Robust Time-Delayed Control of a Class of Uncertain Nonlinear Systems

Robust Time-Delayed Control of a Class of Uncertain Nonlinear Systems

4th International Conference on Advances in Control and Optimization of Dynamical Systems 4th International Conference on Advances in Control and Opti...

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4th International Conference on Advances in Control and Optimization of Dynamical Systems 4th International Conference on Advances in Control and Optimization of Dynamical Systems 4th International Conference on Advances in Control and February 1-5, of 2016. NIT Tiruchirappalli, India Optimization Dynamical Systems February 1-5, 2016. NIT Tiruchirappalli, India online at www.sciencedirect.com Optimization Dynamical Systems Available February 1-5, of 2016. NIT Tiruchirappalli, India February 1-5, 2016. NIT Tiruchirappalli, India

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IFAC-PapersOnLine 49-1 (2016) 736–741 Robust Time-Delayed Control of a Class Robust Time-Delayed Control of a Class Robust Time-Delayed Control of a Class RobustUncertain Time-Delayed Control of a Class Nonlinear Systems Uncertain Nonlinear Systems Uncertain Nonlinear Systems Uncertain Nonlinear Systems

of of of of

Spandan Roy and Indra Narayan Kar Spandan Spandan Roy Roy and and Indra Indra Narayan Narayan Kar Kar Spandan Roy and Indra Narayan Kar Indian Institute of of Technology Delhi, Delhi, New New Delhi, India India (e-mail: Indian Indian Institute Institute of Technology Technology Delhi, New Delhi, Delhi, India (e-mail: (e-mail: [email protected], [email protected]). [email protected], [email protected]). Indian Institute of Technology Delhi, New Delhi, India (e-mail: [email protected], [email protected]). [email protected], [email protected]). Abstract: In this paper, the tracking control problem of class of uncertain systems Abstract: In In this this paper, paper, the the tracking tracking control control problem problem of of aaa class class of of uncertain uncertain nonlinear nonlinear systems systems Abstract: nonlinear is addressed. The conventional Time-Delayed Control (TDC) approach, as reported in literature, is addressed. The conventional Time-Delayed Control (TDC) approach, as reported in literature, Abstract: In this paper, the tracking control problem of a class of uncertain nonlinear systems is addressed. The conventional Time-Delayed Control (TDC) approach, as reported inThis literature, does not provide any selection criterion for sampling interval and controller gains. design does not provide provide any selection selection Time-Delayed criterion for for sampling sampling intervalapproach, and gains. design is addressed. The conventional Control (TDC) as reported inThis literature, does not any criterion interval and controller controller gains. This design issue of TDC is solved in this paper through a new stability approach based on the Lyapunovissue of TDC is solved in this paper through a new stability approach based on the Lyapunovdoes not provide any selection criterion for sampling interval and controller gains. This design issue of TDC is solved inathis paper between through sampling a new stability approach based ongains the LyapunovKrasvoskii method and relation interval and controller has been Krasvoskii method andinaathis relation between sampling interval and controller controller gains has been been issue of TDC is solved paper between through sampling a new stability approach based ongains the LyapunovKrasvoskii method and relation interval and has established. Moreover, a new Robust Time-Delayed Control (RTDC) strategy has been proposed established. Moreover, a new Robust Time-Delayed Control (RTDC) strategy has been proposed Krasvoskii method and a relation between sampling interval and controller gains has been established. Moreover, a new Robust Time-Delayed Control (RTDC) strategyofhasuncertainties been proposed to negotiate the time-delayed error that arises from the approximation in to negotiate negotiateMoreover, the time-delayed time-delayed errorTime-Delayed that arises arises from from the (RTDC) approximation ofhasuncertainties uncertainties in established. a new Robust Control strategyof been proposed to the error that the approximation in TDC. The RTDC methodology, unlike the existing control methodologies, does not require TDC. The RTDC methodology, unlike the existing control methodologies, does not require to negotiate the time-delayed error that arises from the approximation of uncertainties in TDC. The RTDC methodology, unlike the existing control methodologies, does not require any predefined bound of the uncertainties. The switching gain of RTDC attempts to circumvent any predefined bound of the uncertainties. The switching gain of RTDC attempts to circumvent TDC. The RTDC methodology, unlike the existing control methodologies, does not require any predefined bound of the uncertainties. The switching gain of RTDC attempts to circumvent the approximation error through the control input and feedback information of past instances. the approximation approximation error through the control control input and feedback feedback information of past instances. any predefined bound of the uncertainties. The switching gain of RTDC attempts to circumvent the error through the input and information of past instances. The maximum allowable length of past data is evaluated from the stability analysis of TDC. The maximum allowable length of past data is evaluated from the stability analysis of TDC. TDC. the approximation error through the control input and feedback information of past instances. The maximum allowable length of past data is evaluated from the stability analysis of Experimental result of the proposed methodology using a nonholonomic wheeled mobile robot Experimental result of the proposed methodology using a nonholonomic wheeled mobile robot The maximum allowable length of past data is evaluated from the stability analysis of TDC. Experimental a nonholonomic wheeled mobile robot result ofand the improved proposed methodology using of (WMR) is presented tracking accuracy the proposed control law is noted (WMR) is presented and improved tracking accuracy of the proposed control law is noted Experimental result of the proposed methodology using a nonholonomic wheeled mobile robot (WMR) istopresented and control. improved tracking accuracy of the proposed control law is noted compared time-delayed comparedisto topresented time-delayed control. (WMR) and control. improved tracking accuracy of the proposed control law is noted compared time-delayed compared to (International time-delayedFederation control. of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. © 2016, IFAC Keywords: Euler-Lagrange system, robust control, time-delayed control, uniformly Keywords: Euler-Lagrange system, robust robust control, control, time-delayed time-delayed control, control, uniformly uniformly ultimately ultimately Keywords: Euler-Lagrange system, ultimately bounded, wheeled mobile robot. bounded, wheeled mobile robot. Keywords: Euler-Lagrange system, robust control, time-delayed control, uniformly ultimately bounded, wheeled mobile robot. bounded, wheeled mobile robot. 1. In 1. INTRODUCTION INTRODUCTION In general, general, time time delay delay has has pervasive pervasive effect effect on on system system perper1. INTRODUCTION In general, time delay has pervasive effect on system performance and many controllers have been developed to formance and many controllers have been developed to 1. INTRODUCTION In general, time delay has pervasive effect on system performance and many controllers have been developed to compensate that effect (Mukhija et al. (2014)). However, compensate that effect (Mukhija et al. (2014)). However, Design of an efficient controller for nonlinear systems formance and many controllers have been developed to Design of of an an efficient efficient controller controller for for nonlinear nonlinear systems systems compensate that effect(TDC), (Mukhija et al. (2014)). Time-Delayed Control as in and Design Time-Delayed Control as reported reported in Hsia HsiaHowever, and Gao Gao subjected to parametric and uncertainties compensate that effect(TDC), (Mukhija et al. (2014)). However, subjected and nonparametric nonparametric uncertainties Design of toanparametric efficient controller for nonlinear systems Time-Delayed Control (TDC), as reported in Hsia and Gao (1990), Chang et al. (1996), Shin and Kim (2009), Roy (1990), Chang et al. (1996), Shin and Kim (2009), Roy subjected to parametric and nonparametric uncertainties has always been a challenging task. Among many other Time-Delayed Control (TDC), as reported in Hsia and Gao has alwaystobeen a challenging task. Amonguncertainties many other (1990), subjected parametric and nonparametric Chang et al. (1996), Shin and Kim (2009), Roy et al. (2015) represents all the uncertain terms by a single et al. (2015) represents all the uncertain terms by a single has always been a challenging task. Among many other approaches, Adaptive control and Robust control are the (1990), Chang et al. (1996), Shin and Kim (2009), Roy approaches, Adaptive control and Robust control the et al. (2015) represents all thethat uncertain terms by acontrol single has always been a challenging task. Among manyare other function and approximates function using function and represents approximates function using approaches, Adaptive control and Robust controlhave are the two popular control strategies that researchers exet al. (2015) all thethat uncertain terms by acontrol single two popular control strategies that researchers have exapproaches, Adaptive control and Robust control are the function and approximates that function using control input and state information of the immediate past time two popular controlwhile strategies that researchers have ex- function input and state information of the immediate past time tensively employed dealing with uncertain nonlinear and approximates that function using control tensively employed dealingthat withresearchers uncertain nonlinear two popular controlwhile strategies have ex- input and state information of the immediate past time instant. The advantage of this robust control approach instant. The advantage of this robust control approach tensively employed while dealing with uncertain nonlinear systems. In general, adaptive control uses predefined painput and state information of the immediate past time systems. In general,while adaptive control predefined pa- is that itThe tensively employed dealing with uses uncertain nonlinear advantage of this robust controlthe is easy to implement and reduces burden is that that it itThe is easy easy to implement implement and reduces reduces theapproach burden systems. In general, laws adaptive control uses principle predefined pa- instant. rameter adaptation and equivalence based instant. advantage of this robust controlthe approach rameter adaptation adaptation laws and equivalence equivalence principle based is to and burden systems. In general, laws adaptive control uses principle predefined pa- is of tedious modelling of complex system to a great exrameter and based of tedious modelling of complex system to a great excontrol law which adjusts the parameters of the controller is that it is easy to implement and reduces the burden control law law which adjusts adjusts the parameters parameters ofprinciple the controller controller rameter adaptation laws and equivalence of based of tedious modelling of complex system to a great extent. In spite of this, the unattended approximation error, control which the the tent. In spite of this, the unattended approximation error, on the fly according to the pertaining uncertainties (Kristic of tedious modelling of complex system to a great exon the fly according to the pertaining uncertainties (Kristic commonly control law which adjusts the parameters of the controller In spite of this,as unattended approximation error, termed time-delayed error (TDE) causes commonly termed asthe time-delayed error (TDE) (TDE) causes on the (1995)). fly according to thethis pertaining uncertainties (Kristic tent. et However, approach has tent. In spite of this,as thetime-delayed unattended approximation error, et al. al. However, approach has poor poor transient transient termed error causes on the (1995)). fly according to thethis pertaining uncertainties (Kristic commonly detrimental effect to the performance of the closed loop detrimentaltermed effect to toasthe the performance of the the closedcauses loop et al. (1995)). However, this approach has poor transient performance and online calculation of the unknown system commonly time-delayed error (TDE) performance and online calculation of the unknown system detrimental effect performance of closed loop et al. (1995)). However, this approach has poor transient system and its stability. In this front, a few work have performance and online calculation of the unknown system system and its stability. In this front, a few work have parameters and controller gains for complex systems is detrimental effect to the performance of the closed loop parameters and and controller gains for for complex systems is system performance and controller online calculation of the unknown system and its this front, few work have been out to TDE which internal parameters gains complex systems is been carried carried outstability. to tackle tackleIn which aaincludes includes internal computationally intensive et al. (2008)). Whereas, system and its stability. InTDE this front, few work have computationally intensive (Liu (Liu al. complex (2008)).systems Whereas, parameters and controller gainsetfor is been carried out to tackle TDE which includes internal model (Cho et al. (2009)), gradient estimator (Han and modelcarried (Cho et etout al.to(2009)), (2009)), gradient estimator (Han and computationally intensive (Liu etthe al. uncertainties (2008)). Whereas, robust control aims at tackling of the been tackle TDE which includes internal robust control aims at tackling the uncertainties of the model (Cho al. gradient estimator (Han and computationally intensive (Liu et al. (2008)). Whereas, Chang (2010)), ideal velocity feedback (Jin et al. (2008)), Chang (2010)), ideal velocity feedback (Jin et al. (2008)), robust control aims at tackling the uncertainties of the system within an uncertainty bound defined a priori. It model (Cho et al. (2009)), gradient estimator (Han and system control within an an uncertainty bound defined aa priori. priori. It Chang (2010)), ideal(Jin velocity feedback (Jin et al. (2008)), robust aims at tacklingbound the uncertainties of the nonlinear damping et (2013)) and mode system within uncertainty defined It nonlinear damping et al. al.feedback (2013))(Jin andetsliding sliding mode reduces computation complexity to a great extent for Chang (2010)), ideal(Jin velocity al. (2008)), reduces computation complexity to a great extent for system within an uncertainty bound defined a priori. It nonlinear damping (Jin et al. (2013)) and sliding mode based approach (Chang and Park (2003), Roy et al. reduces computation complexity to a great extent for based approach (Chang and Park (2003), Roy et al. complex systems compared to adaptive control as exclusive nonlinear damping (Jin et al. (2013)) and sliding mode complex systems compared to adaptive exclusive reduces computation complexity to acontrol great asextent for based approach (Chang and Park (2003), Roy et al. (2014)). The stability of the closed loop system in Cho (2014)). The stability of the closed loop system in Cho complex systems compared to adaptive control as exclusive online estimation of uncertain parameters is not required based approach (Chang and Park (2003), Roy et al. online estimation of uncertain parameters is not required (2014)). The stability ofChang the closed loop system in Cho complex systems compared to adaptive control as exclusive et al. (2009), Han and (2010) Jin et al. (2008), et al. al. (2009), (2009), Han and andofChang Chang (2010) Jinsystem et al. al. in (2008), online estimation of uncertain parameters isnominal not required (Corless and Leitmann (1981)). However, mod(2014)). The stability the closed loop Cho (Corless and Leitmann (1981)). However, nominal modet Han (2010) Jin et (2008), online estimation of uncertain parameters is not required Chang et al. (1996), Shin and Kim (2009) depends (Corless anduncertainties Leitmann (1981)). However, nominal modChang et al. (1996), Shin and Kim (2009) depends elling of the is necessary to decide upon their et al. (2009), Han and Chang (2010) Jin et al. (2008), elling of the is necessary to decide upon modtheir on the boundedness (Corless anduncertainties Leitmann (1981)). However, nominal et al. (1996), Shin and Kim in (2009) depends of as Hsia Gao on the boundedness of TDE TDE as shown shown Hsia and and Gao elling of the uncertainties is necessary decide to upon their Chang bounds, which is possible. Again, increase Chang et al. (1996), Shin and Kim in(2009) depends bounds, which is not not always always possible.to increase elling of the uncertainties is necessary toAgain, decide to upon their on the boundedness of TDE as shown in Hsia and Gao (1990). This method approximates the continuous time (1990). This method approximates the continuous time bounds, which is not always possible. Again, to increase the operating region of the controller, often higher unceron the boundedness of TDE as shown in Hsia and Gao the operating the controller, higher uncer- (1990). Thissystem methodin approximates thewithout continuous time bounds, whichregion is notofalways possible. often Again, to increase closed aa discrete considerclosed loop loop discrete form form considerthe operating region of the controller, often higher uncer- (1990). tainty bounds are assumed. This in turn leads to problems Thissystem methodin approximates thewithout continuous time tainty bounds are assumed. This in turn leads to problems the operating region of the controller, often higher uncerclosed loop system in a discrete form without considering the effect of discretization error. Again, the stability tainty bounds are assumed. This consequent in turn leadspossibility to problems ing the effect of discretization error. Again, the stability like higher controller gain and of closed loop system in a discrete form without considerlike higher controller gain and of criterion tainty bounds are assumed. This consequent in turn leadspossibility to problems the effect of discretization stability mentioned in Gao (1990) restricts the criterion mentioned in Hsia Hsia and anderror. Gao Again, (1990) the restricts the like higher for controller gain and possibility of ing chattering the law based controller ing the effect of discretization error. Again, the stability chattering the switching switching lawconsequent based robust robust controller like higher for controller gain and consequent possibility of criterion mentioned in Hsia and Gao (1990) restricts the allowable range of perturbation and thus limits controller allowable range of perturbation and thus limits controller chattering for the switching law based robust controller like Sliding Mode Control (SMC). This in effect reduces criterion mentioned in Hsia and Gao (1990) restricts the like Slidingfor Mode (SMC). This robust in effectcontroller reduces allowable rangeStability of perturbation and thus limits controller chattering the Control switching law based working of in and working range. range. of the the system system in Chang Chang and Park Park like Slidingaccuracy Mode Control (SMC). This in effect reduces controller (Lee and Utkin (2007)). Higher order allowable rangeStability of perturbation and thus limits controller controller accuracy (Lee and Utkin (2007)). Higher order like Sliding Mode Control (SMC). This in effect reduces working range. Stability of the system in Chang and Park (2003) is established in frequency domain, which makes controller accuracy (Lee and Utkin (2007)). Higher order (2003) is established in frequency domain, which makes sliding mode (Levant (2003)) can alleviate the chattering working range. Stability of the system in Chang and Park sliding mode (Levant (2003)) can alleviate chattering controller accuracy (Lee and Utkin (2007)).the Higher order (2003) is established in frequency domain, systems. which makes the inapplicable to The the approach approach inapplicable to the the nonlinear nonlinear The sliding (Levant (2003)) can alleviate the still chattering problem but of bound exists. (2003) is established in frequency domain, systems. which makes problemmode but prerequisite prerequisite of uncertainty uncertainty bound still exists. sliding mode (Levant (2003)) can alleviate the still chattering the approach inapplicable to the nonlinear systems. The exists. the approach inapplicable to the nonlinear systems. The problem but prerequisite of uncertainty bound problem but prerequisite of uncertainty bound still exists.

Copyright © 2016 IFAC 730 Copyright © 2016, 2016 IFAC 730Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright ©under 2016 responsibility IFAC 730Control. Peer review of International Federation of Automatic Copyright © 2016 IFAC 730 10.1016/j.ifacol.2016.03.144

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controllers reported by Cho et al. (2009), Han and Chang (2010) can only negotiate slowly varying or constant TDE. Moreover, the controllers designed in Jin et al. (2013) and Roy et al. (2014) require nominal modelling and upper bound of the TDE respectively which is not always possible in practical circumstances. Also, to the best knowledge of the authors, controller design issues such as selection of controller gains and sampling interval to achieve efficient performance is still an open problem. In contrast to TDC, works reported in Kuperman and Zhong (2011), Talole and Phadke (2009) and Suryawanshi et al. (2014) use low pass filter 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 filter. Furthermore, the order of the low pass filter needs to be adjusted according to order of the disturbance to maintain stability of the controller. A neural network based adaptive controller was reported by Purwar et al. (2005) to compensate system uncertainties assuming the bound on the uncertainties are known. Considering the individual limitations of adaptive and robust control, recently global research is reoriented towards adaptive-robust control (ARC) where switching gain of the controller is adjusted online. The series of publications (Liu et al. (2008), Zhu et al. (2008), Zhu et al. (2009), Zhang et al. (2010), Sun et al. (2013), Islam et al. (2015), Chen et al. (2015), Liu et al. (2014)) regarding ARC, estimates the uncertain terms online based on predefined projection function, but predefined bound on uncertainties is still a requirement. The work reported in Chen et al. (2009), Nasiri et al. (2014) attempts to estimate the maximum uncertainty bound but the integral adaptive law makes the controller susceptible to very high switching gain and consequent chattering (Roy et al. (2015)).

1.1 Contributions This paper has two major contributions. First, a new stability analysis for TDC, based on the Lyapunov-Krasvoskii approach, is provided. The proposed stability analysis is carried out in continuous time domain unlike the one in Hsia and Gao (1990), Chang et al. (1996), Shin and Kim (2009), Roy et al. (2015) which approximates the closed loop system in discrete time. Through the proposed stability analysis of TDC, a relation between the sampling interval and controller gain is established as well as an upper bound on the sampling interval is obtained which is necessary to stabilize the system. Towards the second contribution, Robust Time-delayed Control (RTDC) has been devised for a class of uncertain Euler-Lagrange systems. The proposed control law approximates the uncertainties by time-delayed logic and provides robustness against the uncertainties by switching control. The calculation of switching gain for RTDC does not require any prior knowledge of uncertainties. For a proof of concept, experimental result of the proposed control methodology in comparison to TDC (Hsia and Gao (1990), Chang et al. (1996), Shin and Kim (2009)) is provided using the ”PIONEER-3” nonholonomic WMR . 731

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1.2 Organization The article is organized as follows: a new stability analysis of TDC along with its design issues is first discussed in Section 2. This is followed by the proposed robust control methodology and its detailed analysis. Section 3 presents the experimental results of the proposed controller and its comparison with TDC. Section 4 concludes the entire work. 1.3 Notations The following notations are assumed to hold true throughout the paper: any variable µ delayed by an amount h as µ(t − h), is denoted as µh ; λmin (·) and || · || represent minimum eigen value and Euclidean norm of the argument respectively; I represents the identity matrix; Z+ is the set of positive integers. 2. CONTROLLER DESIGN 2.1 Time-Delayed Control In general, an Euler-Lagrange system having second order dynamics can be written as, M (q)¨ q + H(q, q) ˙ = τ (t), (1) n n where, q(t) ∈ R is the system state, τ (t) ∈ R is the control input, M (q) ∈ Rn×n is the mass/inertia matrix and H(q, q) ˙ ∈ Rn denotes combination of other system dynamics terms based on system properties such as Coriolis, gravitational, friction, damping forces etc. In practice, it can be assumed that unmodelled dynamics and disturbances is subsumed by H. The control input is defined to be, ˆ u + H, ˆ τ =M (2) ˆ and H ˆ are where, u is the auxiliary control input, M the nominal values of M and H respectively. To reduce ˆ can be the modelling effort of the complex systems, H approximated from the input-output data of previous instances using the time-delayed logic (Hsia and Gao (1990), Chang et al. (1996), Shin and Kim (2009)) and the system definition (1) as, ˆ (qh )¨ ˆ q) qh , (3) H(q, ˙ ∼ = H(qh , q˙h ) = τh − M where, h > 0 is a fixed small delay time. Substituting (2) and (3) in (1), the system dynamics is converted into an input as well state delayed dynamics as, ˆ (q)¨ ¯ q, M q + H(q, ˙ q¨, qh , q¨h ) = τh . (4) ¯ = (M − M ˆ )¨ ˆ h q¨h − M ˆ u + H and qh denotes where, H q+M q(t − 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 defined in the following way, u(t) = q¨d (t) − K2 e˙ 1 (t) − K1 e1 (t), (5) where, K1 and K2 are two positive definite matrices with appropriate dimensions. Putting (5) and (2) in (4), following error dynamics is obtained, e¨1 = −K2 e˙ 1h − K1 e1h + σ1 , (6) ˆ −1 M ˆ −1 (H ˆ h − H) ¯ + q¨d − q¨d ˆ h − I)uh + M where, σ1 = (M h and can be treated as overall uncertainty. Further, (6) can be written in state space form as, e˙ = A1 e + B1 eh + Bσ1 , (7)

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      e 0 0 0I where, e = 1 , A1 = , B1 = , B = −K1 −K2 0 0 e˙ 1   0 0 e(t ˙ + θ)dθ, where . Noting that, e(t − h) = e(t) − I −h the derivative inside the integral is with respect to θ, the error dynamics (7) is modified as, 0 e(t) ˙ = Ae(t) − B1 e(t ˙ + θ)dθ + Bσ1 , (8) −h

where, A = A1 + B1 . It is assumed that the choice controller gains K1 and K2 makes A Hurwitz which is always possible. Also, it is assumed that the unknown disturbances are bounded. In this paper, a new stability criterion, based on the Lyapunov-Krasvoskii method, is presented through Theorem 1. Theorem 1. The system (4) employing the control input (2), having auxiliary control input (5) is UUB if K1 , K2 , h are selected such that the following condition holds:   h2 D 0 Q − E − (1 + ξ)   β  Ψ= (9) 2 >0  h 0 (ξ − 1) D β where, E = βP B1 (A1 D−1 AT1 + B1 D−1 B1T + D−1 )B1T P , ξ > 1 and β > 0 are scalar, and P > 0 is the solution of the Lyapunov equation AT P + P A = −Q for some Q > 0.

Proof. Let us consider a Lyapunov function candidate of the following form: V (e) = V1 (e) + V2 (e) + V3 (e) + V4 (e), (10) where, V1 (e) = eT P e (11)  0  t h eT (ψ)De(ψ)dψdθ (12) V2 (e) = β −h t+θ   h 0 t T e (ψ − h)De(ψ − h)dψdθ (13) V3 (e) = β −h t+θ  h2 t T e (ψ)De(ψ)dψ (14) V4 (e) = ξ β t−h Using (8), the time derivative of V1 (e) yields,  0 e(t ˙ + θ)dθ + 2sT σ1 (15) V˙ 1 (e) = −eT Qe − 2eT P B1 −h

where, s = B T P e. Again using (7),  0  T T −2e P B1 e(t ˙ + θ)dθ = −2e P B1 −h

0

[A1 e(t + θ)+ −h

B1 e(t − h + θ) + Bσ1 (t + θ)]dθ,

(16)

For any two non zero vectors z1 and z2 , there exists a scalar β > 0 and matrix D > 0 the following inequality holds, ±2z1T z2 ≤ βz1T D−1 z1 + (1/β)z2T Dz2 . (17) Again, using Jensen’s inequality, for a positive definite matrix D > 0 the following inequality holds (Gu et al. (2003)),  0   0 1 0 T T e (ψ)dψD e(ψ)dψ. e (ψ)De(ψ)dψ ≥ h −h −h −h (18) 732

Applying (17) and (18) to (16) the following inequalities are obtained,  0 − 2eT P B1 A1 e(t + θ)dθ ≤ βeT P B1 A1 D−1 AT1 B1T P e −h  h 0 T + e (t + θ)De(t + θ)dθ (19) β −h  0 − 2eT P B1 B1 e(t − h + θ)dθ ≤ βeT P B1 B1 D−1 × −h  h 0 T T T × B 1 B1 P e + e (t − h + θ)De(t − h + θ)dθ (20) β −h  0 − 2eT P B1 [Bσ1 (t + θ)]dθ ≤ βeT P B1 D−1 B1T P e −h  h 0 + (Bσ1 (t + θ))T DBσ1 (t + θ)dθ (21) β −h Assuming the uncertainties to be square integrable within the delay, let the following inequality holds:  0     h T   ≤ Γ1 . (Bσ (t + θ)) DBσ (t + θ) dθ (22) 1 1  β Again,

−h

 h 0 T h2 T ˙ e De − e (t + θ)De(t + θ)dθ (23) V2 (e) = β β −h  h2 T h 0 T eh Deh − e (t − h + θ)× V˙ 3 (e) = β β −h × De(t − h + θ)dθ (24) 2 h (25) V˙ 4 (e) = ξ (eT De − eTh Deh ) β Putting (19)-(22) into (15) and adding it with (23)-(25) yields, V˙ (e) ≤ −¯ eT Ψ¯ e + Γ1 + 2sT σ1 , (26)  T T T where, e¯ = e eh . So, for the stability of the system the first term of (26) is required to be negative definite. Hence, controller gains K1 , K2 and delay time h are required to be selected to make Ψ > 0. Also, since ξ > 1, D > 0 an upper bound of h can be formulated by applying Schur’s complement to (9)as:  β(λmin (Q) − ||E||) h< := hmax , (27) (1 + ξ)||D|| provided, λmin (Q) > ||E||.

Then, V˙ (e) < 0 would be established if λmin (Ψ)||e||2 > Γ1 + 2||σ1 ||||s||. Evaluating the structure of s, one can find a positive scalar  such that ||s|| ≤ ||¯ e||. Thus (4) would be UUB with the ultimate bound,  Γ1 ||¯ e|| = ι1 + + ι21 = 0 . (28) λmin (Ψ) 1 || where, ι1 = λ||σ . Let Ξ denote the smallest level min (Ψ) surface of V containing the ball B0 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. The time required to reach 0 is zero when e¯(t0 ) ∈ Ξ, otherwise, while e¯(t0 ) ∈ / Ξ the finite time tr0 to reach 0 is given by Leitmann (1981), e(t0 )||) − V (0 ))/c0 where V˙ (t) ≤ −c0 . tr0 − t0 ≤ (V (||¯

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Remark 1. : Since E depends on the controller gains, (9) provides a selection criterion for the choice of delay h for given controller gains and Q. This design issue was previously unaddressed in the literature. Moreover, the ˆ − H), as in (3), would reduce for approximation error (H small values of h. It can be noted from (9) that lower value of h would increase the value λmin (Ψ) provided other control parameters are kept unchanged. This in turn would improve controller accuracy by reducing the error bound in (28). However, h cannot be selected smaller than the sampling interval because, the input output data is only available at sampling intervals. So, the lowest possible selection of h is the sampling interval. Again, choice of sampling interval is governed by the corresponding hardware response time, computation time etc. Hence, the proposed stability approach provides a necessary step for the selection of sampling interval for given controller gains or vice-versa.

Case 1 ||s|| ≥ : The error dynamics of RTDC using (29) is found to be, e˙ = A1 e + B1 eh + Bσ2 , (33) where, σ2 = σ1 + ∆u. Utilizing (26) and (30) we have, s + σ1 ), e + Γ2 + 2sT (−γ (34) V˙ (e) ≤ −¯ eT Ψ¯ ||s||  0  ¯ T DBσ ¯ where, Γ2 ≥ βh || −h (Bσ2 (t + θ))T D 2 (t + θ) dθ||. So, using (31) and (32) we have, V˙ (e) ≤ −λmin (Ψ)||¯ e||2 + Γ2 + 2(γ2 − γˆ )||s||. (35)

Hence, (4) would be UUB with the following error bound,  Γ2 ||¯ e|| = ι2 + + ι22 . (36) λmin (Ψ) where, ι2 =

(γ2 −ˆ γ) λmin (Ψ) .

Case 2 ||s|| < : The term sT σ1 can be written as,

2.2 Robust Time-Delayed Control A novel robust control law, named Robust Time-Delayed Control (RTDC) is proposed in this section, which does not need any prior knowledge of the uncertainties for computation of switching gain. The structure of the control input of RTDC is similar to (2), however, the auxiliary control input u is selected as below, u=u ˆ + ∆u. (29) u ˆ is the nominal control input and selected as similar to (5). ∆u is the switching control law which is responsible for negotiating σ1 and it is defined as below,  s −γ(e, t) if  s ≥ ,  s  (30) ∆u = −γ(e, t) s if  s < ,  where,  > 0 is a scalar, γ is the switching gain. Let σ1 be bounded as ||σ1 || ≤ γ1 + γ2 , (31) ˆ −1 M ˆ h − I)uh + (¨ γ1 = ||(M qhd − q¨d )|| ˆ −1 ||||(H ˆ h − H)|| ¯ γ2 = ||M

¯ ˆ h − H) However, it is not exactly possible to predict (H ¯ is unknown. So, γ is selected as: since H γ = γ1 + γˆ , (32)  n−1  n−1   ˆ −1 || || γˆ = ||M αn−i (Ω(i + 1) − Ω(i)) || / αn−i i=1

739

i=1

ˆ (q(t − ih))¨ Ω(i) = τ (t − ih) − M q (t − ih)

Here, h is the sampling interval, n ≤ hmax /h ∈ Z+ ; 0 < α ≤ 1 is a discount factor. Choice of α = 1 puts equal weightage on all past data while, α → 1 emphasizes on recent data. The stability analysis of RTDC is stated through Theorem 2. Theorem 2. The system (4) employing (2), (29) and (30) is UUB, provided the selection of K1 , K2 , h satisfies condition (9). Proof. Using the Lyapunov function candidate (10), the stability aspect of RTDC is investigated for two different cases as follows: 733

sT s ||σ1 ||. ||s|| Exploiting (30) for this Case we have, s s ). (37) e||2 +Γ2 +2sT (−γ +||σ1 || V˙ (e) ≤ −λmin (Ψ)||¯  ||s|| The third term of (37) takes maximum value (||σ1 ||2 )/(2γ) for ||s|| = (||σ1 ||)/(2γ). Thus,(4) would be UUB with the error bound,  Γ2 + (||σ1 ||2 )/(2γ) . (38) ||¯ e|| = λmin (Ψ) sT σ1 ≤ ||s||||σ1 || =

The reaching time to the bounds 1 and 2 are found to be tri ≤ t0 + (V (||¯ e(t0 )||) − V (i ))/c0 i = 1, 2. 3. EXPERIMENTAL RESULTS AND COMPARISON A PIONEER-3 WMR is used as a platform to test the proposed control law since, under practical circumstances a WMR is always subjected to uncertainties like friction, slip, skid etc. These terms are difficult to model and in many cases they are not considered while modelling. Thus, both TDC and RTDC provide us the tool to negotiate these unmodelled terms and in turn reduces the burden of complex system modelling. The dynamic equation of a WMR that have been used to carry out the experimentation has been borrowed from Roy et al. (2014). The performance of RTDC is compared with TDC (Hsia and Gao (1990), Chang et al. (1996), Shin and Kim (2009)) while the robot is directed to track the following circular path: xdc = 1.25sin(0.35t) + .1, φd = 0.25t, ycd = 1.25cos(0.35t) + 1.35, θrd = 3t, θld = 2t. For a choice of K1 = K2 = Q = I, D = 0.5I, ξ = 1.5, β = 0.03, r = 1.5, the maximum allowable delay is found to be hmax = 122.5ms. So, selecting the delay time (which is also the sampling interval) as h = 30ms we have n = 4. Hence, with present choice of control parameters previous data of upto four samples can be used to compute γˆ . Other control parameters are selected as α = 0.9,  = 0.1. Again, to create a dynamic payload variation, a further 3.5kg

IFAC ACODS 2016 740 February 1-5, 2016. NIT Tiruchirappalli, IndiaSpandan Roy et al. / IFAC-PapersOnLine 49-1 (2016) 736–741

Fig. 2. xc position error comparison

Fig. 1. Circular path tracking comparison payload is added and kept for 5sec and then removed. This process is carried out for the entire duration of experimentation. A time gap 5sec is maintained between two successive instances of addition of the payload. However, the payload was added randomly at different places on the robotic platform every time to create variation in center of mass and inertia. The trajectory tracking performance of RTDC in comparison with TDC is depicted in Fig. 1 while following the desired circular path. Tracking performance comparison of the two controllers are illustrated in Fig. 2 and Fig. 3 in terms of xc and yc position error respectively. All the error plots are in absolute value. TDC does not possess any measure to counter the uncertainties that arises from the timedelayed approximation. On the other hand, RTDC have robustness properties against the uncertainties and provides better tracking compared to TDC, which is clearly evident from the error plots. However, It is easier to infer the performance of the individual controllers in terms of absolute average error in xc (AE-xc ) and yc position (AEyc ). These data are represented in Table 1 where the percentage error is calculated with respect to the diameter of the circular path. The tabulated data further establishes the superior performance of RTDC over TDC. The control input requirement for RTDC is depicted through Fig. 4.

Fig. 3. yc position error comparison

Table 1. Position error (mm) comparison between TDC and RTDC Controller TDC RTDC (proposed)

AE-xc 95.09 46.20

% AE-xc 3.80 1.85

AE-yc 89.55 55.47

% AE-yc 3.58 2.21

Fig. 4. Control input for RTDC

4. CONCLUSION The choice of sampling interval and controller gain for time-delayed control is crucial for its performance. This previously unaddressed design issue of TDC has been solved in this paper. An upper bound of the sampling interval and its relation with the controller gain is formulated through a new stability criterion. Also, a robust control 734

law, RTDC has been proposed which approximates unknown dynamics and perturbations through time-delayed logic and compensates the approximation error, that surfaces from TDC, by switching logic without any prior knowledge of uncertainties. Experimentation results with a WMR shows improved path tracking performance of RTDC compared to conventional TDC.

IFAC ACODS 2016 February 1-5, 2016. NIT Tiruchirappalli, IndiaSpandan Roy et al. / IFAC-PapersOnLine 49-1 (2016) 736–741

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