Practical Guidelines for Tuning PD and PI Delay-Based Controllers

15th IFAC Workshop on Time Delay Systems 15th IFAC Workshop on Time9-11, Delay Systems Sinaia, Romania, September 2019 15th IFAC Workshop on Time Delay Systems 15th IFAC Workshop on Time Delay Systems Available online at www.sciencedirect.com Sinaia, Romania, September 9-11, 2019 15th IFAC Workshop on Time Delay Systems 15th IFAC Workshop on Time9-11, Delay Systems Sinaia, Romania, September 2019 Sinaia, Romania, September 9-11, 2019 Sinaia, Romania, Romania, September September 9-11, 9-11, 2019 2019 Sinaia,

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IFAC PapersOnLine 52-18 (2019) 61–66

Practical Guidelines for Tuning P D Practical Guidelines for Tuning P D Practical Guidelines for Tuning P D Practical Guidelines for Tuning P D Controllers Practical Delay-Based Guidelines for Tuning P D Delay-Based Controllers Delay-Based Controllers Delay-Based Controllers ∗,∗∗ Jos´ e-Enrique Hern´ andez-D´ıez ∗,∗∗ Delay-Based Controllers Jos´ e-Enrique Hern´ a∗∗ ndez-D´ıez ∗,∗∗

and and and and and

P II P P I P P II

C´ esar-Fernando M´ endez-Barrios Silviu-Iulian Niculescu ∗∗ ∗,∗∗ Jos´ e -Enrique Hern´ a ndez-D´ ıez ∗∗ ∗,∗∗ Jos´ e -Enrique Hern´ a ndez-D´ ıez ∗,∗∗ C´ esar-Fernando M´ endez-Barrios Silviu-Iulian Niculescu ∗ Jos´ e -Enrique Hern´ a ndez-D´ ıez ∗∗ Jos´ e -Enrique Hern´ a ndez-D´ ıez ∗∗ ∗ C´ e sar-Fernando M´ e ndez-Barrios Niculescu ∗∗ Silviu-Iulian ∗ C´ e sar-Fernando M´ e ndez-Barrios Silviu-Iulian Niculescu ∗∗ ∗ C´ eLaboratoire sar-Fernando M´ e ndez-Barrios Silviu-Iulian Niculescu ∗e C´ sar-Fernando M´ e ndez-Barrios Silviu-Iulian Niculescu des Signaux et Syst`emes (L2S, UMR CNRS 8506), ∗ Laboratoire des Signaux et Syst` e mes (L2S, UMR CNRS 8506), ∗ CNRS-Sup´ elec,des 3, rue Joliotet Curie, 91192, Gif-sur-Yvette, France. ∗ Laboratoire Signaux Syst` eemes (L2S, UMR CNRS 8506), ∗ Laboratoire des Signaux et Syst` mes (L2S, UMR CNRS 8506), ∗ Laboratoire CNRS-Sup´ elec,des 3, rue Joliot Curie, 91192, Gif-sur-Yvette, France. ∗∗ Signaux et Syst` eemes (L2S, UMR CNRS 8506), Laboratoire des Signaux et Syst` mes (L2S, UMR CNRS 8506), Autonomous University of San Luis Potos´ ı (UASLP), Dr. Manuel CNRS-Sup´ e lec, 3, rue Joliot Curie, 91192, Gif-sur-Yvette, France. ∗∗ CNRS-Sup´ e lec, 3, rue Joliot Curie, 91192, Gif-sur-Yvette, France. Autonomous University of San Luis Potos´ ı (UASLP), Dr. Manuel CNRS-Sup´ e lec, 3, rue Joliot Curie, 91192, Gif-sur-Yvette, France. ∗∗ CNRS-Sup´ elec,University 3, rue 8, Joliot Curie, 91192, Gif-sur-Yvette, France. Nava San Luis Potos´ ı, Mexico. ∗∗ Autonomous of San Luis Potos´ ı (UASLP), Dr. Manuel ∗∗ Autonomous University of San Potos´ ııı (UASLP), Dr. Manuel ∗∗ Nava 8, San LuisLuis Potos´ ı, Mexico. Autonomous University University of San Luis Potos´ (UASLP), Dr. Manuel Autonomous of San Luis Potos´ (UASLP), Dr. Manuel Nava 8, San Luis Potos´ ı, Mexico. Nava 8, San Luis Potos´ ı, Mexico. Nava 8, 8, San San Luis Luis Potos´ Potos´ı, ı, Mexico. Mexico. Nava Abstract: This paper focuses on the design of two different delay-based control schemes. Abstract: This paper focuses on theare design of two delay-based control schemes. These two low-complexity controllers proposed as different alternatives for the classical P D and Abstract: This paper focuses on the design of two different delay-based control schemes. Abstract: This paper focuses on the design of two different delay-based control schemes. These two low-complexity controllers are proposed as alternatives for the classical P D and Abstract: This paper focuses on the design of two different delay-based control schemes. Abstract: ThisMore paper focuses on the design ofPtwo different delay-based control schemes. P I controllers. precisely, first, we study a D controller using the Euler approach for These two low-complexity controllers are proposed as alternatives for the classical P D and These two low-complexity controllers are proposed as alternatives for the classical P D P I controllers. More precisely, first, we study a P D controller using the Euler approach for These two low-complexity controllers are proposed as alternatives for the classical P D and These two low-complexity controllers arestudy proposed ascontroller alternatives forthe the classical Pdelay D and and approximating the derivative action. Second, we analyze the implications of imposing a in P I controllers. More precisely, first, we a P D using Euler approach for P III controllers. More precisely, first, we study aaaanalyze P D controller using the Euler approach for approximating the derivative action. Second, we the implications of imposing a delay in P controllers. More precisely, first, we study P D controller using the Euler approach for P controllers. More precisely, first, we study P D controller using the Euler approach for the error signal on the integral action of the P I controller for closed-loop response manipulation approximating the derivative action. Second, we analyze the implications of imposing a delay in approximating the derivative Second, analyze the implications of imposing aaa delay in the error signal on the integralaction. action of in theproposing Pwe I controller forpractical closed-loop response manipulation approximating the derivative action. Second, we analyze the implications of imposing delay in approximating the derivative action. Second, we analyze the implications of imposing delay in purposes. Our main contribution lies some guidelines for the tuning the error signal on the integral action of the P I controller for closed-loop response manipulation the error signal on the integral action of the P I controller for closed-loop response manipulation purposes. Our main contribution lies in proposing some practical guidelines for the tuning the error signal on the integral action of the P I controller for closed-loop response manipulation the error delayed signal oncontrol thecontribution integral action ofthat theproposing Pthe I controller forpractical closed-loop response manipulation of these schemes such closed-loop system is stable. To this end, the purposes. Our main lies in some guidelines for the tuning purposes. Our contribution lies in proposing some the tuning of these developed delayed control such thethe closed-loop system isguidelines stable.method, Tofor end, the purposes. Our main lies in some practical guidelines for the tuning purposes. Our main main contribution lies that in proposing some practical practical guidelines forthis thewith tuning criteria in contribution thisschemes work makes useproposing of well-known D−partition the of these delayed control schemes such that the closed-loop system is stable. To this end, of these delayed control schemes such that the closed-loop system is stable. To this end, criteria developed in this work makes use of the well-known D−partition method, with the of these delayed control schemes such that the closed-loop system is stable. To this end, of these delayed control schemes such that the closed-loop system is stable. To this end, the difference that we propose a simple criterion tothe detect stabilizing regions, avoiding in with this way criteria developed in this work makes use of well-known D−partition method, the criteria developed in work makes use of the well-known D−partition method, with the difference that we propose aa simple criterion detect stabilizing regions, avoiding in this way criteria developed in this this work makes use ofto the well-known D−partition method, with the criteria developed in this work makes use of the well-known D−partition method, with the the crossing direction analysis. Several numerical examples illustrate the effectiveness of difference that we propose simple criterion to detect stabilizing regions, avoiding in this way difference that we propose a simple criterion to detect stabilizing regions, avoiding in this way the crossing direction analysis. Several numerical examples illustrate theavoiding effectiveness of way the difference that we a criterion to stabilizing regions, in difference that we propose propose a simple simple criterion to detect detect stabilizing regions, avoiding in this this way proposed approach. the crossing direction analysis. Several numerical examples illustrate the effectiveness of the the crossing direction analysis. Several numerical examples illustrate the effectiveness of proposed approach. the crossing direction analysis. Several numerical examples illustrate the effectiveness of the the crossing direction analysis. Several numerical examples illustrate the effectiveness of the the proposed approach. proposed approach. Keywords: Delay-Based Control, Stability Crossing Curves, PID controllers. Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights reserved. proposed approach. proposed approach. Keywords: Delay-Based Control, Stability Crossing Curves, PID controllers. Keywords: Delay-Based Control, Stability Crossing Curves, PID controllers. Keywords: Delay-Based Control, Stability Crossing Curves, PID controllers. Keywords: Delay-Based Keywords: Delay-Based Control, Control, Stability Stability Crossing Crossing Curves, Curves, PID PID controllers. controllers. 1. INTRODUCTION can be found in Niculescu et al. (2010)). In addition, it has 1. INTRODUCTION can be found in Niculescu et al. (2010)). In addition, it has 1. INTRODUCTION been reported that there exist situations where an can be found in Niculescu et al. (2010)). In addition, it has It is well recognized that low-order controllers are one also 1. INTRODUCTION can be found in Niculescu et al. (2010)). In addition, it has 1. INTRODUCTION also been reported that there exist situations where an It is well recognized that low-order controllers are one can be found in Niculescu et al. (2010)). In addition, it 1. INTRODUCTION can be found in Niculescu et delay al. (2010)). In addition, it has has appropriate selection of the parameter may improve of the most widely applied strategies to control indusalso been reported that there exist situations where an It is well recognized that low-order controllers are one also been that there exist situations where an It is well recognized that low-order controllers are one selection of the delay parameter may improve of theprocesses most widely applied strategies toagglund controlare indusalso been reported that there situations where an It well recognized that low-order one also system’s been reported reported that there exist situations where an It is is well recognized thatAstr¨ low-order controllers are one appropriate the response (see, forexist instance, Chen (1987)). trial (see, applied e.g., om andcontrollers H¨ (2001); appropriate selection of the delay parameter may improve of the most widely strategies to control indusappropriate selection of the delay parameter may improve of the most widely applied strategies to control industhe system’s response (see, for instance, Chen (1987)). trial processes (see, e.g., Astr¨ o m and H¨ a gglund (2001); appropriate selection of the delay parameter may improve of the most widely applied strategies to control indusappropriate selection of the delay parameter may improve of the most widely applied strategies to control indusInspired by the above observations, the design of lowO’Dwyer (2009)). Such a “popularity” is mainly due to the system’s response (see, for instance, Chen (1987)). trial processes (see, e.g., Astr¨ o m and H¨ a gglund (2001); the system’s response (see, for Chen trial processes (see, e.g., Astr¨ m and H¨ gglund (2001); by the above observations, the design of lowO’Dwyer (2009)). Such “popularity” isaa mainly due to to the system’s response (see, for instance, Chen (1987)). trial (see, e.g., Astr¨ oo and (2001); the system’s response (see, forainstance, instance, Chen (1987)). (1987)). trial processes processes (see, e.g.,aafeatures: Astr¨ om m simplicity and H¨ H¨ agglund gglund (2001); order controllers with delay as control parameter have Inspired by the above observations, the design of lowtheir particular distinct and ease of Inspired O’Dwyer (2009)). Such “popularity” is mainly due Inspired by the above observations, the design of lowO’Dwyer (2009)). Such a “popularity” is mainly due to order controllers with delay as a control parameter have their particular distinct features: simplicity and ease of Inspired by the above observations, the design of lowO’Dwyer (2009)). Such a “popularity” is mainly due to Inspired by the inabove observations, the parameter designNiculescu of have lowO’Dwyer (2009)).distinct Such afeatures: “popularity” is mainly due to been addressed several works, for example, implementation. Among these controllers, those of PIDorder controllers with delay as a control their particular simplicity and ease of order controllers with delay as a control parameter have their particular distinct features: simplicity and ease of been addressed in several works, for example, Niculescu implementation. Among these controllers, those of PIDorder controllers with delay as a control parameter have their particular distinct features: simplicity and ease of order controllers with delay as a control parameter have their are particular distinct features: simplicity and ofease of been and Michiels (2004) (stabilizing chains of integrators by type known to be able to cope with uncertainties, disaddressed in several works, for example, Niculescu implementation. Among these controllers, those PIDbeen addressed in works, for example, Niculescu implementation. Among these controllers, those of PIDand Michiels (2004) (stabilizing chains of(multiple integrators by type are known known toAmong be able able to cope cope with uncertainties, dis- and been addressed in several several works, for example, Niculescu implementation. these controllers, those of been addressed in several works, for example, Niculescu implementation. Among these controllers, those of PIDPIDusing delays), Kharitonov et al. (2005) delay turbances, elimination of steady-state errors and transient Michiels (2004) (stabilizing chains of integrators by type are to be to with uncertainties, disand Michiels (2004) (stabilizing chains of integrators by type are known to be able to cope with uncertainties, disusing delays), Kharitonov et al. (2005) (multiple delay turbances, elimination of steady-state errors and transient and Michiels (2004) (stabilizing chains of integrators by type to to with uncertainties, disMichiels (2004) chains of(multiple integrators by type are are known known to be be able able to cope cope with uncertainties, dis- and blocks), Mazenc et al.(stabilizing (2003)et (bounded input,single delay), using delays), Kharitonov al. (2005) delay response improvement (Astr¨ om and H¨ agglund (1995); turbances, elimination of steady-state errors and transient using delays), Kharitonov et al. (2005) (multiple delay turbances, elimination of steady-state errors and transient blocks), Mazenc et al. (2003) (bounded input,single delay), response improvement (Astr¨ o m and H¨ a gglund (1995); using delays), Kharitonov et al. (2005) (multiple delay turbances, elimination of steady-state errors and transient using delays), Kharitonov et(bounded al. (2005) (multipledelay), delay turbances, elimination of(Astr¨ steady-state errors and transient to mention a few. M´ e ndez-Barrios et al. (2008); Ram´ ırez et al. (2016)). Howblocks), Mazenc et al. (2003) input,single response improvement o m and H¨ a gglund (1995); blocks), Mazenc et response improvement (Astr¨ m and H¨ gglund (1995); mention aa few. M´ ndez-Barrios et al. (2008); (2008); Ram´ ırez etH¨ al. (2016)). How- to blocks), Mazenc et al. al. (2003) (2003) (bounded (bounded input,single input,single delay), delay), response (Astr¨ oo and aa (1995); blocks), Mazenc et al. (2003) (bounded input,single delay), response improvement (Astr¨ om m and H¨ agglund gglund (1995); to mention few. ever, the improvement main et drawbacks ofRam´ PID controllers, reported M´ eeendez-Barrios al. ırez et al. (2016)). Howto mention a few. M´ ndez-Barrios et al. (2008); Ram´ ırez et al. (2016)). HowIn this paper, we present two alternatives using delays ever, the main drawbacks of PID controllers, reported to mention a few. M´ e ndez-Barrios et al. (2008); Ram´ ırez et al. (2016)). Howto mention a few. M´ e ndez-Barrios et al. (2008); Ram´ ırez et al. (2016)). Howin Astr¨ ommain and drawbacks H¨ agglund (1995), lie in the tuning of In this paper, we present two alternatives using delays ever, the of PID controllers, reported ever, the main drawbacks of PID controllers, reported design parameters to thetwo classical P D andusing P I control In this paper, we present alternatives delays in Astr¨ om mmain andterm, H¨ agglund gglund (1995), lie in high-frequency the tuning tuning of as ever, the drawbacks of PID controllers, reported ever, the main drawbacks of PID controllers, reported In this we present two alternatives using delays the derivative which may amplify in Astr¨ o and H¨ a (1995), lie in the of as design parameters to the classical P D and P II control In this paper, paper, wehand, present two alternatives using delays In this paper, we present two alternatives using delays in Astr¨ o m and H¨ a gglund (1995), lie in the tuning of schemes. On one we study the P δ controller, which as design parameters to the classical P D and P control the derivative term, which may amplify high-frequency in Astr¨ o m and H¨ a gglund (1995), lie in the tuning of in Astr¨ om and H¨ agglund (1995), lie in in theAstr¨ tuning of schemes. as design parameters to the classical P D and P I control measurement noise. In fact, as mentioned o m and the derivative term, which may amplify high-frequency On one hand, we study the P δ controller, which as design parameters to the classical P D and P I control as design parameters to the classical P D and P I control the derivative term, which may amplify high-frequency consists in substituting directly the derivative part of a PD schemes. On one hand, we study the P δ controller, which measurement noise. In fact, as mentioned in Astr¨ o m and the derivative term, which may amplify high-frequency the derivative term, which may amplify high-frequency schemes. On one hand, we study the P δ controller, which H¨ a gglund (2001); O’Dwyer (2009) the above arguments measurement noise. In fact, as mentioned in Astr¨ o m and consists in substituting directly the derivative part of aa P D schemes. On one hand, we study the P δ controller, which schemes. On one hand, we study the P δ controller, which measurement noise. In fact, as mentioned in Astr¨ o m and controller by the above mentioned Euler approximation. consists in substituting directly the derivative part of P D H¨ agglund gglund (2001); O’Dwyer (2009) the above arguments measurement noise. In mentioned in Astr¨ oom measurement noise. In fact, fact, as as mentioned in applications. Astr¨ m and and controller consists in substituting directly the derivative part of a P D advise to avoid the derivative action in most H¨ a (2001); O’Dwyer (2009) the above arguments by the above mentioned Euler approximation. consists in substituting directly the derivative part of a P D consists in substituting directly the derivative part of a P D H¨ a gglund (2001); O’Dwyer (2009) the above arguments One of the main benefits in considering such an approxcontroller by the above mentioned Euler approximation. advise to avoid the derivative action in most applications. H¨ a gglund (2001); O’Dwyer (2009) the above arguments H¨ a gglund (2001); O’Dwyer (2009) the above arguments controller by the above mentioned Euler approximation. advise to avoid the derivative action in most applications. One of the main benefits in considering such an approxcontroller by the above mentioned Euler approximation. controller by the above mentioned Euler approximation. advise to avoid the derivative action in most applications. imation is that most control schemes are implemented One of the main benefits in considering such an approxIn order to circumvent such a problem, the Euler approxadvise to avoid the derivative action in most applications. advise toto avoid the derivative in most applications. One of main benefits in such an approxis most control schemes are In order circumvent such aa action problem, the Euler Euler approx- imation One of the the main benefits in aconsidering considering suchimplemented an needs approxOne of the main benefits in considering such an approxdigitally. Asthat a consequence, numerical method to imation is that most control schemes are implemented imation of the derivative: In order to circumvent such problem, the approximation is that most control schemes are implemented In order to circumvent such a problem, the Euler approxdigitally. As a consequence, a numerical method needs to imation is that most control schemes are implemented imation of the derivative: In order to circumvent such a problem, the Euler approximation is that most control schemes are implemented In order to circumvent such a problem, the Euler approxbe considered in order to achieve a derivative action. In digitally. As a consequence, a numerical method needs to y (t) − y (t − ) imation of the derivative: digitally. As aaa in consequence, aaa numerical method needs to imation of the derivative: considered order to achieve aaofderivative action. In (1) be y  (t) ≈ y (t) − y (t − ) , digitally. As consequence, numerical method needs to imation digitally. As consequence, numerical method needs to imation of of the the derivative: derivative: this vein, one of the main features the P δ controller is be considered in order to achieve derivative action. In y (t) − y (t − ) ≈ y (t) − y (t − ) , (1) yy  (t) be considered in order to achieve a derivative action. In this vein, one of the main features of the P δ controller is be considered in order to achieve a derivative action. In y (t) − y (t − ) (t) ≈ , (1) be considered in order to achieve a derivative action. In y (t) − y (t − )  that is easier to implement on such platforms and its model this vein, one of the main features of the P δ controller is ≈ , (1)  (t) for small  > 0, yyyseems to be the simplest way to replace (t) ≈ ≈ (1) that  this one of the main features of controller is (t) ,, way to replace (1) is easier such platforms its model this vein, vein, oneto ofimplement theaccurately main on features of the the P P δδδand controller is for small  > 0, seems to be the simplest this vein, one of the main features of the P controller is  delay-difference approximates more the derivative action alike that is easier to implement on such platforms and its model the derivative action by using its approxifor small > 0, seems to be the simplest way to replace that is easier to implement on such platforms and its model for small  > 0, seems to be the simplest way to replace approximates more accurately the derivative action alike that is easier to implement on such platforms and its model the derivative action by using its delay-difference approxithat is easier to implement on such platforms and its model for small  > 0, seems to be the simplest way to replace applied for small delay values. On the other hand, based approximates more accurately the derivative action alike for small  > 0, seems to be the simplest way to replace mation counterpart (Niculescu and Michiels (2004)). How- applied the derivative action by using its delay-difference approxiapproximates more accurately the derivative action alike the derivative action by using its delay-difference approxismall delay values. On the other based approximates more accurately the derivative action alike mation counterpart and Michiels (2004)). Howapproximates more accurately the derivative action alike the derivative action(Niculescu bypoint usingout its delay-difference approxion a P I for controller, we consider a delay in thehand, error signal applied for small delay values. the derivative action by using its delay-difference approxiOn the other hand, based ever, it is important to that the presence of a demation counterpart (Niculescu and Michiels (2004)). Howapplied for small delay values. On the other hand, based on a P I controller, we consider a delay in the error signal mation counterpart (Niculescu and Michiels (2004)). Howapplied for small delay values. On the other hand, based ever, it is important to point out that the presence of a deapplied for small delay values. On the other hand, based mation counterpart (Niculescu and Michiels (2004)). Howonly in the integral action. This provides an extra degree on a P I controller, we consider a delay in the error signal mation counterpart (Niculescu and Michiels (2004)). Howlay in the feedback loop of continuous-time systems is acever, it is important to point out that the presence of a deon P controller, we consider delay in the error signal ever, it is to out that of aaa deonly in integral action. This on aaa P IIIthe controller, we consider aaaprovides delay in the error signal lay in feedback loop of continuous-time systems acon freedom P controller, we consider delay in an theextra error degree signal ever, it is important to point out that the presence of deof in the tuning of this controller maintaining only in the integral action. This provides an extra degree ever, itthe is important important to point point out that the the presence presence of is decompanied among others with oscillations, instability and lay in the feedback loop of continuous-time systems is aconly in the integral action. This provides an extra degree lay in the feedback loop of continuous-time systems is acof freedom in the tuning of this controller maintaining only in the integral action. This provides an extra degree companied among others with oscillations, instability and only in the integral action. This provides an extra degree lay in the feedback loop of continuous-time systems is acthe most important feature of the P I controller, which of freedom in the tuning of this controller maintaining lay in the feedback loop of continuous-time systems is acbandwidth sensitivity (see,with for oscillations, instance, Niculescu (2001); companied among others instability and of freedom in tuning of this controller maintaining companied among others with oscillations, instability and the most feature of the P II controller, of freedom in the thestate tuning ofin this controller maintaining bandwidth sensitivity (see, for instance, Niculescu (2001); of freedom in the tuning this controller maintaining companied among others with oscillations, instability and the null important steady errorof the regulation of zerowhich type the most important feature of the P controller, which companied among others with oscillations, instability and is Michiels and Niculescu (2014)). It is also worth to mention bandwidth sensitivity (see, for instance, Niculescu (2001); the most important feature of the P I controller, which is the null steady state error in the regulation of zero type bandwidth sensitivity (see, for instance, Niculescu (2001); the most important feature of the P I controller, which Michiels and Niculescu (2014)). It is also worth to mention the most important feature of the P I controller, which bandwidth sensitivity (see, for instance, Niculescu (2001); systems (open-loop systems with no poles at the origin). is the null steady state error in the regulation of zero type bandwidth sensitivity (see, for instance, Niculescu (2001); that there exist some situations when the delay may induce Michiels and Niculescu (2014)). It is also worth to mention is the null steady state error in the regulation of zero type Michiels and Niculescu (2014)). It is also worth to mention systems (open-loop systems with no poles at the origin). is the null steady state error in the regulation of zero type that there exist some situations when the delay may induce is the null steady state error with in thenoregulation of zero type Michiels and Niculescu (2014)). It worth to mention systems (open-loop systems poles at the origin). Michiels and Niculescu (2014)). It is is also also worth toshown mention stability, as explained in the classical example in that there exist some situations when the delay may induce systems (open-loop systems with no poles at the origin). that there exist some situations when the delay may induce systems (open-loop systems with no poles at the origin). stability, as explained in the classical example shown in systems (open-loop systems with no poles at the origin). that there exist some situations when the delay may induce 2. PROBLEM FORMULATION that there exist some situations when the delay may induce Abdallah et al. (1993), where an oscillator is controlled by stability, as explained in the classical example shown in 2. PROBLEM FORMULATION stability, as explained in the classical example shown in Abdallah et al. (1993), where an oscillator is controlled by stability, explained in classical example shown in PROBLEM stability, as explained in the the classical example shownand in Consider the2. one delayas “block”: (gain, delay), with positive gains class of proper FORMULATION SISO open-loop systems given Abdallah et al. (1993), where an oscillator is controlled by 2. PROBLEM FORMULATION Abdallah et al. (1993), where an oscillator is controlled by 2. PROBLEM FORMULATION 2. PROBLEM FORMULATION Consider the class of proper SISO open-loop systems given one delay “block”: (gain, delay), with positive gains and Abdallah et al. (1993), where an oscillator is controlled by Abdallah et al. (1993), where an oscillator is controlled by by small delay values (a detailed analysis of such an approach the transfer function: Consider the class of proper SISO open-loop systems given one delay “block”: (gain, delay), with positive gains and one delay “block”: (gain, delay), with positive gains and Consider the class of proper SISO open-loop systems given by the transfer function: small delay values (a detailed analysis of such an approach one delay “block”: (gain, delay), with positive gains and Consider the class of proper SISO open-loop systems given one delay “block”: (gain, delay), with positive gains and Consider the class of proper SISO open-loop systems given by the transfer function: small delay values (a detailed analysis of such an approach small delay values (a detailed analysis of such an approach by the transfer function: small delay values (a detailed analysis of such an approach by the transfer function: small delay values (a detailed analysis of such an approach by the transfer function: 2405-8963 Copyright © 2019. The Authors. Published by Elsevier Ltd. All rights reserved.

Copyright © 2019 IFAC 132 Copyright 2019 responsibility IFAC 132Control. Peer review©under of International Federation of Automatic Copyright © 2019 IFAC 132 Copyright © 132 10.1016/j.ifacol.2019.12.207 Copyright © 2019 2019 IFAC IFAC 132 Copyright © 2019 IFAC 132

2019 IFAC TDS 62 Jose-Enriqué Hernández-Diez et al. / IFAC PapersOnLine 52-18 (2019) 61–66 Sinaia, Romania, September 9-11, 2019

(2)

where (A, B, C T ) is the state-space representation of the open-loop system, with P and Q polynomials of degree m and n, respectively, whose highest degree coefficients are denoted by pm = 0 and pn = 0. In the remaining part of the paper, we will consider that the following assumption is satisfied. Assumption 1. Polynomials P and Q satisfy the following conditions: (i) deg Q > deg P . (ii) P (s) and Q(s) are coprime polynomials. It is clear that Assumption 1-(i) states that the system is causal. If Assumption 1-(ii) is not fulfilled, this implies that there exist a non constant common factor c(s), such ˜ that P (s) = c(s)P˜ (s) and Q(s) = c(s)Q(s). In such a case, choosing c(s) to be of the highest possible degree, the analysis can be pursued if c(s) is a Hurwitz polynomial, otherwise, the system will remain unstable independently of the control action. The problems considered in this paper can be summarized as follows: Problem 1. Find explicit conditions on the parameters (τ, kp , kδ ) ∈ R+ × R2 , such that the P δ controller: 1 − e−τ s , (3) Cδ (s) = kp + kδ τ asymptotically stabilizes the closed-loop system. Problem 2. Find explicit conditions on the parameters (τ, kp , ki ) ∈ R+ × R2 , such that the P δI controller: e−τ s , (4) Ci (s) = kp + ki s asymptotically stabilizes the closed-loop system. In the following, we consider the vectors kδ := [kp , kδ ]T and ki := [kp , ki ]T referring to the P δ and P δI controllers, respectively. The real functions (σ, ω) (and (σ, ω)) stand for the real (and imaginary part) of G−1 (σ + iω). Moreover, from a geometric point of view, for a fixed τ ∗ ∈ R+ , we can define the collection of all controller gains kδ ∈ R2 as points in the kp -kδ parameters plane. Therefore, Problem 1 can be stated as the task of finding at least one region in the kp -kδ parameters-plane such that, for all kδ −points inside this region, the characteristic equation of the closed-loop system has all of its roots in the LHP (left-half plane) of the complex plane. A region of the kp -kδ parameters-plane with such a feature is defined as a stability region. Without any loss of generality, the same can be stated for the controller P δI.

with two stable poles s1,2 = −0.17 ± 2.1i and a real unstable one s3 = 1.34. Considering the use of the well known P D controller, such case leads to the following closed-loop characteristic equation: ∆(s) = s3 − s2 + (kd + 4)s + (kp − 6) = 0. (6) Using the Routh-Hurwitz stability criterion is easy to prove that a necessary condition for closed-loop stability lies in having a positive second order term. Notice that in this case, the P D controller does not have the necessary impact on the characteristic equation to achieve it. In fact, it is only possible to design the zero and first order terms through this control scheme. In contrast, using the MatLab package DDE-BIFTOOL we compute the location of the rightmost roots of the characteristic equation of the closed-loop system, now by tunning a P δ controller with parameters kδ = [6.4, −3.4]T with a fixed delay τ = 1s. As depicted in Fig. 1, all of these are located inside the LHP, therefore, the system can be stabilized with such controller. Example 2. Consider the following open-loop transfer function: 1 G(s) = 2 , (7) s − 0.1s − 0.02 with poles s1 = 0.2 and s2 = −0.1. Considering the use of the P I controller, such a case leads to the following characteristic equation: ∆(s) = s3 − 0.1s2 + (kp − 0.02)s + ki = 0. (8) In a similar fashion that the first example, using the Routh-Hurwitz stability criterion it arises the necessary condition of having only positive terms in this polynomial in order to achieve stability. Also for this example, the use of the P I controller is not enough for this purpose due to its null impact on the second order negative term. Now, we compute the location of the rightmost roots of the characteristic equation by considering a P δI controller with parameters ki = [80, 200]T with a fixed delay τ = 0.3s. These results are shown in Fig. 1, since the roots are located inside the LHP, therefore, the addition of the delayed action gives the possibility of achieving stability. 40

133

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0 −10

−20

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(kp , ki , τ ) = (80, 200, 0.3)

(kp , kδ , τ ) = (6.4, −3.4, 1) −30 −40 −10

3. MOTIVATING EXAMPLES In this section we depict two motivating examples of the use of each controller (P δ and P Iδ). The main purpose of these is to enhance some advantages regarding the stability of the closed-loop system with respect to their low-order controllers counterparts (P D and P I). Example 1. Consider the following open-loop transfer function: 1 G(s) = 3 , (5) 2 s − s + 4s − 6

40

P δ Controller

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Im

P (s) = C T (sI − A)−1 B, Q(s)

Im

G(s) :=

−30

−8

−6

−4

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−2

0

2

−40 −30

−20

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Fig. 1. Motivating examples – Closed-loop system rightmost roots location. 4. CONTROL SCHEME DESIGN In this section the main results proposed in this paper are presented and derived. First, the tuning methodology is explained in detail. Second, we describe some insights of the P δ controller and we give conditions of the parameters (kp , kδ , τ ) such that the characteristic equation has at least

2019 IFAC TDS Jose-Enriqué Hernández-Diez et al. / IFAC PapersOnLine 52-18 (2019) 61–66 Sinaia, Romania, September 9-11, 2019

one root in s = σ + iω. Third, the same is properly proposed for the P δI controller. Finally, we characterize the delay interval such that the closed-loop system remains stable given a stabilizing triplet (kp , ki , τ ). 4.1 Delayed Controllers Tuning Methodology In this section, without any loss of generality we consider both controllers (Cδ (s) and Ci (s)) as C(s, e−τ s ) with its respective gains k ∈ R2 (kδ and ki ). The characteristic equation of this closed-loop scheme rewrites as follows: (9) ∆τ (s) = C(s, e−τ s ) + G−1 (s) = 0. It is well known that, in order to achieve asymptotic stability, all the roots of (9) have to remain in the LHP of the complex plane. Remark 1. It is clear that if we take the complex conjugate of (9), the following equality holds: ∆τ (σ + iω) = ∆τ (σ − iω). Therefore, in the rest of the paper we will consider only nonnegative frequencies ω. Now, let τ ∈ R+ and σ ∈ R+ ∪ {0} be fixed values, we introduce thefollowing set:  (10) T (σ) := k ∈ R2 |∆τ (σ + iω) = 0, ∀ω ∈ Ω , with Ω ⊂ R+ , which is the set of all ω values such that ∆τ (σ + iω) = 0 for a fixed pair (τ, σ). Such set of frequencies is characterized in Propositions 1 and 2 considering the P δ and P δI controller, respectively. Roughly speaking, the set T (σ) contains all gain vectors k such that the characteristic equation of the closed-loop system has at least one root on a vertical line at σ of the complex plane. In other words, Ω includes all the frequencies for which the gains k ∈ R2 define some crossing points, that is, points located in the complex plane on the line {s} = σ. With this notation, it is clear that all possible gain vectors k such that the system has at least one root in the RHP (right-half plane) or in the imaginary axis of the complex plane can be characterized by:  T¯ + := T (σ). (11) σ∈R+ ∪{0}

Therefore, all stabilizing controllers k are contained in the set T¯ − := R2 \ T¯ + . However, it is worthy to notice that we focus in a particular region of the parameters-space of k ⊂ R2 for computational purposes. This process is explained below.

First of all, it is necessary to enhance the importance of the set T (0). This set contains all possible gain vectors k such that the characteristic equation (9) has at least one root on the imaginary axis. In other words, T (0) is nothing else that the so-called “stability crossing curves” (see, e.g. Gu et al. (2005), for the definition). Notice that any continuous variation of k such that k ∈ T (0) implies that no roots exchange through the imaginary axis is achieved. Taking into account this argument, it is easy to understand how these stability crossing curves partition the parametersspace in regions in which any choice of k implies that (9) has a finite number of roots on the RHP. Second, notice that if some element of T (σ) with σ > 0 is located inside one of this regions implies that the characteristic equation (9) has at least one unstable root in the RHP. Therefore, this can be labeled as an unstable region. Finally, any region which is not unstable is a subset of T¯ − and can be labeled as a stability region. 134

63

4.2 P δ Controller Consider the use of the P δ controller shown in (3). The corresponding control law to be applied can be described as:   e(t) − e(t − τ ) u(t) = kp e(t) + kδ . (12) τ Notice that the delayed action resembles the simplest approximation of a derivative given by the Euler approximation (1) previously discussed in the Introduction. Roughly speaking, for small values of τ this controller approximates to a classical P D controller as Cd (s) = kp + kδ s ≈ Cδ (s). In order to study its stability, the characteristic equation of the closed-loop system can be computed by Cδ (s)G(s)+ 1 = 0, which straightforwardly leads us to:   1 − e−τ s + G−1 (s) = 0. (13) ∆δ (s) = kp + kδ τ The following result shown in this section works as a tool for describing the behavior of the roots of this equation. Proposition 1. Let τ ∈ R+ and σ ∈ R be fixed values. Then, the characteristic equation (13) has at least one root in s = σ + iω, iff:   kp = − (σ, ω)+ e−τ σ csc(τ ω)−cot(τ ω)  (σ, ω) , (14) kδ = −τ eτ σ csc(τ ω) (σ, ω) .

(15)

with ω ∈ Ωδ , where the set Ωδ is defined by:    π  Ωδ := ω ∈ R ω = n, P (σ + iω) = 0 , (16) τ where n ∈ Z. Furthermore, it has a single root in s = σ iff P (σ) = 0 and:   τ kp + G−1 (σ) , for σ = 0, (17) kδ = −τ σ e −1 q0 kp = − , kδ ∈ R, for σ = 0. (18) p0 4.3 P δI Controller Consider the use of the P δI controller shown in (4). The control law corresponding to this scheme can be described by:  t u(t) = kp e(t) + ki e(v − τ )dv. (19) 0

Notice that this is basically a classical P I controller in which the error signal is delayed by finite constant amount of time τ before integrating it. As mentioned before, the main reason for adding this delayed action to this controller is to study the behavior of the closed-loop response as τ is varied. In other words, to have an extra degree of freedom in the tuning of a P I-alike controller. In order to study its stability, the characteristic equation of the closed-loop system rewrites as Ci (s)G(s) + 1 = 0, which leads to:   (20) ∆i (s) = s kp + G−1 (s) + ki e−τ s . The following result summarized in this section works as tools for describing the behavior of the roots of this equation. Proposition 2. Let τ ∈ R+ and σ ∈ R be fixed values. Then, the characteristic equation (20) has at least one root in s = σ + iω, iff:

2019 IFAC TDS 64 Jose-Enriqué Hernández-Diez et al. / IFAC PapersOnLine 52-18 (2019) 61–66 Sinaia, Romania, September 9-11, 2019

ω sin(τ ω) − σ cos(τ ω) (σ,ω), σ sin(τ ω) + ω cos(τ ω) σ2 + ω2 ki = (σ, ω)eτ σ , σ sin(τ ω) + ω cos(τ ω)

kp = −(σ,ω)+

(21) (22)

with ω ∈ Ωi , where the set Ωi is defined by: Ωi := {ω ∈ R |ω cot(τ ω) + σ = 0, P (σ + iω) = 0 } , (23) where n ∈ Z. Furthermore, it has a single root in s = σ iff P (σ) = 0 and:   ki = −σ kp + G−1 (σ) eτ σ . (24)

Finally, in order to test this result we choose three different controllers, being c1 and c2 stable and c3 unstable controllers as is depicted in Fig. 2. Some simulation results using these controller parameters are shown in Fig. 3. Also in this figure, we show the closed-loop response of a P D controller using the same gains as the P δ controller ((kp , kd ) = (kp , kδ )). As expected, these results corroborate the graphical results on figure 2.

Furthermore, we present an additional proposition for computing the stabilizing interval of the delay value given a stabilizing triplet (kp , ki , τ ). Proposition 3. Let (kp , ki , τ ∗ ) be a stabilizing triplet, then, the closed-loop system is asymptotically stable for any delay value τ ∈ [τ ∗ , τc ), where: τc = min {τ ∈ R |τ (ω ∗ ) > 0, ω ∗ ∈ Ωp } , (25) ∗ in which τ (ω ) is computed as:     1 ki P (iω ∗ ) τ (ω ∗ )= ∗ arg +(2n+1)π , (26) ω iω ∗(kp P (iω ∗ )+Q(iω ∗ )) for n ∈ Z and where the set Ωp is defined as the set of all real roots of the following equation:

Considering the open-loop transfer function (28) and the P δ controller shown in (3) the characteristic equation of the closed-loop system can be computed as:   1 − e−τ s ∆δ (s) = s2 − 3s + 5 + kp + kδ = 0. (30) τ Using Proposition 1 with a fixed delay value τ = 0.04s we compute the stability crossing curves (T (0)) as some curves from the set (σ) with σ > 0. These graphical results are shown in Figure 2 on the kp − kδ parametersspace. We use the curves from the set T (σ) with σ > 0 for discriminating the unstable regions to further find a stability region. 135

2

τ = 0.04s

1.5

y(t)

5. ILLUSTRATIVE EXAMPLES In this section, we describe in detail how the methodology explained in Section 4.1 can be applied for two different examples of second-order systems using the P δ and P δI controllers. Example 3. Consider the following transfer function: 1 , (28) G(s) = 2 s − 3s + 5 which two poles lie on s = 1.5 ± 1.65i. Since it has two roots on the RHP, it is an unstable open-loop system. Now, let us consider the application of a P D controller, the characteristic equation of the closed-loop system can be computed as: s2 + (kd − 3)s + kp + 5 = 0. (29) By Hurwitz criterion, it is easy to observe that in order to achieve closed-loop stability the application of a derivative action is mandatory so every coefficient has the same sign. This is the case of a simple P D controller with kp > −5 and kd > 3. To avoid such a derivative action we propose the use of the P δ controller in the following lines.

Fig. 2. Stability analysis in the kp − kδ parameters space c1 = (152, 12)

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(27)

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c3 = (3500, 150)

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y(t)

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|ki P (iω ∗ )|2 − ω ∗ |kp P (iω ∗ ) + Q(iω ∗ )|2 = 0.

0 −1 −2 0

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0.4

0.6

Time(s)

Fig. 3. Closed-loop system response comparison between a P D controller and a P δ controller Example 4. Consider the following open-loop transfer function: 1 , (31) G(s) = 2 s + 2s + 3 with poles s1,2 = −1 ± 1.41i. In this stable open-loop system we are considering the problem of a controller design such that the steady state error is equal to zero. This can be easily achieved by a simple P I controller, however, as stated before we aim to use the delayed action to manipulate the closed-loop response. Considering the open-loop transfer function (31) and the P Iδ controller shown in (4) the characteristic equation of the closed-loop system can be computed as: (32) ∆δ (s) = s3 + 2s2 + (3 + kp )s + ki e−τ s = 0.

2019 IFAC TDS Jose-Enriqué Hernández-Diez et al. / IFAC PapersOnLine 52-18 (2019) 61–66 Sinaia, Romania, September 9-11, 2019

response as the other part of the interval is considered, leading as expected to instability. 1.2

τ=0 τ = 0.2τc

1.1

y(t)

Following the same methodology explained in the last example using a fixed delay τ = 0.5s we find a stability region as shown in Fig. 4. In a similar way, we test its reliance with three different controllers, c4 and c5 stable controllers and c6 an unstable one. We show some simulation results presented in Fig. 5 which corroborates this result in comparison to a simple P I controller τ = 0. From this comparison, we can notice the damping added with controller c5 relative to the controller c4 and also to the simple P I controller. At last, we show another

65

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T ime(s)

Fig. 6. Closed-loop system response using a P δI controller under different time delays with τ ∈ (0, τc ) 5.1 PD Alike Controllers Comparison As mentioned in the Introduction, non-desired high frequency sensors noise can potentially be amplified by the use of a classical P D controller. Even some filtered schemes have been proposed in the literature (see, for instance O’Dwyer (2009)) to circumvent such scenario. Probably the most direct example of this is the P Df controller (P D controller with filtered derivative) shown below:   Td Cf (s) := Kc 1 + , (33) 1 + TNd s

Fig. 4. Stability analysis in the kp − ki parameters space 2

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c6 = (155, 364)

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Fig. 5. Closed-loop system response comparison between a P I controller and a P Iδ controller way to tune the delayed action in the P δI controller. Consider the P δI gains (kp , ki ) = (5, 5) with τ = 0, the roots of the characteristic equation are s1,2 = −0.64 ± 2.58i and s3 = −0.7. Since this is a stable system, we consider Proposition 3 to compute the stabilizing delay interval τ = (0, τc ), obtaining τc = 2.15s. Some simulation results depicting the continuous variation of the closedloop response as τ is varied on this interval are presented in Fig. 6. In this figure, one can notice how from τ = 0 to τ = 0.2τc we are able to inject damping to the closedloop response. Recall that this task is commonly achieved by the use of a derivative action which we are avoiding. Also in this figure, we show the behavior of the closed-loop 136

where Kc , Td and N are real parameters and can be described as a P D controller with a low pas filter in the derivative action with break frequency wo = TNd . In this section, we show an example in which we compare three different schemes using the P D, P Df and P δ controllers applied to the open-loop system shown in (31). Also, we evaluate these considering a tracking problem and under high-frequency noise disturbances due to sensors noise. Example 5. Consider a low frequency reference signal r(t) = sin(2πfR ) with fR = 1Hz and a high frequency noise signal in the error as e(t) = r(t) − [y(t) + n(t)], where n(t) = sin(2πfn ) with fn = 50Hz. Furthermore, in order to make an equivalent comparison regarding controllers tuning we focus on the proportional and derivative gains analogies inside each topology. That is, using the P δ controller gains c7 = (700, 80) with τ = 0.04s, this translates as (kp , kd ) = (kp , kδ ) for the P D controller and (Kc , Td ) = (kp , kkpδ ) for the P Df controller. Finally, for this last we choose N = Td ω0 for achieving a break frequency of wo = 2πfo with fo = 40Hz (below the noise signal frequency). The results of this tests are shown in Fig 7. Now, with the purpose of making a quantitative comparison we propose the following performance indicators: eA -Amplitude of the ripple in the error signal in steady state due to noisy behavior, eM -Maximum peak of the absolute value of the error signal and ts -Settling time. All of this indicators values are shown in Tab. 1. Using this

2019 IFAC TDS 66 Jose-Enriqué Hernández-Diez et al. / IFAC PapersOnLine 52-18 (2019) 61–66 Sinaia, Romania, September 9-11, 2019

6. CONCLUDING REMARKS A methodology for the design of the P δI and P δ controllers such that the closed-loop system is asymptotically stable and some motivating examples of the use of those is presented. We like to enhance the fact that this methodology avoids any crossing direction analysis and is presented as practical guidelines to develop a simple control scheme design. Moreover, simulation results corroborating these ideas are shown to enhance the advantages of using these delayed controllers. Likewise, a P D controllers alike comparison considering a tracking problematic under highfrequency noise is addressed

y(t)

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REFERENCES

PD Controller 1.648 Pδ X: Controller −0.03428 PDY: f Controller

−2 0

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Fig. 7. Tracking performance and noise rejection comparison between the P D, P Df and P δ controllers. Table 1. Performance indicators values Controller PD P Df Pδ

eA 0.44 0.085 0.038

eM 0.802 0.870 2.137

ts 0.1 0.35 0.95

A 0.225 0.129 0.507

information we show in Fig. 8 a radar chart normalized with respect to the worst case for a given indicator. Also, another interesting indicator of this figure is the areas A of the polygons depicted in this figure for each controller. The ideal scenario concerns to the case in which this area is zero, that is no error signal. In other words, as this area is minimized, a controller fulfills more suitably the ideal indicators.

Fig. 8. Radar chart comparison between the P D, P Df and P δ controllers for tracking performance and noise rejection. Finally, from the analysis depicted in Fig. 8 and Tab 1 we can notice that the controller better fulfilling this indicators is the P Df controller. However, even though the P δ controller has the bigger area in this radar chart it is worth noticing that is the one that achieves the better high frequency noise rejection without implementing any additional behavior. 137

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