Closed-loop Data-driven Trade-off PID Control Design

Proceedings of the 3rd IFAC Conference on Proceedings of the 3rd IFAC Conference on Advances in Proportional-Integral-Derivative Proceedings of on Proceedings of the the 3rd 3rd IFAC IFAC Conference Conference on Control Advances in Proportional-Integral-Derivative Control Available online at www.sciencedirect.com Ghent, Belgium, May 9-11, 2018 Advances in Proportional-Integral-Derivative Control Proceedings of the 3rd IFAC Conference on Control Advances in Proportional-Integral-Derivative Ghent, Belgium, May 9-11, 2018 Ghent, 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) 244–249

Closed-loop Closed-loop Data-driven Data-driven Trade-off Trade-off PID PID Closed-loop Control Data-driven Trade-off PID Design Closed-loop Control Data-driven Trade-off PID Design Control Design Control Design Ryo Kurokawa ∗∗ Takao Sato ∗∗ Ramon Vilanova ∗∗ ∗∗

Ryo Kurokawa ∗∗ Takao Sato ∗∗ Ramon Vilanova ∗∗ ∗ Ryo Takao Sato Vilanova ∗ Ryo Kurokawa Kurokawa ∗ Yasuo Takao Konishi Sato ∗ Ramon Ramon Vilanova ∗∗ Yasuo Konishi ∗ ∗ Ryo Kurokawa Yasuo Takao Sato Ramon Vilanova ∗∗ Konishi Yasuo Konishi ∗ ∗ Yasuo Konishi ∗ Department of Mechanical Department of Mechanical Engineering, Engineering, Graduate Graduate School School of of ∗ ∗ DepartmentEngineering, of Mechanical Mechanical Engineering, Graduate School School of of University of Hyogo, Department of Engineering, Graduate Engineering, University of Hyogo, ∗ Department of Mechanical Engineering, Graduate School of Engineering, University of Hyogo, Hyogo, 2167 Shosha, Himeji, Hyogo 671-2280, Japan Engineering, University of 2167 Shosha, Himeji, Hyogo 671-2280, Engineering, University of Hyogo,Japan 2167 Shosha, Himeji, Hyogo Japan (e-mail: {tsato,konishi}@eng.u-hyogo.ac.jp). 2167 Shosha, Himeji, Hyogo 671-2280, 671-2280, Japan (e-mail: {tsato,konishi}@eng.u-hyogo.ac.jp). ∗∗ 2167 Shosha, Himeji, Hyogo 671-2280, Japan (e-mail: {tsato,konishi}@eng.u-hyogo.ac.jp). of Telecommunications and Systems Engineering ∗∗ Department (e-mail: {tsato,konishi}@eng.u-hyogo.ac.jp). Department of Telecommunications and Systems Engineering ∗∗ ∗∗ (e-mail: Department of {tsato,konishi}@eng.u-hyogo.ac.jp). Telecommunications and Systems Engineering Engineering Universitat Aut` o noma de Barcelona Department of Telecommunications and Systems Universitat Aut` o noma de Barcelona ∗∗ Department of de Telecommunications Systems Engineering Universitat Aut` o08193 noma Bellaterra, deand Barcelona Edifici Q-Campus la UAB, Barcelona, Universitat Aut` o noma de Barcelona Edifici Q-Campus de la UAB, 08193 Bellaterra, Barcelona, Spain Spain Universitat Aut` o noma de Barcelona Edifici Q-Campus de la UAB, 08193 Bellaterra, Barcelona, (e-mail: [email protected]). Edifici Q-Campus de la [email protected]). UAB, 08193 Bellaterra, Barcelona, Spain Spain (e-mail: Edifici Q-Campus de la [email protected]). UAB, 08193 Bellaterra, Barcelona, Spain (e-mail: [email protected]). (e-mail: (e-mail: [email protected]). Abstract: Abstract: The The present present study study proposes proposes aa new new approach approach to to aa data-driven data-driven PID PID control control design design Abstract: The present study proposes a new new approach to aaEven data-driven PID control control design method based on one-shot closed-loop input and output data. if the proposed controller is Abstract: The present study proposes a approach to data-driven PID design method based on one-shot closed-loop input and output data. Even if the proposed controller is Abstract: The present study proposes a prescribed new to stability aEven data-driven PID control design method based based on one-shot closed-loop input andapproach output data. Even if and the proposed proposed controller is designed using only one-shot data, both the robust tracking performance method on one-shot closed-loop input and output data. if the controller is designed using only one-shot data, both the prescribed robust stability and tracking performance method based on one-shot closed-loop input and output data. Even if by thesolving proposed controller is designed using only one-shot data, both the prescribed robust stability and tracking performance optimization are attained. The proposed control law is designed a constrained designed using only one-shot data, both the prescribed robust stability and tracking performance optimization are attained. The proposed control law is designed by solving a performance constrained designed usingproblem, only one-shot data, both the prescribed robust stability and tracking are attained. The proposed control law is designed by solving a constrained optimization in which the robust stability as a constraint condition is designed by are attained. The the proposed designed by solving isa designed constrained optimization problem, in which robustcontrol stabilitylaw as is a constraint condition by optimization are attained. Theusing proposed control is designed by solving constrained problem, in which which the robust stabilitylaw asFourier constraint condition isa performance designed by a sensitivity function estimated the discrete-time transform, and the problem, in the robust stability as aa constraint condition is designed by aoptimization sensitivity function estimated using the discrete-time Fourier transform, and the performance optimization problem, in whichusing thereference robust stability asFourier a constraint condition is performance designed by a sensitivity sensitivity function estimated using the discrete-time discrete-time Fourier transform, andproposed the performance function defined using a fictitious is minimized. As a result, the method a function estimated the transform, and the function defined usingestimated a fictitious reference is minimized. As transform, a result, the proposed method a sensitivity function using the discrete-time Fourier and the performance function defined using a fictitious reference is minimized. As a result, the proposed method provides trade-off design between the performance robust stability, where the function defined a fictitious is minimized. Asand a result, proposed method provides trade-offusing design between reference the tracking tracking performance and robustthe stability, where the function defined using a fictitious reference is minimized. Asperturbation. a result, proposed method provides trade-off design between the tracking performance and robust stability, where the robust stability is arbitrarily selected on the plant The effectiveness of provides trade-off design between thedepending tracking performance and robustthe stability, where the robust stability is arbitrarily selected depending on the plant perturbation. The effectiveness of provides trade-off design between thedepending trackingnumerical performance and robust stability, where the robust stability is arbitrarily arbitrarily selected depending on the plant plant perturbation. The effectiveness effectiveness of the proposed method is demonstrated through examples. robust stability is selected on the perturbation. The of the proposed method is demonstrated through numerical examples. robust stability is arbitrarily selected depending on the plant perturbation. The effectiveness of the proposed proposed method is demonstrated demonstrated through numerical numerical examples. the method is through examples. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. the proposed method is iterative demonstrated through numerical examples. Keywords: Keywords: PID PID control, control, iterative method, method, transient transient response, response, robust robust stability, stability, sensitivity sensitivity Keywords: PID control, iterative method, transient response, robust stability, sensitivity function, frequency estimation, multiobjective optimization, sampled-data system Keywords: PID control, iterative method, transient response, robust stability, sensitivity function, frequency estimation, multiobjective optimization, sampled-data system Keywords: PID control, iterativemultiobjective method, transient response,sampled-data robust stability, sensitivity function, estimation, optimization, system function, frequency frequency estimation, multiobjective optimization, sampled-data system function, frequency estimation, multiobjective optimization, sampled-data system 1. the reference reference model model tracking tracking (Kano (Kano et et al., al., 2010; 2010; Saeki Saeki the 1. INTRODUCTION INTRODUCTION 1. the reference model tracking (Kano et al., 2010; Saeki et al., 2013). 1. INTRODUCTION INTRODUCTION the reference model tracking (Kano et al., 2010; Saeki et al., 2013). 1. INTRODUCTION the reference et 2013). et al., al., 2013). model tracking (Kano et al., 2010; Saeki Conventional direct tuning tuning methods methods have have focused focused on on referreferet al., 2013). direct Conventional ˚ Proportional-integral-derivative (PID) control ( A str¨ o m Conventional direct tuning methods have focused on reference model tracking, whereas actual systems also require ˚ Conventional direct tuning methods have focused on referProportional-integral-derivative (PID) control (A str¨ om ence model tracking, whereas actual systems also require Proportional-integral-derivative (PID) control ((˚ A oom ˚ Conventional direct tuning methods have focused onrequire referand H¨ gglund, 2006; 2006; Vilanova Vilanova and and Visioli, 2012; Vilanova Vilanova ence model tracking, whereas actual systems also Proportional-integral-derivative (PID) control Astr¨ str¨ m robust robust stability (Saeki and Sugitani, 2011; Parastvand and ence model tracking, whereas actual systems also require and H¨ aagglund, Visioli, 2012; stability (Saeki and Sugitani, 2011; Parastvand and ˚ Proportional-integral-derivative (PID) control ( A str¨ o m and H¨ a gglund, 2006; Vilanova and Visioli, 2012; Vilanova ence model tracking, whereas actual systems also require et al., 2017) is well known and widely used because of its robust stability (Saeki and Sugitani, 2011; Parastvand and andal., H¨ a2017) gglund, 2006;known Vilanova Visioli, Vilanova Khosrowjerdi, 2014; Koenings et al., 2017) because the robust stability2014; (SaekiKoenings and Sugitani, 2011; Parastvand et is well andand widely used2012; because of its Khosrowjerdi, et al., 2017) because and the and H¨ a2017) gglund, 2006; Vilanova and Visioli, Vilanova et is and widely used because of robust stability (Saeki and Sugitani, 2011; Parastvand and simple structure andknown useful properties. Most2012; of the the tuning Khosrowjerdi, Koenings et 2017) because the et al., al., 2017) is well well known and widely used because of its its stability stability of the the2014; control system is al., critical. The relationKhosrowjerdi, 2014; Koenings etis al., 2017) because the simple structure and useful properties. Most of tuning of control system critical. The relationet al., 2017) is well widely used of its ship simple structure and useful properties. Most of Khosrowjerdi, 2014; Koenings etis 2017) because the approaches for PID controllers are model-based: model-based: mathstability of control system critical. The simple structure andknown useful and properties. Mostbecause of the theaa tuning tuning between the tracking performance and therelationrobust stability of the thethe control system is al., critical. The relationapproaches for PID controllers are mathship between tracking performance and the robust simple structure andcontrollers useful Most of the approaches for model-based: aa tuning mathstability of thethe control system is critical.and The relationematical model (usually ofproperties. loware order) of the the controlled ship tracking performance the robust approaches for PID PID controllers are model-based: math- stability stability experiment a trade-off relationship, so that both ship between between the tracking performance and the robust ematical model (usually of low order) of controlled experiment a trade-off relationship, so that both approaches for PID controllers are model-based: aprocess math- the ematical model (usually of order) of controlled ship between the tracking performance andso robust plant is needed needed and tuning relation relates the stability experiment aa trade-off relationship, that both ematical model (usually of low low order) of the thethe controlled tracking performance and the robust stability stability cannot stability experiment trade-off relationship, sothe that both plant is and aa tuning relation relates process the tracking performance and the robust cannot ematical model (usually of low order) of the controlled plant is needed and a tuning relation relates the process stability experiment a trade-off relationship, so that both model parameters with the controller parameters. One of the tracking performance and the robust stability cannot plant is needed and a tuning relation relates the process be optimized simultaneously. Therefore, a trade-off design theoptimized tracking performance andTherefore, the robusta stability cannot model parameters with the controller parameters. One of be simultaneously. trade-off design plant isparameters neededofand aapproach tuning relation relates the process model with the controller parameters. One of the tracking performance and the robust stability cannot the drawbacks this is the need to encapsulate be optimized simultaneously. Therefore, a trade-off design model parameters with the controller parameters. One of based on the required modeling accuracy represents be optimized simultaneously. Therefore, a trade-off design the drawbacks of this approach is the need to encapsulate based on the required modeling accuracy represents aa model parameters with the order controller parameters. One of feasible drawbacks of this the to be optimized Therefore, a trade-off the process data into low model (O’Dwyer, 2003). based on the required accuracy represents aa drawbacks of into this approach is model the need need to encapsulate encapsulate alternative designmodeling approach. To this this end, bydesign using based on thesimultaneously. required modeling accuracy represents the process data aaapproach low orderis (O’Dwyer, 2003). feasible alternative design approach. To end, by using drawbacks of into this approach isdata-driven the need toapproach encapsulate the process data a low order model (O’Dwyer, 2003). based on the required modeling accuracy represents a In contrast to this approach, the for feasible alternative design approach. To this end, using the process data into a low order model (O’Dwyer, 2003). the sensitivity function, the stability margin is by selected feasible alternative designthe approach. Tomargin this end, by using In contrast to this approach, the data-driven approach for the sensitivity function, stability is selected the process data into a low order model (O’Dwyer, 2003). In contrast to this approach, the data-driven approach for feasible alternative design approach. To this end, by using tuning the PID control has also been approached from the sensitivity function, the stability margin is selected In contrast to this approach, the data-driven approach for to be appropriate for the model uncertainty (Arrieta and thebesensitivity function, stability margin(Arrieta is selected tuning the PID control has also been approached from to appropriate for the the model uncertainty and In contrast to thiscontrol data-driven approach for Vilanova, tuning the PID has also been approached from thebe the stability is trade-off selected different of view. view. Inapproach, unfalsified control (Safonov and Tsao, Tsao, to appropriate for model uncertainty and tuning the PID control has the also been(Safonov approached from 2011, function, 2012; Kurokawa et al., al., margin 2017).(Arrieta In to besensitivity appropriate for the the model uncertainty (Arrieta and different of In unfalsified control and Vilanova, 2011, 2012; Kurokawa et 2017). In trade-off tuning the PIDfeedback control has also been(Safonov approached from design, different of In control to be appropriate for the model performance, uncertainty (Arrieta and 1997), iterative tuning (Hjalmarsson etand al., Tsao, 1998; Vilanova, 2011, 2012; Kurokawa et al., 2017). In trade-off different of view. view. In unfalsified unfalsified control (Safonov and Tsao, the higher the tracking the smaller Vilanova, 2011, 2012; Kurokawa et al., 2017). In trade-off 1997), iterative feedback tuning (Hjalmarsson et al., 1998; design, the higher the tracking performance, the smaller different of view. In unfalsified control (Safonov and Tsao, 1997), iterative feedback tuning (Hjalmarsson et al., 1998; Vilanova, 2011, 2012; Kurokawa et al., 2017). In trade-off Hjalmarsson, 2002) and correlation-based tuning (CbT) design, the higher the tracking performance, the smaller 1997), iterative feedback tuning (Hjalmarsson et al., 1998; the stability margin, and vice versa. versa. Since conventional conventional design, the higher theand tracking performance, the smaller Hjalmarsson, 2002) and correlation-based tuning (CbT) the stability margin, vice Since 1997), iterative feedback et al.,(CbT) 1998; Hjalmarsson, and correlation-based design, the higher tracking performance, the smaller (Karimi et al., al.,2002) 2004; Miˇsstuning kovi´ et(Hjalmarsson al., 2005), 2005),tuning although the methods the margin, and vice Since Hjalmarsson, 2002) and correlation-based tuning (CbT) require a the plant model, data-driven trade-off dethe stability stability margin, and vice versa. versa. Since conventional conventional (Karimi et 2004; Miˇ kovi´ cc et al., although the methods require a plant model, data-driven trade-off deHjalmarsson, 2002) and correlation-based tuning (CbT) (Karimi et al., 2004; Miˇ s kovi´ c et al., 2005), although the the stability margin, and vice versa. Since conventional controller parameters are decided directly by the conmethods require a plant model, data-driven trade-off de(Karimi et al., 2004; Miˇ s kovi´ c et al., 2005), although the sign is the motivation of the present study. methods require a plant model, data-driven trade-off decontroller parameters are decided directly by the con- sign is the motivation of the present study. (Karimi et parameters al., 2004; skovi´ c etexperiments al.,directly 2005), although the sign controller are decided by conmethods require a plant model, data-driven trade-off detrolled process data,Miˇ iteration arethe needed is the motivation of the present study. controller parameters are decided directly by the consign is the motivation of the present study. trolled process data, iteration experiments are needed As an open loop data-driven data-driven robust tuning, tuning, van Heusden Heusden controller parameters are decided directly con- As trolled process data, iteration are needed signan is open the motivation of the present study. van for optimization. In addition addition to experiments these ones, by noniterative loop robust trolled process data, iteration experiments arethe needed for optimization. In to these ones, noniterative As an open loop data-driven robust tuning, van Heusden et al. (2011) discussed closed-loop stability, and Rojas and As an open loop data-driven robust tuning, van Heusden trolled process data, iteration experiments are needed for In to ones, noniterative tuning methods, such such as virtual virtual reference feedback tun- et al. (2011) discussed closed-loop stability, and Rojas and for optimization. optimization. In addition addition to these these ones,feedback noniterative tuning methods, as reference tunAs an(2011) open loop data-driven robust tuning, vanRojas Heusden et al. discussed closed-loop stability, and Vilanova (2011, 2012) studied robust stability using the et al. (2011) discussed closed-loop stability, and Rojas and for optimization. In addition to these ones, noniterative tuning methods, such as virtual reference feedback tuning (VRFT) (Campi etasal., al., 2002;reference Masuda feedback et al., al., 2017), 2017), (2011, 2012) studied robust stability using and the tuning methods, suchet virtual tun- Vilanova ing (VRFT) (Campi 2002; Masuda et et al. (2011) discussed closed-loop stability, and Rojas and Vilanova (2011, 2012) studied robust stability using the empirical transfer-function estimate (Keesman, 2011). On Vilanova (2011, 2012) studied robust stability using the tuning methods, such as virtual reference feedback tuning (VRFT) (Campi et al., 2002; Masuda et al., 2017), fictitious reference iterative tuning (FRIT) (Souma et al., empirical transfer-function estimate (Keesman, 2011). On ing (VRFT) (Campi et al.,tuning 2002; (FRIT) Masuda (Souma et al., 2017), fictitious reference iterative et al., the Vilanova (2011, 2012) studied robust stability using the empirical transfer-function estimate (Keesman, 2011). On other hand, the present study proposes a closed-loop empirical transfer-function estimate (Keesman, 2011). On ing (VRFT) (Campi etand al.,noniterative 2002; (FRIT) Masuda et al., 2017), fictitious iterative tuning (Souma et 2004; Jengreference et al., al., 2017), correlation-based other hand, the present study proposes a closed-loop fictitious reference iterative tuning (FRIT) (Souma et al., al., the 2004; Jeng et 2017), and noniterative correlation-based empirical transfer-function estimate (Keesman, 2011). On the other hand, the present study proposes a closed-loop data-driven robust tuning method subject to prescribed the other hand, the present study proposes a closed-loop fictitious reference iterative tuning (FRIT) (Souma et al., 2004; et 2017), noniterative correlation-based tuning (NCbT) (Karimi et al., al., 2007), have have been proposed. proposed. data-driven robust tuning method subject to prescribed 2004; Jeng Jeng et al., al.,(Karimi 2017), and and noniterative correlation-based tuning (NCbT) et 2007), been the other hand, the present study proposes a closed-loop data-driven robust tuning method subject to prescribed robust stability. Totuning this end, end, the subject maximum sensitivity data-driven robust method to prescribed 2004; et al., 2017), and noniterative correlation-based tuning (NCbT) (Karimi et al., 2007), been proposed. Furthermore, VRFT and FRIT have have been extended such robust stability. To this the maximum sensitivity tuningJeng (NCbT) (Karimi etFRIT al., 2007), have been proposed. Furthermore, VRFT and have been extended such data-driven robust tuning method subject to estimation prescribed robust stability. To this end, the maximum sensitivity is obtained based on frequency characteristic robust stability. To this end, the maximum tuning (NCbT) (Karimi et al., 2007), have been proposed. Furthermore, VRFT and FRIT have been extended such that the reference model is also tuned in order to improve is obtained based on frequency characteristic estimation Furthermore, VRFT andisFRIT have been extended such robust stability. To this end, the maximum sensitivity that the reference model also tuned in order to improve sensitivity is obtained obtained based based on on frequency frequency characteristic characteristic estimation estimation Furthermore, VRFT andis have been extended such is that model also in to that the the reference reference model isFRIT also tuned tuned in order order to improve improve is obtained based on frequency characteristic estimation that the reference model is also tuned in order to improve 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Copyright © 2018 244 Copyright 2018 IFAC IFAC 244Control. Peer review© responsibility of International Federation of Automatic Copyright ©under 2018 IFAC IFAC 244 Copyright © 2018 244 10.1016/j.ifacol.2018.06.073 Copyright © 2018 IFAC 244

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245

-1

Md(z ) rd(k)

+ ε (k) d -

-1

Gcl(z )

Fig. 2. Error system

Fig. 1. Sampled-data control system

2.2 Constrained Optimization Problem

(Matsui et al., 2010). Furthermore, the tracking performance is optimized using extended FRIT (E-FRIT) (Kano et al., 2010). As a result, trade-off design between tracking performance and robust stability is achieved. Generally, the higher the robust stability, the worse the tracking performance. Therefore, in the proposed method, the robust stability is selected depending on the plant perturbation, and the tracking performance is then optimized subject to the prescribed robust stability. Consequently, the tracking performance is better when the plant perturbation is small enough. The remainder of the present paper is organized as follows. Section 2 presents the control system to be designed and the design objective of the present study. In Section 3, a data-driven constrained optimization problem is designed, and the optimal PID parameters are decided. In Section 4, the effectiveness of the proposed method is evaluated through numerical examples. Finally, concluding remarks are presented in Section 5.

The design objective of the present study is to have the closed-loop plant output follow the reference model output subject to an assigned stability margin. A block-diagram of the error system is illustrated in Fig. 2. In this figure, Md (z −1 ) is the reference model which is the discrete-time model of a continuous-time reference model M (s), and Gcl (z −1 ) is a closed-loop system from rd (k) to yd (k) in Fig. 1. Hence, εd (k) is defined as follows: εd (k) = (Md (z −1 ) − Gcl (z −1 ))rd (k)

2. DESIGN OBJECTIVE

(2)

In the present study, the reference model Md (z ) is defined by the following second-order plus dead-time transfer function: ω02 M (s) = 2 e−L0 s (3) s + 2ω0 s + ω02 where ω0 and L0 are the natural angular frequency and dead-time, respectively. −1

Using Eq. (2), a performance function is defined as follows: J=

N 1  εd (k)2 N

(4)

k=1

where N denotes the control horizon.

2.1 PID Control Law Consider the control system illustrated in Fig. 1, where P (s) is a controlled continuous-time plant, and H and S denote the zero-th holder and sampler, respectively. A discrete-time control input ud (k) is decided by a discretetime controller and is converted by the holder to a continuous-time signal u(t), which is to be sent to the plant. A continuous-time plant output y(t) is converted to a discrete-time signal yd (k) by the sampler, which is used in a discrete-time controller to decide the discrete-time control input. The discrete-time control input is calculated by a PID control law given as follows: ud (k) = Ce (z −1 )ed (k) − Cy (z −1 )yd (k) ed (k) = rd (k) − yd (k) Ts Ce (z −1 ) = Kp + Ki Δ Δ −1 Cy (z ) = Kd Ts Δ = 1 − z −1

yd(k)

(1)

where rd (k) is a reference input, Ts denotes the sampling interval, and z denotes the forward shift operator. Moreover, Kp , Ki , and Kd are the proportional gain, the integral gain, and the derivative gain, respectively, and are referred to collectively as PID gains. The PID gains are tuned according to the design objective described in 2.2. 245

The design objective is not only to minimize the performance function but also to guarantee robust stability. To this end, the following sensitivity function is introduced: 1 Sf (e−jω ) = (5) 1 + Cd (e−jω )Pd (e−jω ) Cd (e−jω ) = Ce (e−jω ) + Cy (e−jω ) (6) where Pd (z −1 ) denotes a plant model in discrete time. Using the sensitivity function, the maximum sensitivity is also obtained as follows: Ms = max |Sf (e−jω )| (7) ω

In the present study, in order to obtain the assigned stability margin, the following constraint condition is defined: |Ms − Msd | = 0 (8)

Msd is the desired maximum sensitivity and is assigned by the designer. The recommended range of Msd is known to be between 1.4 and 2.0 (˚ Astr¨ om and H¨ agglund, 2006). Here, Ms is defined such that as the stability margin increases, Ms decreases. In contrast, as the tracking performance decreases, Ms increases. As a result, the constrained optimization problem is defined as follows: min J (9) Kp ,Ki ,Kd

subject to |Ms − Msd | = 0

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In the present study, this problem is solved without conducting an iterative experiment and without previous knowledge of a plant model. However, the plant model is included in Eq. (5), and Eq. (4) must be minimized non-iteratively. In the next section, constraint condition Eq. (7) is first estimated using one-shot data in 3.1, and the performance function Eq. (4), which can be minimized based on the one-shot data in 3.2, is then rewritten.

3. PROPOSED DESIGN USING ONE-SHOT DATA 3.1 Estimation of the maximum sensitivity In order to obtain Ms without the plant model, the gain and phase characteristic curves are estimated using the available data as in Matsui et al. (2010). In this estimation method, a bandpass filter is introduced so that a Discrete Fourier Transform (DFT) can be used to estimate the frequency characteristics. As a result, the data is absolutely integrable, and thus the discontinuity between the start and end data is resolved. The bandpass filter is used as follows: T1 s B(s) = (10) (T1 s + 1)(T2 s + 1) T 1 = ωl T s T 2 = ωh T s where ωl and ωh are design parameters, and the estimated frequency range is decided by selecting them. The bandpass filter is converted to a discrete-time filter Bd (z −1 ), and initial closed-loop data u0d (k) and yd0 (k) are transformed as follows: Uf (ω) Yf (ω)

= F [Bd (z )u0d (k)] = F[Bd (z −1 )yd0 (k)] −1

(11) (12)

where F [·] denotes the DFT. Using Eq. (11) and Eq. (12), the frequency characteristic is estimated as follows: Yf (ω) Pˆd (e−jω ) = Uf (ω)

(13)

The estimated frequency characteristics are used in Eq. (5), and hence the sensitivity function can be estimated as follows, although the high-frequency area is removed by the bandpass filter: 1 Sˆf (e−jω ) = (14) −jω 1 + Cd (e )Pˆd (e−jω ) Hence, in the present study, the maximum sensitivity is also estimated as follows: ˆ s = max |Sˆf (e−jω )| M ω

3.2 Tracking Performance Optimization Using E-FRIT A closed-loop data-based noniterative tuning is designed using E-FRIT (Kano et al., 2010). This method evaluates the error between the initial output yd0 (k) and its fictitious reference output y˜d (k), and hence the performance function is defined as follows: 246

N 1  (˜ εd (k)2 + λΔ˜ ud (k)2 ) J˜ = N

(15)

k=1

ε˜d (k) = yd0 (k) − y˜d (k) rd (k) y˜d (k) = Md (z −1 )˜ −1 −1 0 r˜d (k) = Ce (z ) (ud (k) + Cd (z −1 )yd0 (k)) u ˜d (k) = Ce (z −1 )rd0 (k) − Cd (z −1 )Md (z −1 )rd0 (k) where λ is a weighting factor for the input deviation, and rd0 (k) denotes the initial reference input. The minimization of Eq. (15) corresponds to that of Eq. (4) when λ = 0. Hence, the controller parameters that optimize Eq. (4) are obtained by minimizing Eq. (15) using the one-shot closedloop data. In the design of E-FRIT, ω0 and L0 are also estimated for minimizing Eq. (15). In other words, ω0 and L0 are selected so that the best reference model is selected. Consequently, the optimization problem to be solved is redefined as follows: min J˜ (16) Kp ,Ki ,Kd ,ω0 ,L0

ˆ s − M d| = 0 subject to |M s Solving this problem, the PID and reference model parameters are decided such that the tracking performance is optimized subject to the prescribed robust stability. In the original E-FRIT, the control system stability is achieved by weighting the input deviation, although the stability condition is not used. On the other hand, in the proposed method, since the PID parameters are decided such that the prescribed stability margin is achieved, the weight for the input deviation is not used, namely, λ = 0. Therefore, the coordination of the weighting factor is not needed. 4. NUMERICAL EXAMPLE Consider the following transfer function: 1 −0.6s P (s) = e s+1

(17)

As this model is assumed to be unknown, a control system is designed. The sampling interval (Ts ) is set to 0.01 s, and the reference input is defined as a unit step function. The initial controller parameters are set as Kp = 0.05, Ki = 0.1, and Kd = 0.01 by trial and error such that the closed-loop system is stabilized. The obtained initial control input and output responses are plotted by dashed lines in the upper and lower panels, respectively, of Fig. 3. The parameters are decided based on this data. In order to realize robust stability as the constraint condition, the frequency characteristic is identified using the estimation method given in Section 3.1. The filtered input and output data are plotted by solid lines in the upper and lower panels, respectively, of Fig. 3. In the present study, in order to estimate the frequency characteristic in the range from 10−1 rad/s to 102 rad/s, ωl = 100 and ωh = 10. The identified gain and phase characteristics are then shown in Fig. 4. From Fig. 4, the frequency characteristics are identified approximately correctly without the high-frequency area. Using the identified frequency characteristics, the constraint condition is implemented using the selected Msd . The optimization problem stated in Eq. (16) is solved using the initial data and estimated frequency characteristics. In

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Table 1. PID and reference model parameters of the proposed method for Msd varied as 1.4, 1.6, 1.8, and 2.0

1.5

Output

1 Initial output response Filtered output response

0.5 0 −0.5 0

10

20

30

40

50

Time [sec]

60

70

80

90

247

100

1.5

Msd

ω0

L0

Kp

Ki

Kd

1.4 1.6 1.8 2.0

2.3806 3.7213 4.9016 5.7176

0.4872 0.5499 0.5830 0.5671

0.8899 1.2406 1.4559 1.6457

0.7508 0.9157 1.0041 1.0830

0.1253 0.2396 0.3112 0.3437

Input

1

Table 2. PID and reference model parameters of E-FRIT with control weight

Initial input response Filtered input response

0.5 0 −0.5 0

10

20

30

40

50

Time [sec]

60

70

80

90

100

Fig. 3. Initial and filtered input/output signals

λ

ω0

L0

Kp

Ki

Kd

10−2 10−3

0.4277 1.6619

0.5819 0.4484

0.0013 0.5951

0.1900 0.6036

9.7584×10−4 2.0164×10−4

10

1.2

−20

1

−30 −40 −50

Estimated True

−60 −2 10

10

0.8 −1

0

10

1

10

Frequency [rad/s]

2

10

10

3

Output

Gain [dB]

0 −10

2000

0.6

Phase [deg]

0

Proposed output: Mds =2.0

−2000

s

Proposed output: Mds =1.6

−6000 −8000 −10000

10

Proposed output: Mds =1.4

0.2

Estimated True

−12000 −2 10

−1

0

10

1

10

Frequency [rad/s]

2

10

10

Conventional output: λ=10−2 Conventional output: λ=10−3 Initial output response

3

0 0

Fig. 4. Original and estimated gain and phase characteristics the present study, the problem is solved using “fmincon” with option “interior point” 1 . In the present study, the control system is designed with Msd varied as 1.4, 1.6, 1.8, and 2.0, respectively. For comparison with the proposed method, E-FRIT is also designed. Since E-FRIT can guarantee closed-loop stability by weighing the control input deviation, the control system is designed using λ varied as 10−2 and 10−3 , respectively, where the robust stability constraint is not considered in E-FRIT. The parameters obtained using the proposed method are shown in Table 1. Note that as Msd increases, the natural angular frequency of the reference model increases, and hence the trade-off design is achieved because the stability margin is inversely proportional to Ms . The parameters obtained using E-FRIT are shown in Table 2, in which λ decreases as the natural angular frequency increases. The tracking performance and robust stability are illustrated in Section 4.1 and Section 4.2, respectively.

1

2

3

4

5

Time [sec]

6

7

8

9

10

Fig. 5. Output responses of the proposed and conventional methods cyan and orange solid lines indicate the output responses obtained using E-FRIT with λ = 10−2 and 10−3 , respectively. The magenta dashed-dotted line indicate the initial output response using the initial PID parameters. In the proposed method, the tracking performance depends on Msd because Msd increases as the reference model response becomes faster. On the other hand, the output response obtained using E-FRIT with a small value of λ is faster than that obtained using E-FRIT with a large value of λ, where the output result obtained using λ = 0 is not shown because it is unstable. Note that the tracking performance of E-FRIT depends on the selection of λ. The tracking performance for Msd is evaluated quantitatively. The control performance of the plant output for the reference input is evaluated by Eq. (18). Jcontrol =

4.1 Tracking Performance

N 1  (rd (k) − yd (k))2 N

(18)

k=1

The closed-loop control results obtained using the obtained parameters are shown in Fig. 5. In the figure, the red, green, blue, and black solid lines indicate the output responses obtained using the proposed method with Msd varied as 1.4, 1.6, 1.8, and 2.0, respectively, and the 1

Proposed output: Md=1.8

0.4

−4000

Mathworks, Inc.

247

The evaluated values of the proposed method are shown in Table 3, and trade-off design is achieved by selecting Msd because the tracking performance depends on Msd . The evaluated values of the conventional E-FRIT are also shown in Table 3. As shown in this table, the convergence to the reference input is worse than that of the proposed method.

IFAC PID 2018 248 Ghent, Belgium, May 9-11, 2018

Ryo Kurokawa et al. / IFAC PapersOnLine 51-4 (2018) 244–249

Table 3. Tracking performance evaluation in Fig. 5 and obtained Ms

1.5 1.4

Msd = Proposed method Conventional method

λ=

Jcontrol

Ms

0.0102 0.0089 0.0085 0.0082 0.0351 0.0121

1.4000 1.6000 1.8000 2.0000 1.2436 1.3886

1.4 1.6 1.8 2.0 10−2 10−3

1.3 1.2 1.1

Output

Design parameter

1 0.9 0.8 0.7

Proposed output: Mds =1.8 Proposed output: Md=1.6 s

Proposed output: Md=1.4 s

1.5

0.6

Conventional output: λ=10−2 Conventional output: λ=10−3

0.5 60

80

90

100

110

Time [sec]

120

130

140

150

Fig. 7. Enlarged view of Fig. 6

1

5. CONCLUSION

Proposed output: Md=2.0

Output

70

s

Proposed output: Mds =1.8 0.5

Proposed output: Mds =1.6 Proposed output: Mds =1.4 −2

Conventional output: λ=10

−3

Conventional output: λ=10 Initial output response

0 0

50

Time [sec]

100

150

Fig. 6. Output responses with plant perturbation 4.2 Robust Stability The values of Ms obtained using the proposed method for Msd are shown in Table 3. The prescribed robust stability is attained. Furthermore, as Msd decreases, the robust stability increases because the robust stability has a tradeoff relationship with respect to the tracking performance. In order to clarify this trade-off relationship, the robust stability corresponding to Msd is confirmed. The simulation results for selected Msd are shown in Fig. 6. In this figure, the plant is not perturbed and is Eq. (17) from the start until 100 s. Then, after 100 s, the plant is perturbed as follows: 1.5 e−0.7s P � (s) = (19) 0.73s + 1 The output response designed using Msd = 2.0 diverges after 100 s. An enlarged view of Fig. 6 is shown in Fig. 7, where, for clarity, the output response using Msd = 2.0 and the initial response are not shown. This figure shows that the output responses using a small Msd are stabilized. Therefore, the trade-off relationship between the tracking performance and the robust stability is confirmed. The values of Ms obtained using the conventional E-FRIT are also shown in Table 3. Since Ms obtained using the conventional E-FRIT is smaller than that obtained using the proposed method, the system designed using the conventional E-FRIT is sufficiently robust. Thus, the output responses obtained using the conventional E-FRIT are also stabilized with non-zero λ. Although trade-off design using the conventional E-FRIT is possible by selecting λ, the robust stability is not explicitly assigned. 248

In the present study, we proposed a new data-driven approach. In the proposed method, a constrained optimization problem is solved, in which the performance function was designed by E-FRIT, and robust stability was achieved by using the sensitivity function as a constraint condition. As a result, using one-shot closed-loop data, controller parameters were decided such that the tracking performance was optimized subject to the prescribed robust stability. Since the robust stability was selected, a trade-off design between tracking performance and robust stability was achieved. Hence, the tracking performance was improved by selecting a small stability margin when the plant perturbation is small enough. Finally, the effectiveness of the proposed design method was demonstrated through numerical examples. The proposed method focuses on servo tracking, although regulation control for a disturbance has been also investigated (Ishii et al., 2015). Therefore, in the future, we intend to investigate regulation control for disturbance attenuation. Furthermore, in the present study, a non-convex problem was solved in order to obtain the trade-off design, and so the obtained control system was influenced by the initial conditions. As such, in the future, this influence must be reduced. ACKNOWLEDGEMENTS The present study was supported by JSPS KAKENHI Grant Number JP16K06425. REFERENCES ˚ Astr¨ om, K.J. and H¨ agglund, T. (2006). Advanced PID Control. Instrumentation, Systems, and Automation Society. Arrieta, O. and Vilanova, R. (2011). Simple PID tuning rules with guaranteed Ms robustness achievement. In Preprints of the 18th IFAC World Congress, 12042– 12046. Milano. Arrieta, O. and Vilanova, R. (2012). Simple servo/regulation proportional-integral-derivative (PID) tuning rules for arbitrary Ms -based robustness achievement. Industrial & Engineering Chemistry Research, 51, 2666–2674.

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Ryo Kurokawa et al. / IFAC PapersOnLine 51-4 (2018) 244–249

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