Pole placement for delay differential equations with time-periodic delays using Galerkin approximations ⁎

5th International Conference on Advances in Control and Optimization of Systems 5th International Conference on Advances in Control and Optimization of Dynamical Dynamical Systems 5th International Conference on in and February 18-22, 2018. Hyderabad, India 5th International Conference on Advances Advances in Control Control Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India Available onlineand at www.sciencedirect.com Optimization of Dynamical Dynamical Systems 5th International Conference on Advances in Control and Optimization of Systems February 18-22, 2018. Hyderabad, India February 18-22, 2018. Hyderabad, India Optimization of Dynamical Systems February 18-22, 2018. Hyderabad, India February 18-22, 2018. Hyderabad, India

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IFAC PapersOnLine 51-1 (2018) 560–565 Pole Pole placement placement for for delay delay differential differential Pole placement for delay differential Pole placement for delay equations with time-periodic delays equations with time-periodicdifferential delays using using Pole Galerkin placement for delay differential equations with time-periodic delays using   approximations equations with time-periodic delays using Galerkin approximations  equations with time-periodic delays using Galerkin approximations  Galerkin approximations ∗ ∗  ∗ C. P. Vyasarayani Shanti Swaroop Kandala Galerkin approximations Shanti Swaroop Kandala C. P. Vyasarayani ∗

Shanti Swaroop Kandala ∗∗∗ C. P. Vyasarayani ∗∗∗ Shanti Kandala C. ∗ Shanti Swaroop Swaroop Kandala C. P. P. Vyasarayani Vyasarayani ∗ Department of Mechanical and Aerospace Engineering, ∗ ∗ Department of Mechanical and Aerospace Engineering, Indian Indian Shanti Swaroop Kandala C. P. Vyasarayani ∗ Institute of Technology Technology Hyderabad, Kandi, Sanga Sanga Reddy, 502285, 502285, Department of Mechanical and Aerospace Engineering, Indian ∗ Institute of Hyderabad, Kandi, Reddy, ∗ Department of Mechanical and Aerospace Engineering, Indian Department of Mechanical and Aerospace Engineering, Indian Telangana, India (e-mail: Institute of Technology Hyderabad, Kandi, Sanga Reddy, 502285, ∗ Telangana, India (e-mail: [email protected]). [email protected]). Institute of Technology Hyderabad, Kandi, Sanga Reddy, 502285,

Department of Mechanical and Aerospace Engineering, Indian Institute of Technology Hyderabad, Kandi, Sanga Reddy, 502285, Telangana, India (e-mail: [email protected]). Telangana, India (e-mail: [email protected]). [email protected]). InstituteTelangana, of Technology Hyderabad, Kandi, Sanga Reddy, 502285, India (e-mail: Abstract: a is to Telangana, India (e-mail: [email protected]). Abstract: In In this this work, work, a new new methodology methodology is proposed proposed to obtain obtain the the feedback feedback gains gains for for the the closed-loop control systems having time-periodic delays. A new pseudo-inverse method combined Abstract: In this work, a new methodology is proposed to obtain the feedback gains for the closed-loop control systems having time-periodic delays. A new pseudo-inverse method combined Abstract: In this work, a new methodology is proposed to obtain the feedback gains for the Abstract: In approximations this systems work, a having newis methodology is delays. proposed to obtain the feedback gains for the with developed which approximates the differential equations closed-loop control time-periodic A new pseudo-inverse method combined with Galerkin Galerkin approximations is methodology developed which approximates the delay delay differential equations closed-loop control systems having time-periodic delays. A new pseudo-inverse method combined Abstract: In this work, a new is proposed to obtain the feedback gains for the closed-loop control systems having time-periodic delays. A new pseudo-inverse method combined (DDEs) with time-periodic delays to a system of time-periodic ordinary differential equations with Galerkin approximations is developed which approximates the delay differential equations (DDEs) with time-periodic delays to a system of time-periodic ordinary with Galerkin approximations is developed which approximates the delay differential equations closed-loop control systems having time-periodic delays. A new pseudo-inverse method combined with Galerkin approximations is developed which approximates the delay differential differential equations (ODEs). Floquet theory to the of resulting time-periodic ODEs. (DDEs) with time-periodic delays to system ofstability time-periodic ordinary equations (ODEs). Floquet theory is is applied applied to aaobtain obtain theof stability of the the the resulting time-periodic ODEs. (DDEs) with time-periodic delays to system time-periodic ordinary differential equations with Galerkin approximations is developed which approximates delay (DDEs) with time-periodic delays to aobtain system of time-periodic ordinary differential equations Later, an optimization approach is used to find suitable feedback gains to stabilize the system. (ODEs). Floquet theory is applied to the stability of the resulting time-periodic ODEs. Later, an optimization approach is used to find suitable feedback gains to stabilize the system. (ODEs). Floquet theory is applied to obtain the stability of the resulting time-periodic ODEs. (DDEs) time-periodic delays to aobtain system time-periodic equations (ODEs). Floquet theory is applied to theof stability of the ordinary resulting time-periodic ODEs. The so result in of transition matrix to Later, anwith optimization approach is spectral used to radius find suitable feedback gains todifferential stabilize the system. The gains gains so obtained obtained result in the theis spectral radius of the the Floquet Floquet transition matrix (FTM) (FTM) to be be Later, an optimization approach used to find suitable feedback gains to stabilize the system. (ODEs). Floquet theory is applied to obtain the stability of the resulting time-periodic ODEs. Later, an optimization approach is used to find suitable feedback gains to stabilize the system. less than unity. The proposed pseudo-inverse method is validated by comparing the results so The gainsunity. so obtained result in pseudo-inverse the spectral radius of theis Floquet transition matrixthe (FTM) to be less than The proposed method validated by comparing results so The gains so obtained result in the spectral radius of the Floquet transition matrix (FTM) to Later, an optimization approach is used to find suitable feedback gains to stabilize the system. The gainsfor so the obtained result in pseudo-inverse the spectral radius of theis Floquet transition matrixThe (FTM) to be be obtained examples from literature for both first and second-order systems. proposed less than unity. The proposed method validated by comparing the results so obtained for the examples from literature for both first and second-order systems. The proposed less than unity. The proposed proposed pseudo-inverse method is Floquet validated by comparing comparing the results so The gainsfor so the obtained result in pseudo-inverse the spectral of theis transition matrixThe (FTM) to be less than unity. The method validated by the results so optimization approach in combination with the pseudo-inverse method was to stabilize obtained examples from literature forradius both first and second-order systems. proposed optimization approach in from combination with the pseudo-inverse method was found found toresults stabilize obtained for the examples literature for both first and second-order systems. The proposed less than unity. The proposed pseudo-inverse method is validated by comparing the so obtained for the examples from literature for both first and second-order systems. The proposed the that were from and satisfactory results were obtained. optimization approach in combination withliterature the pseudo-inverse method was found to stabilize the systems systems that were considered considered from the the literature and satisfactory results wereThe obtained. optimization approach in combination with the pseudo-inverse method was found to stabilize obtained for the examples from literature for both first and second-order systems. proposed optimization approach in combination withliterature the pseudo-inverse method was found to stabilize the systems that were considered from the and satisfactory results were obtained. the systems that were considered from the and satisfactory results were obtained. optimization approach in combination withliterature the pseudo-inverse method was found to stabilize the systems that were considered from the literature and satisfactory results were obtained. © 2018, (International Federation of Automatic Control) by Elsevier Ltd. and All rights reserved. 1. INTRODUCTION W (Yi Asl (2003); the systems that were considered from the literature andHosting satisfactory results obtained. 1.IFAC INTRODUCTION W function function (Yi (2010); (2010); Asl were and Ulsoy Ulsoy (2003); Jarlebring Jarlebring 1. INTRODUCTION and Damm (2007); Yi et al. (2007)), Galerkin W function (Yi (2010); Asl and Ulsoy (2003);approximaJarlebring and Damm (2007); Yi et al. (2007)), Galerkin approxima1. INTRODUCTION W function (Yi (2010); Asl and Ulsoy (2003); Jarlebring 1. time-delayed INTRODUCTION W function (Yi (2010); Asl and Ulsoy (2003);approximaJarlebring tions (Wahi and Chatterjee (2005); Vyasarayani (2012); and Damm (2007); Yi et al. (2007)), Galerkin The applications of systems (TDS) over the tions (Wahi and Chatterjee (2005); Vyasarayani (2012); The applications of systems (TDS) over the W 1. time-delayed INTRODUCTION and Damm (2007); Yi et al. (2007)), Galerkin approximafunction (Yi (2010); Asl and Ulsoy (2003); Jarlebring and Damm (2007); Yi et al. (2007)), Galerkin approximaSadath and Vyasarayani (2015)), Laplace transforms tions (Wahi and Chatterjee (2005); Vyasarayani (2012); past few years is steadily expanding to various The applications of time-delayed systems (TDS)domains. over the Sadath and Vyasarayani (2015)), Laplace transforms past few years is steadily expanding to various domains. tions (Wahi and Chatterjee (2005); Vyasarayani (2012); and Damm (2007); Yi etsemi-discretization al. (2015)), (2007)), approximaThe applications of time-delayed time-delayed systems (TDS) lasers over the the tions (Wahi and Chatterjee (2005); Galerkin Vyasarayani (2012); The applications of systems (TDS) over (Kalm´ a r-Nagy (2009)), (Insperger and Sadath and Vyasarayani Laplace transforms From control systems to machining processes, to past few years is steadily expanding to various domains. ar-Nagy (2009)), semi-discretization (Insperger and Fromapplications control systems to expanding machining processes, lasers to (Kalm´ Sadath and Vyasarayani (2015)), Laplace transforms tions (Wahi and Chatterjee (2005); Vyasarayani (2012); past few years is steadily to various domains. The of time-delayed systems (TDS) over the Sadath and Vyasarayani (2015)), Laplace transforms past few yearssystems isnetworks steadily expanding to variousand domains. St´ e p´ a n (2011)), spectral-tau methods (Wahi and Chat(Kalm´ a r-Nagy (2009)), semi-discretization (Insperger and biology, neural to fluid mechanics more. From control to machining processes, lasers to St´ e p´ a n (2011)), spectral-tau methods (Wahi and Chatbiology, neural networks to fluid mechanics and more. (Kalm´ a r-Nagy (2009)), semi-discretization (Insperger and Sadath and Vyasarayani (2015)), Laplace transforms From control systems to expanding machining processes, lasers to (Kalm´ past few yearssystems isnetworks steadily to various domains. ar-Nagy (2009)), semi-discretization (Insperger and From control to machining processes, lasers to terjee (2003)), pseudo-spectral collocation (Butcher et al. St´ e p´ a n (2011)), spectral-tau methods (Wahi and ChatIn all of these applications, the presence of delay results biology, neural to fluid mechanics and more. terjee (2003)), pseudo-spectral collocation (Butcher et al. In all of these systems applications, the presence of delay results St´ e p´ a n (2011)), spectral-tau methods (Wahi and Chat(Kalm´ a r-Nagy (2009)), semi-discretization (Insperger and biology, neural networks to fluid mechanics and more. From control to machining processes, lasers to St´ e p´ a n (2011)), spectral-tau methods (Wahi and Chatbiology, neural networks to fluid mechanics and more. (2004); Breda et al. (2005); Wu and Michiels (2012)). terjee (2003)), pseudo-spectral collocation (Butcher et al. in the equations of the system to be delay In all ofgoverning these applications, the presence of delay results (2004); Breda et al. (2005); Wu and Michiels (2012)). in the governing equations of the system to be delay terjee (2003)), pseudo-spectral collocation (Butcher et al. al. St´ ep´an(2003)), (2011)), spectral-tau methods (Wahi and(2012)). ChatIn all of of neural these applications, presence of delay delay results biology, networks to the fluid mechanics and more. terjee pseudo-spectral collocation (Butcher et In all these applications, the presence of results Determining the stability of time-periodic delay systems (2004); Breda et al. (2005); Wu and Michiels differential equations (DDEs). The stability behavior of in the governing equations of the system to be delay Determining the stability of time-periodic delay (2012)). systems differential equations (DDEs). The stability behavior of (2004); Breda et al. (2005); Wu and Michiels terjee (2003)), pseudo-spectral collocation (Butcher et al. in the governing equations of the system to be delay In all of these applications, the presence of delay results (2004); Breda et al. (2005); Wu and Michiels (2012)). in the governing equations of The the stability system to be which delay (TPDS) is as straightforward as determining the stability of time-periodic delay systems the DDEs depends on the time differential equations (DDEs). behavior of Determining (TPDS) Breda is not notthe as straightforward asand determining the stastathethe DDEs depends on(DDEs). the previous previous time instant instant which Determining the stability of time-periodic delay systems (2004); et al. (2005); Wu Michiels (2012)). differential equations The stability behavior of in governing equations of the system to be delay Determining the stability of time-periodic delay systems differential equations (DDEs). The stability behavior of bility of constant delay systems because Floquet multipliis not as delay straightforward as determining the staadds to complexity of the DDEs. to it the DDEs depends on the previous timeDue instant which bility of constant systems because Floquet multipliaddsDDEs to the thedepends complexity of previous theThe DDEs. to this, this, it (TPDS) (TPDS) is not as straightforward as the staDetermining stability ofdetermined. time-periodic delay systems the on(DDEs). the timeDue instant which differential equations stability behavior of issystem notthe asneed straightforward as determining determining thestatestathe DDEs on the time instant which ers of the to be Some of the bility of constant delay systems because Floquet multiplibecomes to analyze and understand the typical adds to imperative thedepends complexity of previous the DDEs. Due to this, it (TPDS) ers of the system need to be determined. Some of the statebecomes imperative to analyze and understand the typical bility of constant delay systems because Floquet multipli(TPDS) is not as straightforward as determining the staadds to the complexity of the DDEs. Due to this, it the DDEs depends on the previous time instant which bility of constant delay systems because Floquet multipliadds to imperative the complexity of the DDEs. Due to this, it ers of-the-art methods for analyzing stability of TPDS include of the system need to be determined. Some of the stateparameter space in which the TDS can be stable. becomes to analyze and understand the typical of-the-art methods for to analyzing stabilitySome of TPDS include parameter space in which thethe TDS can beDue stable. ers of the system need be determined. of the statebility of constant delay systems because Floquet multiplibecomes imperative to analyze analyze andDDEs. understand the this, typical adds to the complexity of to it ers of the system need to be determined. Some of the statebecomes imperative to and understand the typical Galerkin approximations (Wahistability and (2005); of-the-art methods for analyzing of TPDS include parameter space in which the TDS can be stable. Galerkin approximations and Chatterjee Chatterjee (2005); of-the-art methods for analyzing stability of of the system need to be(Wahi determined. Some of theinclude stateIf are not aa system, essentially the parameter space in which the in TDS can be be stable. stable. becomes imperative topresent analyze and understand the typical of-the-art methods forSadath analyzing stability of TPDS TPDS include parameter space which the TDS can If the the delays delays are in not present in system, essentially the ers Vyasarayani (2012); and Vyasarayani (2015); AhGalerkin approximations (Wahi and Chatterjee (2005); Vyasarayani (2012); Sadath and Vyasarayani (2015); AhGalerkin approximations (Wahi and Chatterjee (2005); of-the-art methods for analyzing stability of TPDS include governing equations of it are ordinary differential equaIf the delays are not present in a system, essentially the parameter space in which the TDS can be stable. Galerkin approximations (Wahi and Chatterjee (2005); governing equations of it are ordinary differential equasan et al. (2015)), direct numerical simulation (AltinVyasarayani (2012); Sadath and Vyasarayani (2015); AhIf the delays are not present in a system, essentially the san et al. (2015)), direct numerical simulation (AltinIf the delays are not present in a system, essentially the Vyasarayani (2012); Sadath and Vyasarayani Vyasarayani (2015); Ahapproximations (Wahi and Chatterjee (2005); tions (ODEs). When compared between TDS with discrete governing equations of it are ordinary differential equa- Galerkin Vyasarayani (2012); Sadath and (2015); Ahtions (ODEs). When compared between TDS with discrete tas and Chan (1992); Radulescu et al. (1997)), semisan et al. (2015)), direct numerical simulation (Altingoverning equations of it are ordinary differential equaIf the delays are not present inordinary a system, essentially the san tas and Chan (1992); Radulescu et simulation al. (1997)), semigoverning equations of itthe arestability differential equaet al. (2015)), direct numerical (AltinVyasarayani (2012); Sadath and Vyasarayani (2015); Ahand time-periodic delays, of the discrete delay tions (ODEs). When compared between TDS with discrete san et al. (2015)), direct numerical simulation (Altinand time-periodic delays, the stability ofTDS the discrete delay tas discretization et al. (2009); Long and Balachandran and Chan(Long (1992); et al. semitions (ODEs). When compared between with discrete governing equations of the itthe are ordinary differential equadiscretization (Long et al.Radulescu (2009); Long and(1997)), Balachandran tions (ODEs). When compared between with But, discrete tas and Chan (1992); Radulescu et al. (1997)), semisan et al. (2015)), direct numerical simulation (Altinsystems depends upon rightmost eigenvalue. for and time-periodic delays, stability ofTDS the discrete delay tas and Chan (1992); Radulescu et al. (1997)), semisystems depends upon the rightmost eigenvalue. But, for (2010)), full-discretization (Sastry et al. (2001); Yilmaz discretization (Long et al. (2009); Long and Balachandran and time-periodic delays, the stability ofTDS theisdiscrete discrete delay (2010)), tions (ODEs). When compared between with discrete full-discretization (SastryLong et (2001); Yilmaz and time-periodic delays, stability of the delay discretization (Long et (2009); and Balachandran and Chanand (1992); Radulescu et al. al. (1997)), semiTDS with time-periodic delays, stability determined systems depends upon thethe rightmost eigenvalue. But, for tas discretization (Long et al. al. (2009); and(2001); Balachandran TDStime-periodic with time-periodic delays, stability isdiscrete determined et al. (2002)), using Lyapunov functions (Liu and Liao (2010)), full-discretization (SastryLong et al. Yilmaz systems depends upon the rightmost eigenvalue. But, for and delays, the stability of the delay et al. (2002)), and using Lyapunov functions (Liu and Liao systems depends upon the rightmost eigenvalue. But, for (2010)), full-discretization (Sastry et al. (2001); Yilmaz discretization (Long et al. (2009); Long and Balachandran using the Floquet theory. If the spectral radius of Floquet TDS with time-periodic delays, stability is determined (2010)), full-discretization (Sastry et al. (2001); Yilmaz using the Floquet theory. If the spectral radius of Floquet (2004); Jiang et al. (2006)). An issue with determining the et al. (2002)), and using Lyapunov functions (Liu and Liao TDS with time-periodic delays, stability is determined systems depends upon the rightmost eigenvalue. But, for (2004); Jiang et al. (2006)). An issue with determining the TDS with time-periodic delays, stability is determined et al. (2002)), and using Lyapunov functions (Liu and (2010)), full-discretization (Sastry et al. (2001); Yilmaz transition matrix (FTM) less than unity, such using the Floquet theory. Ifis the spectral radiusthen of Floquet al. (2002)), and using Lyapunov functions (Liu and Liao Liao transition matrix (FTM) If is the lessspectral than unity, then such aa et stability of TPDS TPDS is that that the construction of Lyapunov Lyapunov (2004); Jiang et al. (2006)). An issue with determining the using the Floquet theory. radius of Floquet TDS with time-periodic delays, stability is determined stability of is the construction of using the Floquet theory. If the spectral radius of Floquet (2004); Jiang et al. (2006)). An issue with determining the et al. (2002)), and using Lyapunov functions (Liu and Liao system is stable. This implies that, for a closed loop control transition matrixThis (FTM) is less than unity, then such a (2004); Jiang et al. is (2006)). An(Liu issue with determining the systemthe is stable. implies that, for a unity, closed loop control functions for them difficult and Liao (2004); Jiang stability of TPDS that the construction of Lyapunov transition matrix (FTM) is less than then such a using Floquet theory. If the spectral radius of Floquet functions for them is difficult (Liu and Liao (2004); Jiang transition matrix (FTM) is less than unity, then such a (2004); stability of TPDS is that the construction of Lyapunov Jiang et al. (2006)). An issue with determining the system with time-periodic terms to be stable, the spectral is stable. This implies that, for a closed loop control stability of TPDS is that the construction of Lyapunov system with time-periodic terms to be stable, the spectral et al. (2006)). Not many standardized optimization framefunctions for them is difficult (Liu and Liao (2004); Jiang system is stable. stable. This implies that, for aaunity. closed then loop control control transition matrix (FTM) is less than unity, such a et al. (2006)). Not many standardized optimization framesystem is This implies that, for closed loop functions for them is difficult (Liu and Liao (2004); Jiang stability of TPDS that the construction of Lyapunov radius of the FTM must be less than system with time-periodic terms to be stable, the spectral functions for them is difficult (Liu and Liao (2004); Jiang radius of thetime-periodic FTM must beterms less than unity. works are available for standardized the and control of al. (2006)). Not many optimization framewith to be stable, the system is stable. This implies that, closed loop control et works arefor available the stabilization stabilization and control of system with tofor beaunity. stable, the spectral spectral et al. Not many standardized optimization framefunctions them is for difficult (Liu and Liao (2004); Jiang radius of thetime-periodic FTM must beterms less than et al. (2006)). (2006)). Notattempts many standardized optimization frameTPDS. Only few arestabilization made to stabilize and conworks are available for the and control of DDEs with constant delays have been extensively studied radius of the FTM must be less than unity. system with time-periodic terms to be stable, the spectral TPDS. Only few attempts are made to stabilize and conradius of theconstant FTM must be have less than DDEs with delays beenunity. extensively studied et works are available for the stabilization and control of al. (2006)). Not many standardized optimization frameworks are available for theareoptimization stabilization and control of trol the TPDS which include based (Butcher TPDS. Only few attempts made to stabilize and conin the literature and there are some standardized methods DDEs with constant delays have been extensively studied radius of the FTM must be less than unity. trol the TPDS which include optimization based (Butcher in the literature and there are some standardized methods TPDS. Only few attempts are made to stabilize and conworks are available for the stabilization and control of DDEs with constant delays have been extensively studied TPDS. Only few attempts are made to stabilize and conand Mann (2009)) and non-optimization based (Ma et al. DDEs with constant delays have been extensively studied trol the TPDS which include optimization based (Butcher to determine their determine the of in the literature andstability. there areTo some standardized methods Mann (2009)) and non-optimization based (Ma etconal. to the determine their stability. Tosome determine the stability stability of and trol the TPDS which include optimization based (Butcher TPDS. Only few attempts are made to stabilize and in literature and there are standardized methods DDEs with constant delays have been extensively studied trol the TPDS which include optimization based (Butcher (2003, 2005); Nazari et al. approaches. in the literature andstability. there areTosome standardized methods Mann (2009)) and non-optimization based (Ma et al. DDEs with delays, the knowledge its charto determine their determine theofstability of and (2003, 2005); Nazari et al. (2013)) (2013)) approaches. DDEs with constant constant delays, the knowledge its charand Mann (2009)) and non-optimization based et the TPDS which include optimization based(Ma (Butcher to determine their stability. Tosome determine theofstability stability of in the literature and there areTo standardized methods and Mann (2009)) and non-optimization based (Ma et al. al. to determine their stability. determine the of trol (2003, 2005); Nazari et al. (2013)) approaches. acteristic roots is important. Several methods are availDDEs with constant delays, the knowledge of its characteristic roots is important. Several methods are availIn developing Galerkin approximations, the DDE is (2003, 2005); Nazari et al. (2013)) approaches. and Mann (2009)) and non-optimization based (Ma etconal. DDEs with constant delays, the knowledge of its charto determine their stability. To determine the stability of (2003, 2005); Nazari et al. (2013)) approaches. In developing Galerkin approximations, the DDE is conDDEs with constant delays, the knowledge of its charable the literature for the characteristic roots acteristic roots is important. Several methods are available in in with theroots literature for finding finding the characteristic roots verted into a set of partial differential equations (PDEs) In developing Galerkin approximations, the DDE is con(2003, 2005); Nazari et al. (2013)) approaches. acteristic is important. Several methods are availDDEs constant delays, the knowledge of its charverted into a set of partial differential equations (PDEs) acteristic roots is important. Several methods are availdeveloping Galerkin approximations, the DDE is conof constant which include able in thewith literature fordelays, finding the characteristic roots In In developing Galerkin approximations, the DDE istimeconof DDEs DDEs constant delays, which include LambertLambertalong with boundary conditions. Since there are into the a set of partial differential equations (PDEs) able in the thewith literature for finding the roots acteristic roots is important. Several methods availalong with the boundary conditions. Since there are timeable in literature fordelays, finding the characteristic characteristic roots verted verted into a set of partial differential equations (PDEs) In developing Galerkin approximations, the DDE is conof DDEs with constant which include are Lambertverted into a set of partial differential equations (PDEs) periodic terms, the PDEs and boundary conditions so along with the boundary conditions. Since there are timeof DDEs DDEs with constant delays, which include LambertLambertable in thewith literature fordelays, findingwhich the characteristic roots periodic terms, the PDEs and boundary conditions so  of constant include along with the boundary conditions. Since there are timeverted into a set of partial differential equations (PDEs) CPV gratefully acknowledges the Department of Science and  along with the boundary conditions. Since there are timedeveloped will also be time-periodic in nature. The PDEs periodic terms, the PDEs and boundary conditions so CPV gratefully acknowledges the Department of Science and of DDEs with constant delays, which include Lambertdeveloped will also be time-periodic in nature. The PDEs  periodic terms, the PDEs and boundary conditions so along with the also boundary conditions. Since there areODEs. timeTechnology for this through fellowship gratefully acknowledges the Department of Science and periodic terms, thebePDEs and boundary conditions so Technology for funding funding this research research through Inspire Inspire fellowship are then approximated to a system of time-periodic  CPV developed will time-periodic in nature. The PDEs CPV gratefully acknowledges the Department of Science and  are then approximated to a system ofin time-periodic ODEs. (DST/INSPIRE/04/2014/000972) CPV gratefully acknowledges the Department of Science and developed will also be time-periodic nature. The PDEs periodic terms, the PDEs and boundary conditions so Technology for funding this research through Inspire fellowship developed will also be time-periodic in nature. The PDEs (DST/INSPIRE/04/2014/000972) are then approximated to a system of time-periodic ODEs.  Technology for this through fellowship CPV gratefully acknowledges the Department of Science and Technology for funding funding this research research through Inspire Inspire fellowship (DST/INSPIRE/04/2014/000972) are then approximated to a system of time-periodic ODEs. developed will also be time-periodic in nature. The PDEs are then approximated to a system of time-periodic ODEs. (DST/INSPIRE/04/2014/000972) Technology for funding this research through Inspire fellowship (DST/INSPIRE/04/2014/000972) Copyright © 2018 IFAC 592 are then approximated to a system of time-periodic ODEs. (DST/INSPIRE/04/2014/000972) Copyright © 2018, 2018 IFAC 592Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright ©under 2018 responsibility IFAC 592Control. Peer review of International Federation of Automatic Copyright © © 2018 2018 IFAC IFAC 592 Copyright 592 10.1016/j.ifacol.2018.05.094 Copyright © 2018 IFAC 592

5th International Conference on Advances in Control and Optimization of Dynamical Systems Shanti Swaroop Kandala et al. / IFAC PapersOnLine 51-1 (2018) 560–565 February 18-22, 2018. Hyderabad, India

Using the Floquet theory, the stability of time-periodic ODEs is obtained. In this work, a new pseudo-inverse method, that uses the Galerkin approximations is proposed to overcome the complexities in incorporating the boundary conditions for time-periodic DDEs. Using the proposed pseudo-inverse method and with the help of Floquet theory, spectral radius of FTM is obtained and the stability characteristics of the TPDS are studied. Then, an optimization framework combined with the pseudo-inverse method is proposed for pole placement, where the objective is to reduce the spectral radius of FTM to less than unity. In this method, an objective function containing the information of spectral radius of the FTM is minimized. Using the proposed approach, time-periodic DDEs are stabilized by reducing the spectral radius of FTM to less than unity. The paper is organized as follows: section 2 describes the problem definition along with the proposed optimization framework for the pole placement of TPDS. A detailed account of finding the stability regions of the TPDS using the pseudo-inverse method is presented in section 3. The proposed pseudo-inverse method is validated in section 4. In section 5, test cases from the literature, Sadath and Vyasarayani (2015), are considered for stabilizing the TPDS. In section 6 the work is summarized. 2. POLE PLACEMENT FOR TIME-PERIODIC DDES The eigenvalues of time-periodic TDS do not exist explicitly. It is for this reason, Floquet theory is used and the spectral radius of FTM is obtained. The spectral radius of FTM has a direct impact in determining the stability characteristics of time-periodic systems. Using the proposed pole placement technique both SISO and MIMO systems can be stabilized. 2.1 Problem definition Consider the following system of DDEs represented in the state-space form: m  x(t) ˙ + Ax(t) + Bq KTq x(t − τq (t)) = 0, τq > 0. (1) q=1

where, x(t) = [x1 (t) x2 (t) · · · xP (s)]T are states, A is a square matrix of dimensions P × P , Bq and Kq are column matrices of dimensions P × 1, and τq (t) are the time-periodic delays. Given A, Bq and τq (t), suitable Kq needs to be obtained to stabilize the system. In other words, by finding a suitable Kq , the spectral radius of the FTM (described in section 3) of the system should be less the unity, so that the system becomes stable. In the current work, an optimization-based framework is developed to reduce the spectral radius of FTM to below unity. Therefore, the following optimization problem is posed for finding the feedback gains: K∗q = arg min(abs (λmax. (Kq )))2 . (2) Here, abs (λmax. ) is the spectral radius of the FTM, and K∗q is the optimal feedback gain matrix that results in the desired behavior of Eq. (1). To find the optimal feedback gain matrix, K∗q that achieves the desired system behavior, firstly the spectral radius of the FTM should be obtained. 593

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Once the spectral radius is known, the system can be stabilized by reducing the spectral radius of the FTM to less than unity in the complex plane. 3. MATHEMATICAL MODELING A new method called the pseudo-inverse method is developed in this section. With the proposed method, the stability of DDEs with time-periodic delays is determined using Galerkin approximations. Consider the system represented by Eq. (1) as described in section 2. It is the presence of time-periodic terms, τq (t), that makes it complex and more difficult to find the stability characteristics of such a system. The stability behavior of such systems is obtained by converting the system of DDEs into a system of PDEs with time dependent boundary conditions using suitable transformation. The resulting system of PDEs are converted into a system of time-periodic ODEs using Galerkin approximations combined with pseudo-inverse method. Then, as described later in this section, using the Floquet theory, the stability of DDEs with time-periodic delays is determined. First, introduce the following transformation: y(s, t) = x(t + τ (t)s), (3) where, y is a function of s and t, s ∈ [−1, 0] and τ (t) = max [τ1 (t), τ2 (t), ..., τq (t)]. It is assumed that the fundamental time-period of the time-periodic delays is T . Now, Eq. (3) is partially differentiated with respect to t and s respectively. The resulting PDEs are as follows: ∂x(t + τ (t)s) ∂y(s, t) = (1 + τ˙ (t)s), s ∈ [−1, 0], (4a) ∂t ∂(t + τ (t)s) ∂x(t + τ (t)s) ∂y(s, t) = τ (t), s ∈ [−1, 0]. (4b) ∂s ∂(t + τ (t)s) Combining the PDEs represented by Eq. (4), we get, ∂y(s, t) 1 ∂y(s, t) τ˙ (t) s∂y(s, t) = + , s ∈ [−1, 0]. (5) ∂t τ (t) ∂s τ (t) ∂s Substituting the transformation given by Eq. (3) into Eq. (1), we get: m  ∂y(s, t) ∂y(s, t) + Ay(0, t) + = 0. (6) Bq KTq ∂t ∂t q=1 Now, the boundary conditions required are obtained by evaluating Eq. (6) at the boundaries, which results in the following:   m   ∂y(s, t)  T ∂y(s, t)  + Ay(0, t) + Bq K q = 0.  τq (t) ∂t ∂t  s=0

q=1

s=−

τ (t)

(7) A series solution of the following form is assumed for Eq. (5): ∞  φij (s)ηij (t), (8) yi (s, t) = j=1

where, i = 1, 2, ..., P ; j = 1, 2, ..., ∞; φij (s) and ηij (t) are the basis functions and the time dependent coordinates respectively, i represents the corresponding state in x(t) and j is the corresponding term of each state in each of the basis functions. Since it is practically impossible to consider the infinite series, the number of terms in the

5th International Conference on Advances in Control and 562 Optimization of Dynamical Systems Shanti Swaroop Kandala et al. / IFAC PapersOnLine 51-1 (2018) 560–565 February 18-22, 2018. Hyderabad, India

series is truncated at a finite value, which, here is given by ‘N ’. For a finite series, Eq. (8) can be rewritten as: yi (s, t) = φi (s)T η i (t); i = 1, 2, ..., P, (9) where, φi (s) = [φi1 (s) φi2 (s) · · · φiN (s)]T and η i (t) = [ηi1 (t) ηi2 (t) · · · ηiN (t)]T . In vector form, Eq. (8) can be written as: y(s, t) = [φ1 (s)T η 1 (t) φ2 (s)T η 2 (t) · · · φP (s)T η P (t)]T . (10) By defining Ψ(s) = diag.[φ1 (s)T φ2 (s)T · · · φP (s)T ]TP ×N P , (11) and T  (12) β(t) = η T1 (t) η T2 (t) · · · · · · η TP (t) P N ×1 , Eq. (10) can be written as: y(s, t) = Ψ(s)T β(t). Substituting Eq. (13) in Eq. (5), we get,   ˙ τ (t) 1 T ˙ +s Ψ (s)T β(t) Ψ(s) β(t) = τ (t) τ (t)

(13)

(14)

Here, the symbol  denotes the derivative with respect to s. Pre-multiplying Eq. (14) with Ψ(s) and integrating it over the domain s ∈ [−1, 0] and simplifying it, we get, ˙ Mβ(t) = Kβ(t) (15) where, τ˙ (t) 1 C+ D τ (t) τ (t) M = diag. [M1 M2

(16a)

K=

C = diag. [C

1 1

C

2 2

···

···

MP ]TP N ×N P , CP ]TP N ×N P , DP ]TP N ×N P .

··· D = diag. [D D i i i Here, M , C and D are defined as follows:  0 i M = φi (s)φi (s)T ds, −1  0 φi (s)φi (s)T ds, Ci = −1  0 sφi (s)φi (s)T ds. Di =

(16b) (16c) (16d)

(17a) (17b) (17c)

−1

Boundary conditions can be derived by substituting Eq. (13) in Eq. (7). The corresponding boundary conditions for the states in x(t) are given by:  m  −τ (t) T   q T T T ˙ Bq K q Ψ β(t). Ψ(0) β(t) = −AΨ(0) − τ (t) q=1 (18) Now, Eq. (15) and Eq. (18) can be combined and written as follows: ˙ = KP I (t)β(t). (19) MP I β(t) Here, MP I and KP I (t) are given by:     K(t) M , KP I = ¯ , MP I = m ¯ k(t) P (N +1)×N P

P (N +1)×N P

(20)

where,

m ¯ = Ψ(0)TP ×N P , m  −τ (t) T   q T T ¯ = −AΨ(0) − Bq K q Ψ k(t) τ (t) q=1 

(21) . P ×N P

(22) 594

Equation (19) represents an over determined system of ˙ P × (N + 1) equations having P × N unknowns in β(t). Equation (19) can also be represented as follows: ˙ β(t) = G(t)β(t). (23) Where, G(t) = M+ P I KP I (t).

(24)

M+ PI

Here, is the pseudo-inverse of MP I . In this section, we have essentially converted a system of DDEs, given by Eq. (1), into a system of PDEs, Eq. (5). Then, the system of PDEs are approximated into a set of ODEs, described by Eq. (23) using Galekin approximations. Boundary conditions are later integrated into the system of ODEs to arrive at the final set of the equations whose stability behavior is to be determined. It can be clearly seen that G(t) is a periodic matrix with period T. Hence, with the help of Floquet theory, the stability behavior of Eq. (23) can be determined by determining the spectral radius of the FTM. The FTM is defined as follows: x(T ) = Φ(T )x(0). (25) If the spectral radius of the FTM, Φ(T ), is less than unity, (|λmax |(Φ(T )) < 1), then it can be safely assumed that the system is stable, else unstable. The FTM, Φ(T ), is obtained by integrating Eq. (26) over the domain t ∈ [0, T ] with initial conditions Φ(0) = I. ˙ ) = G(t)Φ(T ). Φ(T (26) The basis functions used for the current study is the shifted Legendre polynomial as they show better convergence properties (Vyasarayani et al. (2014)). Using the Shifted Legendre polynomial it is relatively easier to express the entries of matrices Mp , Cp and Dp in closed form as follows:   1 , if i = j p , i, j = 1, 2, ..., N, (27a) Mij = 2i − 1 0, otherwise  2, if i + j is odd Cpij = , i, j = 1, 2, ..., N, (27b) 0, otherwise  i−1   , if i = j  2i − 1 i, j = 1, 2, ..., N. (27c) Dpij = (−1)i+j , if i ≤ j   0, otherwise 4. VALIDATION OF PSEUDO-INVERSE METHOD

Using the procedure described in the section 3, stability diagrams are obtained and validated for first and secondorder TPDS. 4.1 First-order DDE with a single time-periodic delay Consider the following first-order DDE with a single timeperiodic delay (Sadath and Vyasarayani (2015)): x(t) ˙ + k1 x(t) + k2 sin(ω1 t)x(t − τ (t)) = 0. (28) Here, τ (t) = 0.6 − 0.4 sin(ω2 t). It is clear that the system has only a single state, therefore, from the methodology described in section 3, DDE represented by Eq. (28) is approximated to the system of ODEs as follows: ˙ = KP I (t)β(t). (29) MP I β(t)

5th International Conference on Advances in Control and Optimization of Dynamical Systems Shanti Swaroop Kandala et al. / IFAC PapersOnLine 51-1 (2018) 560–565 February 18-22, 2018. Hyderabad, India

563

6

2 spectral tau pseudo inverse

spectral tau pseudo inverse

4

1 2 0

0

-2 -1 -4 -2 -1

-0.5

0

-6 -3

0.5

-2

-1

0

1

2

Fig. 1. Stability diagram of the system given by Eq. (28).

Fig. 2. Stability diagram of the system given by Eq. (33).

Here, MP I and KP I are given as follows:     K(t) M MP I = , KP I = ¯ . ¯ (N +1)×N m k(t) (N +1)×N

using the proposed pseudo-inverse method are in close agreement with the results obtained in literature. (30)

¯ are given ¯ and k Where, M and K are given by Eq. (16), m as: ¯ = φT (0)1×N , (31) m   T T ¯ = −k1 φ (0) − k2 sin(ω1 t)φ (−1) . (32) k(t) 1×N By comparing Eq. (29) with Eq. (24), it can be seen that G(t) has three time-periodic terms i.e., tp1 = sin(ω1 t), ˙ τ (t) τ (t) .

1 tp2 = τ (t) and tp3 = Considering ω1 = π/2 and ω2 = π the time periods for tp1 , tp2 and tp3 are evaluated. Using the time periods of tp1 , tp2 and tp3 , the obtained value of fundamental time period (T ) of G(t) is T = 4. Once T is obtained, the FTM, Φ(T ), is calculated by integrating Eq. (26) over the domain t ∈ [0, 4] with initial conditions Φ(0) = I. To obtain the stability diagram of the system, k1 and k2 are varied in the domain of [−1, 0.5] and [−2, 2] respectively. Figure 1 shows the stability diagram obtained using the spectral-tau (Sadath and Vyasarayani (2015)) and proposed pseudo-inverse methods for the system described by Eq. (28). Red dots represent the stability points obtained using the spectral-tau method and blue circles represent the stability points obtained using the pseudo-inverse method. It can clearly be seen that the stability points obtained using the proposed pseudo-inverse method are in close agreement with the results obtained in literature.

4.2 Delayed damped Mathieu equation with a single time periodic delay and a constant delayed feedback gain Consider a delayed damped Mathieu equation with a single time periodic delay and a constant delayed feedback gain as follows (Sadath and Vyasarayani (2015)): ... x (t) + cx(t) ˙ + (δ +  cos(ω1 t))x(t) + k x(t ˙ − τ (t)) = 0. (33) Here, τ (t) = 0.6 + 0.2 cos(t). Figure 2 shows the stability diagram obtained using spectral-tau and proposed pseudoinverse methods for the system described by Eq. (33) by varying δ ∈ [−3, 2] and  ∈ [−3, 6]. In Fig. 2, red dots represent the stability points obtained using the spectraltau method and blue circles represent the stability points obtained using the pseudo-inverse method. It can once again be clearly seen that the stability points obtained 595

5. RESULTS AND DISCUSSIONS In this section, the proposed pseudo-inverse method on the examples from literature is applied to obtain the stability regions of TPDS. Then, the proposed optimization based framework is applied to stabilize them. Example 1: Consider the first-order system given by Eq. (34) (Sadath and Vyasarayani (2015)): x(t) ˙ − ax(t) − bx(t − τ1 (t)) − cx(t − τ2 (t)) = 0, (34) Considering N = 7, the system represented by Eq. (34) is converted into a system of ODEs of the form given by Eq. (21), which results in a G(t) matrix of size 7 × 7. Figure 3 shows the stability diagram obtained using the spectral-tau (Sadath and Vyasarayani (2015)) and pseudoinverse methods for the system described by Eq. (34) by considering τ1 (t) = 0.6 − 0.4 sin(ωt), τ2 (t) = 0.6 − 0.2 sin(ωt), c = −3.5 and varying a ∈ [−10, 2] and b ∈ [−10, 15]. Red dots represent the stability points obtained using the spectral-tau method and blue circles represent the stability points obtained using the pseudoinverse method. Now, consider a = −4, b = 5, c = −3.5, τ1 (t) = t1 + t2 cos(t), τ2 (t) = t1 + t3 cos(t), t2 = t1 /3 and t3 = 2t1 /3. The above values of a and b are considered from the stability region shown in Fig. 3. Red line in Fig. 4 shows the obtained cut-off delay for the system using above parameters by varying t1 , which is t1cutof f = 1.18. Cut-off delay is defined as the delay at which the spectral radius of the FTM exceeds unity. By varying c, the system represented by Eq. (34) is stabilized for t1 = 1.2 by minimizing the objective function given by Eq. (2). Optimization is performed using the Matlab function “f minsearch”, based on the Nelder-Mead algorithm. The optimization process took 20 iterations to converge, at which the value of objective function is 0.5359. Figure 4 shows that for t1 = 1.2 and c∗ = −1.4896, |λmax | < 1, implying that the system is stabilized using the proposed optimization procedure for the defined objective function. Blue line in Fig. 4 shows the obtained cut-off delay for the system using parameters obtained after optimization by varying t1 , which is t1cutof f = 1.57. A critical observation that can be made from Fig. 4 is that the value of c obtained for the given set of parameters using the proposed

5th International Conference on Advances in Control and 564 Optimization of Dynamical Systems Shanti Swaroop Kandala et al. / IFAC PapersOnLine 51-1 (2018) 560–565 February 18-22, 2018. Hyderabad, India

5

15 spectral tau pseudo inverse

10

4

5

3

0

2

-5

1

-10 -10

-8

-6

-4

-2

0

0

2

Fig. 3. Stability diagram of Eq. (34) in the [a, b] plane for c = −3.5. 2.5

1.5 1 0.5 0

0

0.5

1

1.5

2

Fig. 4. Cut-off delay of Eq. (34) by considering a = −4, b = −5 for c = −3.5 and optimal c∗ = −1.4896 respectively. 0.49

j6max j

0.485

0.48

0.475

0.47

0

2

0

1

2

3

4

5

6

Fig. 6. Cut-off delay of Eq. (35) by considering  = 7, δ = 7 for k = 2 and optimal k ∗ = 0.0523 respectively. method is dependent on N . ξ is defined as the difference in the spectral radius of FTM obtained by considering N terms and N − 1 terms in the series. Now, that particular N is chosen for which ξ < 10−3 .

before optimization after optimization

2

before optimization after optimization

4

6

8

10

N

Fig. 5. Variation of spectral radius of FTM of Eq. (34). optimization procedure stabilizes the system not only for the specified time delay of t1 = 1.2 but for a delay of t1 ∈ [0.1, 1.57]. A new cut-off delay that is ≈ 30% more than the non-optimized system’s cut-off delay is obtained. Next, the influence of N is studied for the proposed example by considering a = −4, b = −5 and c∗ = −1.4896. Figure 5 shows the variation in the spectral radius of FTM with respect to N . As it can be seen clearly Fig. 5 that the spectral radius of the FTM obtained using the proposed 596

Example 2: Consider a delayed damped Mathieu equation with a single time periodic delay and a constant delayed feedback gain as given by Eq. (33) (Sadath and Vyasarayani (2015)): ... x (t) + cx(t) ˙ − τ (t)) = 0. (35) ˙ + (δ +  cos(ω1 t))x(t) + k x(t Considering N = 7, the system represented by Eq. (35) is converted into a system of ODEs of the form given by Eq. (21), which results in a G(t) matrix of size 14 × 14. Figure 3 shows the stability diagram obtained using the spectral-tau (Sadath and Vyasarayani (2015)) and pseudoinverse methods for the system described by Eq. (34) by considering τ1 (t) = 0.6 + 0.2 cos(t) and by varying δ ∈ [−3, 2] and  ∈ [−3, 6]. Now, consider, δ = 7,  = 7, k = 2, τ (t) = t1 + t2 cos(t) and t2 = t1 /3. Red line in Fig. 6 shows the obtained cut-off delay for the system using above parameters by varying t1 is t1cutof f = 4.14. By varying k, the system represented by Eq. (35) is stabilized for t1 = 4.14 by minimizing the objective function given by Eq. (2). Optimization is performed using the Matlab function “f minsearch”. The optimization process took 19 iterations to converge, at which the value of the objective function is 0.0120. Figure 6 shows that for t1 = 4.14 and k ∗ = 0.0092, |λmax | < 1, implying that the system is stabilized using the proposed optimization procedure for the defined objective function. Blue line in Fig. 6 shows the obtained cut-off delay for the system using parameters obtained after optimization by varying t1 , which is t1cutof f = 5.587. A critical observation that can once again be made from Fig. 6 is that the value of k obtained for the given set of parameters using the proposed optimization procedure stabilizes the system not only for the specified time delay of t1 = 4.14 but for a delay of t1 ∈ [0.1, 5.587]. A new cut-off delay that is ≈ 30% more than the non-optimized system’s cut-off delay is obtained. 6. CONCLUSIONS In this work, we have addressed the problem of pole placement of TPDS. For TPDS, it is shown that by using an optimization framework combined with an algorithm that

5th International Conference on Advances in Control and Optimization of Dynamical Systems Shanti Swaroop Kandala et al. / IFAC PapersOnLine 51-1 (2018) 560–565 February 18-22, 2018. Hyderabad, India

can find the spectral radius of the FTM can be successfully stabilized. The developed pseudo-inverse based Galerkin approximation method for finding the spectral radius of FTM was validated by comparing the stability charts of different DDEs obtained using spectral-tau method. The pole placement capabilities of the proposed technique were illustrated on some example problems (first and second order systems) taken from literature. These systems were not only stabilized for the considered delay but also for a range of delay that is more than the delay considered for stabilizing the system using the proposed method. REFERENCES Ahsan, Z., Sadath, A., Uchida, T.K., and Vyasarayani, C.P. (2015). Galerkin–arnoldi algorithm for stability analysis of time-periodic delay differential equations. Nonlinear Dynamics, 82(4), 1893–1904. Altintas, Y. and Chan, P.K. (1992). In-process detection and suppression of chatter in milling. International Journal of Machine Tools and Manufacture, 32(3), 329– 347. Asl, F.M. and Ulsoy, A.G. (2003). Analysis of a system of linear delay differential equations. Journal of Dynamic Systems, Measurement, and Control, 125(2), 215–223. Breda, D., Maset, S., and Vermiglio, R. (2005). Pseudospectral differencing methods for characteristic roots of delay differential equations. SIAM Journal on Scientific Computing, 27(2), 482–495. Butcher, E. and Mann, B. (2009). Stability analysis and control of linear periodic delayed systems using chebyshev and temporal finite element methods. Delay differential equations: recent advances and new directions. Springer, New York/London, 93–129. Butcher, E.A., Ma, H., Bueler, E., Averina, V., and Szabo, Z. (2004). Stability of linear time-periodic delaydifferential equations via chebyshev polynomials. International Journal for Numerical Methods in Engineering, 59(7), 895–922. Insperger, T. and St´ep´ an, G. (2011). Semi-discretization for time-delay systems: stability and engineering applications, volume 178. Springer Science & Business Media. Jarlebring, E. and Damm, T. (2007). The lambert w function and the spectrum of some multidimensional time-delay systems. Automatica, 43(12), 2124–2128. Jiang, M., Shen, Y., and Liao, X. (2006). Global stability of periodic solution for bidirectional associative memory neural networks with varying-time delay. Applied Mathematics and Computation, 182(1), 509–520. Kalm´ ar-Nagy, T. (2009). Stability analysis of delaydifferential equations by the method of steps and inverse laplace transform. Differential Equations and Dynamical Systems, 17(1-2), 185–200. Liu, Z. and Liao, L. (2004). Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays. Journal of Mathematical Analysis and Applications, 290(1), 247–262. Long, X. and Balachandran, B. (2010). Stability of up-milling and down-milling operations with variable spindle speed. Journal of Vibration and Control, 16(78), 1151–1168. Long, X., Insperger, T., and Balachandran, B. (2009). Systems with periodic coefficients and periodically varying delays: semidiscretization-based stability analysis. 597

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Delay Differential Equations, Springer, New York, 131– 153. Ma, H., Deshmukh, V., Butcher, E., and Averina, V. (2003). Controller design for linear time-periodic delay systems via a symbolic approach. In American Control Conference, 2003, volume 3, 2126–2131. IEEE. Ma, H., Deshmukh, V., Butcher, E., and Averina, V. (2005). Delayed state feedback and chaos control for time-periodic systems via a symbolic approach. Communications in Nonlinear Science and Numerical Simulation, 10(5), 479–497. Nazari, M., Bobrenkov, O.A., and Butcher, E.A. (2013). Feedback control of periodic delayed systems. In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers. Radulescu, R., Kapoor, S., and DeVor, R. (1997). An investigation of variable spindle speed face milling for tool-work structures with complex dynamics, part 1: simulation results. Journal of Manufacturing Science and Engineering, 119, 266–272. Sadath, A. and Vyasarayani, C.P. (2015). Galerkin approximations for stability of delay differential equations with distributed delays. Journal of Computational and Nonlinear Dynamics, 10(6), 061024. Sastry, S., Kapoor, S.G., DeVor, R.E., and Dullerud, G.E. (2001). Chatter stability analysis of the variable speed face-milling process. Journal of manufacturing science and engineering, 123(4), 753–756. Vyasarayani, C.P. (2012). Galerkin approximations for higher order delay differential equations. Journal of Computational and Nonlinear Dynamics, 7(3), 031004. Vyasarayani, C.P., Subhash, S., and Kalm´ar-Nagy, T. (2014). Spectral approximations for characteristic roots of delay differential equations. International Journal of Dynamics and Control, 2(2), 126–132. Wahi, P. and Chatterjee, A. (2003). Galerkin projections for delay differential equations. In ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2211–2220. American Society of Mechanical Engineers. Wahi, P. and Chatterjee, A. (2005). Asymptotics for the characteristic roots of delayed dynamic systems. Journal of applied mechanics, 72(4), 475–483. Wu, Z. and Michiels, W. (2012). Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method. Journal of Computational and Applied Mathematics, 236(9), 2499–2514. Yi, S. (2010). Time-delay systems: analysis and control using the Lambert W function. World Scientific. Yi, S., Nelson, P., and Ulsoy, A. (2007). Survey on analysis of time delayed systems via the lambert w function. differential equations, 25, 28. Yilmaz, A., Emad, A.R., and Ni, J. (2002). Machine tool chatter suppression by multi-level random spindle speed variation. Journal of manufacturing science and engineering, 124(2), 208–216.