PID Posicast Control for Uncertain Oscillatory Systems: A Practical Experiment

Proceedings of the 3rd IFAC Conference on Proceedings of Proceedings of the the 3rd 3rd IFAC IFAC Conference Conference on on Advances in Proportional-Integral-Derivative Proceedings of the 3rd IFAC Conference on Control Advances in Proportional-Integral-Derivative Control Advances in Proportional-Integral-Derivative Control Available online at www.sciencedirect.com Ghent, Belgium, May 9-11, 2018 Proceedings of the 3rd IFAC Conference on Control Advances in Proportional-Integral-Derivative Ghent, May 9-11, 2018 Ghent, Belgium, Belgium, May 9-11, 2018 Advances in Proportional-Integral-Derivative Control Ghent, Belgium, May 9-11, 2018 Ghent, Belgium, May 9-11, 2018

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IFAC PapersOnLine 51-4 (2018) 416–421

PID PID Posicast Posicast Control Control for for Uncertain Uncertain PID Posicast Control for Uncertain Oscillatory Systems: A Practical Oscillatory A Practical PID Posicast Systems: Control for Uncertain OscillatoryExperiment Systems: A Practical OscillatoryExperiment Systems: A Practical Experiment ∗,∗∗∗ ∗∗,∗∗∗ Experiment Josenalde ∗,∗∗∗ Paulo ∗∗,∗∗∗ Josenalde Oliveira Oliveira ∗,∗∗∗ Paulo Moura Moura Oliveira Oliveira ∗∗,∗∗∗

Josenalde Oliveira Paulo Moura Oliveira ∗∗,∗∗∗ ∗∗,∗∗∗ ∗,∗∗∗Jos´ ∗∗,∗∗∗ ∗,∗∗∗ ∗∗,∗∗∗ Tatiana M. Pinho e Boaventura Cunha ∗∗,∗∗∗ ∗∗,∗∗∗ Josenalde Paulo Moura Oliveira ∗∗,∗∗∗ Tatiana M. Pinho Jos´ e Boaventura Cunha Tatiana M. Oliveira Pinho ∗∗,∗∗∗ Jos´ e Boaventura Cunha ∗,∗∗∗ ∗∗,∗∗∗ ∗∗,∗∗∗ ∗∗,∗∗∗ Josenalde Paulo Moura Oliveira Tatiana M. Oliveira Pinho ∗∗,∗∗∗ Jos´ e Boaventura Cunha ∗∗,∗∗∗ ∗∗,∗∗∗ ∗∗,∗∗∗ ∗ Tatiana M. Pinho Jos´ e Boaventura Cunha Agricultural School of Jundiai, Federal University of Rio Grande ∗ ∗ Agricultural School of Jundiai, Federal University of Rio Grande do do School of Jundiai, Federal University of Rio Grande do ∗ Agricultural ∗ Norte, 59280000, Brazil (e-mail: [email protected]). Agricultural School of Jundiai, Federal University of Rio Grande do Norte, 59280000, Brazil (e-mail: [email protected]). Norte, 59280000, Brazil (e-mail: [email protected]). ∗∗∗ University of Tr´ a s-os-Montes and Alto Douro (UTAD), School of ∗∗ Agricultural School of Jundiai, Federal University of Rio Grande do Norte, 59280000, Brazil (e-mail: [email protected]). ∗∗ University of Tr´ a and Douro (UTAD), School of University of Technology, Tr´ as-os-Montes s-os-Montes anddeAlto Alto Douro (UTAD), School of ∗∗Sciences ∗∗ Quinta Prados 5001-801 Vila Real, Norte,and 59280000, Brazil (e-mail: [email protected]). University of Technology, Tr´ as-os-Montes anddeAlto Douro (UTAD), School of Sciences and Quinta Prados 5001-801 Vila Real, and Technology, Quinta de Prados 5001-801 Vila Real, ∗∗Sciences Portugal (e-mail: [email protected], [email protected], University of Technology, Tr´ as-os-Montes anddeAlto Douro (UTAD), School Sciences and Quinta Prados 5001-801 Vila Real,of Portugal [email protected], Portugal (e-mail: (e-mail:[email protected]). [email protected], [email protected], [email protected], Sciences and Technology, Quinta de Prados 5001-801 Vila Real, Portugal (e-mail: [email protected], [email protected], [email protected]). [email protected]). ∗∗∗ TEC -- (e-mail: INESC Technology and Science, Campus da FEUP, ∗∗∗ Portugal [email protected], [email protected], [email protected]). ∗∗∗ INESC INESC TEC INESC Technology and Science, Campus da Technology and Science, Campus da FEUP, FEUP, ∗∗∗ INESC TEC - INESC [email protected]). ∗∗∗ Porto,Portugal. INESC TEC - INESC Technology and Science, Campus da FEUP, Porto,Portugal. Porto,Portugal. ∗∗∗ INESC TEC - INESC Technology and Science, Campus da FEUP, Porto,Portugal. Porto,Portugal. Abstract: Half-cycle Posicast Control is currently used in a vast range of applications. Abstract: Half-cycle Posicast Control is used in vast applications. Abstract: Half-cycle Posicast Control is currently currently used in aadisadvantages vast range range of of applications. Although the proved benefits of this technique, one of its major concerns model Abstract: Half-cycle Posicast Control is currently used in a vast range of applications. Although the proved benefits of this technique, one of its major disadvantages concerns model Although the proved benefits of this technique, one of its major disadvantages concerns model uncertainties. This has motivated the development and integration of robust methods to Abstract: Half-cycle Posicast Control is currently used in a vast range of applications. Although the proved benefits of this technique, one of its major disadvantages concerns model uncertainties. This has motivated the development and integration of robust methods to uncertainties. This has motivated the development and integration of robust methods to overcome this issue. In this paper, a practical experiment for auto-tuning of a two degrees Although the proved benefits of this technique, one of its major disadvantages concerns model uncertainties. This has motivated the development and integration of robust methods to overcome this issue. In this paper, aa practical experiment for auto-tuning of aa two degrees overcome this issue. In this paper, practical experiment for auto-tuning of two degrees of freedom control configuration using a Half-Cycle Posicast pre-filter (or input-shaping), and uncertainties. This has motivated the development and integration of robust methods to overcome this issue. In this paper, a practical experiment for auto-tuning of a two degrees of freedom control configuration using aa Half-Cycle Posicast pre-filter (or input-shaping), and ofPID freedom control configuration using Half-Cycle Posicast pre-filter (or input-shaping), and a controller under parametric variations is presented. The proposed method requires using overcome this issue. In this paper, a practical experiment for auto-tuning of a two degrees of freedom control configuration using a Half-Cycle Posicast pre-filter (or input-shaping), and aa PID controller under parametric variations presented. The method requires using PID controller under parametric variations is iscontrol presented. The proposed proposed method requires using an oscillatory system model in an auto-tuning structure. The error derivative among the of freedom control configuration using a Half-Cycle Posicast (or input-shaping), and a PID controller under parametric variations iscontrol presented. Thepre-filter proposed method requires using an oscillatory system model in an auto-tuning structure. The error derivative among the an oscillatory system model in an auto-tuning control structure. The error derivative among the model and system output is used to trigger both the identification and retuning procedure. The a PID controller under parametric variations is presented. The proposed method requires using an oscillatory system model in an auto-tuning control structure. The error derivative among the model and system output is used to trigger both the identification and retuning procedure. The model and system output is used to trigger both the identification and retuning procedure. The proposed method is flexible for choosing identification plus optimization methods. Practical an oscillatory system model in an auto-tuning control structure. The error derivative among the model and system output is used to trigger both the identification and retuning procedure. The proposed method is flexible choosing identification plus methods. Practical proposed methodfor is electronic flexible for forfilter choosing identification plus optimization optimization methods. Practical results obtained plants suggest improved performance for the considered model and system output is used to trigger both the identification and retuning procedure. The proposed method is flexible for choosing identification plus optimization methods. Practical results obtained for for electronic electronic filter improved performance performance for for the the considered considered results obtained filter plants plants suggest suggest improved cases. proposed method is flexible for choosing identification plus optimization methods. Practical results cases. cases. obtained for electronic filter plants suggest improved performance for the considered results cases. obtained for electronic filter plants suggest improved performance for the considered © 2018, IFAC (International Federation of Automatic Control)Oscillatory Hosting by systems. Elsevier Ltd. All rights reserved. cases. Keywords: PID control, Posicast control, Robustness, Keywords: Keywords: PID PID control, control, Posicast Posicast control, control, Robustness, Robustness, Oscillatory Oscillatory systems. systems. Keywords: PID control, Posicast control, Robustness, Oscillatory systems. Keywords: PID control, Posicast control, Robustness, Oscillatory systems. 1. INTRODUCTION sign of both the input command shaper and the feedback 1. INTRODUCTION INTRODUCTION sign of of both both the the input input command command shaper shaper and and the the feedback feedback 1. sign controller is also known as concurrent design (Kenison 1. INTRODUCTION sign of both the input command shaper and the feedback controller is also known as concurrent design (Kenison controller is also known as concurrent design (Kenison and (2000), Kenison and Singhose (2002), Chang 1. INTRODUCTION sign Singhose of both input command and the (Kenison feedback Posicast Control (PC) was originally proposed by Smith is the also known as concurrent design and Singhose (2000), Kenison andshaper Singhose (2002), Chang Posicast Control Control (PC) (PC) was originally originally proposed proposed by by Smith Smith controller and Singhose (2000), Kenison and Singhose (2002), Chang Posicast was Park (2001)). While some of these works addressed controller is also known as concurrent design (Kenison (1957) to achieve dead-beat responses for underdamped and Singhose (2000), Kenison and Singhose (2002), Chang Posicast Control (PC) was originally proposed by Smith and Park Park (2001)). (2001)). While While some some of of these these works works addressed addressed (1957) to to achieve achieve dead-beat dead-beat responses responses for for underdamped underdamped and (1957) the design of PD, the concurrent of input shapers Singhose Kenison and Singhose Chang second-order systems. Since then, many other input comPark (2001)). While some of design these works addressed PosicasttoControl (PC) was originally proposed by Smith (1957) achieve dead-beat responses for underdamped the design design of (2000), PD, the the concurrent design of (2002), input shapers second-order systems. Since then, many other input com- and the of PD, concurrent design of input shapers second-order systems. Since then, many other input comwith PID controllers is also addressed by Huey (2006). and Park (2001)). While some of these works addressed mand shaping techniques were derived from the pioneerthe design of PD, the concurrent design of input shapers (1957) to achieve dead-beat responses for underdamped second-order systems. Since then, many from otherthe input com- with with PID controllers is also addressed by Huey (2006). mand shaping techniques were derived pioneerPID controllers is alsostructures addressedusing by Huey (2006). mand shapingconcept techniques were derived from the(ZV) pioneerThe design PID control aa switching the design ofof concurrent designusing of input shapers ing Posicast such as: Zero Vibration and PID controllers is alsostructures addressed by Huey (2006). second-order Since then, many otherthe input command shapingsystems. techniques were derived from pioneerThe design ofPD, PIDthe control switching ing Posicast concept such as: Zero Vibration (ZV) and with The design of PID control structures using a switching ing Posicast concept such as: Zero Vibration (ZV) and technique between feedforward control using half-cycle PC with PID controllers is also addressed by Huey (2006). Zero Vibration and Derivative (ZVD) (Singer and Seering The design of PID control structures using a switching mand shaping techniques were derived from the pioneering Posicast concept such as: Zero Vibration (ZV) and technique between between feedforward feedforward control control using using half-cycle half-cycle PC Zero Vibration Vibration and and Derivative Derivative (ZVD) (ZVD) (Singer (Singer and and Seering Seering technique PC Zero and PID control was proposed by Oliveira and Vranˇ ci´ The design of PID control structures using a switching (1990)). This type of control technique has wide range of technique between feedforward control using half-cycle PC ing Posicast concept such as: Zero Vibration (ZV) and Zero Vibration and Derivative (ZVD) (Singer and Seering and PID control was proposed by Oliveira and Vranˇ i´ccc (1990)). This This type type of of control control technique technique has has wide wide range range of of and PID control was proposed by Oliveira and Vranˇcci´ (1990)). (2012). A technique to design PID controllers with halftechnique between feedforward control using half-cycle PC practical applications such as in: crane control (Sorensen and PID control was proposed by Oliveira and Vranˇ c i´ Zero Vibration and Derivative (ZVD) (Singer and Seering (1990)). type of control wide(Sorensen range of (2012). (2012). A A technique technique to to design design PID PID controllers controllers with with halfhalf-c practicalThis applications such as astechnique in: crane crane has control practical applications such in: control (Sorensen cycle PC within the feedback loop based on the magnitude and PID control was proposed by Oliveira and Vranˇ ci´c et al. (2007)), vibration control (Singhose (2009)), robot (2012). A technique to design PID controllers with half(1990)). This type of control technique has wide range of practical applications such as in: crane control (Sorensen cycle PC PC within within the the feedback feedback loop loop based based on on the the magnitude magnitude et al. al. (2007)), (2007)), vibration vibration control control (Singhose (Singhose (2009)), (2009)), robot robot cycle et optimum method was proposed by Oliveira and Vranˇ i´ (2012). A technique to design PID controllers with halfcontrol (Singhose and Seering (2005)), electronics (Ahucycle PC within the feedback practical applications such as in: crane control (Sorensen et al. (2007)), vibration control (Singhose (2009)), robot loop based on the magnitude optimum method method was was proposed proposed by by Oliveira Oliveira and and Vranˇ Vranˇccci´ i´ccc control (Singhose (Singhose and and Seering Seering (2005)), (2005)), electronics electronics (Ahu(Ahu- optimum control (2012). More recently, the gravitational search algorithm cycle PC within the feedback loop based on the magnitude mada et al. (2016)) etc. A major problem with the origoptimum method was proposed by Oliveira and Vranˇ c i´ et al. (2007)), vibration control (Singhose (2009)), robot control (Singhose and Seering (2005)), electronics (Ahu(2012). More recently, the gravitational search algorithm mada et et al. al. (2016)) (2016)) etc. etc. A A major major problem problem with with the the origorig- (2012). More recently, the gravitational search algorithmc mada was proposed to design 2DOF control structures with halfoptimum method was proposed by Oliveira and Vranˇ ci´c inal half-cycle PC (ZV) is the high sensitivity More recently, the gravitational search algorithm control and (2005)), electronics (Ahumada et(Singhose al. (2016)) etc.Seering A major problem with to themodel orig- (2012). was proposed to design 2DOF control structures with halfinal half-cycle PC (ZV) is the high sensitivity to model proposed to design 2DOF control structures with halfinal half-cycle This PC (ZV) is thethe high sensitivity of to robust model was cycle PC in Oliveira et al. (2015). However, 2DOF config(2012). More recently, the gravitational search algorithm uncertainties. motivated development was proposed to design 2DOF control structures with halfmada et al. (2016)) etc. A major problem with the original half-cycle PC (ZV) is the high sensitivity to model cycle PC PC in in Oliveira Oliveira et et al. al. (2015). (2015). However, However, 2DOF 2DOF configconfiguncertainties. This This motivated motivated the the development development of of robust robust cycle uncertainties. urations half-cycle PC as pre-filter do not perform was proposed toaa design structures halfcommand shaping to control flexible structures PCusing in Oliveira et 2DOF al. (2015). However, configinal half-cycle PC techniques (ZV) is the high sensitivity to robust model cycle uncertainties. This motivated development of urations using half-cycle PC control as pre-filter pre-filter do2DOF notwith perform command shaping techniques tothe control flexible structures structures urations using a half-cycle PC as do not perform command shaping techniques to control flexible well to model parametric cyclewhen PCusing inthe Oliveira et is al. subjected (2015). However, config(Singhose (2009), et al. (2017)). Most urations asystem half-cycle PC as pre-filter do2DOF not perform uncertainties. ThisChatlatanagulchai motivatedtothe development of robust command shaping techniques control flexible structures well when the system is subjected to model parametric (Singhose (2009), Chatlatanagulchai et al. (2017)). Most when the This system is subjected to model parametric (Singhose (2009), Chatlatanagulchai et al. (2017)). Most well uncertainties. is mostly to the input shaper urations using half-cycle PC asdue pre-filter not perform of these techniques require obtaining several sequence when the aThis system subjected parametric command shaping techniques control structures (Singhose (2009), Chatlatanagulchai et flexible al. (2017)). Most well uncertainties. is is mostly due to model thedoinput input shaper of these techniques techniques require to obtaining several sequence uncertainties. This is mostly due to the shaper of these require obtaining several sequence sensitivity to model uncertainties, but also depends on the well when the system is subjected model parametric impulses (or steps) input commands amplitudes and reuncertainties. This is mostly due to the input shaper (Singhose (2009), Chatlatanagulchai et al. (2017)). Most of these techniques require obtaining several sequence sensitivity to to model model uncertainties, uncertainties, but but also also depends depends on on the the impulses (or (or steps) steps) input input commands commands amplitudes amplitudes and and rere- sensitivity impulses selected PID gains. Thus, in Oliveira et al. (2017) an autouncertainties. This is mostly due to the input shaper spective occurrence time instants. Often, many oscillatory sensitivity to model uncertainties, but also depends on the of these techniques require obtaining several sequence impulses (or steps) input commands amplitudes and reselected PID gains. Thus, in Oliveira et al. (2017) an autospective occurrence occurrence time time instants. instants. Often, Often, many many oscillatory oscillatory selected PID gains. Thus, in Oliveira et al. (2017) an autospective tuning technique was proposed based on adepends plant model, sensitivity to gains. model uncertainties, butetalso on the systems require the use of optimization methods to obPID Thus, in Oliveira al. (2017) autoimpulses occurrence (or steps) input amplitudes and re- selected spective Often, many oscillatory tuning technique was proposed based on plantanmodel, model, systems require thetime use instants. ofcommands optimization methods to obobtuning technique was proposed based on aa plant systems require the use of optimization methods to which retunes the system controllers. This technique is selected PID gains. Thus, in Oliveira et al. (2017) an autotain both amplitudes and time instants (Singh (2002)). tuning technique was proposed based on a plant model, spective occurrence time instants. Often, many oscillatory systems require the use of optimization methods to obwhich retunes retunes the the system system controllers. controllers. This This technique technique is is tain both both amplitudes amplitudes and and time time instants instants (Singh (Singh (2002)). (2002)). which tain based on the particle swarm optimization (PSO) algorithm tuning technique was proposed based on a plant model, PC shapers can be combined with feedback control using which retunes the system controllers. This technique is systems require the use of optimization methods to obtain both amplitudes and time instants (Singh (2002)). based on on the the particle particle swarm swarm optimization optimization (PSO) (PSO) algorithm algorithm PC shapers shapers can can be be combined combined with with feedback feedback control control using using based PC (Kennedy and Eberhart which is deployed as which on retunes the system controllers. This is different configurations. Two of the the particle swarm(1995)), optimization (PSO) algorithm tain shapers bothfeedforward/feedback amplitudes and time instants (Singh (2002)). PC can be combined with feedback control using (Kennedy and Eberhart (1995)), which is technique deployed as different feedforward/feedback configurations. Two of the based (Kennedy and Eberhart (1995)), which is deployed as different feedforward/feedback configurations. Two of the optimizer both for the model parameter identification based on the particle swarm optimization (PSO) algorithm main strategies are: i) as an input reference signal pre-filter (Kennedy and Eberhart (1995)), which is deployed as PC shapers canare: be combined with feedback control using different feedforward/feedback configurations. Two of the optimizer optimizer both for the model parameter identification as main strategies i) as an input reference signal pre-filter both forPC the and model parameter identification as main strategies are: i)of asfreedom an input(2DOF) referenceconfiguration; signal pre-filter well as half-cycle PID controller (Kennedy and for Eberhart (1995)), which identification isretuning. deployedThe within a two-degrees ii) both the and model parameter as different feedforward/feedback configurations. Two of the main strategies are: i)of asfreedom an input referenceconfiguration; signal pre-filter well as half-cycle half-cycle PC PID controller retuning. The within a two-degrees two-degrees (2DOF) ii) optimizer well as PC and PID controller retuning. The within a of freedom (2DOF) configuration; ii) controlled methodology was shown to function well in optimizer both for the model parameter identification inside the control loop often in series with the feedback well as half-cycle PC and PID controller retuning. The main strategies are: i) as an input reference signal pre-filter within a two-degrees of freedom (2DOF) configuration; ii) controlled methodology methodology was was shown shown to to function function well well as in inside the the control control loop loop often often in in series series with with the the feedback feedback controlled in inside Oliveira et al. (2017) achieving good simulation results. In well as half-cycle PC and PID controller retuning. The controller (e.g. see Hung (2007)). The simultaneous decontrolled methodology was shown to function well in within a two-degrees of freedom (2DOF) configuration; ii) inside the control loop often in series with the feedback Oliveira et al. (2017) achieving good simulation results. In controller (e.g. (e.g. see see Hung Hung (2007)). (2007)). The The simultaneous simultaneous dede- Oliveira et al. (2017) achieving good simulation results. In controller controlled methodology was shown to function well In in et al. (2017) achieving good simulation results. inside the (e.g. control often in series the feedback controller seeloop Hung (2007)). Thewith simultaneous de- Oliveira Oliveira et al. (2017) achieving good simulation results. In controller (e.g. see Hung (2007)). The simultaneous deCopyright © 2018, 2018 IFAC 416Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018 IFAC 416 Copyright 2018 responsibility IFAC 416Control. Peer review©under of International Federation of Automatic Copyright © 2018 IFAC 416 10.1016/j.ifacol.2018.06.130 Copyright © 2018 IFAC 416

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this paper, a practical experiment is proposed to validate the simulation results. The selected practical experience is based on oscillatory systems implemented as low pass filters with operational amplifiers. The overall practical experiment goal is to implement simple electronic circuits which can be used to validate the technique proposed in Oliveira et al. (2017) using real-time synchronization for R control. All the models are implemented in Simulink R  and the algorithms in Matlab . The interface with the computer was accomplished using a National Instruments AD/DC acquisition board (PCIe 6361). The remaining of the paper is organized as follows: Section II states the problem of auto-tuning requirement under uncertainties, presents system and controller structures and describes the methodology and practical experiment set-up, followed by results and discussion at Section III. In the end, Section IV presents some conclusions along with recommendations for future works. Fig. 2. Effect of uncertain time delay: tuned response (solid curve) × without retuning (dashed line)

2. PROBLEM STATEMENT Consider the classical 2DOF control configuration presented in Figure 1, with the following variable correspondence: r represents the reference input, y the controlled variable, u the controller output, d a load perturbation, Gf , Gc and Gp represent respectively the transfer function for the pre-filter (or input shaper), PID controller and process to be controlled. In simple terms, the design

Gp1 and Gp2 , and two second order systems with a first order low pass filter in series, Gp3 and Gp4 , to represent a third order system. All second order filters are implemented based on the Sallen-Key topology, as depicted in Figure 3, with LM741 general purpose operational amplifiers. The switches for C11 and C12 allow the system variation at execution time. C1 in (3)−(6) refers to C11 or C12 in Figure 3. With this topology the following transfer

Fig. 1. Two-degrees of freedom control configuration used problem consists in tuning both the pre-filter and PID controller parameters in order to achieve good set-point tracking and load disturbance rejection. There are multiple approaches to design this type of 2DOF control systems. Here, the same approach presented in Oliveira et al. (2015) is deployed, in which both the pre-filter and PID controller are designed simultaneously. The pre-filter considered in this study is the half-cycle Posicast (HC-PC) represented by: (1) Gf (s) = A1 + A2 e−t1 s , with A1 and A2 = 1 − A1 representing the first and second steps amplitude and t1 the delay time applied to the second step, relatively to the first step. When parametric uncertainties and/or unmodeled dynamics affects the actual system Gp , HC-PC and PID controller may require to be retuned to continue providing the performance requirements. The HC prefilter is much more sensitive to plant parametric variations than PID. For motivation purpose, Figure 2 shows this behavior when an unmodeled significant time delay is inserted into the system for t > 12s. 2.1 Methodology and Materials In order to evaluate different real scenarios, four systems are considered: two canonical second order system models, 417

Fig. 3. Second order active low pass filter - Sallen-Key topology function can be considered: Gp(s) = K

s2

ω2 a0 = 2 ω00 + a1 s + a0 s + Q + ω02

(2)

R4 4 where K = 1 + R R3 is the gain, but here the relation R3 was chosen to get almost unitary gain. To get a better tuning, R3 = 2.2kΩ and R4 is a 22kΩ potentiometer. The parameters are function of the physical components (resistors and capacitors) as follows:

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1 , (3) C 1 C 2 R 1 R2 R1 + R2 , (4) a1 = C 1 R1 R2 and, therefore, specifications can be extracted from (2), namely, the natural oscillation frequency ω0 , quality factor Q and the damping coefficient ζ, a0 =



1 , (5) C 1 C 2 R1 R2  1 1 C 1 R1 R2 Q= . (6) , ζ= R1 + R2 C2 2Q Based on (2)-(6), Table 1 summarizes the component values and parameters for Gp1 and Gp2 , with fixed R1 = R2 = 100kΩ and C2 = 680nF : ω0 =

system identification stage. Since the focus is on oscillatory systems, usually open loop step responses are enough to provide an accurate model, from overshoot and peak times estimation. However, due to measurement noise, a simple software filtering before identification improves the results. Once the identified model is available, the optimization routine gets new controller parameters for non-stop operation. For Gp1 and Gp2 PSO (Kennedy and Eberhart (1995)) is applied for 2DOF controllers, whereas R a Matlab command line pidtool is applied only for the PID controllers for Gp3 , Gp4 . The general block diagram

Table 1. Parameters for canonical second order systems Gp1 and Gp2

a0 a1 P oles ω0 rad/s ζ Q

Gp1 C11 14.7059 2 −1 ± 3.7021j 3.8348 0.2608 1.9174

Gp2 C12 3.1289 0.4255 −0.2128 ± 1.7560j 1.7689 0.1200 4.16

Fig. 4. Second order plus low pass first order filter

and, therefore, such systems are modeled as: 14.7059 , (7) + 2s + 14.7059 3.1289 Gp2 (s) = 2 . (8) s + 0.4255s + 3.1289 Systems Gp3 and Gp4 are built from the scheme in Figure 4, with a fixed R5 = 100kΩ and a switching C3 , which modifies the cutoff frequency ωc of the low pass filter (second stage). If necessary, a similar gain loop can be added, as in Figure 3. The first stage is the same Gp1 from Table 1. If a simple first order system transfer function is assumed to model a time delay, this configuration can be seen as a time delayed second order model. For other delay approximations (Pade, for instance), additional components and circuits would be necessary, depending on approximation order. For Gp3 , C3 = 4.7µF and therefore ωc = R5 C3−1 = 10 rad/s. For Gp4 , C3 is changed to 1µF , ωc = 2.13 rad/s. Therefore, nominal models can be described as: Gp1 (s) =

s2

14.7059 1 + 2s + 14.7059 s + 10 14.7059 1 Gp4 (s) = 2 s + 2s + 14.7059 s + 2.13 Gp3 (s) =

s2

(9) (10)

The same approach proposed by Oliveira et al. (2017) is applied in a practical set-up (Figure 5). However, each stage is open to be implemented using any tools/algorithms. In this paper, uncertainty detection is based on nominal model output comparison with the actual output (a common fault detection strategy). When the error derivative exceeds a pre-defined threshold which depends on the uncertainty type/magnitude, a detection signal starts the 418

is given in Figure 6 and it is noteworthy that for accurate R Matlab continuous model simulation and uncertainty detection, the Digital Analog output (DA) was emulated with a quantizer and ZOH block (Figure 7). For the practical controller/plant, besides data acquisition blocks (analog input/analog output), a real time synchronization block guarantees the sampling time Ts = 0.01s, which is compatible with this feature. Non filtered discrete PID blocks (11) for Gp1 , Gp2 and with derivative filter (12) for Gp3 , Gp4 were added, using trapezoidal approximation and sampling time Ts = 0.01 seconds: Ts z + 1 1 z−1 + Kd 2 z−1 Ts z Ts z + 1 N + Kd Kp + K i z+1 2 z−1 1 + N T2s z−1 Kp + Ki

(11) (12)

where Kp , Ki and Kd represent respectively the proportional, integrative and derivative gains, and N is the filter constant. The controller output is limited in the interval −10V ≤ u(t) ≤ +10V , according to the used AD/DA card specifications (PCIe 6361, National Instruments, Austin, USA). Next section presents the practical results and give implementation details. 3. RESULTS AND DISCUSSION To evaluate the method in Oliveira et al. (2017) (Figure 8), two experiments were carried out. Experiment I: starting with Gp1 and a previously tuned 2DOF controller for it (Ctrl1 ), the objective is to trigger identification and optimization stages for tuning both Posicast and PID control (Ctrl2 ) when for 8s < t < 10s the system is switched to Gp2 . The start time for Figures 2, 9-13 was set next to t = 5s to avoid the initial oscillation every time the acquisition card is initiated. For a real world scenario,

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estimated model output. For this case T hdem > 5V . Since only one variation is considered, a global counter controls the identification trigger. When a parametric variation occurs, its effect in HC-PC is felt in the next reference change. In Experiment II the same procedure is applied to Gp3 with Ctrl3 after changing to Gp4 and Ctrl4 . The main objective is to compare and analyse this effect when the system changes while the previous controller is kept. For both experiments, an unit step load disturbance is applied at t = 25s.

Fig. 5. Proposed methodology

Fig. 9. Gp1 and Gp2 variation detection: system output (top plot), uncertainty detection trigger when the threshold exceeds (middle plot) and model mismatch error derivative (bottom plot)

Fig. 6. General block diagram - experiment

3.1 Optimization

Fig. 7. DA blocks for continuous model simulation this is not necessary, since the card is initiated just a single time. The effect of this parametric variation on tracking

Fig. 8. General block diagram as in Oliveira et al. (2017) of reference changes can be seen in Figure 9, suggesting that the HC parameters need to be optimized, whereas the PID suggests robustness to this particular variation. Several tests were conducted to set a threshold T hdem for the modeling error em = y −ym derivative, where ym is the 419

To highlight the flexibility of this approach, two different R pidtune methods are used, PSO for Gp1 , Gp2 and Matlab tool for Gp3 and Gp4 , which was configured to get an overshoot as small as possible. The PSO configuration was c1 , c2 = 2 and inertia factor ω linearly decreasing from 0.9 − 0.4, with 150 iterations and 50 particles. The optimization is subject to the following constraints: 0.1 ≤ A1 ≤ 0.8, 0.2s ≤ t1 ≤ 5s and 0.1 ≤ Kp , Ki , Kd ≤ 5, which define the search interval. However, other intervals may be considered. Alternatively, an informed initialization procedure can be used, based on classical tuning methods (Vranˇci´c et al. (2001)), to decide on the search interval. After the identification stage, the optimization runs for ˆ 2 estimated model: Gp 2.819 ˆ 2= Gp , (13) 2 s + 0.4082s + 2.819 with a suitable approximation of R2 = 0.9894 when compared to the data in Table 1. Since the practical step responses for both filters Gp3 , Gp4 represented common second order responses/curves, the identification method got: 1.836 , R2 = 0.8463, (14) + 1.962s + 1.836 1.55 ˆ 4 (s) = Gp , R2 = 0.9305 (15) 2 s + 0.7598s + 1.55

ˆ 3 (s) = Gp

s2

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However, if necessary, other identification methods can be used to get other dynamics. The statistical index R2 is a common way to check how the model explains the actual measurements and, for the proposed technique, R2 > 0.8 was found suitable. For these estimated models, Tables 2 and 3 summarize the 2DOF controllers: Table 2. 2DOF controller parameters - PSO Controller Ctrl1 Ctrl2

Plant Gp1 Gp2

A1 0.7 0.7

A2 0.3 0.3

t1 0.2s 0.8s

Kp 5 5

Ki 5 5

Kd 1.0497 1.0956

Table 3. 2DOF controller parameters - pidtune tool Controller Plant A1 Ctrl3 Gp3 0.8 Ctrl4 Gp4 0.8

A2 0.2 0.2

t1 0.8s 0.8s

Kp 0.0047 0.5

Ki 0.95 1.1

Kd 0.1 0.3

N 100 100

Fig. 11. Gp1 and Gp2 responses

From Table 2 and Figures 10-11, the influence of the parameter t1 to improve tracking is highlighted and the other parameters were enough to maintain performance after optimization. A slightly increase in Kd provided a better disturbance rejection (11). A detail on the control signal can be seen in Figure 13, which corroborates a higher Tu index in Table 5.

Fig. 12. Gp3 and Gp4 responses

Fig. 10. Gp2 output improvement with retuned 2DOF controller For delayed-like systems Gp3 , Gp4 , the optimization retuned all PID gains in Table 3 and a better disturbance rejection is presented in Figure 12. Common performance indexes used in process control are used to analyse the auto tuning methodology, namely, Integral of Absolute Error (IAE), Integral of Time Weighted Absolute Error (IT AE) and Integral of Squared Error  (ISE). Besides, the total control effort Tu = |u|, maximum overshoot Mp to measure  load disturbance effect and output total variation T Vy = |(y(t) − y(t − 1)| give insight for performance comparison. Tables 4 and 5 present the overall improvement when the retuned controller is used for each system. 420

Fig. 13. Gp3 and Gp4 responses: (top plot) and control signal (bottom plot) 4. CONCLUSIONS A practical experiment with electronics circuits was conducted to validate the simultaneous 2DOF optimization

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Table 4. Performance canonical second order systems IAE IT AE ISE Tu Mp T Vy

Gp1 ,Ctrl1 24.3 547.6 0.7 2500 14.39 20.5

Gp2 ,Ctrl1 108.5 2200 17.7 2700 26.17 19.3

Gp2 ,Ctrl2 51.1 845.5 6.8 2800 13.55 8.6

Table 5. Performance second order systems plus first order filter IAE IT AE ISE Tu Mp T Vy

Gp3 , Ctrl3 170.1 3900 23.7 2470 85.16 7.3

Gp4 ,Ctrl3 135.0 3200 24.7 2460 102.7 7.1

Gp4 ,Ctrl4 115.6 2700 10.3 2480 62.8 6.3

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