [t0, ∞) → Rdenotes a time-varying unknown positive time delay,1 where t0is the initial time. Throughout the paper, delayed functions aredenoted as

hτ ,

h (t − τ) t − τ ≥ t00 t − τ < t0.

The dynamic model of the system in (1) can be rewritten as

x(n)1 = f (X, t) + d + u (t − τ) , (2)

where the superscript (n) denotes the nth time derivative. Inaddition, the dynamic model of the system in (1) satisfies thefollowing assumptions.

Assumption 1. The function f and its first and second partialderivatives are bounded on each subset of their domain of the formΞ × [t0, ∞), where Ξ ⊂ Rmn is compact and for any given Ξ , thecorresponding bounds are known.2

Assumption 2 (Fischer, Kan, &Dixon, 2012). The nonlinear additivedisturbance term and its first time derivative (i.e., d, d) exist and arebounded by known positive constants.

Assumption 3. The reference trajectory xr ∈ Rm is designed suchthat the derivatives x(i)

r , ∀i = 0, 1, . . . , (n + 2) exist and arebounded by known positive constants.

1 The developed method can be extended to the case of multiple delays.Assumption 4 can be modified for the case of multiple delays by redefining thedelayed input vector and using themaximum input delay instead of the actual delaybound such that max{τ1, τ2, . . . , τm} < Υ . To obviate the requirement of exactknowledge of the time delay dynamics in the stability analysis and introducing newLyapunov–Krasovskii functionals for each input delay, the closed-loop dynamicscan be revised in terms of uτ , (uτ − uτ ) instead of the terms uτ , uτ , : (uτ − uτ ).In this paper, single time-varying input delay is considered for ease of exposition.2 Given a compact set Ξ ⊂ Rmn , the bounds of f , ∂ f (X,t)

∂X , ∂ f (X,t)∂t , ∂2 f (X,t)

∂2X, ∂2 f (X,t)

∂X∂t ,

and ∂2 f (X,t)∂2 t

over Ξ are assumed to be known.

224 S. Obuz et al. / Automatica 76 (2017) 222–229

Assumption 4. The input delay is bounded such that τ (t) < Υ forall t ∈ R, differentiable, and slowly varying such that |τ | < ϕ < 1for all t ∈ R, where ϕ, Υ ∈ R are known positive constants.Additionally, a sufficiently accurate constant estimate τ ∈ R ofτ is available such that |τ | ≤ τ , for all t ∈ R, where τ , τ − τ , and¯τ ∈ R is a known positive constant.3 Furthermore, it is assumedthat the system in (1) does not escape to infinity during the timeinterval [t0, t0 + Υ ].

3. Control development

The objective of the control design is to develop a continuouscontroller which ensures that the state x1 of the delayed system in(2) tracks a reference trajectory, xr .

To quantify the control objective, a tracking error, denoted bye1 ∈ Rm, is defined as

e1 , xr − x1. (3)To facilitate the subsequent analysis, auxiliary tracking errorsignals, denoted by ei ∈ Rm, i = 2, 3, . . . , n, are defined as Xian,Dawson, de Queiroz, and Chen (2004)

e2 , e1 + e1, (4)e3 , e2 + e2 + e1, (5)

...

en , en−1 + en−1 + en−2. (6)

A general expression of ei for i = 2, 3, . . . , n can be written as

ei =

i−1j=0

ai,je(j)1 , (7)

where ai,j ∈ R are defined as4 Tatlicioglu (2007)

ai,0 ,1

√5

1 +√5

2

i

−

1 −

√5

2

i , ∀i = 2, 3, . . . , n,

ai,j ,

i−1p=1

ai−p−j+1,0ap+j−1,j−1,

∀i = 3, 4, . . . , n, ∀j = 1, 2, . . . , (i − 2) .

To obtain a delay-free control expression for the input in theclosed-loop error system, an auxiliary tracking error signal,denoted by eu ∈ Rm, is defined as

eu , −

t

t−τ

u (θ) dθ. (8)

It should be emphasized that the estimate of τ , denoted by τ , isrequired instead of exact knowledge of τ in the control design.For example, the constant estimate τ may be selected to bestapproximate the mean of τ . Based on the subsequent stabilityanalysis, the following continuous robust controller is designed as

u , k (en − en (t0)) + υ, (9)where en (t0) ∈ Rm is the initial error signal and υ ∈ Rm is thesolution to the differential equation

υ = k (Λen + αeu) , (10)

where k, Λ, α ∈ Rm×m are constant, diagonal, positive definitegain matrices.

3 Since the maximum tolerable error, ¯τ , and the estimate of actual delay, τ , areknown, themaximum tolerable input delay can be determined. Because the boundson the input delay are feasible to obtain in many applications (Richard, 2003),Assumption 4 is reasonable.4 It should be noted that ai,i−1 = 1, ∀i = 1, 2, . . . , n.

4. Stability analysis

To facilitate the stability analysis an auxiliary tracking errorsignal, denoted by r ∈ Rm, is defined as5

r , en + Λen + αeu. (11)

The open loop dynamics for r can be obtained by substituting thefirst time derivatives of (2) and (8), the second time derivative of(7) with i = n, and the (n + 1)th time derivative of (3) into (11) as

r = −fX, X, t

− d +

n−2j=0

an,je(j+2)1 + x(n+1)

r

− αu + αuτ − (1 − τ ) uτ + Λen. (12)

Substituting the first time derivative of the controller in (9) into(12), the closed-loop error system for r can be obtained as

r = −fX, X, t

− d +

n−2j=0

an,je(j+2)1 + x(n+1)

r + Λen

− αkr + (α − I + τ I) krτ + α (uτ − uτ ) , (13)

where I ∈ Rm×m is the identity matrix. The stability analysis canbe facilitated by segregating the terms in (13) that can be upperbounded by a state-dependent function and terms that can beupper bounded by a constant, such that

r = −αkr + (α − I + τ I) krτ + α (uτ − uτ ) + N + Nr − en. (14)

The auxiliary functions N ∈ Rm and Nr ∈ Rm are defined as

N , −fX, X, t

+ f

Xr , Xr , t

+

n−2j=0

an,je(j+2)1

+ Λen + en, (15)

Nr , −fXr , Xr , t

− d + x(n+1)

r , (16)

where Xr ,

xTr , xTr , . . . ,

x(n−1)r

TT∈ Rmn.

Remark 1. Based on Assumptions 2 and 3, Nr is upper bounded as

supt∈R

∥Nr∥ ≤ ζNr , (17)

where ζNr ∈ R is a known positive constant.

Remark 2. An upper bound can be obtained for (15) using As-sumption 1 and the Lemma 5 in Kamalapurkar, Rosenfeld, Klotz,Downey, and Dixon (2014) asN ≤ ρ (∥z∥) ∥z∥ , (18)

where ρ is a positive, radially unbounded,6 and strictly increasingfunction, and z ∈ R(n+2)m is a vector of error signals defined as

z ,eT1, eT2, . . . , e

Tn, eTu, rT

T. (19)

To facilitate the subsequent stability analysis, auxiliary bound-ing constants σ , δ ∈ R are defined as

5 Since en is not measurable, r cannot be used in the control design.6 For some classes of systems, the bounding function ρ can be selected as a

constant. For those systems, a global uniformly ultimately bounded result can beobtained as described in Remark 3.

S. Obuz et al. / Automatica 76 (2017) 222–229 225

σ , min1,1 −

ϵ2

2

,

Λ −

α

2ϵ1+

12ϵ2

,

k α

8,ω2

4τ− αϵ1

, (20)

δ ,12min

σ

2,

ω2k3 α (1 − ϕ)

4k (α + ϕ − 1)

2 ,ω2k2αϵ1

4ω21 k2

,1

4¯τ + τ

,

(21)

where Λ, k, α ∈ R denote the minimum eigenvalues of Λ, k, α,respectively, k, α ∈ R denote the maximum eigenvalues of k andα, respectively, and ωi, ϵi ∈ R, i = 1, 2, are known, selectable,positive constants.

Let the functions Q1, Q2, Q3 ∈ R be defined as

Q1 ,

ω1k

2αϵ1

t

t−τ

∥r (θ)∥2 dθ, (22)

Q2 ,

k (α + ϕ − 1)

2kα (1 − ϕ)

t

t−τ

∥r (θ)∥2 dθ, (23)

Q3 ,ω2

t

t−( ¯τ+τ)

t

s∥u (θ)∥2 dθds, (24)

and let y ∈ R(n+2)m+3 be defined as

y ,z,Q1,

Q2,

Q3T

. (25)

For use in the following stability analysis, let

D1 ,y ∈ R(n+2)m+3

| ∥y∥ < χ1, (26)

where χ1 , infρ−1

[

σk α

2 , ∞)

. Provided ∥z (η)∥ <

γ , ∀ η ∈ [t0, t], (14) and the fact that u = kr can be used toconclude that u < M , where γ and M7 are positive constants. LetD , D1∩

Bγ ∩ R(n+2)m+3

where Bγ denotes a closed ball of radius

γ centered at the origin and let

SD , {y ∈ D| ∥y∥ < χ2} (27)

denote the domain of attraction, where8 χ2 ,

min

12 ,

ω12

max

1, ω1

2

infρ−1

[

σk α

2 , ∞)

.

Theorem 1. Given the dynamics in (1), the controller given in (9) and(10) ensures uniformly ultimately bounded tracking in the sense that

lim supt→∞

∥e1 (t)∥

≤

max1, ω1

2

min

12 ,

ω12

·

2ζ 2

Nr+ α kα ¯τ

2M2

2α kδ

, (28)

provided that y (t0) ∈ SD and that the control gains are selectedsufficiently large based on the initial conditions of the system such that

7 The subsequent analysis does not assume that the inequality u < M holds forall time. The subsequent analysis only exploits the fact that provided ∥z (η)∥ <

γ , ∀η ∈ [t0, t], then u < M .8 For a set A, the inverse image ρ−1 (A) is defined as ρ−1 (A) , {a | ρ (a) ∈ A}.

the following sufficient conditions are satisfied910

ω2 > 4αϵ1τ ,

Λ >α

2ϵ1+

12ϵ2

, 2 > ϵ2, α k8 −

2(ω1 k)2

αϵ1−

(k(α+ϕ−1))2

α k(1−ϕ)

ω2k2−

ω2τ k2 +α2

ω2k2

≥ ¯τ ,

χ2 >

2ζ 2Nr

+ α kα ¯τ2M2

2α kδ

12

. (29)

Proof. Let V : D → R be a continuously differentiable Lyapunovfunction candidate defined as

V ,12

ni=1

eTi ei +12rT r +

ω1

2eTueu +

3i=1

Qi. (30)

In addition, the following upper bound can be provided for Q3

Q3 ≤ ω2

¯τ + τ

sup

sϵt−( ¯τ+τ),t

t

s∥u (θ)∥2 ds,

≤ ω2

¯τ + τ

t

t−( ¯τ+τ)∥u (θ)∥2 dθ. (31)

By applying Leibniz Rule, the time derivatives of (22)–(24) can beobtained as

Q1 =

ω1k

2αϵ1

∥r∥2

− ∥rτ∥2 , (32)

Q2 =

k (α + ϕ − 1)

2k α (1 − ϕ)

∥r∥2

− (1 − τ ) ∥rτ∥2 , (33)

Q3 = ω2

¯τ + τ

k ∥r∥2

−

t

t−( ¯τ+τ)∥u (θ)∥2 dθ

. (34)

Based on (30), the following inequalities can be developed:

min12,ω1

2

∥y∥2

≤ V (y) ≤ max1,

ω1

2

∥y∥2 . (35)

The time derivative of the first term in (30) can be obtained byusing (4)–(6), (11), and the definition of ei in (7) for i = n, as

ni=1

eTi ei = −

n−1i=1

eTi ei − eTnΛen + eTn−1en − eTnαeu + eTnr. (36)

By using (8), (14), (32)–(34), and (36), the time derivative of (30)can be determined as

V = −

n−1i=1

eTi ei − eTnΛen + eTn−1en − eTnαeu + eTnr

+ rT (−αkr + (α − I + τ I) krτ + α (uτ − uτ ))

+ rTN + Nr − en

+ ω1eTu (krτ − kr)

9 To achieve a small tracking error for the case of a large value of ζNr (i.e., fastdynamics with large disturbances), large gains, small delay, and a better estimateof the delay are required.10 By choosingα close to 1−ϕ, sufficiently smallω1 and ϵ1 , and a sufficiently largeΛ, the gain conditions can be expressed in terms of k, τ , ¯τ , ϕ. The gain k can thenbe selected provided τ , ¯τ , ϕ are small enough.

226 S. Obuz et al. / Automatica 76 (2017) 222–229

+

ω1k

2αϵ1

∥r∥2

− ∥rτ∥2+

k (α + ϕ − 1)

2kα (1 − ϕ)

∥r∥2

− (1 − τ ) ∥rτ∥2+ ω2

¯τ + τ

k2 ∥r∥2

−

t

t−( ¯τ+τ)∥u (θ)∥2 dθ

. (37)

After canceling common terms and using Assumption 4, theexpression in (37) can be upper bounded as

V ≤ −

n−1i=1

∥ei∥2− Λ ∥en∥2

+eTn−1en

+ αeTneu

+ αrT (uτ − uτ )

− α k ∥r∥2

+ rT N + ∥r∥ ζNr + k |α + ϕ − 1|rT rτ

+ ω1k (∥eu∥ ∥rτ∥ + ∥eu∥ ∥r∥) +

ω1k

2αϵ1

∥r∥2

− ∥rτ∥2+

k (α + ϕ − 1)

2k α (1 − ϕ)

∥r∥2

− (1 − τ ) ∥rτ∥2+ ω2

¯τ + τ

k2 ∥r∥2

−

t

t−( ¯τ+τ)∥u (θ)∥2 dθ

. (38)

UsingYoung’s Inequality the following inequalities can be obtainedeTneu ≤12ϵ1

∥en∥2+

ϵ1

2∥eu∥2 , (39)eTn−1en

≤ϵ2

2∥en−1∥

2+

12ϵ2

∥en∥2 , (40)rT (uτ − uτ ) ≤

12

∥r∥2+

12

∥uτ − uτ∥2 . (41)

After completing the squares for the cross terms containing r andrτ , substituting the time derivative of (9) and (18), (39)–(41) into(38), and using Assumption 4, the following upper bound can beobtained

V ≤ −

n−2i=1

∥ei∥2−

1 −

ϵ2

2

∥en−1∥

2

−

Λ −

α

2ϵ1+

12ϵ2

∥en∥2

+ αϵ1 ∥eu∥2−

α k8

∥r∥2

−

α k8

− κ

∥r∥2

+1

α kρ2(∥z∥) ∥z∥2

+1

α kζ 2Nr

+α ∥uτ − uτ∥

2

2− ω2

t

t−( ¯τ+τ)∥u (θ)∥2 dθ, (42)

where κ ,2(ω1 k)

2

αϵ1+

(k(α+ϕ−1))2

α k(1−ϕ)+ ω2

¯τ + τ

k2 +

α2 . The

Cauchy–Schwarz inequality is used to develop the following upperbound

∥eu∥2≤ τ

t

t−τ

∥u(θ)∥2 dθ. (43)

Note that using Assumption 4, the inequalities tt−τ

∥u (θ)∥2 dθ ≤

k2 tt−( ¯τ+τ) ∥r (θ)∥2 dθ and

tt−τ

∥u (θ)∥2 dθ ≤ k2 tt−( ¯τ+τ) ∥r (θ)∥2

dθ can be obtained. In addition, using the expressions in (22), (23),(31) and (43), the following inequalities can be obtained

−ω2

4τ∥eu∥2

≥ −ω2

4

t

t−( ¯τ+τ)∥u (θ)∥2 dθ, (44)

−ω2k2αϵ1

4ω21 k2

Q1 ≥ −ω2

4

t

t−( ¯τ+τ)∥u (θ)∥2 dθ, (45)

−ω2k3 α (1 − ϕ)Q2

4k (α + ϕ − 1)

2 ≥ −ω2

4

t

t−( ¯τ+τ)∥u (θ)∥2 dθ, (46)

−1

4¯τ + τ

Q3 ≥ −ω2

4

t

t−( ¯τ+τ)∥u (θ)∥2 dθ. (47)

By using (44)–(47), (42) can be upper bounded as

V ≤ −

n−2i=1

∥ei∥2−

1 −

ϵ2

2

∥en−1∥

2

−

Λ −

α

2ϵ1+

12ϵ2

∥en∥2

−

ω2

4τ− αϵ1

∥eu∥2

−αk8

∥r∥2

−

α k8

− κ

∥r∥2

+1

α kρ2(∥z∥) ∥z∥2

+1

α kζ 2Nr

+α ∥uτ − uτ∥

2

2

−ω2k2αϵ1

4ω21 k2

Q1 −ω2k3 α (1 − ϕ)

4k (α + ϕ − 1)

2Q2

−1

4¯τ + τ

Q3. (48)

Note that the Mean Value Theorem can be used to obtain the in-equality ∥uτ − uτ∥ ≤

u Θ t, τ

|τ |, where Θt, τ

is a point

in time between t − τ and t − τ . Furthermore, using the gainconditions in (29), the definition of σ in (20), and the inequality∥y∥ ≥ ∥z∥, the following upper bound can be obtained

V ≤ −

σ

2−

1α k

ρ2 (∥y∥)

∥z∥2−

σ

2∥z∥2

+1

α kζ 2Nr

+α ¯τ

2 u Θ t, τ

22

−ω2k2αϵ1

4ω21 k2

Q1 −ω2k3 α (1 − ϕ)

4k (α + ϕ − 1)

2Q2 −1

4¯τ + τ

Q3.

(49)

Provided y (η) ∈ D ∀η ∈ [t0, t], then from the definition of δ in(21), the expression in (49) reduces to

V ≤ −δ ∥y∥2 , ∀ ∥y∥ ≥

2ζ 2Nr

+ α kα ¯τ2M2

2α kδ

12

. (50)

Using techniques similar to Theorem 4.18 in Khalil (2002) it canbe concluded that y is uniformly ultimately bounded in the sense

that lim supt→∞ ∥y (t)∥ ≤

max

1, ω1

2

2ζ 2

Nr+α kα ¯τ

2M2

min12 ,

ω12

2α kδ

provided

y (t0) ∈ SD , where uniformity in initial time canbe concluded fromthe independence of δ and the ultimate bound from t0.

Remark 3. If the system dynamics are such that N is linear in z,then the function ρ can be selected to be a constant, i.e., ρ (∥z∥) =

ρ, ∀z ∈ R(n+2)m for some known ρ > 0. In this case, the last

S. Obuz et al. / Automatica 76 (2017) 222–229 227

sufficient condition in (29) reduces to

k ≥2ρ2

σα, (51)

and the result is global in the sense that D = SD = R(n+2)m+3.

Since ei, r, eu ∈ L∞, i = 1, 2, 3, . . . , n, from (2), u ∈ L∞.An analysis of the closed-loop system shows that the remainingsignals are bounded.

5. Simulation results

To illustrate performance of the developed controller, numer-ical simulations were performed on a two-link revolute, direct-drive robot11 with the following dynamicsu1τ

u2τ

=

p1 + 2p3c2 p2 + p3c2p2 + p3c2 p2

x1x2

+

−p3s2x2 −p3s2 (x1 + x2)p3s2x1 0

x1x2

+

fd1 00 fd2

x1x2

+

d1d2

, (52)

where x, x, x ∈ R2. Additive disturbances are applied as d1 = 0.2sin (0.5t) and d2 = 0.1 sin (0.25t). Additionally, p1 = 3.473 kgm2,p2 = 0.196 kg m2, p3 = 0.242 kg m2, p4 = 0.238 kg m2,p5 = 0.146 kg m2, fd1 = 5.3 Nm s, fd2 = 1, 1 Nm s, and s2, c2denote sin (x2), and cos (x2), respectively.

The initial conditions for the system are selected as x1, x2 = 0.The desired trajectories are selected as

xd1 (t) = (30 sin (1.5t) + 20)1 − e−0.01t3

,

xd2 (t) = − (20 sin (t/2) + 10)1 − e−0.01t3

.

Several simulation results were obtained using various time-varying delays and different estimated delays, shown in Table 1,to demonstrate performance of the developed continuous robustcontroller. Cases 1 and 2 use a high-frequency, low-amplitudeoscillating delay for different delay upper-bound estimates. Cases3 use a low-frequency, high-amplitude oscillating delay for knownand unknown delay. Case 4 uses random, uniformly distributeddelay between 0 and 120 ms for each time instance.

The controller in (9) and (10) is implemented for each case. Thecontrol gains are selected for Cases 1 and 2 as α = diag{1, 1}, Λ =

diag{50, 20.5}, and k = diag{60, 6}, and for Cases 3 and 4 asα = diag{1, 1}, Λ = diag{23, 8}, and k = diag{140, 2.75}. Theroot mean square (RMS) errors obtained for each case are listedin Table 1. By comparing the RMS error for Cases 1 and 2, it isclear that selecting a delay estimate closer to the actual upperbound of the unknown delay yields better tracking performance.Case 3 demonstrates that the performance of the developedcontroller using a constant estimate of the large unknown delayis comparable to the controller in Fischer, Kamalapurkar et al.(2012) which uses exact knowledge of the time-varying delay.12Additionally, the developed controller is reasonably robust evenfor a constant estimate of the delay when the actual delay islong and time-varying. Although the stability analysis in Section 4assumes a slowly time-varying input delay, Case 4 shows that the

11 Provided the inertia matrix is known, the dynamics in (52) can be describedusing (1) (Fischer et al., 2013).12 The RMS errors for x1 and x2 obtained using the controller in Fischer,Kamalapurkar et al. (2012) for the same delay as Case 3 were 0.7544° and 0.9722°,respectively.

Table 1RMS errors for time-varying time-delay rates and magnitudes.

τi (t) (ms) τ (t) (ms) RMS Errorx1 x2

Case 1 10 sin (5t) + 10 15 0.1315° 0.1465°Case 2 10 sin (5t) + 10 100 0.4774° 0.2212°Case 3 60 sin (t) + 60 75 0.7479° 0.9928°Case 4 rand [0, 120] 75 0.7476° 1.0068°

Fig. 1. Tracking errors, control effort and time-varying delays vs. time for Case 1.

Fig. 2. Tracking errors, control effort and time-varying delays vs. time for Case 3.

developed controller still provides good tracking performance fora discontinuous, high frequency input delay. Results in Figs. 1–3depict the tracking errors, control effort, time-varying delays andestimated delays for Cases 1, 3 and 4, respectively.

6. Conclusion

Novelty of the controller comes from the fact that a continuousrobust controller is developed for a class of uncertain nonlinearsystems with additive disturbances subject to uncertain time-varying input time delay. A filtered tracking error signal is designedto facilitate the control design and analysis. A Lyapunov-basedanalysis is used to prove ultimate boundedness of the errorsignals through the use of Lyapunov–Krasovskii functionals that

228 S. Obuz et al. / Automatica 76 (2017) 222–229

Fig. 3. Tracking errors, control effort and time-varying delays vs. time for Case 4.

are uniquely composed of an integral over the estimated delayrange rather than the actual delay range. Simulation resultsindicate the performance of the controller over a range of timevarying delays and estimates. The results even illustrate robustnessto random delays up to 120 ms. Improved performance maybe obtained by altering the design to allow for a time-varyingestimate of the delay. Result such as Herrera, Ibeas, Alcántara, dela Sen, and Serna-Garcés (2013) provides possible insights intoestimating/identifying delays in future work.

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Serhat Obuz received Bachelors degree in Electrical andElectronics Engineering from Inonu University, Turkey, in2007 and his Masters degree in Electrical and ComputerEngineering from Clemson University in 2012. His workhas been recognized by IEEEMulti-Conference on Systemsand Control (M.Sc.) and was awarded Best Student PaperAward in 2015. He was also awarded a scholarship fromthe Turkish Ministry of National Education for masterand doctoral studies in the US. His research is focusedon designing robust controller for a class of uncertainnonlinear systems subject to unknown time-varying input

delay. Serhat is currently pursuing a Ph.D. at the University of Florida under theadvisement of Dr. Dixon.

Justin R. Klotz received the Ph.D. degree in mechanicalengineering from the University of Florida, Gainesville,FL, USA, in 2015, where he was awarded the Science,Mathematics and Research for Transformation (SMART)Scholarship, sponsored by the Department of Defense. Hisresearch interests include the development of Lyapunov-based techniques for reinforcement learning-based con-trol, switching control methods, delay-affected control,and trust-based cooperative control.

Rushikesh Kamalapurkar received his M.S. and his Ph.D.degree in 2011 and 2014, respectively, from the Mechani-cal and Aerospace Engineering Department at the Univer-sity of Florida. After working for a year as a postdoctoralresearch fellow with Dr. Warren E. Dixon, he was selectedas the 2015–16MAE postdoctoral teaching fellow. In 2016he joined the School of Mechanical and Aerospace Engi-neering at the Oklahoma State University as an Assistantprofessor. His primary research interest has been intel-ligent, learning-based control of uncertain nonlinear dy-namical systems. His work has been recognized by the

2015 University of Florida Department of Mechanical and Aerospace EngineeringBest Dissertation Award, and the 2014University of Florida Department ofMechan-ical and Aerospace Engineering Outstanding Graduate Research Award.

Warren Dixon received his Ph.D. in 2000 from theDepartment of Electrical and Computer Engineering fromClemson University. He was selected as an Eugene P.Wigner Fellow at Oak Ridge National Laboratory (ORNL).In 2004, he joined the University of Florida in theMechanical and Aerospace Engineering Department. Hismain research interest has been the development andapplication of Lyapunov-based control techniques foruncertain nonlinear systems. He has co-authored severalbooks, over a dozen chapters, and approximately 130journal and 230 conference papers. His work has been

recognized by the 2015 & 2009 American Automatic Control Council (AACC) O.Hugo Schuck (Best Paper) Award, the 2013 Ferd Ellersick Award for Best OverallMILCOM Paper, a 2012–2013 University of Florida College of Engineering DoctoralDissertationMentoring Award, the 2011 American Society of Mechanical Engineers(ASME) Dynamics Systems and Control Division Outstanding Young InvestigatorAward, the 2006 IEEE Robotics and Automation Society (RAS) Early AcademicCareer Award, an NSF CAREER Award (2006–2011), the 2004 Department of EnergyOutstandingMentor Award, and the 2001 ORNL Early Career Award for EngineeringAchievement. He is an ASME Fellow and IEEE Fellow, an IEEE Control SystemsSociety (CSS) Distinguished Lecturer, and served as the Director of Operationsfor the Executive Committee of the IEEE CSS Board of Governors (2012–2015).He currently serves as a member of the US Air Force Science Advisory Board.He is currently or formerly an associate editor for ASME Journal of DynamicSystems, Measurement and Control, Automatica, IEEE Control Systems Magazine,IEEE Transactions on Systems Man and Cybernetics: Part B Cybernetics, and theInternational Journal of Robust and Nonlinear Control.