Robust PID Controller Design for Both Delay-Free and Time-Delay Systems⁎

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

ScienceDirect

IFAC PapersOnLine 51-4 (2018) 930–935

Robust PID Controller Design for Both Robust PID Controller Design for Both Robust PID Controller Design for Both  Robust PID Controller Design for Both Delay-Free and Time-Delay Systems  Delay-Free and Time-Delay Systems Robust PID Controller Design for Both Delay-Free and Time-Delay Systems Delay-Free and Time-Delay Systems  ∗ ∗∗ Delay-Free Time-Delay Systems Eitaku and Nobuyama Yasushi Kami

∗ ∗∗ ∗ ∗∗ Eitaku Eitaku Nobuyama Yasushi Kami Eitaku Nobuyama Nobuyama ∗∗ Yasushi Yasushi Kami Kami ∗∗ ∗∗ Eitaku Nobuyama Yasushi Kami ∗ ∗ Informatics, Kyushu ∗∗ Department of Systems Design and Institute of Eitaku Nobuyama Yasushi Kami ∗ ∗ Department of Systems Design and Informatics, Kyushu Institute of ∗ Department of Systems Design and Informatics, Kyushu Institute of of Department of Systems Design and Informatics, Kyushu Institute Technology, Kawazu, Iizuka, Fukuoka 820-8502 Japan (e-mail: ∗ Department of Kawazu, Systems Iizuka, Design Fukuoka and Informatics, Kyushu Institute of Technology, 820-8502 Japan (e-mail: Technology, Kawazu, Iizuka, Fukuoka 820-8502 Japan (e-mail: ∗ Technology, Kawazu, Iizuka, Fukuoka 820-8502 Japan (e-mail: [email protected]). Department of Kawazu, Systems Design Fukuoka and Informatics, Institute of Technology, Iizuka, 820-8502Kyushu Japan (e-mail: [email protected]). ∗∗ [email protected]). [email protected]). Department of Electrical and Computer Engineering, National ∗∗ Technology, Kawazu, Iizuka, Fukuoka 820-8502 Japan (e-mail: ∗∗ [email protected]). Department of Electrical and Computer Engineering, National ∗∗ Department of Electrical and Computer Engineering, National of [email protected]). Electrical and Computer Engineering, National Institute of Technology, Akashi College, Uozumi-cho, Akashi, Hyogo ∗∗ Department Department of Electrical and Computer Engineering, National Institute of Technology, Akashi College, Uozumi-cho, Akashi, Hyogo Institute of 674-8501, Technology, Akashi College, Uozumi-cho, Akashi, Hyogo ∗∗ Institute of Technology, Akashi College, Uozumi-cho, Akashi, Hyogo Japan (e-mail: [email protected]) Department of Electrical and Computer Engineering, National Institute of 674-8501, Technology, Akashi College, Uozumi-cho, Akashi, Hyogo 674-8501, Japan (e-mail: [email protected]) Japan (e-mail: [email protected]) 674-8501, Japan (e-mail: [email protected]) Institute of 674-8501, Technology, Akashi College, Uozumi-cho, Akashi, Hyogo Japan (e-mail: [email protected]) 674-8501, Japan (e-mail: [email protected]) Abstract: Robust PID controller design methods are proposed for linear single-input singleAbstract: Robust Robust PID PID controller controller design design methods methods are are proposed proposed for for linear linear single-input single-input singlesingleAbstract: Abstract: Robust PID controller design methods are proposed for linear single-input singleoutput systems. Non-parametric models represented by a finite number of frequency responses Abstract: Robust PID controller designrepresented methods are proposed for linear single-input singleoutput systems. Non-parametric models represented by a finite number of frequency responses output systems. Non-parametric models by a finite number of frequency responses output systems. Non-parametric models represented by a finiteare number ofbased frequency responses are used for them. Sufficient conditions for closed-loop stability derived on the Nyquist Abstract: PID controller design methods are proposed for linear single-input singleoutput models by a finiteare number ofbased frequency are used usedsystems. for Robust them.Non-parametric Sufficient conditions forrepresented closed-loop stability are derived based on the theresponses Nyquist are used for them. Sufficient conditions for closed-loop stability derived on the Nyquist are for them. Sufficient conditions for closed-loop stability are derived based on Nyquist stability criterion and the sufficient conditions are reduced to convex constraints. Together with output systems. Non-parametric models represented by a finite number of frequency responses are used for them. and Sufficient conditions for closed-loop stability are derived based on the Nyquist stability criterion and the sufficient sufficient conditions are reduced reduced to convex convex constraints. Together with stability criterion the sufficient conditions are reduced to convex constraints. Together with stability criterion and the conditions are to constraints. Together with the convex constraints and closed-loop model matching problems the robust PID controller are used for them. Sufficient conditions for closed-loop stability are derived based on the Nyquist stability criterion and the sufficient conditions are reduced problems to convex the constraints. Together with the convex constraints and closed-loop model matching problems the robust PID controller the convex constraints and closed-loop model matching robust PID controller the convex constraints and closed-loop model matching problems the robust PID controller design problems are as convex optimization important feature of the stability criterion andformulated the sufficient are reducedproblems. to convexA constraints. Together the convex constraints and closed-loop model matching problems robust PID controller design problems are formulated as conditions convex optimization problems. Athe important feature ofwith the design problems are formulated as convex optimization problems. A important feature of the design problems are formulated as convex optimization problems. A important feature of the proposed design methods is that they can be applied to delay-free and time-delay systems in the convex constraints and closed-loop model matching problems the robust PID controller design problems are formulated as convex optimization problems. A important feature of the proposed design methods is that they can be applied to delay-free and time-delay systems in proposed design methods methods is that that they canmethods be applied applied to delay-free and time-delay systems in proposed design is they can be to delay-free and time-delay systems in the same manner. Moreover, the proposed are extended to two-degree-of-freedom PID design problems are formulated as convex optimization problems. A important feature of the proposed design methods is that they can be applied to delay-free and time-delay systems in the same manner. Moreover, the proposed methods are extended to two-degree-of-freedom PID the same manner. manner. Moreover, the proposed proposed methods are to extended to two-degree-of-freedom two-degree-of-freedom PID the same Moreover, the are extended to PID controller design methods. proposed design is that they canmethods be applied delay-free and time-delay systems in the same manner. Moreover, the proposed methods are extended to two-degree-of-freedom PID controller design methods methods. controller design methods. controller design methods. the same manner. Moreover, the proposed methods are extended to two-degree-of-freedom PID controller design methods. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. controller design methods. Two-degree-of-freedom PID controllers, Robust control, Keywords: PID controllers, Keywords: PID controllers, Two-degree-of-freedom PID controllers, Robust control, Keywords: controllers, Two-degree-of-freedom Keywords: PID PID controllers, Two-degree-of-freedom PID PID controllers, controllers, Robust Robust control, control, Time-delay, Convex optimization, Keywords: PID controllers, Two-degree-of-freedom PID controllers, Robust control, Time-delay, Convex optimization, Time-delay, Convex optimization, Time-delay,PID Convex optimization, Keywords: controllers, Two-degree-of-freedom PID controllers, Robust control, Time-delay, Convex optimization, Time-delay, Convex optimization, 1. INTRODUCTION design methods and two-degree-of-freedom PID controller 1. INTRODUCTION INTRODUCTION design and two-degree-of-freedom PID controller 1. INTRODUCTION design methods methods and two-degree-of-freedom PID results controller 1. methods and two-degree-of-freedom PID controller design methods are proposed by using the of 1. INTRODUCTION design methods and two-degree-of-freedom PID results controller methods are proposed by using the of design methods are proposed by using the results of design methods methods aretwo-degree-of-freedom proposed by using the results of Robust PID controller design methods have been proposed Section 3. In Section 5, some numerical examples are 1. INTRODUCTION design and PID controller design methods are proposed by using the results are of Robust PID controller design methods have been proposed Section 3. In Section 5, some numerical examples Robust PID controller design methods have been proposed Section 3. In Section 5, some numerical examples are Robust PID controller design methods have been proposed Section methods 3. In Section 5, some by numerical examples are by many researchers so far. In particular, the convex demonstrated. design are proposed using the results of Robust PID controller design methods have been proposed Section 3. In Section 5, some numerical examples are by many researchers so far. In particular, the convex demonstrated. by many researchers so far. far.methods Ininparticular, particular, the convex demonstrated. demonstrated. by many researchers so In the convex optimization methods proposed Karimi et al. (2008), Robust PID controller design have been proposed Section 3. In Section 5, some numerical examples are by many researchers so far. Inin convex demonstrated. optimization methods proposed proposed inparticular, Karimi et et the al. (2008), (2008), An important feature of the proposed design methods is optimization methods proposed Karimi et al. (2008), optimization methods in Karimi al. Karimi et al. (2010), Galdos (2010), and Nagasaka important feature of the proposed design methods is by many researchers so far.et Inal. the convex An demonstrated. An important feature of the proposed design methods is optimization methods proposed inparticular, Karimi et al.Nagasaka (2008), Karimi et al. (2010), Galdos et al. (2010), and Nagasaka An important feature of the proposed design methods is that they can be applied to delay-free and time-delay sysKarimi et al. (2010), Galdos et al. (2010), and Karimi et al. methods (2010), Galdos et and al. (2010), andal.Nagasaka An important feature of the proposed design methods is et al. (2011) are interesting important from the that they can be applied to delay-free and time-delay sysoptimization proposed in Karimi et (2008), that they can be applied to delay-free and time-delay sysKarimi et al. (2010), Galdos et al. (2010), and Nagasaka et al. (2011) are interesting and important from the that they can be applied to delay-free and time-delay systems in the same manner, because FNFR models (defined An important feature of the proposed design methods is et al. (2011) are interesting and important from the et al. et (2011) are interesting and important from the that they can be applied to delay-free and time-delay syspractical viewpoint. In these paper, a non-parametric tems in the same manner, because FNFR models (defined Karimi al. (2010), Galdos et al. (2010), and Nagasaka tems inare the same manner, because FNFR models (defined et al. (2011) are interesting and important from the below) practical viewpoint. In these these paper, paper, a non-parametric non-parametric tems in the same manner, because FNFR models (defined used for the derived optimization problems. that they can be applied to delay-free and time-delay syspractical viewpoint. In these paper, a non-parametric practical viewpoint. In a tems in the same manner, because FNFR models (defined model a spectral model (it is called an FNFR model below) are used for the derived optimization problems. et al. called (2011) are interesting and important from the below) are used for the optimization problems. practical viewpoint. Inmodel these(it a an non-parametric model called spectral model (itpaper, is called called an FNFR model below) are used for the derived derived optimization problems. tems in thesome samebasic manner, because FNFR models (defined model called a is FNFR model model called aa spectral spectral model (it is called an FNFR model below) are used for theideas derived optimization problems. in this paper) is used for reducing robust performance Note that for robust stability conditions practical viewpoint. In these paper, a non-parametric model called a spectral model (it is called an FNFR model in this paper) is used for reducing robust performance Note that some basic ideas for robust stability conditions below) are used for the derived optimization problems. in this paper) is used for reducing robust performance Note that some basic ideas for robust stability conditions in this paper) is used for reducing robust performance problems to convex optimization With use of Note that some basic ideas for robust stability conditions in this paper are found in Karimi et al. (2008), model a spectral model (it is problems. called an FNFR model in thiscalled paper) is used for reducing robust performance problems to convex convex optimization problems. With use use of derived Note that some basic ideas for robust stability conditions derived in this paper are found in Karimi et al. (2008), problems to convex optimization problems. With use of derived in this paper are found in Karimi et al. (2008), problems to optimization problems. With of such non-parametric models treat delay-free and timederived in this paper are found in Karimi et al. (2008), Karimi et al. (2010), and Galdos et al. (2010). In particin this paper) is used for reducing robust performance Note that some basic ideas for robust stability conditions problems to convex optimization of derived such non-parametric non-parametric models treat treatproblems. delay-freeWith and use timein al. this(2010), paper and are found inet Karimi et al.In (2008), Karimi et Galdos al. (2010). particsuch non-parametric models treat delay-free and timeKarimi et al. (2010), and Galdos et al. (2010). In particsuch models delay-free and timedelay systems can be treated in the same manner, which Karimi et al. (2010), and Galdos et al. (2010). In particular, the idea of the stability condition that Nyquist diaproblems to convex optimization problems. With use of derived in this paper are found in Karimi et al. (2008), such non-parametric models in treat delay-free and which time- Karimi delay systems can be treated in the same manner, which et al. (2010), and Galdos et al. (2010). In particular, the idea of the stability condition that Nyquist diadelay systems can be treated the same manner, ular, the idea of the stability condition that Nyquist diadelay systems can be treated in the same manner, which is a very important feature from practical viewpoint. ular, the idea of the stability condition that Nyquist diagrams (or uncertainty discs) exist below a line is originated such non-parametric models treat delay-free and which time- Karimi et al. (2010), and Galdos et al. (2010). In particdelay systems can be treated in the same manner, is a very important feature from practical viewpoint. ular, the idea of the stability condition that Nyquist diagrams (or uncertainty discs) exist below a line is originated is a very important feature from the practical viewpoint. grams (or uncertainty discs) exist below a line is originated is a very important feature from the same practical viewpoint. grams (or uncertainty discs) exist below a line is originated in these papers. In this sense, most of robust stability delay systems can be treated in the manner, which ular, the idea of the stability condition Nyquist diais athis very important feature from themodel practical viewpoint. grams (orpapers. uncertainty discs) exist belowof athat line is originated In paper, such a non-parametric is also used and in these In this sense, most robust stability in these papers. In this sense, most of robust stability In athis this paper, such a non-parametric model is also also used and and grams in these In inthis sense, ofabeline robust stability conditions derived this paper can interpreted as is very important feature fromcontroller themodel practical viewpoint. (orpapers. uncertainty discs) existmost below is originated In paper, such non-parametric is used In this paper, such aarobust non-parametric model is also used and in these papers. In this sense, most of robust stability alternative simple PID design methods conditions derived in this paper can be interpreted as conditions derived inthis thissense, paper can of berobust interpreted as In this paper, such arobust non-parametric modeldesign is also used and conditions alternative simple robust PID controller design methods derived in this paper can be interpreted as special cases or modifications of these papers. From this in these papers. In most stability alternative simple PID controller methods alternative simple robust PID methods controller design methods conditions derived in this paper can be interpreted as are proposed. All the design are described by special cases or modifications of these papers. From this In this paper, such a non-parametric model is also used and special cases or modifications of these papers. From this alternative simple robust PID controller design methods are proposed. All the design methods are described by special cases or modifications of these papers. From this viewpoint, the explicit contributions of this paper can be derived in this paper canof papers. be interpreted as are proposed. All robust the design methods are described by conditions are proposed. All the design methods are described by special cases or modifications of these From this convex optimization problems which can easily be solved viewpoint, the explicit contributions this paper can be alternative simple PID controller design methods viewpoint, the explicit contributions of this paper can be are proposed. All the design methods described by mentioned convex optimization problems which can canare easily be solved solved viewpoint, theor explicit contributions of papers. this paper canthis be as follows: special cases modifications of these From convex optimization problems which can easily be solved convex optimization problems which easily be viewpoint, the explicit contributions of this paper can be numerically. Note that the derived convex optimization mentioned as follows: are proposed. All that the design methods described by mentioned convex optimization problems which convex canare easily be solved numerically. Note the derived derived convex optimization mentioned as as follows: thefollows: explicit contributions of this paper can be numerically. Note that the optimization numerically. Note that the derived convex optimization mentioned as follows: problems are based on sufficient for •• Closed-loop model matching problems are treated, convex optimization problems whichconditions can easily be robust solved viewpoint, numerically. Note that the derived convex optimization problems are based on sufficient conditions for robust Closed-loop model matching problems are treated, mentioned as follows: problems are based on sufficient conditions for robust • Closed-loop model matching problems are treated, problems are hence based on proposed sufficient conditions for include robust stability, and the design methods • while Closed-loop model matching problems are treated, loop-shaping problems or open-loop model numerically. that the derived convex optimization problems areNote based on proposed sufficient conditions for include robust stability, and and hence the proposed design methods include • Closed-loop model matching problems are treated, while loop-shaping problems or open-loop model stability, and hence the design methods while loop-shaping problems or open-loop model stability, hence the proposed design methods include some conservativeness. while loop-shaping problems or open-loop model matching problems are treated in Karimi et al. problems are based on sufficient conditions for robust • Closed-loop model matching problems are treated, stability, and hence the proposed design methods include some conservativeness. while loop-shaping problems or open-loop model matching problems are treated in Karimi et some conservativeness. matching problems are treated in Karimi et al. al. some conservativeness. matching problems are treated in Karimi et al. (2008), Karimi et al. (2010), Galdos et al. (2010). stability, and hence the proposed design methods include while loop-shaping problems or open-loop model some conservativeness. matching problems are treated in et Karimi et al. In Section 2, an FNFR model of the plant is introduced, (2008), Karimi et al. (2010), Galdos al. (2010). (2008), Karimi et al. (2010), Galdos et al. (2010). In Section 2, an FNFR model of the plant is introduced, (2008), Karimi et al. (2010), Galdos et al. (2010). • A necessary and sufficient condition for a uncertainty some conservativeness. matching problems treated infor et al. In Section 2,controller an FNFR FNFRand model of the the plant plant is is introduced, introduced, In Section 2, an model of (2008), Karimi etsufficient al. are (2010), Galdos etKarimi (2010). and aa PID a two-degree-of-freedom PID A necessary and sufficient condition uncertainty ••• A necessary and condition for aaaal. uncertainty In an FNFRand model of the plant is introduced, andSection PID2,controller controller and a two-degree-of-freedom two-degree-of-freedom PID A necessary and sufficient condition for uncertainty disc to exist below aa line is derived using the coordi(2008), Karimi et al. (2010), Galdos et al. (2010). and a PID controller a two-degree-of-freedom PID and a PID and a PID • A necessary and sufficient condition for a uncertainty controller used in this paper are described. In Section 3, disc to exist below line is derived using the coordiIn Section 2,controller aninFNFR model of described. the plant isInintroduced, to exist below aa line derived using the coordiand a PIDused a are two-degree-of-freedom PID controller used this and paper are Section 3, disc to exist below line is iscondition derived using the coordinate of the nearest point 3), while a polygon • disc A necessary and sufficient for a uncertainty controller in this paper described. In 3, controller used indesign this paper are described. In Section Section 3, disc to exist below a line (Lemma is derived using the coordiaa PID method for a fixed plant model is nate of the nearest point (Lemma 3), while a polygon and a controller PIDused controller and a are two-degree-of-freedom PID nate of the nearest point (Lemma 3), while a polygon controller in this paper described. In Section 3, PID controller design method for a fixed plant model is nate of the nearest point (Lemma 3), while a polygon approximation and a sufficient condition are derived disc to exist below a line is derived using the coordia PID controller design method for a fixed plant model is aderived PID controller design method for a fixedtoplant model 3, is nate of the nearest point (Lemma 3), while a polygon and then the method is extended a two-degreeapproximation and a sufficient condition are derived controller used in this paper are described. In Section approximation and sufficient condition are derived aderived PID controller method for a fixedto model is derived and then then design the method method is extended extended toplant a two-degreetwo-degreeapproximation and aa sufficient condition are derived in Karimi et al. (2010). nate of the nearest point (Lemma 3), while a polygon and then the method is extended a two-degreederived and the is to a approximation and a sufficient condition are derived of-freedom PID controller design In Section in Karimi et al. (2010). a PID controller design method formethod. a fixedtoplant model 4, is in (2010). derived andPID then controller the method is extended a two-degreeof-freedom PID controller design method. In Section 4, in Karimi Karimi et et al. al.and (2010). approximation a sufficient condition are derived of-freedom design method. In Section 4, of-freedom PID controller design method. In Section 4, in paper, KarimiRetand al. C(2010). two types of plant models which have model variations In this denote the set of real numbers and derived and then the method is extended to a two-degreeof-freedom PID controller design method. In Section 4, two types of plant models which have model variations In this paper, R and C denote the set of real numbers in Karimi et al. (2010). two types of plant models which have model variations In this this paper, R and and C denote denoterespectively. the set set of of real real numbers numbers and and two types of plant models which have model variations and uncertainties are considered for robust PID controller In paper, R C the and set of complex numbers, of-freedom PID controller design method. In Section 4, the two types of plant models which have model variations and uncertainties are considered for robust PID controller In this paper, R and C denote the set of real numbers and set of complex numbers, respectively. and uncertainties are considered for robust PID controller the the set of complex numbers, respectively. and uncertainties are considered formodels robust PID variations design. For both types of plant PID controller the set of complex numbers, respectively. two types of plant models which have model In this paper, R and C denote the set of real numbers and and uncertainties are considered for robust PID design. For both types of plant models PID controller the set of complex numbers, respectively. design. For both both are types of plant plantformodels models PID controller design. For types of PID controller  and uncertainties considered robust PID controller the set of complex numbers, respectively. This work supported (16K14286). design. Forwas both types by ofKAKENHI plant models PID controller  This supported (16K14286).   This work work was supported by KAKENHI (16K14286). design. Forwas both types by ofKAKENHI plant models PID controller  This work was supported by KAKENHI (16K14286).

This work was supported by KAKENHI (16K14286).  This work was supported by KAKENHI (16K14286). Copyright © 2018, 2018 IFAC 930Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2018 IFAC 930 Copyright ©under 2018 responsibility IFAC 930Control. Peer review© of International Federation of Automatic Copyright 2018 IFAC 930 Copyright © 2018 IFAC 930 10.1016/j.ifacol.2018.06.103 Copyright © 2018 IFAC 930

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

Eitaku Nobuyama et al. / IFAC PapersOnLine 51-4 (2018) 930–935

2. SYSTEM DESCRIPTION In this section, an FNFR model of the plant is introduced, and a PID controller and a two-degree-of-freedom PID controller used in this paper are described. 2.1 Plant Description

931

r(s)   K(s) + −

 P (s)

y(s)  

Fig. 1. A feedback PID controller.

Let P (s) be the transfer function of the plant to be controlled, and consider the following set of pairs (1) {(ωk , P (jωk )) , k = 1, . . . , M } √ −1, M is a positive integer, ωk (≥ 0) where j = (k = 1, . . . , M ) are specified frequencies, and P (jωk )(∈ C) denotes the frequency response of the system at the frequency ω = ωk . In this paper, (1) is called an FNFR model of the system.

 F (s) +  r(s)     K(s)    P (s) + + −

y(s)  

Note that “FNFR” is an abbreviation for a finite number of frequency responses, and a non-parametric model represented by an FNFR (such as (1)) is called an FNFR model in this paper.

Fig. 2. A two-degree-of-freedom PID controller.

In the sequel, stability conditions and controller design problems for the transfer function P (s) will be derived theoretically, and then the FNFR model (1) is used for the practical numerical calculation. In this paper, assume that the plant is stable.

In this section, PID controller design methods is derived for the plant which is described by a transfer function or an FNFR model (1), and in the next section robust PID controller design methods is derived for the plant which has model variations or uncertainties.

2.2 PID controller

3. BASIC PID CONTROLLER DESIGN

3.1 Feedback PID Controller Design

A feedback PID controller is given by 1 s = ρT φ(s) (2) K(s) = ρP + ρI + ρD s τs + 1 where τ > 0 and     ρP 1 3 1/s ρ = ρI ∈ R , φ(s) = . (3) s/(τ s + 1) ρD Note that s/(τ s+1) is a pseudo-derivative and τ is selected as a small number (e.g., τ = 0.01). Moreover, two-degree-of-freedom PID controller is given by a combination of a feedback PID controller (2) and the following feedforward PD controller: s F (s) = σP + σD = σ T ψ(s) (4) τs + 1 where     σP 1 σ= . (5) ∈ R2 , ψ(s) = s/(τ s + 1) σD In the sequel, a two-degree-of-freedom PID controller is called a TDF PID controller.

A feedback PID controller and a TDF PID controller are shown in Figs. 1 and 2, respectively. The closed-loop transfer functions from r(s) to y(s) using a feedback PID controller and a TDF PID controller are given by K(s)P (s) (6) Tfb (s) = 1 + K(s)P (s) and (K(s) + F (s))P (s) , (7) Ttd (s) = 1 + K(s)P (s) respectively. 931

Let’s consider the feedback system in Fig. 1 where K(s) is a PID controller given by (2). The design parameter is ρ ∈ R3 .

First, let’s derive a convex stability condition for the feedback system. Let Lfb (s) be Lfb (s) = K(s)P (s) = ρT φ(s)P (s) (8) then it follows from the Nyquist stability criterion that the closed-loop system is stable if the following condition holds: (C1) The locus Lfb (jω), ω ≥ 0 does not circle the point (−1, 0) clockwise.

y = α(x + β) (−1, 0) 

(9)

Im y

O

Re  x

Lf b (jω)

Fig. 3. Stability condition for the feedback system. From this it is easy to see that the closed-loop system is stable if the locus (9) exists below the line y = α(x + β)

IFAC PID 2018 932 Ghent, Belgium, May 9-11, 2018

Eitaku Nobuyama et al. / IFAC PapersOnLine 51-4 (2018) 930–935

in the complex plane (see Fig. 3) , which is described by the condition (10) ρT Iω ≤ α(ρT Rω + β), ∀ω ≥ 0 where α > 0, 0 < β < 1 and Iω = Im{φ(jω)P (jω)}, Rω = Re{φ(jω)P (jω)}. (11) Hence, the next result is obtained. Lemma 1. If the condition (12) ρT (Iω − αRω ) ≤ αβ, ∀ω ≥ 0 holds the closed-loop system in Fig. 1 is stable. Note that (12) is equivalent to (10) and convex with respect to ρ. Next, let’s derive a closed-loop model matching problem for good performance. Let Tm (s) be an ideal model transfer function which has desirable properties. Then the difference between the closed-loop transfer function and the model becomes K(s)P (s) − Tm (s) Tfb (s) − Tm (s) = 1 + K(s)P (s) ρT φ(s)P (s) − (1 + ρT φ(s)P (s))Tm (s) = 1 + ρT φ(s)P (s) ρT φ(s)P (s)(1 − Tm (s)) − Tm (s) . (13) = 1 + ρT φ(s)P (s) Note that the right-hand side is not convex with respect to ρ because the denominator includes ρ, while the numerator is convex with respect to ρ. Moreover, Tfb (s) − Tm (s) approaches to zero if the numerator approaches to zero. Hence, as a closed-loop model matching problem an optimization problem which minimizes the weighted numerator in (13) is employed. That is, the model matching problem for good performance is formulated as   min max W (jω) ρT φ(jω)P (jω) (1 − Tm (jω)) ρ ω≥0  − Tm (jω)  (14) where W (s) is a frequency weight function. Finally, together with the model matching problem (14) and the stability condition (12) a PID controller design problem is formulated as follows:    min max W (jω) ρT φ(jω)P (jω) (1 − Tm (jω)) ρ ω≥0  − Tm (jω)  (15)

s.t. ρT (Iω − αRω ) ≤ αβ, ∀ω ≥ 0. Note that this problem is convex with respect to ρ. However, since it includes the infinite constraint (∀ω ≥ 0), the problem (15) cannot be solved numerically in practice.

To reduce the problem (15) to a solvable optimization problem a practical approximation by FNFR models is used, i.e., the infinite frequency range ω ≥ 0 is approximated by a finite number of frequencies ωk (k = 1, . . . , M ). Then the next PID controller design problem is obtained. (PIDP) Feedback PID Controller Design Problem:    (16) min max Wk ρT Ak − Tmk  ρ

k=1,...,M

s.t. ρT (Ik − αRk ) ≤ αβ (k = 1, . . . , M )

932

where Wk (≥ 0) is a weight at the frequency ω = ωk and Ak = φ(jωk )P (jωk ) (1 − Tm (jωk )) , Tmk = Tm (jωk ), Ik = Im{φ(jωk )P (jωk )}, Rk = Re{φ(jωk )P (jωk )}.

Note that the problem (PIDP) is convex with respect to ρ and the number of constraints is finite. Hence it can easily be solved numerically. Moreover, since the problem (PIDP) is approximation of (15), the solution of (PIDP) cannot guarantee the stability of the feedback system theoretically. However, from the practical viewpoint, the solution can guarantee the stability of the feedback system except very special cases, because (PIDP) can fully approximate (15) by increasing the number of frequencies. 3.2 Two-Degree-of-Freedom PID Controller Design Consider the TDF PID controller in Fig. 2 Then the difference between the closed-loop transfer function and the model transfer function becomes (K(s) + F (s))P (s) − Tm (s) Ttd (s) − Tm (s) = 1 + K(s)P (s)   T ρ φ(s) + σ T ψ(s) P (s) − (1 + ρT φ(s)P (s))Tm (s) = 1 + ρT φ(s)P (s) T ρ φ(s)P (s)(1 − Tm (s)) + σ T ψ(s)P (s) − Tm (s) (17) = 1 + ρT φ(s)P (s) As in the previous subsection the model matching problem which minimizes the weighted numerator of (17) is employed in this subsection. Moreover, together with the model matching problem, the stability condition (12) and approximation of ω > 0 by ωk (k = 1, . . . , M ) a TDF PID controller design is formulated as follows: (TDFP) TDF PID Controller Design Problem:    min max Wk ρT Ak + σ T Bk − Tmk  ρ,σ k=1,...,M s.t. ρT (Ik − αRk )

where Ak , Tmk

(18)

≤ αβ (k = 1, . . . , M ) are defined in the problem (PIDP) and Bk = ψ(jωk )P (jωk ).

Note that the problem (TDFP) is convex with respect to both ρ and σ and the number of constraints is finite. Hence, the problem can easily be solved numerically. 4. ROBUST PID CONTROLLER DESIGN In this section, (PIDP) and (TDFP) are extended to robust controller design problems for the plant which has model variations or uncertainties. In the following, two types of plant models are treated. 4.1 Plant model 1: Multi-Transfer-Function Model In practical identification experiments, different transfer functions are derived in different experiment conditions or even in the same experiment conditions. In this subsection, such cases are considered. Now, suppose the different many transfer functions P (s) ( = 1, . . . , N ) are obtained for the plant by identification experiments. Then the objective of robust control is to stabilize all the transfer functions simultaneously and

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

Eitaku Nobuyama et al. / IFAC PapersOnLine 51-4 (2018) 930–935

hence robust stability is defined as the simultaneous closedloop stability for P (s) ( = 1, . . . , N ). That is, the closed-loop system is said to be robustly stable when K(s) stabilizes P (s) ( = 1, . . . , N ) simultaneously. The next lemma can be obtained immediately from Lemma 1. Lemma 2. If the condition ρT (Iω − αRω ) ≤ αβ, ∀ω ≥ 0,  = 1, . . . , N (19) holds K(s) stabilizes P (s) ( = 1, . . . , N ) simultaneously, i.e. the feedback system is robustly stable where Iω = Im{φ(jω)P (jω)}, Rω = Re{φ(jω)P (jω)}. (20) Note that if the condition (19) holds the feedback system in Fig. 1 is stable for the plant whose transfer function P (s) is given by the convex combination of P (s) i.e., P (s) =

N  =1

ν P (s), ν ≥ 0,

N  =1

ν ≤ 1,

(21)

because the locus of P (jω) (ω ≥ 0) exists below the line y = α(x + β) in the complex plane if all the loci of P (jω) (ω ≥ 0;  = 1, . . . , N ) exist below the line. In this paper, (21) is called a multi-transfer-function model for the plant. Moreover, the same model matching method for good performance and practical approximation are employed as in the previous section. Then a robust PID controller design problem is formulated as follows: (RPIDP1) Robust PID Controller Design Problem 1:    min max Wk ρT Ak − Tmk  (22) ρ

k=1,...,M =1,...,N

T

s.t. ρ (Ik − αRk ) ≤ αβ, k = 1, . . . , M,  = 1, . . . , N where Wk is a weight at the frequency ω = ωk and Ak = φ(jωk )P (jωk ) (1 − Tm (jωk )) , Ik = Im{φ(jωk )P (jωk )}, Rk = Re{φ(jωk )P (jωk )}. Note that the problem (RPIDP1) is convex with respect to ρ, and the number of constraints is finite, which means that the problem can easily be solved numerically. In a similar manner, a robust TDF PID controller design problem for multi-transfer-function models is formulated as follows: (RTDFP1) Robust TDF PID Controller Design Problem 1:    min max Wk ρT Ak + σ T Bk − Tmk  (23) ρ,σ k=1,...,M =1,...,N

T

s.t. ρ (Ik − αRk ) ≤ αβ, k = 1, . . . , M,  = 1, . . . , N where Ak , Ik , Rk are defined in (RPIDP1) and Bk = ψ(jωk )P (jωk ). Note that the problem (RTDFP1) is convex with respect to both ρ and σ and the number of constraints is finite, which means that the problem can easily be solved numerically. 4.2 Plant model 2: Multiplicative Uncertainty Model In this subsection, suppose the plant is given by the following multiplicative uncertainty model: 933

933

˜ (s)∆(s)) P (s) = Pn (s)(1 + W

(24) ˜ (s) where Pn (s) is the nominal model transfer function, W is the weight function, and ∆(s) is the uncertainty. Note that (24) is a plant model used in the well-known H∞ control design, and the feedback system is said to be robustly stable if K(s) stabilizes P (s) for all ∆(s) which satisfies |∆(jω)| ≤ 1, ∀ω ∈ R. Moreover, it is known well that the closed-loop system is robustly stable if the following condition holds: (C2) The locus of the disc Dω (ω ≥ 0) with the center ˜ (jω)| does not cross nor circle Ln (jω) and the radius |W the point (−1, 0) in the complex plane where Ln (s) = K(s)Pn (s). It is easy to see as in the previous section that the condition (C2) holds if the locus of the disc Dω (ω ≥ 0) exists below the line y = α(x + β) in the complex plane where α > 0, 0 < β < 1. Moreover, let Qω be the nearest point from the disc Dω to the line y = α(x + β), then the disc Dω exists below the line if the point Qω does so. Hence, if the locus Qω (ω ≥ 0) exists below the line the condition (C2) holds. Im y  

y = α(x + β)     Re       x (−1, 0)      −1  tan α    Qω        Ln (jω) Dω    ˜ (jω)| |W Fig. 4. Robust stability condition. It follows from geometrical consideration in Fig. 4 that the coordinate of Qω is given by ˜ (jω)| sin θ, Im{Ln (jω)} + |W ˜ (jω)| cos θ) (Re{Ln (jω)} − |W

where θ = tan−1 α. Hence the next lemma is obtained. Lemma 3. If the condition ˜ (jω)|(cos θ + α sin θ) (25) ρT (Inω − αRnω ) ≤ αβ − |W holds the feedback system is robustly stable where Inω = Im{φ(jω)Pn (jω)}, Rnω = Re{φ(jω)Pn (jω)}.

Proof: The point Qω exists below the line y = α(x + β) if ˜ (jω)| cos θ Im{Ln (jω)} + |W   ˜ (jω)| sin θ + β . (26) ≤α Re{Ln (jω)} − |W

Since Ln (s) = ρT φ(s)Pn (s) the condition (25) is obtained from (26).  For the plant model (24) a nominal model matching which minimizes the difference between the nominal closed-loop transfer function Tn (s) = K(s)Pn (s)/(1 + K(s)Pn (s)) and the model transfer function Tm (s) is employed.

IFAC PID 2018 934 Ghent, Belgium, May 9-11, 2018

Eitaku Nobuyama et al. / IFAC PapersOnLine 51-4 (2018) 930–935

Then a robust PID controller design problem is formulated as follows: (RPIDP2) Robust PID Controller Design Problem 2:     (27) min max Wk ρT Ank − Tmk  ρ

s.t.

k=1,...,M ρT (Ink −

˜ (jωk )|(cos θ + α sin θ), αRnk ) ≤ αβ − |W k = 1, . . . , M

1.5

1

0.5

0 0

where Wk is a weight at the frequency ω = ωk and

4

8

12

16

20

Time

Ank = φ(jωk )Pn (jωk ) (1 − M (jωk )) ,

Fig. 5. Step responses of the plant models.

Ink = Im{φ(jωk )Pn (jωk )}, Rnk = Re{φ(jωk )Pn (jωk )}. Note that the problem (RPIDP2) is convex with respect to ρ, and the number of constraints is finite, which means that the problem can easily be solved numerically. In a similar manner, a robust TDF PID controller design is formulated as follows: (RTDFP2) Robust TDF PID Controller Design Problem 2:    (28) min max Wk ρT Ank + σ T Bnk − Tmk  ρ,σ k=1,...,M s.t. ρT (Ink −

˜ (jωk )|(cos θ + α sin θ), αRnk ) ≤ αβ − |W k = 1, . . . , M

where Ank , Ink , Rnk are defined in the problem (RPIDP2) and Bnk = ψ(jωk )Pn (jωk )

2

(29)

Note that the problem (28) is convex with respect to both ρ and σ and the number of constraints is finite, which means that the problem can easily be solved numerically.

1 e−d1 s , + 0.2s + 0.9 1 P2 (s) = 2 e−d2 s , s + 0.2s + 1.1 1 P3 (s) = 2 e−d3 s , s + 0.5s + 0.9 1 P4 (s) = 2 e−d4 s , s + 0.5s + 1.1 1 P5 (s) = 2 e−d5 s , s + 0.8s + 0.9 1 P6 (s) = 2 e−d6 s , s + 0.8s + 1.1 where d1 , . . . , d6 are delay durations. P1 (s) =

s2

For this plant, let M = 100 and take ωk (k = 1, . . . , M ) as 100 points from ω = 10−2 ∼ 101 with equal intervals in logarithmic scale. Moreover, let τ = 0.01, α = 1, β = 0.8, and set the model transfer function as 1 Tm (s) = e−dm s . (30) (τm s + 1)2 5.1 Delay-Free Plant Case First, let’s consider the delay-free case, i.e., d1 = d2 = · · · = d6 = 0. In this case, the step responses of P1 (s), . . . , P6 (s) are shown in Fig. 5.

4.3 Choice of the Line for Robust Stability Conditions The choice of the parameters α and β of y = α(x + β) is important, because it affects the performance of the obtained closed-loop system. Our empirical method is as follows: possible, • The first choice of α is α = tan 45◦ = 1. If √ try α = tan 30◦ = √12 and α = tan 60◦ = 2, then pick up the best one. • The first choice of β is β = 0.8, in which case the gain margin is about 2 dB. If stronger stability is required, choose β = 0.7 or β = 0.5, in which case the gain margin is about 3 dB or 6 dB, respectively.

5. NUMERICAL EXAMPLES In this section, the case of Plant Model 1 is considered where the plant is described by a second-order delay-free or time-delay system whose damping factor and stationary gain change in the different experiment conditions. For the plant suppose the following 6 transfer functions are obtained from some identification experiments: 934

For this plant τm = 0.6 and dm = 0.7 are taken for the model transfer function (30), and the frequency weight to focus on the intermediate frequencies are chosen as  10000, 41 ≤ k ≤ 80 (31) Wk = 1, othewise. Then the following PID controller and TDF controller are obtained as the numerical solutions to the problems (RPID1) and (RTDF1), respectively:   0.3536 PID : ρ = 0.7651 0.6931     2.5028 −0.0193 TDF PID : ρ = 0.9048 , σ = −1.7769 2.1380 Figs. 6 and 7 show the step responses of the closedloop systems with the PID controller and the TDF PID controller, respectively. 5.2 Time-Delay Plant Case Next, let’s consider the plant has a time-delay of 0.4 ∼ 0.7 second. In this example, let

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

Eitaku Nobuyama et al. / IFAC PapersOnLine 51-4 (2018) 930–935

1.2

1.2

1

1

0.5

0.5

0

935

0 0

4

8

12

16

20

0

4

8

Time

12

16

20

Time

Fig. 6. Delay-free case with the PID controller.

Fig. 8. Time-delay case with the PID controller. 1.2

1.2

1 1

0.5 0.5

0 0

0 0

4

8

12

16

4

8

12

16

20

Time

20

Time

Fig. 9. Time-delay case with the TDF PID controller.

Fig. 7. Delay-free case with the TDF PID controller.

ACKNOWLEDGEMENTS

d1 = 0.7, d2 = 0.5, d3 = 0.6, d4 = 0.4, d5 = 0.5, d6 = 0.7. For this plant, τm = 0.4 and dm = 0.7 are taken (the maximum value of the plant time-delay) for (30), and the same frequency weight as in (31) are used. Then the following PID controller and TDF controller are obtained as the numerical solutions to the problems (RPID1) and (RTDF1), respectively:   0.2139 PID : ρ = 0.3784 0.3701     0.1696 0.1697 TDF PID : ρ = 0.2991 , σ = −0.0961 0.4448 Figs. 8 and 9 show the step responses of the closedloop systems with the PID controller and the TDF PID controller, respectively. 6. CONCLUSION In this paper, some numerical methods have been proposed for robust PID controller and two-degree-of-freedom PID controller design. A important feature of these method is that treat delay-free and time-delay systems can be treated in the same manner, In fact, the effectiveness of our proposed method for time-delay systems was shown by the numerical examples. As mentioned in Introduction, the proposed design methods include some conservativeness. A possible way for reducing the conservativeness is to employ non-constant α and β. Our future work is to develop a systematic method for determining non-constant α and β for good performance. 935

Special thanks to Shogo Iwasaki for his valuable comments on the stability conditions. REFERENCES Karimi, A., Galdos, G., and Longchamp, R. (2008). Robust fixed-order H∞ controller design for spectral models by convex optimization. 47th IEEE Conference on Decision and Control, 921–926. Karimi, A., and Galdos, G. (2010). Fixed-order H∞ controller design for nonparametric models by convex optimization. Automatica, 46:1388–1394. Galdos, G., Karimi, A., and Longchamp, R. (2010). H∞ controller design for spectral MIMO models by convex optimization. Journal of Process Control, 20:1175–1182. Nagasaka, T., Yubai, K., and Hirai, J. (2011). Controller design on the Nyquist diagram by convex optimization satisfying stability and performance constraint. The 54th Japan Joint Automatic Control Conference, 1784– 1789 (in Japanese).