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 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT Adaptive Sliding Mode Control of a Class of Nonlinear Systems with Articial Delay Spandan Roy ∗, Indra Narayan Kar a, a CR IP T a Electrical Engineering, Indian Institute of Technology Delhi, New Delhi, India Abstract In this paper, an adaptive-robust control (ARC) strategy, christened as Adap- AN US 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 M switching gain of the conventional ARC methodologies suer from over- and under-estimation problem. The novel adaptive law of ATSMC alleviates the ED 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- PT sults of the proposed methodology using a nonholonomic wheeled mobile robot (WMR) is presented and improved tracking accuracy of the proposed control CE law is noted compared to time-delayed control and conventional adaptive sliding mode control. Keywords: Adaptive sliding mode control, time-delayed control, Razumikhin AC 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 ACCEPTED MANUSCRIPT 1. Introduction Adaptive control and Robust control are the two popular control strategies to deal with uncertain nonlinear systems. In general, adaptive control 5 CR IP T 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. AN US 10 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 M 15 accuracy [4]. Higher order sliding mode [5] can alleviate the chattering problem but prerequisite of uncertainty bound still exists. ED 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- PT 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- CE 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 AC 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 ACCEPTED MANUSCRIPT 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 35 CR IP T 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] AN US 40 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. M Considering the constraints of adaptive and robust control, recently global research is reoriented towards adaptive-robust control (ARC). The series of pub- ED 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 PT 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 CE 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 AC 55 increasing (resp. decreasing) even if tracking error decreases (resp. increases) and this creates overestimation (resp. underestimation) problem of switching gain [35]. 3 ACCEPTED MANUSCRIPT CR IP T 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. AC CE 80 AN US 75 M 70 ED 65 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. PT 60 85 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; 4 ACCEPTED MANUSCRIPT 90 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 CR IP T identity matrix of appropriate dimension. In general, an Euler-Lagrange system with second order nonlinear dynamics can be written as (1) AN US 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] M unmodelled dynamics and disturbances are subsumed by H . The control input τ = M̂ u + Ĥ, (2) ED 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 PT the time-delayed logic ([6], [31], [37]) and the system denition (1) as Ĥ(q, q̇) ∼ = H(qh , q̇h ) = τh − M̂h q̈h , (3) CE where h is a xed small delay time and τh = τ (t − h), M̂h = M̂ (q(t − h)), qh = AC 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), 5 (5) ACCEPTED MANUSCRIPT 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) CR IP T 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 AN US 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 ED M 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 95 PT 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. CE The time-delayed error (TDE) σ1 remains bounded if M̂ is selected such that [6], [31]: AC 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]. 6 ACCEPTED MANUSCRIPT 100 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. CR IP T 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 110 AN US 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. M Assumption 1. Let V (e) be a Lyapunov function candidate. Then following the Razumikhin-type theorem, the following inequality holds [38]: ED Assumption 2. V (e(ν)) < rV (e(t)), t − 2h ≤ ν ≤ t, (11) PT where r > 1 is a constant. Assumption 2 from the Razumikhin theorem [38] is a well established tool for 115 CE 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) AC V (e(ν)) < rV (e(t)). These two conditions are depicted in Fig. 1. 120 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 7 ACCEPTED MANUSCRIPT V (e(t )) CR IP T V (e(t )) V (e( )) V (e( )) V (e( )) rV (e(t )) AN US 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 125 analysis. M 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)) ED 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) 130 then the error trajectories will be contained within the ball V (e(t)) and the PT system would remain stable. The detailed physical interpretation of Razumikhin theorem and its relation with conventional Lyapunov stability is provided in [38]. CE 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 AC Theorem 1. h< λmin (Q) := h̄max , ||E|| (12) ∀h, (13) if controller gains K1 and K2 are selected such that λmin (Q) > h||E|| 8 ACCEPTED MANUSCRIPT 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 CR IP T 135 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 AN US −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) M 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 ED 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 PT 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. CE AC 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 + 9 (20) ACCEPTED MANUSCRIPT −2 Z 0 −h eT P B1 [Bσ1 (t + θ)] dθ ≤ Z 0 [βeT P B1 P −1 −h (21) CR IP T × 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 AN US 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|| M be found from (23) as h< (24) Let Ψ = Q − hE > 0 and the uncertainties are bounded. So, V̇ (e) < 0 would be ED established if λmin (Ψ)||e||2 > 2Γ1 + 2||B T P ||||σ1 ||||e||. Thus (4) would be UUB with the error bound PT ||e|| = µ0 + ||B T P ||||σ1 || λmin (Ψ) 2Γ1 + µ20 := $0 , λmin (Ψ) (25) . Let Ξ denote the smallest level surface of V containing CE 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. AC 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. 10 ACCEPTED MANUSCRIPT 150 155 CR IP T AN US 145 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. M 140 2.2. Adaptive Time-delayed Sliding Mode Controller ED 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 160 PT 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- CE 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], AC [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 11 ACCEPTED MANUSCRIPT 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, CR IP T the adaptive law of ATSMC addresses the overestimation-underestimation issue of the switching gain. Before stating the control structure of ATSMC, the 175 following assumption is made: Let there exist constant matrices P̄ > 0, Q̄ > 0 and K > 0 such that the following conditions hold: Assumption 3. 180 AN US P̄1 P̄2T , where P̄1 > 0, P̄2 > 0, P̄3 > 0, (i) P̄ = P̄2 P̄3 0 I is Hurwitz, where Ω = P̄3−1 P̄2 , (ii) Ā = −K −2Ω (iii) ĀT P̄ + P̄ Ā = −Q̄. M To facilitate the ATSMC controller design, the system dynamics (1) is rewritten as ED 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), PT as (27) where û is the equivalent control input and ∆u is the switching control input CE which negotiates the uncertainties. These terms would be dened later individ- AC 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 . 12 (29) ACCEPTED MANUSCRIPT The equivalent control û in (27) is designed as (30) û = q̈ d − Ωė1 − fˆ, CR IP T 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) AN US 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) M 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: ED ë1 = −Ke1 − Ωė1 + P̄3−1 ṡ + Ke1 = −Ke1 − Ωė1 − Ωė1h + ∆u + ∆uh + σ2 + Ke1 . (33) PT 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 IP T ||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 14 ACCEPTED MANUSCRIPT ||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) CR IP T ∀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 . 15 ACCEPTED MANUSCRIPT 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) CR IP T 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 16 ACCEPTED MANUSCRIPT Case (i): ĉ > γ, Using (32) we have ||s|| ≥ and G(e) > 0 (43) CR IP T ˙ 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. M 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) CR IP T 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|| 18 ACCEPTED MANUSCRIPT 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 US 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. M 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 CR IP T $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 IP T 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 20 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 CR IP T 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 21 ACCEPTED MANUSCRIPT 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 US CR IP T 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) CR IP T ACCEPTED MANUSCRIPT m̄ 0 AN US 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. 23 ACCEPTED MANUSCRIPT 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 T 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 US 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 ACCEPTED MANUSCRIPT 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 IP T 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 ED 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 PT 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 CE 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 AC 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 ACCEPTED MANUSCRIPT 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 US ||τ|| (Nm) 0.3 CR 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 M 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. ED 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 PT 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 %̂ CE 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 ACCEPTED MANUSCRIPT 0.25 0.15 0.1 0 5 10 15 20 time (sec) 25 30 0.06 CR IP 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 ED 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 ACCEPTED MANUSCRIPT 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 CR IP T 0.1 15 20 time (sec) 25 30 35 Figure 8: Switching gain properties for ATSMC. M 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 ED 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 PT 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, CE 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 ACCEPTED MANUSCRIPT 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) CR IP T 0.098 0.1 0.018 0 1 time (sec) 0.017 22 2 23 time (sec) 24 M Figure 9: Switching gain evaluation of ATSMC for right wheel (magnied). 0.104 0.1 0.103 PT 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 ACCEPTED MANUSCRIPT 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 CR IP T APE 350 AN US 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 M eect on the error bounds. The choice of sampling interval and controller gains are crucial for the per- ED 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- PT 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. 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