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Model Reference Based Tuning for Fractional-Order 2DoF PI Controllers with a Robustness Consideration
12th IFAC Symposium on Dynamics and Control of 12th IFAC Symposium on Dynamics and Control of Process including Biosystems 12th IFACSystems, Symposium on Dynamics and Control of Process Systems, including Biosystems Available online at www.sciencedirect.com Florianópolis - SC,including Brazil, April 23-26,and 2019 Process Systems, Biosystems 12th IFAC Symposium on Dynamics Control of Florianópolis - SC, Brazil, April 23-26, 2019 Florianópolis - SC,including Brazil, April 23-26, 2019 Process Systems, Biosystems Florianópolis - SC, Brazil, April 23-26, 2019
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IFAC PapersOnLine 52-1 (2019) 207–212
Model Model Reference Reference Based Based Tuning Tuning for for Model Reference Based Tuning for Fractional-Order 2DoF PI Controllers with Model Reference Tuning for Fractional-Order 2DoFBased PI Controllers with a a Fractional-Order 2DoF PI Controllers with a Robustness Consideration Fractional-Order 2DoF PI Controllers with a Robustness Consideration Robustness Consideration ∗,∗∗∗ ∗Consideration ∗∗ ∗∗∗ O. Arrieta Robustness ∗,∗∗∗ ∗,∗∗∗ , D. Castillo ∗ ∗ , R. Vilanova ∗∗ ∗∗ , J.D. Rojas ∗∗∗ ∗∗∗
O. Arrieta ∗,∗∗∗ , D. Castillo ∗ , R. Vilanova ∗∗ , J.D. Rojas ∗∗∗ O. Arrieta , D. Castillo , R. Vilanova , J.D. Rojas ∗O. Arrieta ∗,∗∗∗ , D. Castillo ∗ , R. Vilanova ∗∗ , J.D. Rojas ∗∗∗ ∗ Instituto de Investigaciones en Ingeniería, Facultad de Ingeniería, ∗ ∗ Instituto de Investigaciones en Ingeniería, Facultad de Ingeniería, Instituto de Investigaciones Ingeniería,San Facultad de Ingeniería, Universidad de Costa Rica,en 11501-2060 José, Costa Rica. ∗ Universidad de Costa Rica, 11501-2060 San José, Costa Rica. Instituto de Investigaciones Ingeniería,San Facultad de Ingeniería, email: [email protected] Universidad deemail: [email protected] Rica,en 11501-2060 José, Costa Rica. ∗∗ Universidad de Costa Rica, 11501-2060 San José, Costa Rica. Departament de Telecomunicació i d’Enginyeria de Sistemes email: [email protected] ∗∗ ∗∗ de Telecomunicació i d’Enginyeria de Sistemes ∗∗ Departament email: [email protected] Escola d’Enginyeria, Universitat Autònoma de Barcelona Departament de Telecomunicació i d’Enginyeria de Sistemes ∗∗ Escola d’Enginyeria, Universitat Autònoma de Barcelona Departament de Telecomunicació i d’Enginyeria de Sistemes Escola d’Enginyeria, Universitat Autònoma de Barcelona 08193 Bellaterra, Barcelona, Spain. 08193 Bellaterra, Barcelona, Spain. ∗∗∗ Escola d’Enginyeria, Universitat Autònoma de Barcelona Departamento de Automática, Escuela de Ingeniería 08193 Bellaterra, Barcelona, Spain. ∗∗∗ ∗∗∗ Departamento de Automática, Escuela de Ingeniería Eléctrica, Eléctrica, ∗∗∗ 08193 Bellaterra, Barcelona, Universidad de Costa Rica, 11501-2060 11501-2060 San José, Costa Costa Rica. Departamento de Automática, Escuela San deSpain. Ingeniería Eléctrica, Universidad de Costa Rica, José, Rica. ∗∗∗ Departamento de Automática, Escuela San de Ingeniería Eléctrica, Universidad de Costa Rica, 11501-2060 José, Costa Rica. Universidad de Costa Rica, 11501-2060 San José, Costa Rica. Abstract: The The paper paper proposes proposes the the tuning tuning of of aa two-degree-of-freedom two-degree-of-freedom (2-DoF) Abstract: (2-DoF) fractional-order fractional-order Abstract: The paper proposes the tuning of a two-degree-of-freedom (2-DoF) fractional-order PI (F OP I ) controller based on the Model Reference Robust Tuning (MoReRT) methodology 2 PI (F OP I22 ) controller based on the Model Reference Robust Tuning (MoReRT) methodology Abstract: The paperand proposes tuning ofReference a two-degree-of-freedom (2-DoF) fractional-order presented Alfaro Vilanova (2012, 2016). The rules are for aa first PI (F OP Iby based onthe the Model Robust Tuning (MoReRT) 2 ) controller presented by Alfaro and Vilanova (2012, 2016). The tuning tuning rules are optimized optimized formethodology first order order PI (F OP I ) controller based on the Model Reference Robust Tuning (MoReRT) methodology plus dead time (FOPDT) model and for an over-damped servo and regulatory closed-loop presented by Alfaro and Vilanova (2012, 2016). The tuning rules are optimized for a firsttarget order 2 plus dead time (FOPDT) model and for an over-damped servo and regulatory closed-loop target presented by Alfaro and Vilanova (2012, 2016). The tuning rules are optimized for a first responses. Moreover, the design takes into account a robustness consideration, expressed as plus dead time (FOPDT) model and for an over-damped servo and regulatory closed-loop target responses. Moreover, the design takes into account a robustness consideration, expressedorder as aa plus dead time (FOPDT) model and for an over-damped servo and regulatory closed-loop target responses. Moreover, the design takes into account a robustness consideration, expressed as a maximum sensitivity target value of 2.0, dealing with the performance/robustness trade-off maximum sensitivity target value of 2.0, dealing with the performance/robustness trade-off responses. Moreover, the design takes into account a robustness consideration, expressed as is a of the closed-loop control system. A set of controller tuning equations (four parameters) maximum sensitivity target value of 2.0, dealing with the performance/robustness trade-off of the closed-loop control system. A set of controller tuning equations (four parameters) is maximum sensitivity target value of 2.0, dealing with the performance/robustness trade-off provided for FOPDT models with normalized dead-times from 0.1 to 2.0. Also, an evaluation of of the closed-loop control system. A set of controller tuning equations (four parameters) is provided for FOPDT models with normalized dead-times from 0.1 to 2.0. Also, an evaluation of of closed-loop control system. Arobustness set of controller tuning equations (fourthe parameters) is the control performance and is order to effectiveness provided forsystem FOPDT models with normalized dead-times fromin to 2.0. Also, an evaluation of thethe control system performance and robustness is presented, presented, in0.1 order to show show the effectiveness provided for FOPDT models with normalized dead-times from 0.1 to 2.0. Also, an evaluation of the control system performance and robustness is presented, in order to show the effectiveness of the proposed tuning method, that is compared with other tuning rules. of the proposed tuning method, that is compared with other tuning rules. thethe control system performance is with presented, in orderrules. to show the effectiveness of proposed tuning method, and that robustness is compared other tuning © the 2019,proposed IFAC (International Federation ofisAutomatic Control) Hosting by Elsevier Ltd. All rights reserved. of tuning method, that compared with other tuning rules. Keywords: Keywords: PI PI Control, Control, Model Model Reference, Reference, Fractional Fractional Control, Control, Tuning Tuning Rules Rules Keywords: PI Control, Model Reference, Fractional Control, Tuning Rules Keywords: PI Control, Model Reference, Fractional Control, Tuning 1. INTRODUCTION However, it has also Rules been recognize that another important 1. INTRODUCTION However, it has also been been recognize recognize that that another another important 1. INTRODUCTION However, it has also issue to be addressed is the trade-off between important servo and issue to be addressed is the trade-off between servo and and 1. INTRODUCTION However, it operation has also been another important issue to be addressed is recognize the of trade-off between servo regulation mode the system (Arrieta regulation operation mode of thethat system (Arrieta and Proportional-Integrative-Derivative (PID) controllers are regulation issue to be2007, addressed is the of trade-off between servo and operation mode the system (Arrieta Vilanova, 2010; Arrieta et al., 2010, 2011). Proportional-Integrative-Derivative (PID) controllers are Vilanova, 2007, 2010; Arrieta et al., 2010, 2011). Proportional-Integrative-Derivative (PID) controllers are Vilanova, with no doubt the most extensive option that can be found regulation operation mode ofetand the system (Arrieta and 2007, 2010; in Arrieta al.,Vilanova 2010, 2011). To face that problem, Alfaro (2012, 2016), with no doubt the most extensive option that can be found To face that that problem, in Alfaroetand and Vilanova (2012, 2016), 2016), Proportional-Integrative-Derivative (PID) controllers are To with no doubt the most extensive option that can be found on industrial control applications (Åström and Hägglund, Vilanova, 2007, 2010; Arrieta al.,Vilanova 2010, 2011). face problem, in Alfaro (2012, it was presented a Model Reference Robust Tuning (MoRon industrial control applications (Åström and Hägglund, it was presented Model Reference Robust Tuning Tuning (MoRwith noTheir doubtsuccess the most extensive option canHägglund, be found it on industrial control applications (Åström and 2001). is mainly due to itsthat simple structure To face that problem, in controllers. Alfaro and That Vilanova (2012,consid2016), was presented aa Model Reference Robust (MoReRT) for 2-DoF PI/PID approach 2001). Their success is mainly due to its simple structure eRT) for 2-DoF PI/PID controllers. That approach considon industrial controlmeaning applications and Hägglund, 2001). success is mainly to its simple structure and to Their the physical of due the(Åström corresponding parame- eRT) it was presented a Model Reference Robust Tuning (MoRfor 2-DoF PI/PID controllers. That approach considers the above mentioned trade-offs for the control system. and to the meaning of the corresponding parameers the above mentioned trade-offs for the control system. 2001). Their success is manual mainly to its simple structure and to the physical physical meaning of due the corresponding parameters (therefore making tuning possible). This fact ers eRT) 2-DoF PI/PID controllers. That approach considthefor mentioned trade-offs control system. What isabove provided in this paper isfor anthe extension of this ters (therefore making manual tuning possible). This fact What isabove provided in thistrade-offs paper is isforan anthe extension of this this and to the meaning the corresponding parameters (therefore making manual tuning possible). fact What makes PID control easier to by control ers the is mentioned control system. provided paper of methodology using ina this fractional-order PIextension controller with makes PIDphysical control easier toof understand understand by the theThis control methodology using a fractional-order PI controller with ters (therefore making manual tuning possible). This fact methodology makes PIDthan control toadvanced understand by the control engineers othereasier more control techniques. What provided ina the this paperdistinctive is anPIextension this fractional-order controller 2-DoF, is one major features the engineers othereasier more control techniques. 2-DoF,isthat that isusing one of of the major distinctive featuresofof ofwith the makes PIDthan control understand by the control engineers than more toadvanced advanced control techniques. In addition, theother PID controller provides satisfactory per- 2-DoF, methodology using a fractional-order PI controller with that is one of the major distinctive features of the proposal. The design considers the performance (servo and In addition, the PID controller provides satisfactory perproposal. The design considers the performance (servo and engineers than other more control techniques. In addition, PIDrange controller provides satisfactory per- proposal. formance in athe wide of advanced practical situations. 2-DoF, that isdesign one of considers the major distinctive features of and the The the performance (servo regulation) and robustness of the control system. formance in aathe wide range of practical situations. regulation) and robustness of the the control system. addition, PID controller provides satisfactory per- regulation) formance in wide range of practical situations. In recent years, there has also been a significant interproposal. design considers the control performance (servo robustness of system. The paperThe isand organized as follows. Next section states and the In recent years, there has also been a significant interThe paper is isand organized as follows. follows. Next section section states the the formance a wide range of also practical In recent years, there has been situations. a significant interest from in the academic and industrial communities for The regulation) robustness of the control system. paper organized as Next states general framework for the problem formulation. In Section est from the academic and industrial communities for general framework for the problem formulation. In Section In recent years, there has been a significant interest from the academic andalso industrial communities for general fractional-order-proportional-integral-derivative (FOPID) paper isthe organized as problem follows. Next states the framework for the formulation. In Section 3The describes controller procedure forsection the design. An fractional-order-proportional-integral-derivative (FOPID) 3general describes the controller controller procedure for the the design. design. An est from the academic industrial communities for 3example fractional-order-proportional-integral-derivative (FOPID) controllers because they are to (as framework forintheSection problem formulation. In ends Section describes the procedure for An is provided 4 and the paper in controllers because they and are capable capable to provide provide (as there there example is provided in Section 4 and the paper ends in fractional-order-proportional-integral-derivative controllers because to they arefor to three provide (as are tune PID for PI) more 3 describes thesome controller procedure design. provided in Section 4 andfor thethe paper endsAn in Section 5iswith conclusions. are five five parameters parameters to tune forcapable PID and and three for (FOPID) PI) there more example Section with some conclusions. conclusions. controllers they are to three provide there are five parameters to tune forcapable PID and PI) more Section flexibility inbecause the control system design (see for for (as example example55iswith provided in Section 4 and the paper ends in some flexibility in the control system design (see for example are five parameters to tune for PID and three for PI) more flexibility the control system (seeBiswas for example Podlubny in (1999); Vinagre et al.design (2007); et al. Section 5 with some conclusions. Podlubny (1999); Vinagre et al. (2007); et al. flexibility in theet system (seeBiswas for example Podlubny (1999); Vinagre etMany al.design (2007); Biswas etrules al. (2009); al. (2009)). different tuning (2009); Zamani Zamani etcontrol al. (2009)). Many different tuning rules Podlubny (1999); Vinagre et al. (2007); Biswas et al. (2009); Zamani et al. (2009)). Many different tuning rules have been proposed in the literature to facilitate their have been proposed in the literature to facilitate their 2. PROBLEM FORMULATION (2009); Zamani et al. (2009)). Many different tuning 2011, rules have been proposed in the literature to facilitate their use (Monje et al., 2004, 2008; Padula and Visioli, 2. PROBLEM PROBLEM FORMULATION FORMULATION 2011, use (Monje et al., 2004, 2008; Padula and Visioli, 2. have been proposed in the literature to facilitate their use Padula and2018). Visioli, 2011, 2014;(Monje Arrietaetetal., al.,2004, 2015;2008; Meneses et al., Most of 2014; Arrieta et al., 2015; Meneses et al., 2018). Most of 2. PROBLEM FORMULATION use Padula and Visioli, 2011, 2014; Arrieta etal., al.,2004, 2015;2008; Meneses et al., 2018). Most of them are for 1-DoF controllers, therefore there is for them(Monje are for et 1-DoF controllers, therefore there is room room for 2.1 Control System Configuration 2014; are Arrieta et 2-DoF al.,controllers, 2015; Meneses et al.,there 2018). Mostfor of 2.1 Control System Configuration them for 1-DoF therefore is room fractional-order controllers. 2.1 Control System Configuration fractional-order 2-DoF controllers. them are for 1-DoF controllers, is room for 2.1 Control System Configuration fractional-order 2-DoF It is well known that acontrollers. properlytherefore designedthere control system It is well known that a properly designed control system fractional-order It is well known that acontrollers. properly designed control system The considered control system is shown in Fig. 1, where must provide an2-DoF effective trade-off between performance must provide an effective trade-off between performance The considered considered control control system is is shown shown in in Fig. Fig. 1, 1, where where It is well known that a properly designed control system must provide an effective trade-off between performance and robustness (Arrieta and Vilanova, 2012; Arrieta, O. P process and robustness (Arrieta and Vilanova, 2012; Arrieta, O. The P (s) (s) is is the the controlled controlled system process and and C(s) C(s) is is the the 2-DoF 2-DoF mustVilanova, provide an effective trade-off between The control system isand shown in is Fig. where and robustness and 2012;performance Arrieta, O. P (s) considered is the controlled process C(s) the1, 2-DoF R.(Arrieta and Rojas, J.Vilanova, D. and Meneses, M., 2016). fractional order PI (F OP I2 ) controller. and and D. M., fractional order PI (F (F OP OPprocess robustnessR. andJ. 2012; Arrieta, O. P (s) is the controlled and C(s) is the 2-DoF and Vilanova, Vilanova, R.(Arrieta and Rojas, Rojas, J.Vilanova, D. and and Meneses, Meneses, M., 2016). 2016). fractional order PI II222 )) controller. controller. and Vilanova, R.IFAC and (International Rojas, J. D.Federation and Meneses, M., 2016). order PI I2 )reserved. controller. 2405-8963 © 2019 2019, IFAC of Automatic Control) Hosting by Elsevier Ltd.(F AllOP rights Copyright © 207 fractional Copyright © under 2019 IFAC 207 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 207 10.1016/j.ifacol.2019.06.063 Copyright © 2019 IFAC 207
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and in compact form as follows u(s) = Cr (s)r(s) − Cy (s)y(s) where
Figure 1. The considered control system.
1 Ti sλ
(6)
is the F OP I2 set-point controller transfer function, which operates on r, and 1 Cy (s) = Kp 1 + (7) Ti sλ
The variables described in Fig. 1 are: • • • •
Cr (s) = Kp β +
(5)
y(s) is the process controlled variable. r(s) is the set-point for the process output. u(s) is the controller output signal. d(s) is the load-disturbance of the system.
The closed-loop control system output y(s), in response to changes in its inputs, r(s) and d(s), is given by the following (1) y(s) = Myr (s)r(s) + Myd (s)d(s) where Myr (s) is the transfer function from the set-point to the process controlled variable, and Myd (s) is that from the load-disturbance to the process controlled variable. These are known as the servo-control closed-loop transfer function and the regulatory-control closed-loop transfer function, respectively.
is the F OP I2 feedback controller transfer function, which operates on y. The closed-loop transfer functions of the servo-control and the regulatory-control in (1) are then given by Cr (s)P (s) (8) Myr (s) = 1 + Cy (s)P (s) and Myd (s) =
P (s) 1 + Cy (s)P (s)
which are related as Myr (s) = Cr (s)Myd (s)
(9)
(10)
3. CONTROLLER DESIGN
2.2 Controlled Process Model The controlled process P (s) is assumed to be modelled by a first-order-plus-dead-time (FOPDT) transfer function of the form Ke−Ls (2) P (s) = Ts + 1 where K is the process gain, T is the time constant and L is the dead-time. The parameters of the controlled process model (2) are θ¯p = {K, T, L}. The availability of FOPDT models in the process industry is a well known fact. The generation of such model just needs for a very simple step-test experiment to be applied to the process. From this point of view, to maintain the need for plant experimentation to a minimum is a key point when considering the applicability of a technique in an industrial environment. 2.3 Fractional Order 2-DoF PI Controller (F OP I2 ) The process will be controlled with a fractional-order twodegree-of-freedom proportional integral controller (F OP I2 ), whose output is defined as 1 u(s) = Kp βr(s) − y(s) + [r(s) − y(s)] (3) Ti sλ where Kp is the controller proportional gain, Ti is the integral time constant, β is the set-point proportional weight and λ is the fractional order for the integral term. For the purposed of analysis, the controller output (3) will be expressed as 1 1 u(s) = Kp β + r(s) − Kp 1 + y(s) (4) Ti sλ Ti sλ 208
Usually the design of 2-DoF controllers is performed in two stages. First, it is designed the regulatory-control behaviour with the control system robustness, that means the feedback controller Cy (s) (7). Second, the set-point controller Cr (s) (6) is used to improve the servo-control performance. Here, a different approach is taken. The controller design is based on the Model Reference Robust Tuning (MoReRT) methodology presented by Alfaro and Vilanova (2012, 2016). The application of that to the 2-DoF fractional order PI controller, means to obtain the complete set of the F OP I2 controller parameters θ¯c = {Kp , Ti , β, λ}, considering a combination of the regulatory and servocontrol performance of a targeted closed-loop dynamic with a desired robustness level. 3.1 Target closed-loop transfer functions From the MoReRT methodology it is important to have the least amount of design parameters. Consequently, the desired control system response for load-disturbance and set-point step changes will involve only one design parameter, the closed-loop time constant Tc . The closedloop response is designed with no steady-state error and an over-damped dynamic. From control system described in Section 2, the desired regulatory-control closed-loop transfer function is defined as Ko se−Ls t (11) (s) = Myd (Tc s + 1)2 where Ko is the static gain and Tc is the time constant of the regulatory control closed-loop transfer function. For a
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PI controller, the static gain in (11) can be expressed as Ko = Ti /Kp , then t (s) = Myd
(Ti /Kp )se−Ls (Tc s + 1)2
(12)
From (10), the servo-control closed-loop transfer function is given by (βTi s + 1)e−Ls t Myr (s) = (13) (Tc s + 1)2 In order to have a set-point step change response without oscillation, overshoot or steady-state error, the servocontrol closed-loop transfer function is selected as e−Ls t Myr (14) (s) = (Tc s + 1) The zero/pole cancellation for (14), βTi = Tc , will not be forced, but taken into account in the optimization procedure to match the closed-loop transfer functions. t Therefore, Myd (s) is the target regulatory-control closedt (s) is the target servoloop transfer function, and Myr control closed-loop transfer function. If Tc is expressed as a function of the time constant of the controlled process, then τc = Tc /T can now be used as a dimensionless design parameter. The closed-loop performance specification requires only the definition of τc parameter, which is an indication of the closed-loop system response speed in relation to the controlled process speed. Replacing the parameter in (12) and (14), the global target output of the control system is (Ti /Kp )se−Ls e−Ls y t (s) = r(s) + d(s) (15) (τc T s + 1) (τc T s + 1)2 The target closed-loop response of the control system expressed in (15) can be rewritten in the time domain as follows (16) y t (t) = yrt (t) + ydt (t) where yrt (t) is the target servo-control step response and ydt (t) is the target regulatory-control step response. 3.2 Controller optimization For the regulatory-control response, the cost function to be minimized is defined as Jd (τc , θ¯c , θ¯p )= ˙
∞
0
[ydt (τc , θ¯c , θ¯p , t) − yd (θ¯c , θ¯p , t)]2 dt (17)
where ydt (τc , θ¯c , θ¯p , t) is the step response of the regulatory control target transfer function (12) and yd (θ¯c , θ¯p , t) is that of the regulatory control system Myd (s) (9), with the FOPDT model for the controlled process P (s) (2) and the F OP I2 controller Cy (s) (7). In a similar way, the servo-control cost function to be minimized is defined as Jr (τc , θ¯c , θ¯p )= ˙
∞
0
[yrt (τc , θ¯p , t) − yr (θ¯c , θ¯p , t)]2 dt
(18)
where yrt (τc , θ¯c , θ¯p , t) is the step response of the servocontrol target transfer function (14) and yr (θ¯c , θ¯p , t) is that of the servo-control system Myr (s) (10), with the controlled process P (s) (2) and controller Cr (s) (6).
209
209
Then, for the 2-DoF FOPI controller design, the following overall cost functional is optimized JT (τc , θ¯c , θ¯p )=J ˙ r (τc , θ¯c , θ¯p ) + Jd (τc , θ¯c , θ¯p ) (19) to obtain the optimum controller parameters θ¯co = {Kpo , Tio , β o , λo } such that JTo =J (20) ˙ T (τc , θ¯co , θ¯p ) = min JT (τc , θ¯c , θ¯p ) θ¯c
Also, note that θ¯co = θ¯co (θ¯p , τc ). Robustness is an important attribute for control systems, because the design procedures are usually based on the use of low-order linear models identified at the closed-loop operation point. Due to the non-linearity found in most of the industrial process, it is necessary to consider the expected changes in the process characteristics assuming certain relative stability margins, or robustness requirements, for the control system. As an indication of the system robustness (relative stability) the Sensitivity Function peak value will be used. Then, for each θ¯co set obtained, the closed-loop control system robustness is measured using control system Maximum Sensitivity Ms defined as 1 ˙ max |S(jω)| = max (21) Ms = ω ω |1 + C(jω)P (jω)|
The recommended values for Ms are typically within the range 1.4 - 2.0 (Åström and Hägglund (2006)). The use of the maximum sensitivity as a robustness measure, has the advantage that lower bounds to the gain, Am , and phase, φm , margins (Åström and Hägglund (2006)) can be assured according to Ms 1 ; φm > 2 sin−1 Am > Ms − 1 2Ms Therefore, ensuring Ms = 2.0 provides what is commonly considered minimum robustness requirement (that translates to Am > 2 and φm > 29o ). Using the controlled process parameters θ¯p as well as the transformation sˆ = T s, the controlled process (2) and the FOPI controller transfer functions (6) and (7) can be expressed in a normalized for as Ke−τo sˆ Pˆ (ˆ s) = (22) sˆ + 1
where τo = L/T is the model normalized dead-time, that was selected in the range from 0.1 to 2.0 to consider both lag dominant and dead-time dominant controlled processes. Moreover, 1 1 ˆ ˆ , Cy (ˆ (23) s) = κp β + s) = κp 1 + Cr (ˆ τi sˆλ τi sˆλ
and therefore the normalized proportional gain and the normalized integrating time of the controller are Ti κp = Kp K, τi = (24) T
From the optimization results, it is possible to obtain the normalized controller parameters as functions of the model parameters θ¯p and the performance specification τc . During the optimization process the design parameter τc
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was selected in such a way that the robustness level of the resulting closed-loop system was raised to a specific target Mst = 2.0 (considered the minimum acceptable value). The achieved optimal controller normalized parameters are shown along the entire dead-time range in Fig. 2 for κp , Fig. 3 for τi , Fig. 4 for β and Fig. 5 for λ.
3.3 Tuning equations The optimal controller parameters obtained from the optimization procedure were used to fit equations of each one of the F OP I2 parameters, as follows (25) κp = 0.4244 + 0.5443τo−1.04 τi = 0.1707 + 1.755τo0.733 β = 0.04707 + 0.8025τo0.2745 λ = 1.009 + 0.03584τo0.8191
(26) (27) (28)
The robustness obtained from (25) to (28) for each normalized dead-time in the analyzed range is shown in Fig. 6. As it can be seen, the proposed tuning equations provide a control system with the selected target robustness Ms = 2.0.
Figure 2. κp parameter of the F OP I2 for the FOPDT.
Figure 6. Robustness for the F OP I2 controller using the proposed tuning. 4. EXAMPLES
Figure 3. τi parameter of the F OP I2 for the FOPDT
Consider the fourth-order controlled process proposed as benchmark in (Åström and Hägglund, 2000) and given by the transfer function Pα (s) = 3
1
n=0 (α
ns
+ 1)
(29)
with α ∈ {0.10, 0.25, 0.50, 1.00}.
Using the three-point identification procedure 123c (Alfaro (2006)) the FOPDT models were obtained an the parameters are in Table 1. Table 1. Example - Parameters of the FOPDT model for Pα (s).
Figure 4. β parameter of the F OP I2 for the FOPDT
α
K
T
L
0.10 0.25 0.50 1.00
1.000 1.000 1.000 1.000
1.003 1.049 1.247 2.343
0.112 0.298 0.691 1.860
From the FOPDT models for the controlled process in Table 1 and using the tuning equations (25)-(28), the F OP I2 parameters can be calculated. In addition and for comparison purposes the parameters for an integer order P I2 have been obtained using the MoReRT methodology (Alfaro and Vilanova, 2012, 2016). All these parameters are shown in Table 2.
Figure 5. λ parameter of the F OP I2 for the FOPDT 210
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Table 2. Parameters for F OP I2 and P I2 controllers. Parametes
F OP I2
Åström and Hägglund (2006); Kristiansson and Lennartson (2006); Tan et al. (2006)). The formulation of the criterion is stated as
P I2
α = 0.10 Kp Ti β λ
5.7455 0.5236 0.4867 1.0149
IAE = ˙
5.5763 0.4590 0.5114 -
2.4393 0.9103 0.6152 1.0218
2.3532 0.8670 0.5629 -
1.4301 1.6319 0.7295 1.0311 1.1164 3.8704 0.8003 1.0387
|e(t)|dt =
∞
k=1
1.3359 1.4134 0.6351 -
0
∞
|r(t) − y(t)|dt
(30)
|uk+1 − uk |
(31)
and can be used as a measure of the smoothness of the control action for both input changes, T Vur and T Vud . Table 3 has the quantitative evaluation for the control system, including the resulting obtained robustness, as well as the performance measures IAE and T Vu , for changes in the set-point and the load-disturbance inputs.
α = 1.00 Kp Ti β λ
0
∞
˙ T Vu =
α = 0.50 Kp Ti β λ
where the index can be measure for changes in the setpoint IAEr or in the load-disturbance IAEd . Moreover, the control signal total variation T Vu is given by
α = 0.25 Kp Ti β λ
211
1.0128 3.0537 0.6906 -
The control system output responses for servo-control and regulatory-control operation and two values of α for process Pα (s) (29), are shown in Fig. 7 and Fig. 8.
Table 3. Example - Performance and Robustness Evaluation for F OP I2 and P I2 controllers. Metric
F OP I2
P I2
α = 0.10 Ms IAEd T V ud IAEr T V ur
1.9780 0.0941 1.5899 0.3726 4.6058
1.9999 0.0827 1.5140 0.3074 3.5620
α = 0.25 Ms IAEd T V ud IAEr T V ur
Figure 7. Example - Control System responses comparison α = 0.5 processes.
2.0153 0.3722 1.5153 0.7299 1.5153
2.0048 0.3685 1.4451 0.7476 1.7004
α = 0.50 Ms IAEd T V ud IAEr T V ur
2.0187 1.0341 1.3856 1.4483 1.4609
2.0065 1.0577 1.4749 1.5856 1.2860
α = 1.00 Ms IAEd T V ud IAEr T V ur
Figure 8. Example - Control System responses for α = 1.0 processes. For the evaluation of the control system performance, it can be found in the literature that the Integral-AbsoluteError (IAE) is the most useful and suitable index to quantify the performance of the system. (Skogestad (2003); 211
2.0270 2.8169 1.3801 3.5607 1.2325
2.0062 3.1120 1.5560 4.1110 1.2410
From this information it is evident the robustness/performance trade-off and also the compromise between servo-control and regulatory-control performance operation. Moreover, it can be seen that for small values of α (i.e. α = 0.1) the performance and robustness for P I2 and F OP I2 are very similar. However, when α increases the is an improvement for the F OP I2 against the regular P I2 controller.
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5. CONCLUSIONS The proposed tuning is based on the Model Reference Robust Tuning (MoReRT) methodology (Alfaro and Vilanova, 2012, 2016). The approach includes into the design problem formulation, the use of two-degree-offreedom fractional-order proportional-integral controllers (F OP I2 ). The obtained tuning rules, use the information of a first order plus dead time (FOPDT) model in the range for normalized dead-times from 0.1 to 2.0, to compute the calculation of the four controller parameters. The design deals with the performance/robustness tradeoff of the closed-loop control system, providing values for robustness close to Ms = 2.0, as well as good evaluation for the performance. It can be seen the improvement that can be achieve with the use of a F OP I2 controller against the integer order P I2 . As future work, the MoReRT methodology can be extended also to 2-DoF fractional order PID controllers. ACKNOWLEDGEMENTS The financial support from the University of Costa Rica, under the grant 731-B4-213, is greatly appreciated. Also, this work has received financial support from the Spanish CICYT program under grant DPI2016-77271-R. REFERENCES Alfaro, V.M. and Vilanova, R. (2012). Model-reference robust tuning of 2DoF PI controllers for first- and second-order plus dead-time controlled processes. Journal of Process Control, 22(2), 359 – 374. doi: https://doi.org/10.1016/j.jprocont.2012.01.001. Alfaro, V.M. and Vilanova, R. (2016). Model-Reference Robust Tuning of PID Controllers. Springer-Verlag Advances in Industrial Control Series. Alfaro, V. (2006). Low-order models identification form the process reaction curve. Ciencia y TecnologÃa (Costa Rica), 24(2), 197–216. Arrieta, O., Urvina, L., Visioli, A., Vilanova, R., and Padula, F. (2015). Servo/Regulation Intermediate Tuning for Fractional Order PID Controllers. In IEEE Multi-Conference on Systems and Control, September 21-23, Sydney - Australia. Arrieta, O. and Vilanova, R. (2007). Performance degradation analysis of Optimal PID settings and Servo/Regulation tradeoff tuning. In Conference on Systems and Control (CSC07), May 16-18, MarrakechMorocco. Arrieta, O. and Vilanova, R. (2010). Performance degradation analysis of controller tuning modes: Application to an optimal PID tuning. International Journal of Innovative Computing, Information and Control, 6(10), 4719–4729. 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. DOI: 10.1021/ie201655c. Arrieta, O., Vilanova, R., and Visioli, A. (2011). Proportional-Integral-Derivative Tuning for Servo/Regulation Control Operation for Unstable 212
and Integrating Processes. Industrial & Engineering Chemistry Research, 50(6), 3327–3334. Arrieta, O., Visioli, A., and Vilanova, R. (2010). PID autotuning for weighted servo/regulation control operation. 20(4), 472–480. Arrieta, O. and Vilanova, R. and Rojas, J. D. and Meneses, M. (2016). Improved PID controller tuning rules for performance degradation/robustness increase trade-off. Electrical Engineering, 98(3), 233–243. Åström, K. and Hägglund, T. (2000). Benchmark Systems for PID control. In IFAC Digital Control: Past, Present and Future of PID Control. Terrassa, Spain. Åström, K. and Hägglund, T. (2001). The future of PID control. 9, 1163–1175. Åström, K. and Hägglund, T. (2006). Advanced PID Control. ISA - The Instrumentation, Systems, and Automation Society. Biswas, A., Das, S., Abraham, A., and Dasgupta, S. (2009). Design of fractional-order PIλ Dµ controllers with an improved differential evolution. Engineering Applications of Artificial Intelligence, 22, 343–350. Kristiansson, B. and Lennartson, B. (2006). Evaluation and simple tuning of PID controllers with highfrequency robustness. 16, 91–102. Meneses, H., Guevara, E., Arrieta, O., Padula, F., Vilanova, R., and Visioli, A. (2018). Improvement of the Control System Performance based on Fractional-Order PID Controllers and Models with Robustness Considerations. In PID18, 3rd IFAC Conference on Advances in Proportional-Integral-Derivative Control, May 09-11, Ghent-Belgium. Monje, C., Vinagre, B., Calderon, A., Feliu, V., and Chen, Y.Q. (2004). On fractional PIλ controllers: some tuning rules for robustness to plan uncertainties. Nonlinear Dynamics, 38, 369–381. Monje, C., Vinagre, B., Feliu, V., and Chen, Y.Q. (2008). Tuning and auto-tuning of fractional order controllers for industry applications. 16, 798–812. Padula, F. and Visioli, A. (2011). Tuning rules for optimal PID and fractional-order PID controllers. 21(1), 69–81. Padula, F. and Visioli, A. (2014). Advances in Robust Fractional Control. Springer Verlag. Podlubny, I. (1999). Fractional-order systems and PIλ Dµ controllers. IEEE Transactions on Automatic Control, 44, 208–214. Skogestad, S. (2003). Simple analytic rules for model reduction and PID controller tuning. 13, 291–309. Tan, W., Liu, J., Chen, T., and Marquez, H.J. (2006). Comparison of some well-known PID tuning formulas. Computers & Chemical Engineering, 30, 1416–1423. Vinagre, B., Monge, C., Calderon, A., and Suarez, J. (2007). Fractional PID controllers for industry application. a brief introduction. Journal of Vibration and Control, 13, 1419–1429. Zamani, M., Karimi-Ghartemani, M., Sadati, N., and Parniani, M. (2009). Design of fractional order PID controller for an AVR using particle swarm optimization. 17, 1380–1387.
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Model Model Reference Reference Based Based Tuning Tuning for for Model Reference Based Tuning for Fractional-Order 2DoF PI Controllers with Model Reference Tuning for Fractional-Order 2DoFBased PI Controllers with a a Fractional-Order 2DoF PI Controllers with a Robustness Consideration Fractional-Order 2DoF PI Controllers with a Robustness Consideration Robustness Consideration ∗,∗∗∗ ∗Consideration ∗∗ ∗∗∗ O. Arrieta Robustness ∗,∗∗∗ ∗,∗∗∗ , D. Castillo ∗ ∗ , R. Vilanova ∗∗ ∗∗ , J.D. Rojas ∗∗∗ ∗∗∗
O. Arrieta ∗,∗∗∗ , D. Castillo ∗ , R. Vilanova ∗∗ , J.D. Rojas ∗∗∗ O. Arrieta , D. Castillo , R. Vilanova , J.D. Rojas ∗O. Arrieta ∗,∗∗∗ , D. Castillo ∗ , R. Vilanova ∗∗ , J.D. Rojas ∗∗∗ ∗ Instituto de Investigaciones en Ingeniería, Facultad de Ingeniería, ∗ ∗ Instituto de Investigaciones en Ingeniería, Facultad de Ingeniería, Instituto de Investigaciones Ingeniería,San Facultad de Ingeniería, Universidad de Costa Rica,en 11501-2060 José, Costa Rica. ∗ Universidad de Costa Rica, 11501-2060 San José, Costa Rica. Instituto de Investigaciones Ingeniería,San Facultad de Ingeniería, email: [email protected] Universidad deemail: [email protected] Rica,en 11501-2060 José, Costa Rica. ∗∗ Universidad de Costa Rica, 11501-2060 San José, Costa Rica. Departament de Telecomunicació i d’Enginyeria de Sistemes email: [email protected] ∗∗ ∗∗ de Telecomunicació i d’Enginyeria de Sistemes ∗∗ Departament email: [email protected] Escola d’Enginyeria, Universitat Autònoma de Barcelona Departament de Telecomunicació i d’Enginyeria de Sistemes ∗∗ Escola d’Enginyeria, Universitat Autònoma de Barcelona Departament de Telecomunicació i d’Enginyeria de Sistemes Escola d’Enginyeria, Universitat Autònoma de Barcelona 08193 Bellaterra, Barcelona, Spain. 08193 Bellaterra, Barcelona, Spain. ∗∗∗ Escola d’Enginyeria, Universitat Autònoma de Barcelona Departamento de Automática, Escuela de Ingeniería 08193 Bellaterra, Barcelona, Spain. ∗∗∗ ∗∗∗ Departamento de Automática, Escuela de Ingeniería Eléctrica, Eléctrica, ∗∗∗ 08193 Bellaterra, Barcelona, Universidad de Costa Rica, 11501-2060 11501-2060 San José, Costa Costa Rica. Departamento de Automática, Escuela San deSpain. Ingeniería Eléctrica, Universidad de Costa Rica, José, Rica. ∗∗∗ Departamento de Automática, Escuela San de Ingeniería Eléctrica, Universidad de Costa Rica, 11501-2060 José, Costa Rica. Universidad de Costa Rica, 11501-2060 San José, Costa Rica. Abstract: The The paper paper proposes proposes the the tuning tuning of of aa two-degree-of-freedom two-degree-of-freedom (2-DoF) Abstract: (2-DoF) fractional-order fractional-order Abstract: The paper proposes the tuning of a two-degree-of-freedom (2-DoF) fractional-order PI (F OP I ) controller based on the Model Reference Robust Tuning (MoReRT) methodology 2 PI (F OP I22 ) controller based on the Model Reference Robust Tuning (MoReRT) methodology Abstract: The paperand proposes tuning ofReference a two-degree-of-freedom (2-DoF) fractional-order presented Alfaro Vilanova (2012, 2016). The rules are for aa first PI (F OP Iby based onthe the Model Robust Tuning (MoReRT) 2 ) controller presented by Alfaro and Vilanova (2012, 2016). The tuning tuning rules are optimized optimized formethodology first order order PI (F OP I ) controller based on the Model Reference Robust Tuning (MoReRT) methodology plus dead time (FOPDT) model and for an over-damped servo and regulatory closed-loop presented by Alfaro and Vilanova (2012, 2016). The tuning rules are optimized for a firsttarget order 2 plus dead time (FOPDT) model and for an over-damped servo and regulatory closed-loop target presented by Alfaro and Vilanova (2012, 2016). The tuning rules are optimized for a first responses. Moreover, the design takes into account a robustness consideration, expressed as plus dead time (FOPDT) model and for an over-damped servo and regulatory closed-loop target responses. Moreover, the design takes into account a robustness consideration, expressedorder as aa plus dead time (FOPDT) model and for an over-damped servo and regulatory closed-loop target responses. Moreover, the design takes into account a robustness consideration, expressed as a maximum sensitivity target value of 2.0, dealing with the performance/robustness trade-off maximum sensitivity target value of 2.0, dealing with the performance/robustness trade-off responses. Moreover, the design takes into account a robustness consideration, expressed as is a of the closed-loop control system. A set of controller tuning equations (four parameters) maximum sensitivity target value of 2.0, dealing with the performance/robustness trade-off of the closed-loop control system. A set of controller tuning equations (four parameters) is maximum sensitivity target value of 2.0, dealing with the performance/robustness trade-off provided for FOPDT models with normalized dead-times from 0.1 to 2.0. Also, an evaluation of of the closed-loop control system. A set of controller tuning equations (four parameters) is provided for FOPDT models with normalized dead-times from 0.1 to 2.0. Also, an evaluation of of closed-loop control system. Arobustness set of controller tuning equations (fourthe parameters) is the control performance and is order to effectiveness provided forsystem FOPDT models with normalized dead-times fromin to 2.0. Also, an evaluation of thethe control system performance and robustness is presented, presented, in0.1 order to show show the effectiveness provided for FOPDT models with normalized dead-times from 0.1 to 2.0. Also, an evaluation of the control system performance and robustness is presented, in order to show the effectiveness of the proposed tuning method, that is compared with other tuning rules. of the proposed tuning method, that is compared with other tuning rules. thethe control system performance is with presented, in orderrules. to show the effectiveness of proposed tuning method, and that robustness is compared other tuning © the 2019,proposed IFAC (International Federation ofisAutomatic Control) Hosting by Elsevier Ltd. All rights reserved. of tuning method, that compared with other tuning rules. Keywords: Keywords: PI PI Control, Control, Model Model Reference, Reference, Fractional Fractional Control, Control, Tuning Tuning Rules Rules Keywords: PI Control, Model Reference, Fractional Control, Tuning Rules Keywords: PI Control, Model Reference, Fractional Control, Tuning 1. INTRODUCTION However, it has also Rules been recognize that another important 1. INTRODUCTION However, it has also been been recognize recognize that that another another important 1. INTRODUCTION However, it has also issue to be addressed is the trade-off between important servo and issue to be addressed is the trade-off between servo and and 1. INTRODUCTION However, it operation has also been another important issue to be addressed is recognize the of trade-off between servo regulation mode the system (Arrieta regulation operation mode of thethat system (Arrieta and Proportional-Integrative-Derivative (PID) controllers are regulation issue to be2007, addressed is the of trade-off between servo and operation mode the system (Arrieta Vilanova, 2010; Arrieta et al., 2010, 2011). Proportional-Integrative-Derivative (PID) controllers are Vilanova, 2007, 2010; Arrieta et al., 2010, 2011). Proportional-Integrative-Derivative (PID) controllers are Vilanova, with no doubt the most extensive option that can be found regulation operation mode ofetand the system (Arrieta and 2007, 2010; in Arrieta al.,Vilanova 2010, 2011). To face that problem, Alfaro (2012, 2016), with no doubt the most extensive option that can be found To face that that problem, in Alfaroetand and Vilanova (2012, 2016), 2016), Proportional-Integrative-Derivative (PID) controllers are To with no doubt the most extensive option that can be found on industrial control applications (Åström and Hägglund, Vilanova, 2007, 2010; Arrieta al.,Vilanova 2010, 2011). face problem, in Alfaro (2012, it was presented a Model Reference Robust Tuning (MoRon industrial control applications (Åström and Hägglund, it was presented Model Reference Robust Tuning Tuning (MoRwith noTheir doubtsuccess the most extensive option canHägglund, be found it on industrial control applications (Åström and 2001). is mainly due to itsthat simple structure To face that problem, in controllers. Alfaro and That Vilanova (2012,consid2016), was presented aa Model Reference Robust (MoReRT) for 2-DoF PI/PID approach 2001). Their success is mainly due to its simple structure eRT) for 2-DoF PI/PID controllers. That approach considon industrial controlmeaning applications and Hägglund, 2001). success is mainly to its simple structure and to Their the physical of due the(Åström corresponding parame- eRT) it was presented a Model Reference Robust Tuning (MoRfor 2-DoF PI/PID controllers. That approach considers the above mentioned trade-offs for the control system. and to the meaning of the corresponding parameers the above mentioned trade-offs for the control system. 2001). Their success is manual mainly to its simple structure and to the physical physical meaning of due the corresponding parameters (therefore making tuning possible). This fact ers eRT) 2-DoF PI/PID controllers. That approach considthefor mentioned trade-offs control system. What isabove provided in this paper isfor anthe extension of this ters (therefore making manual tuning possible). This fact What isabove provided in thistrade-offs paper is isforan anthe extension of this this and to the meaning the corresponding parameters (therefore making manual tuning possible). fact What makes PID control easier to by control ers the is mentioned control system. provided paper of methodology using ina this fractional-order PIextension controller with makes PIDphysical control easier toof understand understand by the theThis control methodology using a fractional-order PI controller with ters (therefore making manual tuning possible). This fact methodology makes PIDthan control toadvanced understand by the control engineers othereasier more control techniques. What provided ina the this paperdistinctive is anPIextension this fractional-order controller 2-DoF, is one major features the engineers othereasier more control techniques. 2-DoF,isthat that isusing one of of the major distinctive featuresofof ofwith the makes PIDthan control understand by the control engineers than more toadvanced advanced control techniques. In addition, theother PID controller provides satisfactory per- 2-DoF, methodology using a fractional-order PI controller with that is one of the major distinctive features of the proposal. The design considers the performance (servo and In addition, the PID controller provides satisfactory perproposal. The design considers the performance (servo and engineers than other more control techniques. In addition, PIDrange controller provides satisfactory per- proposal. formance in athe wide of advanced practical situations. 2-DoF, that isdesign one of considers the major distinctive features of and the The the performance (servo regulation) and robustness of the control system. formance in aathe wide range of practical situations. regulation) and robustness of the the control system. addition, PID controller provides satisfactory per- regulation) formance in wide range of practical situations. In recent years, there has also been a significant interproposal. design considers the control performance (servo robustness of system. The paperThe isand organized as follows. Next section states and the In recent years, there has also been a significant interThe paper is isand organized as follows. follows. Next section section states the the formance a wide range of also practical In recent years, there has been situations. a significant interest from in the academic and industrial communities for The regulation) robustness of the control system. paper organized as Next states general framework for the problem formulation. In Section est from the academic and industrial communities for general framework for the problem formulation. In Section In recent years, there has been a significant interest from the academic andalso industrial communities for general fractional-order-proportional-integral-derivative (FOPID) paper isthe organized as problem follows. Next states the framework for the formulation. In Section 3The describes controller procedure forsection the design. An fractional-order-proportional-integral-derivative (FOPID) 3general describes the controller controller procedure for the the design. design. An est from the academic industrial communities for 3example fractional-order-proportional-integral-derivative (FOPID) controllers because they are to (as framework forintheSection problem formulation. In ends Section describes the procedure for An is provided 4 and the paper in controllers because they and are capable capable to provide provide (as there there example is provided in Section 4 and the paper ends in fractional-order-proportional-integral-derivative controllers because to they arefor to three provide (as are tune PID for PI) more 3 describes thesome controller procedure design. provided in Section 4 andfor thethe paper endsAn in Section 5iswith conclusions. are five five parameters parameters to tune forcapable PID and and three for (FOPID) PI) there more example Section with some conclusions. conclusions. controllers they are to three provide there are five parameters to tune forcapable PID and PI) more Section flexibility inbecause the control system design (see for for (as example example55iswith provided in Section 4 and the paper ends in some flexibility in the control system design (see for example are five parameters to tune for PID and three for PI) more flexibility the control system (seeBiswas for example Podlubny in (1999); Vinagre et al.design (2007); et al. Section 5 with some conclusions. Podlubny (1999); Vinagre et al. (2007); et al. flexibility in theet system (seeBiswas for example Podlubny (1999); Vinagre etMany al.design (2007); Biswas etrules al. (2009); al. (2009)). different tuning (2009); Zamani Zamani etcontrol al. (2009)). Many different tuning rules Podlubny (1999); Vinagre et al. (2007); Biswas et al. (2009); Zamani et al. (2009)). Many different tuning rules have been proposed in the literature to facilitate their have been proposed in the literature to facilitate their 2. PROBLEM FORMULATION (2009); Zamani et al. (2009)). Many different tuning 2011, rules have been proposed in the literature to facilitate their use (Monje et al., 2004, 2008; Padula and Visioli, 2. PROBLEM PROBLEM FORMULATION FORMULATION 2011, use (Monje et al., 2004, 2008; Padula and Visioli, 2. have been proposed in the literature to facilitate their use Padula and2018). Visioli, 2011, 2014;(Monje Arrietaetetal., al.,2004, 2015;2008; Meneses et al., Most of 2014; Arrieta et al., 2015; Meneses et al., 2018). Most of 2. PROBLEM FORMULATION use Padula and Visioli, 2011, 2014; Arrieta etal., al.,2004, 2015;2008; Meneses et al., 2018). Most of them are for 1-DoF controllers, therefore there is for them(Monje are for et 1-DoF controllers, therefore there is room room for 2.1 Control System Configuration 2014; are Arrieta et 2-DoF al.,controllers, 2015; Meneses et al.,there 2018). Mostfor of 2.1 Control System Configuration them for 1-DoF therefore is room fractional-order controllers. 2.1 Control System Configuration fractional-order 2-DoF controllers. them are for 1-DoF controllers, is room for 2.1 Control System Configuration fractional-order 2-DoF It is well known that acontrollers. properlytherefore designedthere control system It is well known that a properly designed control system fractional-order It is well known that acontrollers. properly designed control system The considered control system is shown in Fig. 1, where must provide an2-DoF effective trade-off between performance must provide an effective trade-off between performance The considered considered control control system is is shown shown in in Fig. Fig. 1, 1, where where It is well known that a properly designed control system must provide an effective trade-off between performance and robustness (Arrieta and Vilanova, 2012; Arrieta, O. P process and robustness (Arrieta and Vilanova, 2012; Arrieta, O. The P (s) (s) is is the the controlled controlled system process and and C(s) C(s) is is the the 2-DoF 2-DoF mustVilanova, provide an effective trade-off between The control system isand shown in is Fig. where and robustness and 2012;performance Arrieta, O. P (s) considered is the controlled process C(s) the1, 2-DoF R.(Arrieta and Rojas, J.Vilanova, D. and Meneses, M., 2016). fractional order PI (F OP I2 ) controller. and and D. M., fractional order PI (F (F OP OPprocess robustnessR. andJ. 2012; Arrieta, O. P (s) is the controlled and C(s) is the 2-DoF and Vilanova, Vilanova, R.(Arrieta and Rojas, Rojas, J.Vilanova, D. and and Meneses, Meneses, M., 2016). 2016). fractional order PI II222 )) controller. controller. and Vilanova, R.IFAC and (International Rojas, J. D.Federation and Meneses, M., 2016). order PI I2 )reserved. controller. 2405-8963 © 2019 2019, IFAC of Automatic Control) Hosting by Elsevier Ltd.(F AllOP rights Copyright © 207 fractional Copyright © under 2019 IFAC 207 Control. Peer review responsibility of International Federation of Automatic Copyright © 2019 IFAC 207 10.1016/j.ifacol.2019.06.063 Copyright © 2019 IFAC 207
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and in compact form as follows u(s) = Cr (s)r(s) − Cy (s)y(s) where
Figure 1. The considered control system.
1 Ti sλ
(6)
is the F OP I2 set-point controller transfer function, which operates on r, and 1 Cy (s) = Kp 1 + (7) Ti sλ
The variables described in Fig. 1 are: • • • •
Cr (s) = Kp β +
(5)
y(s) is the process controlled variable. r(s) is the set-point for the process output. u(s) is the controller output signal. d(s) is the load-disturbance of the system.
The closed-loop control system output y(s), in response to changes in its inputs, r(s) and d(s), is given by the following (1) y(s) = Myr (s)r(s) + Myd (s)d(s) where Myr (s) is the transfer function from the set-point to the process controlled variable, and Myd (s) is that from the load-disturbance to the process controlled variable. These are known as the servo-control closed-loop transfer function and the regulatory-control closed-loop transfer function, respectively.
is the F OP I2 feedback controller transfer function, which operates on y. The closed-loop transfer functions of the servo-control and the regulatory-control in (1) are then given by Cr (s)P (s) (8) Myr (s) = 1 + Cy (s)P (s) and Myd (s) =
P (s) 1 + Cy (s)P (s)
which are related as Myr (s) = Cr (s)Myd (s)
(9)
(10)
3. CONTROLLER DESIGN
2.2 Controlled Process Model The controlled process P (s) is assumed to be modelled by a first-order-plus-dead-time (FOPDT) transfer function of the form Ke−Ls (2) P (s) = Ts + 1 where K is the process gain, T is the time constant and L is the dead-time. The parameters of the controlled process model (2) are θ¯p = {K, T, L}. The availability of FOPDT models in the process industry is a well known fact. The generation of such model just needs for a very simple step-test experiment to be applied to the process. From this point of view, to maintain the need for plant experimentation to a minimum is a key point when considering the applicability of a technique in an industrial environment. 2.3 Fractional Order 2-DoF PI Controller (F OP I2 ) The process will be controlled with a fractional-order twodegree-of-freedom proportional integral controller (F OP I2 ), whose output is defined as 1 u(s) = Kp βr(s) − y(s) + [r(s) − y(s)] (3) Ti sλ where Kp is the controller proportional gain, Ti is the integral time constant, β is the set-point proportional weight and λ is the fractional order for the integral term. For the purposed of analysis, the controller output (3) will be expressed as 1 1 u(s) = Kp β + r(s) − Kp 1 + y(s) (4) Ti sλ Ti sλ 208
Usually the design of 2-DoF controllers is performed in two stages. First, it is designed the regulatory-control behaviour with the control system robustness, that means the feedback controller Cy (s) (7). Second, the set-point controller Cr (s) (6) is used to improve the servo-control performance. Here, a different approach is taken. The controller design is based on the Model Reference Robust Tuning (MoReRT) methodology presented by Alfaro and Vilanova (2012, 2016). The application of that to the 2-DoF fractional order PI controller, means to obtain the complete set of the F OP I2 controller parameters θ¯c = {Kp , Ti , β, λ}, considering a combination of the regulatory and servocontrol performance of a targeted closed-loop dynamic with a desired robustness level. 3.1 Target closed-loop transfer functions From the MoReRT methodology it is important to have the least amount of design parameters. Consequently, the desired control system response for load-disturbance and set-point step changes will involve only one design parameter, the closed-loop time constant Tc . The closedloop response is designed with no steady-state error and an over-damped dynamic. From control system described in Section 2, the desired regulatory-control closed-loop transfer function is defined as Ko se−Ls t (11) (s) = Myd (Tc s + 1)2 where Ko is the static gain and Tc is the time constant of the regulatory control closed-loop transfer function. For a
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PI controller, the static gain in (11) can be expressed as Ko = Ti /Kp , then t (s) = Myd
(Ti /Kp )se−Ls (Tc s + 1)2
(12)
From (10), the servo-control closed-loop transfer function is given by (βTi s + 1)e−Ls t Myr (s) = (13) (Tc s + 1)2 In order to have a set-point step change response without oscillation, overshoot or steady-state error, the servocontrol closed-loop transfer function is selected as e−Ls t Myr (14) (s) = (Tc s + 1) The zero/pole cancellation for (14), βTi = Tc , will not be forced, but taken into account in the optimization procedure to match the closed-loop transfer functions. t Therefore, Myd (s) is the target regulatory-control closedt (s) is the target servoloop transfer function, and Myr control closed-loop transfer function. If Tc is expressed as a function of the time constant of the controlled process, then τc = Tc /T can now be used as a dimensionless design parameter. The closed-loop performance specification requires only the definition of τc parameter, which is an indication of the closed-loop system response speed in relation to the controlled process speed. Replacing the parameter in (12) and (14), the global target output of the control system is (Ti /Kp )se−Ls e−Ls y t (s) = r(s) + d(s) (15) (τc T s + 1) (τc T s + 1)2 The target closed-loop response of the control system expressed in (15) can be rewritten in the time domain as follows (16) y t (t) = yrt (t) + ydt (t) where yrt (t) is the target servo-control step response and ydt (t) is the target regulatory-control step response. 3.2 Controller optimization For the regulatory-control response, the cost function to be minimized is defined as Jd (τc , θ¯c , θ¯p )= ˙
∞
0
[ydt (τc , θ¯c , θ¯p , t) − yd (θ¯c , θ¯p , t)]2 dt (17)
where ydt (τc , θ¯c , θ¯p , t) is the step response of the regulatory control target transfer function (12) and yd (θ¯c , θ¯p , t) is that of the regulatory control system Myd (s) (9), with the FOPDT model for the controlled process P (s) (2) and the F OP I2 controller Cy (s) (7). In a similar way, the servo-control cost function to be minimized is defined as Jr (τc , θ¯c , θ¯p )= ˙
∞
0
[yrt (τc , θ¯p , t) − yr (θ¯c , θ¯p , t)]2 dt
(18)
where yrt (τc , θ¯c , θ¯p , t) is the step response of the servocontrol target transfer function (14) and yr (θ¯c , θ¯p , t) is that of the servo-control system Myr (s) (10), with the controlled process P (s) (2) and controller Cr (s) (6).
209
209
Then, for the 2-DoF FOPI controller design, the following overall cost functional is optimized JT (τc , θ¯c , θ¯p )=J ˙ r (τc , θ¯c , θ¯p ) + Jd (τc , θ¯c , θ¯p ) (19) to obtain the optimum controller parameters θ¯co = {Kpo , Tio , β o , λo } such that JTo =J (20) ˙ T (τc , θ¯co , θ¯p ) = min JT (τc , θ¯c , θ¯p ) θ¯c
Also, note that θ¯co = θ¯co (θ¯p , τc ). Robustness is an important attribute for control systems, because the design procedures are usually based on the use of low-order linear models identified at the closed-loop operation point. Due to the non-linearity found in most of the industrial process, it is necessary to consider the expected changes in the process characteristics assuming certain relative stability margins, or robustness requirements, for the control system. As an indication of the system robustness (relative stability) the Sensitivity Function peak value will be used. Then, for each θ¯co set obtained, the closed-loop control system robustness is measured using control system Maximum Sensitivity Ms defined as 1 ˙ max |S(jω)| = max (21) Ms = ω ω |1 + C(jω)P (jω)|
The recommended values for Ms are typically within the range 1.4 - 2.0 (Åström and Hägglund (2006)). The use of the maximum sensitivity as a robustness measure, has the advantage that lower bounds to the gain, Am , and phase, φm , margins (Åström and Hägglund (2006)) can be assured according to Ms 1 ; φm > 2 sin−1 Am > Ms − 1 2Ms Therefore, ensuring Ms = 2.0 provides what is commonly considered minimum robustness requirement (that translates to Am > 2 and φm > 29o ). Using the controlled process parameters θ¯p as well as the transformation sˆ = T s, the controlled process (2) and the FOPI controller transfer functions (6) and (7) can be expressed in a normalized for as Ke−τo sˆ Pˆ (ˆ s) = (22) sˆ + 1
where τo = L/T is the model normalized dead-time, that was selected in the range from 0.1 to 2.0 to consider both lag dominant and dead-time dominant controlled processes. Moreover, 1 1 ˆ ˆ , Cy (ˆ (23) s) = κp β + s) = κp 1 + Cr (ˆ τi sˆλ τi sˆλ
and therefore the normalized proportional gain and the normalized integrating time of the controller are Ti κp = Kp K, τi = (24) T
From the optimization results, it is possible to obtain the normalized controller parameters as functions of the model parameters θ¯p and the performance specification τc . During the optimization process the design parameter τc
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was selected in such a way that the robustness level of the resulting closed-loop system was raised to a specific target Mst = 2.0 (considered the minimum acceptable value). The achieved optimal controller normalized parameters are shown along the entire dead-time range in Fig. 2 for κp , Fig. 3 for τi , Fig. 4 for β and Fig. 5 for λ.
3.3 Tuning equations The optimal controller parameters obtained from the optimization procedure were used to fit equations of each one of the F OP I2 parameters, as follows (25) κp = 0.4244 + 0.5443τo−1.04 τi = 0.1707 + 1.755τo0.733 β = 0.04707 + 0.8025τo0.2745 λ = 1.009 + 0.03584τo0.8191
(26) (27) (28)
The robustness obtained from (25) to (28) for each normalized dead-time in the analyzed range is shown in Fig. 6. As it can be seen, the proposed tuning equations provide a control system with the selected target robustness Ms = 2.0.
Figure 2. κp parameter of the F OP I2 for the FOPDT.
Figure 6. Robustness for the F OP I2 controller using the proposed tuning. 4. EXAMPLES
Figure 3. τi parameter of the F OP I2 for the FOPDT
Consider the fourth-order controlled process proposed as benchmark in (Åström and Hägglund, 2000) and given by the transfer function Pα (s) = 3
1
n=0 (α
ns
+ 1)
(29)
with α ∈ {0.10, 0.25, 0.50, 1.00}.
Using the three-point identification procedure 123c (Alfaro (2006)) the FOPDT models were obtained an the parameters are in Table 1. Table 1. Example - Parameters of the FOPDT model for Pα (s).
Figure 4. β parameter of the F OP I2 for the FOPDT
α
K
T
L
0.10 0.25 0.50 1.00
1.000 1.000 1.000 1.000
1.003 1.049 1.247 2.343
0.112 0.298 0.691 1.860
From the FOPDT models for the controlled process in Table 1 and using the tuning equations (25)-(28), the F OP I2 parameters can be calculated. In addition and for comparison purposes the parameters for an integer order P I2 have been obtained using the MoReRT methodology (Alfaro and Vilanova, 2012, 2016). All these parameters are shown in Table 2.
Figure 5. λ parameter of the F OP I2 for the FOPDT 210
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Table 2. Parameters for F OP I2 and P I2 controllers. Parametes
F OP I2
Åström and Hägglund (2006); Kristiansson and Lennartson (2006); Tan et al. (2006)). The formulation of the criterion is stated as
P I2
α = 0.10 Kp Ti β λ
5.7455 0.5236 0.4867 1.0149
IAE = ˙
5.5763 0.4590 0.5114 -
2.4393 0.9103 0.6152 1.0218
2.3532 0.8670 0.5629 -
1.4301 1.6319 0.7295 1.0311 1.1164 3.8704 0.8003 1.0387
|e(t)|dt =
∞
k=1
1.3359 1.4134 0.6351 -
0
∞
|r(t) − y(t)|dt
(30)
|uk+1 − uk |
(31)
and can be used as a measure of the smoothness of the control action for both input changes, T Vur and T Vud . Table 3 has the quantitative evaluation for the control system, including the resulting obtained robustness, as well as the performance measures IAE and T Vu , for changes in the set-point and the load-disturbance inputs.
α = 1.00 Kp Ti β λ
0
∞
˙ T Vu =
α = 0.50 Kp Ti β λ
where the index can be measure for changes in the setpoint IAEr or in the load-disturbance IAEd . Moreover, the control signal total variation T Vu is given by
α = 0.25 Kp Ti β λ
211
1.0128 3.0537 0.6906 -
The control system output responses for servo-control and regulatory-control operation and two values of α for process Pα (s) (29), are shown in Fig. 7 and Fig. 8.
Table 3. Example - Performance and Robustness Evaluation for F OP I2 and P I2 controllers. Metric
F OP I2
P I2
α = 0.10 Ms IAEd T V ud IAEr T V ur
1.9780 0.0941 1.5899 0.3726 4.6058
1.9999 0.0827 1.5140 0.3074 3.5620
α = 0.25 Ms IAEd T V ud IAEr T V ur
Figure 7. Example - Control System responses comparison α = 0.5 processes.
2.0153 0.3722 1.5153 0.7299 1.5153
2.0048 0.3685 1.4451 0.7476 1.7004
α = 0.50 Ms IAEd T V ud IAEr T V ur
2.0187 1.0341 1.3856 1.4483 1.4609
2.0065 1.0577 1.4749 1.5856 1.2860
α = 1.00 Ms IAEd T V ud IAEr T V ur
Figure 8. Example - Control System responses for α = 1.0 processes. For the evaluation of the control system performance, it can be found in the literature that the Integral-AbsoluteError (IAE) is the most useful and suitable index to quantify the performance of the system. (Skogestad (2003); 211
2.0270 2.8169 1.3801 3.5607 1.2325
2.0062 3.1120 1.5560 4.1110 1.2410
From this information it is evident the robustness/performance trade-off and also the compromise between servo-control and regulatory-control performance operation. Moreover, it can be seen that for small values of α (i.e. α = 0.1) the performance and robustness for P I2 and F OP I2 are very similar. However, when α increases the is an improvement for the F OP I2 against the regular P I2 controller.
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5. CONCLUSIONS The proposed tuning is based on the Model Reference Robust Tuning (MoReRT) methodology (Alfaro and Vilanova, 2012, 2016). The approach includes into the design problem formulation, the use of two-degree-offreedom fractional-order proportional-integral controllers (F OP I2 ). The obtained tuning rules, use the information of a first order plus dead time (FOPDT) model in the range for normalized dead-times from 0.1 to 2.0, to compute the calculation of the four controller parameters. The design deals with the performance/robustness tradeoff of the closed-loop control system, providing values for robustness close to Ms = 2.0, as well as good evaluation for the performance. It can be seen the improvement that can be achieve with the use of a F OP I2 controller against the integer order P I2 . As future work, the MoReRT methodology can be extended also to 2-DoF fractional order PID controllers. ACKNOWLEDGEMENTS The financial support from the University of Costa Rica, under the grant 731-B4-213, is greatly appreciated. Also, this work has received financial support from the Spanish CICYT program under grant DPI2016-77271-R. REFERENCES Alfaro, V.M. and Vilanova, R. (2012). Model-reference robust tuning of 2DoF PI controllers for first- and second-order plus dead-time controlled processes. Journal of Process Control, 22(2), 359 – 374. doi: https://doi.org/10.1016/j.jprocont.2012.01.001. Alfaro, V.M. and Vilanova, R. (2016). Model-Reference Robust Tuning of PID Controllers. Springer-Verlag Advances in Industrial Control Series. Alfaro, V. (2006). Low-order models identification form the process reaction curve. Ciencia y TecnologÃa (Costa Rica), 24(2), 197–216. Arrieta, O., Urvina, L., Visioli, A., Vilanova, R., and Padula, F. (2015). Servo/Regulation Intermediate Tuning for Fractional Order PID Controllers. In IEEE Multi-Conference on Systems and Control, September 21-23, Sydney - Australia. Arrieta, O. and Vilanova, R. (2007). Performance degradation analysis of Optimal PID settings and Servo/Regulation tradeoff tuning. In Conference on Systems and Control (CSC07), May 16-18, MarrakechMorocco. Arrieta, O. and Vilanova, R. (2010). Performance degradation analysis of controller tuning modes: Application to an optimal PID tuning. International Journal of Innovative Computing, Information and Control, 6(10), 4719–4729. 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. DOI: 10.1021/ie201655c. Arrieta, O., Vilanova, R., and Visioli, A. (2011). Proportional-Integral-Derivative Tuning for Servo/Regulation Control Operation for Unstable 212
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