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Magnitude Optimum Design of PID Control Loop with Delay
Proceedings of the 12th IFAC Workshop on Time Delay Systems Proceedings of 12th IFAC Workshop on Proceedings of the the 12th IFACMI, Workshop on Time Time Delay Delay Systems Systems June 28-30, 2015. Ann Arbor, USA Proceedings of 12th IFAC Workshop on Delay Proceedings of the the 12th IFACMI, Workshop on Time Time DelayatSystems Systems online www.sciencedirect.com June Ann Arbor, USA June 28-30, 28-30, 2015. 2015. Ann Arbor, MI, USAAvailable June June 28-30, 28-30, 2015. 2015. Ann Ann Arbor, Arbor, MI, MI, USA USA
ScienceDirect IFAC-PapersOnLine 48-12 (2015) 446–451
Magnitude Magnitude Optimum Optimum Design Design of of PID PID Control Control Loop Loop with with Delay Delay Magnitude Optimum Design of PID Control Loop with Delay 1 1 1,2
Vyhlídal Jaromír Fišer Fišer11,, Pavel 1,2 1,2 Jaromír Pavel Zítek Zítek1111,,, Tomáš Tomáš Vyhlídal Tomáš Vyhlídal Jaromír Fišer 1 1,2 1,, Pavel 1,2 Zítek Pavel Zítek , Tomáš Vyhlídal Jaromír Fišer , Pavel Zítek , Tomáš Vyhlídal Jaromír Fišer 1 11Department of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, Department of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, 11Department Czech Technical University in Prague, Technická Str. 4, CZ 166 07 Czech Republic Department of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, Department of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, Czech Technical University in Prague, Technická Str. 4, CZ 166 07 Czech Republic Czech Technical University in Prague, Technická Str. 4, CZ 166 07 Czech Republic (Tel: +420-22435-3953; e-mail:{ jaromir.fiser, pavel.zitek, tomas.vyhlidal} @fs.cvut.cz) Czech Technical University in Prague, Technická Str. 4, CZ 166 07 Czech Republic Czech Technical University in Prague, Technická Str. 4, CZ 166 07 Czech Republic (Tel: +420-22435-3953; e-mail:{ jaromir.fiser, pavel.zitek, tomas.vyhlidal} @fs.cvut.cz) (Tel: (Tel: +420-22435-3953; +420-22435-3953; e-mail:{ e-mail:{ jaromir.fiser, jaromir.fiser, pavel.zitek, pavel.zitek, tomas.vyhlidal} tomas.vyhlidal} @fs.cvut.cz) @fs.cvut.cz) (Tel: +420-22435-3953; e-mail:{ jaromir.fiser, pavel.zitek, tomas.vyhlidal} @fs.cvut.cz) 2 Institute of Informatics, Robotics and Cybernetics (CIIRC), 2 2Czech Czech Institute Institute of of Informatics, Informatics, Robotics Robotics and and Cybernetics Cybernetics (CIIRC), (CIIRC), 2 2Czech Czech Technical University in Prague Czech of Robotics Cybernetics Czech Institute Institute of Informatics, Informatics, Robotics and and Cybernetics (CIIRC), (CIIRC), Czech Technical University in Prague Czech Technical University in Prague Czech Czech Technical Technical University University in in Prague Prague Abstract: Abstract: The The paper paper deals deals with with aaa magnitude-optimum magnitude-optimum based based synthesis synthesis of of PID PID control control loops loops with with respect respect Abstract: The paper deals with magnitude-optimum based synthesis of PID control loops with respect to the disturbance rejection. To obtain as general as possible formulation of the control synthesis the Abstract: The paper deals with a magnitude-optimum based synthesis of PID control loops with Abstract: The paper deals with aobtain magnitude-optimum based synthesis of PID control loopssynthesis with respect respect to the disturbance rejection. To as general as possible formulation of the control the to the disturbance rejection. To obtain as general as possible formulation of the control synthesis the dimensional analysis is applied resulting in the use of similarity numbers of the so-called swingability to the disturbance rejection. To obtain as general as possible formulation of the control synthesis the to the disturbance rejection. To resulting obtain asin general asofpossible formulation ofthe theso-called control synthesis the dimensional analysis is applied the use similarity numbers of swingability dimensional analysis is applied resulting the use similarity the so-called swingability and laggardness laggardness of the the time delay delay plant.in Just this of approach hasnumbers made it itof possible to work work with the the dimensional analysis is applied resulting in the use of similarity numbers of the so-called swingability dimensional analysis is applied resulting in the use of similarity numbers of the so-called swingability and of time plant. Just this approach has made possible to with and laggardness of the time delay plant. Just this approach has made it possible to work with the transcendental form of the optimized magnitude function. From the meromorphic description of the and laggardness of the time delay plant. Just this approach has made it possible to work with and laggardnessform of the timeoptimized delay plant. Just thisfunction. approach has the made it possible description to work with the transcendental of the magnitude From meromorphic of the transcendental form of the optimized magnitude function. the meromorphic description of control loop loop with with delay the PID setting then then can be be evaluated evaluated by means means of Levenberg-Marquardt iteration transcendental form of the thePID optimized magnitude function. From From the of meromorphic descriptioniteration of the the transcendental form of the optimized magnitude function. From the meromorphic description of the control delay setting can by Levenberg-Marquardt control delay the setting can by Levenberg-Marquardt iteration method.loop Thewith obtained PID settings arethen systematically compared fromof the viewpoint of of the the dominant dominant control loop with delay PID the PID PID setting then can be be evaluated evaluated by means means ofthe Levenberg-Marquardt iteration control loop with delay the PID setting then can be evaluated by means of Levenberg-Marquardt iteration method. The obtained settings are systematically compared from viewpoint method. The obtained PID settings are systematically compared from the viewpoint of the dominant modes of the control loop responses. method. The obtained PID settings are systematically compared from the viewpoint of the dominant method. The obtained PID settings are systematically compared from the viewpoint of the dominant modes of the control loop responses. modes modes of of the the control control loop loop responses. responses. modes the control loop responses. © 2015,ofIFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. Alldominant rights reserved. Keywords: Magnitude optimum criterion, dimensional analysis, disturbance rejection, poles. Keywords: Magnitude optimum criterion, dimensional analysis, disturbance rejection, dominant poles. Keywords: Magnitude optimum criterion, dimensional analysis, disturbance rejection, dominant poles. Keywords: Magnitude optimum criterion, dimensional analysis, disturbance rejection, dominant poles. Keywords: Magnitude optimum criterion, dimensional analysis, disturbance rejection, dominant poles.
1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION The magnitude optimum (MO) method of control loop The magnitude optimum (MO) method of control loop The magnitude optimum (MO) method of control loop synthesis emerged as early as in 1954, introduced by R.C. The magnitude optimum (MO) method of control loop The magnitude optimum (MO) method of control loop synthesis emerged as early as in 1954, introduced by R.C. synthesis emerged as early as in 1954, introduced by R.C. Oldenbourg and H. Sartorius. More specific procedure in synthesis emerged as early as in 1954, introduced by R.C. synthesis emerged as early as in 1954, introduced by R.C. Oldenbourg and H. Sartorius. More specific procedure in Oldenbourg and H. Sartorius. More specific procedure in finding the controller setting fulfilling the requirements of Oldenbourg and H. Sartorius. More specific procedure in Oldenbourg and H. Sartorius. More specific procedure in finding the controller setting fulfilling the requirements of finding the controller setting fulfilling the requirements of this design approach was proposed by C. Kessler (1955). The finding the controller setting fulfilling the requirements of finding the controller setting fulfilling the requirements of this design approach was proposed by C. Kessler (1955). The this design approach was by (1955). original aim was to unify various approaches to optimization this design approach was proposed proposed by C. C. Kessler Kessler (1955). The The this design approach was proposed by C. Kessler (1955). The original aim was to unify various approaches to optimization original aim was to unify various approaches to optimization of the feedback control loops. Classical optimization methods original aim was to unify various approaches to optimization original aim was to unify various approaches to optimization of the feedback control loops. Classical optimization methods of the control loops. optimization methods based on finding the minimum extreme of various integral of the feedback feedback control loops. Classical Classical optimization methods of the feedback control loops. Classical optimization methods based on finding the minimum extreme of various integral based on finding the minimum extreme of various integral criteria are of an indirect character – the result of this design based on finding the minimum extreme of various integral based on finding the minimum extreme of various integral criteria are of an indirect character –– the result of this design criteria are of an indirect character the result of this design is fully subject to the chosen criterion and the obtained criteria are of an indirect character – the result of this design criteria are of an indirect character – the result of this design is fully subject to the chosen criterion and the obtained is fully subject to the chosen criterion and the obtained control loop may not always be the most suitable for practical is fully subject to the chosen criterion and the obtained is fully subject to the chosen criterion and the obtained control loop may not always be the most suitable for practical control loop may not always be the most suitable for practical implementation. For instance the quadratic integral control loop always be most suitable for control loop may may not not always be the the most suitableerror for practical practical implementation. For instance the quadratic error integral implementation. For instance the quadratic error integral (ISE) optimum may lead to insufficiently damped oscillating implementation. For instance the quadratic error integral implementation. For instance the quadratic error integral (ISE) optimum may lead to insufficiently damped oscillating (ISE) optimum may lead to insufficiently damped oscillating control process, while the linear optimum criterion with its (ISE) optimum may lead to insufficiently damped oscillating (ISE) optimum may lead to insufficiently damped oscillating control process, while the linear optimum criterion with its control process, while the linear optimum criterion with its constraint to the aperiodic step response results in overcontrol process, while the linear optimum criterion with its control process, while the linear optimum criterion with its constraint to the aperiodic step response results in overconstraint to the aperiodic step response results in overdamped and rather sluggish control behaviour. On the other constraint to the aperiodic step response results in overconstraint to the aperiodic step response results in overdamped and rather sluggish control behaviour. On the other damped and rather sluggish control behaviour. On the some other hand the immediate approaches consist in prescribing damped and rather control behaviour. On other damped and rather sluggish sluggish control behaviour. On the the some other hand the immediate approaches consist in prescribing hand the immediate approaches consist in prescribing some behaviour properties required to be reached by the design. In hand the immediate approaches consist in prescribing some hand the immediate approaches consist in prescribing some behaviour properties required to be reached by the design. In behaviour properties required to be reached by the design. In the synthesis one can prescribe the frequency and damping of behaviour properties required to be reached by the design. In behaviour properties required to be reached by the design. In the synthesis one can prescribe the frequency and damping of the synthesis one can prescribe the frequency and damping of control step response, the phase margin of the open loop the synthesis one can prescribe the frequency and damping of the synthesis one can prescribe the frequency and damping of the control step response, the phase margin of the open loop the control step the phase margin of the loop frequency or to assign the system the controlresponse step response, response, the phase margin ofdominant the open openpoles loop the control step response, the phase margin of the open loop frequency response or to assign the system dominant poles frequency response or to assign the system dominant poles etc. frequency response or to assign the system dominant poles frequency response or to assign the system dominant poles etc. etc. etc. etc. The The magnitude magnitude optimum optimum approach approach is is related related to to both both these these The magnitude optimum approach is related to both these directions of control loop synthesis. As regards the conditions The magnitude optimum approach is related to both these The magnitude optimum approach is related to both these directions of control loop synthesis. As regards the conditions directions of control loop synthesis. As regards the conditions of zero valued derivatives it represents an extreme seeking directions of control loop synthesis. As regards the conditions directions of control loop synthesis. As regards the conditions of zero zero valued valued derivatives derivatives it it represents represents an an extreme extreme seeking seeking of task but on the other hand directly prescribes aa specific of zero valued derivatives it represents an seeking of zero valued derivatives itit represents an extreme extreme seeking task but on the other hand it directly prescribes specific task but on the other hand it directly prescribes a specific dynamic property of the control loop – the flattest possible task but on the other hand it directly prescribes a specific task but on the other hand it directly prescribes a specific dynamic property of the control loop – the flattest possible dynamic property of the control loop –– the flattest possible shape of the frequency response in the dynamic property of loop flattest possible dynamic property of the the control control loop – the thelowest flattestfrequency possible shape of the frequency response in the lowest frequency shape of the frequency response in the lowest frequency band. Essentially this property is considered to be claimed for shape of the frequency response in the lowest frequency shape of the frequency response in the lowest frequency band. Essentially this property is considered to be claimed for band. Essentially this property is considered to be claimed for the reference tracking frequency response for which the band. Essentially this property is considered to be claimed for band. Essentially this property is considered to be claimed for the reference tracking frequency response for which the the reference tracking frequency response for which the lowest frequency limit value is one. In principle this aim the reference tracking frequency response for which the the reference tracking frequency response for which the lowest frequency limit value is one. In principle this aim lowest frequency limit is In this aim brings to some extent tendency to cancel the closed lowest frequency limitaa value value is one. one. In principle principle this loop aim lowest frequency limit value is one. In principle this aim brings to some extent tendency to cancel the closed loop brings to some extent a tendency to cancel the closed loop poles by the controller parameters, but this kind of control brings to some extent a tendency to cancel the closed loop brings to some extent a tendency to cancel the closed loop poles by poles by the the controller controller parameters, parameters, but but this this kind kind of of control control poles poles by by the the controller controller parameters, parameters, but but this this kind kind of of control control
loop tends to result in over-damped control process, loop tends to result in over-damped control process, loop tends to result in over-damped control process, particularly as regards the load disturbance rejection loop tends to result in over-damped control process, loop tends to result in over-damped control process, particularly as regards the load disturbance rejection particularly as regards the load disturbance rejection (Papadopoulos et al., 2012). As to the reference tracking the particularly as regards the load disturbance rejection particularly as regards the load disturbance rejection (Papadopoulos et et al., 2012). As to the reference tracking the (Papadopoulos al., 2012). As to the reference tracking the MO based synthesis provides relatively fast and non(Papadopoulos et al., 2012). As to the reference tracking the (Papadopoulos et al., 2012). As to the reference tracking the MO based synthesis provides relatively fast and nonMO based synthesis provides relatively fast and nonoscillatory responses well satisfying the demands in most MO based synthesis provides relatively fast and nonMO based synthesis provides relatively fast and nonoscillatory responses well satisfying the demands in most oscillatory responses well satisfying the demands in most applications. As the weak points of the synthesis are oscillatory responses well the demands in oscillatory responses well satisfying satisfying the MO demands in most most applications. As the weak points of the MO synthesis are applications. As the weak points of the MO synthesis are usually mentioned the following. The method is demanding applications. As the weak points of the MO synthesis are applications. As thethe weak points The of the MO is synthesis are usually mentioned following. method demanding usually mentioned the following. The method is demanding for the required assessment of a large number of parameters usually mentioned the following. The method is demanding usually mentioned the following. The method is demanding for the required assessment of a large number of parameters for the required assessment of a large number of parameters which its practical implementation rather A for the required assessment of of parameters for the makes required assessment of aa large large number number of difficult. parameters which makes its practical implementation rather difficult. A which makes its practical implementation rather difficult. A general weak point is the primary concentration on the which makes its practical implementation rather difficult. A which makes its practical implementation rather difficult. A general weak point is the primary concentration on the general weak point is the primary concentration on the reference tracking since the load disturbance compensation is general weak point is the primary concentration on the general weak point is the primary concentration on the reference tracking since the load disturbance compensation is reference tracking since the load disturbance compensation is more significant in most industrial process control. Also the reference tracking since the load disturbance compensation is reference tracking since the load disturbance compensation is more significant in most industrial process control. Also the more significant in most industrial process control. Also the sticking to the rational functions as frequency responses is an more significant in most industrial process control. Also the more significant in most industrial process control. Also the sticking to the rational functions as frequency responses is an sticking to the rational functions as frequency responses is an unnecessary constraint. Somehow more than two decades sticking to rational as frequency responses is sticking to the the rational functions functions asfor frequency responses is an an unnecessary constraint. Somehow for more than two decades unnecessary constraint. Somehow for more than two decades the research of MO based synthesis was attenuated. Only unnecessary constraint. Somehow for more than two decades unnecessary constraint. Somehow for more than two decades the research of MO based synthesis was attenuated. Only the research of MO based synthesis was attenuated. Only towards the end century revival of interest of magnitude the research of MO based was attenuated. Only the research of of MO basedaaa synthesis synthesis was attenuated. Only towards the end of century revival of interest of magnitude towards the end of century revival of interest of magnitude optimum principle may be observed (Umland and Safiuddin, towards the end of century a revival of interest of magnitude towards the end of century a revival of interest of magnitude optimum principle may be observed (Umland and Safiuddin, optimum principle may be observed (Umland and 1990), (Vrančić al., 1999, 2001), where either the Padé optimum principleet may be observed (Umland and Safiuddin, Safiuddin, optimum principle may be observed (Umland and Safiuddin, 1990), (Vrančić et al., 1999, 2001), where either the Padé 1990), (Vrančić et al., 1999, 2001), where either the Padé approximation or Taylor series expansion of time delay is 1990), (Vrančić et al., 1999, 2001), where either the Padé 1990), (Vrančić et al., 1999, 2001), where either the Padé approximation or Taylor series expansion of time delay is approximation or Taylor series expansion of time delay is considered. approximation or Taylor series expansion of time delay is approximation or Taylor series expansion of time delay is considered. considered. considered. considered. Just Just the the aim aim to to overcome overcome the the mentioned mentioned disadvantages disadvantages has has Just the aim to overcome the mentioned disadvantages has revived the interest in further development of the MO based Just the aim to overcome the mentioned has Just the the aiminterest to overcome thedevelopment mentioned disadvantages disadvantages has revived in further of the MO based revived the interest in further development of the MO based synthesis. Several attempts have been worked out in order to revived the interest in further development of the MO based revived the interest in further development of the MO based synthesis. Several attempts have been worked out in order to synthesis. Several attempts have been worked out in order to obtain a better performance in the load disturbance rejection synthesis. Several attempts have been worked out in order to synthesis. Several attempts have been worked out in order to obtain a better performance in the load disturbance rejection obtain a better performance in the load disturbance rejection (Vrančić et al., 2004, 2010). But most of these modifications obtain a better performance in the load disturbance rejection obtain a better performance in the load disturbance rejection (Vrančić et al., 2004, 2010). But most of these modifications (Vrančić et al., 2004, But most of modifications keep on unit value of the frequency (Vrančić etthe al.,requirement 2004, 2010). 2010).of But most of these these modifications (Vrančić et al., 2004, 2010). But most of these modifications keep on the requirement of unit value of the frequency keep on the requirement of unit value of the frequency response function to be optimized (e.g. by means of multiple keep on the requirement of unit value of the frequency keep on the requirement of unit value of the frequency response function to be optimized (e.g. by means of multiple response function to be optimized (e.g. by means of multiple integrations) even though it does not fall in with the system response function to be optimized (e.g. by means of multiple response function to be optimized (e.g. by means of multiple integrations) even though it does not fall in with the system integrations) even though it does not fall in with the system behaviour. Also the keeping on the Padé approximation of integrations) even though it does not fall in with the system integrations) even though it does not fall in with the system behaviour. Also the keeping on the Padé approximation of behaviour. Also the keeping on the Padé approximation of the time delay effect is not necessary with respect to behaviour. Also the keeping on the Padé approximation of behaviour. Also the keeping on necessary the Padé approximation of the time delay effect is not with respect to the time delay effect is not necessary with respect to contemporary potentials of the theory of meromorphic respect to necessary with the time delay effect is not the time delaypotentials effect is of notthe necessary withmeromorphic respect to contemporary theory of contemporary potentials of the theory of meromorphic functions. A modification for non-oscillating plants with contemporary potentials of the theory of meromorphic contemporary potentials offor thenon-oscillating theory of meromorphic functions. A modification plants with functions. A for plants delay and with guaranteed stability margin is proposed by functions. A modification modification for non-oscillating non-oscillating plants with with functions. A modification for non-oscillating plants with delay and with guaranteed stability margin is proposed by delay and with guaranteed stability margin is proposed by Cvejn (2013). delay and with guaranteed stability margin is proposed by delay and with guaranteed stability margin is proposed by Cvejn (2013). Cvejn (2013). Cvejn (2013). (2013). Cvejn
2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015, IFAC 2015 446Hosting by Elsevier Ltd. All rights reserved. Copyright IFAC 2015 446 Peer review© of International Federation of Automatic Copyright ©under IFAC responsibility 2015 446Control. Copyright © 446 Copyright © IFAC IFAC 2015 2015 446 10.1016/j.ifacol.2015.09.419
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In the following sections a novel meromorphic modification of the MO design is proposed for optimizing the load disturbance rejection for the time delay plants. To get more generic kind of the results a generalized control loop model based on the dimensional analysis is applied.
447
principle of MO, i.e. to the Parseval theorem providing the equality between a square error integral in time domain and the integral of its square Fourier transform magnitude in the frequency domain. Proposition 1. Suppose an asymptotically stable control loop as in Fig. 1 with disturbance frequency response D ( j ) ( j ) and with the disturbance step response e D (t ) for which a convergent Fourier transform E ( j ) exists
2. DISTURBANCE REJECTION MODIFICATION OF MAGNITUDE OPTIMUM DESIGN Unlike the original formulation of MO based design by Whiteley (1946), Oldenbourg and Sartorius (1954) or Kessler (1955), explicitly tied with the reference tracking, our intent is to optimize the load disturbance rejection in the control loop in Fig. 1. The PID controller transfer function R(s) is considered as
E D ( j )
e D ( t ) exp( j t ) dt .
(5)
If for optimizing the load disturbance rejection the integral square error criterion
2
Q e D ( t ) dt ,
(6)
0
in time domain and the square magnitude integral of the frequency response
Fig. 1 The considered PID control loop R(s)
rP s rI rD s
2
s
~ R (s)
,
(1)
0
~
K a 0 exp( s ) a 0 a1 s s
2
,
D (s)
1 R ( s )G ( s )
limit of D ( j ) /( j )
2
2
.
,
(4a)
lim e D ( t ) lim D ( j ) 0
.
(4b)
2
to zero, the closer to
zero is also the square control error in disturbance rejection. The frequency response integrand function in (7) is real and even, containing exclusively even powers of and harmonic functions of multiples of . To simplify further elaboration of the modified magnitude optimization the notation
(3)
lim e D ( t ) lim D ( j ) 0
integral is non-zero. With respect to
The meaning of the equality Q F is clear: The closer the
For this type of transfer function both the step response and the module of frequency response start from zero for t 0 and 0 respectively. Also for t and both these characteristics vanish to zero. So for the step response and the module it holds
t 0
2
square magnitude response D ( j )
s (1 a 1 s a 2 s ) ( rP s rI rD s ) K exp( s )
(7)
d
the properties (4a) and (4b) for t 0 , and t , as well as for 0 , and , both the improper integrals Q and F are convergent. The equality Q F then results just from the well-known Parseval theorem (Angot, 1952), (Oldenbourg and Sartorius, 1955). █
0
j
2
Proof. Although D ( j ) 0 for 0 the corresponding
(2)
s K exp( s )
t
0
D ( j )
in the frequency domain are evaluated then between these criteria the equality Q F holds.
for both the disturbance and the actuating variable. The grounds for this choice of the plant model are discussed in the next Section. The disturbance response of the controlled variable is then given by the closed loop meromorphic transfer function G (s)
F E D ( j ) d
s
where R ( 0 ) rI 0 . With regard to the ability of PID controller to influence no more than three low order terms of the control loop model assume a stable linear plant described by second order model with delay Zítek et al. (2013b) G (s)
2
D ( j ) j
2
D ( j ) D ( j ) j ( j )
2
K M ( )
(8)
let be introduced. Our aim is to achieve this function as flat as possible, i.e. M ( ) M ( 0 ) , over a largest possible bandwidth. Proposition 2. To select controller R(s) such that the flattest possible shape of M ( ) in the lowest frequency bandwidth is achieved the following conditions are to be satisfied
The classical MO criterion is essentially connected with the tracking performance, i.e. with the unit value of the frequency magnitude in the lowest frequency band. But in the area of disturbance rejection this magnitude is zero and due to this we propose to get to the core of the theoretical 447
IFAC TDS 2015 448 June 28-30, 2015. Ann Arbor, MI, USA
lim
d
2i
2
M ( )
d
0
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i 1, 2 ,...,
0,
2i
,
d P ( )
(9)
d
2
(14)
4
M ( ) M ( 0 )
k 1
d k M ( ) . k k! d 0
d
(10)
Due to the even character of M ( ) the odd derivatives in (10) are zero for 0 . Hence in fact the expansion (10) consists of the even terms with k 2 i , i 1, 2 ,... , only and only these terms are relevant in the low frequency band. Therefore only the even derivatives in (10) can influence the shape of M ( ) and the more of them are zero the more flat
(1 9 s )(1 s )
.
(11)
Find the PI controller parameters providing the flattest possible magnitude optimum performance of the loop in its load disturbance compensation. The closed loop disturbance frequency response function in this case is D ( j )
j j (1 9 j )( 1 j ) (1 rP ) j rI
.
2
a 0 /( a1 ) ,
D ( j ) D ( j ) j ( j )
1 P ( )
1 6
4
4
2
2
2
81 18(1 rP ) 100 (1 rP ) 20 rI rI
2
(15)
a1 .
2
(16)
2
.
(17)
The plants with the same and are referred to as dynamically similar. It means that for a pair of such plants whose u ( t ), d ( t ) are identical their responses y ( t ) are identical as well (e.g. the step responses considered in the common relative time t ). The advantage of model (17) is the reduced number of parameters: instead of four parameters in (2) only three numbers , , K determine the set of dynamically similar plants, i.e. the plants with different a 0 , a 1 , , K but with the same , , K .
remain and where
2
. In applying conditions (9) it is apparent that the requirement of the flattest possible shape of M ( ) , i.e. M ( ) M ( 0 ) , is equivalent with the zero valued even derivatives of P ( ) . With regard to only two controller parameters it is possible to satisfy only two equalities from (9). Hence the PI controller setting making M ( ) as close to M ( 0 ) as possible is given by the following two derivatives M ( 0 ) 1 /( rI )
,
y ( t ) y ( t ) y ( t ) K [ u ( t 1) d ( t 1)]
,
(13) where only even powers of
432 (1 rP ) 2400
The dimensionless and are referred to as the swingability and laggardness numbers of the plant respectively. The static gain K as the third parameter of (2) is considered as joined to the controller gains in the control loop model. The time t is replaced by the ratio t t / and the Laplace operator is substituted by s s . Then applying these variables and parameters to model (2) all the considered plants are described by a common dimensionless model
(12)
The square magnitude of the frequency response function M ( ) then acquires the following form M ( )
2
In order to obtain a broader comparison of the controller tunings obtained by means of the synthesis proposed in Section 2 a sufficiently generic model of the control loop is needful. The model given by (1) and (2) is able to describe particular cases of control loop dynamics from a rather wide class of stable plants free of RHP zero effects. A really generic model can be obtained from the application of dimensional analysis and the rules of physical similarity. In Zítek et al. (2013a) a generic dimensionless model of the control loop is introduced by means of the following similarity numbers
█
Example. To start with a simple case suppose a control loop with PI controller only ( rD 0 ) and with a delayless plant transfer function 1
29160
3. DIMENSIONLESS MODEL OF THE PID CONTROL LOOP
and closer to zero is the frequency response module D ( j )
G (s)
4
By equating their values to zero for 0 the controller setting rP ( 2400 / 432 ) 1 4 . 55 , rI 2 (1 4 . 55 ) 2 / 40 =1.54, is obtained. It is obvious that in case of using the PID controller the zero-valued derivatives could be provided up to the sixth derivative of P ( ) . The example does not contain the time delay, 0 , but in case of 0 it is easy to see that the derivatives and the controller parameters may be evaluated too.
k
in the lowest frequency bandwidth.
2
4
d P ( )
Proof. The square module function may be expanded in the Taylor series about the point 0
2
2430 216(1 rP ) 1200 2 (1 rP ) 40 rI ,
where is equal to the number of the controller adjustable parameters.
2
Also the standard form of the PID controller (1) with the gains r0 , rI , rD is modified in an analogous way by means of introducing the dimensionless parameters
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P Kr P ,
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D K rD / ,
I K rI ,
The MO setting of the PID is given by the solution of this set of equations for P , I , D .
(18)
so that the dimensionless controller model is in the form u ( t ) P e ( t ) I e ( t ) D e ( t )
.
Proof. The numerator of M ( ) in (21) is constant A 2 since exp( j ) exp( j ) 1 and also H ( 0 ) A I is constant and therefore for the zero value of the M ( ) derivatives with respect to it is sufficient to achieve the zero value of the
(19)
After closing the control feedback and applying the dimensionless Laplace transform with s s (corresponding to t ) the following characteristic quasi-polynomial of the control loop is obtained 3
2
2
2
2
even derivatives of H ( j ) P ( ) for 0 . The M value for
2
H ( s ) s s s exp( s )[ P s I D s ] ,
2
3
H ( j )
2
2
A
P ( )
2
2
,
2
lim
0
n
H ( j ) d
n
2
2
2
d
n
0,
A 2 A A A 2
2
,
2
2
2
2
2
2
2
2
(26)
2
2
4
2
2
6
4
5
I ( ) 2 A 2 A D sin A 3
2
2
2
2 A ( I A D ) sin 2 A I D sin 2
2
2
2
2
2
2 A P ( A ) cos 2 A I sin A I sin 2
2 P
2
2
2
2
4
2
2
(27)
2
A cos 2 A P ( D I ) sin cos A D sin .
The denominator function of (21) to be differentiated is the sum P ( ) R 2 ( ) I 2 ( ) . The squares (26) and (27) have 25 terms together but in their summations the number of the P ( ) terms is reduced to 17 by the cancellations and due to the identity cos 2 sin 2 1 . The sum can be considered in the form ,
P ( ) P0 ( ) PS ( ) sin PC ( ) cos
(22)
(28)
6
4
,
(23)
2
2
2
/ 3 0 , 2
I ( 4 A A 0.2 A ) 0 .
2
2
4
2
2
2
2
2
2
2
2
5
3
4
4
3
2
2
2
2
2
The second, fourth and sixth derivatives of P ( ) result in rather long and complicated expressions but only a little part of their terms has a non-zero value for 0 . Just these values with 0 result in the left-hand sides of (23), (24) and (25), given by the second, fourth and sixth P ( ) derivatives, respectively. The equations set (23), (24) are nonlinear only the third (25) is linear.
(24)
12 A 4 A A P
4
PC ( ) 2 A D 2 A P 2 A P 2 A I .
2
I
2
P0 ( ) 2 A A D A A P
PS ( ) 2 A D 2 A I 2 A P 2 A ( D I ) ,
D
12 D 24 A 12 A 4 A
2
A I 2 A D I ,
A (1 P 2 P 2 I 2 D I ) 2 I 0
P 2 A 2 A A
2
A D cos ,
(21)
n 2, 4, 6
2
2
2
3
2
2
2
2 A P ( I D ) sin cos A P sin
which result in the set of equations
2 A D 2 A 2 A
4
where
d P ( )
0
4
3
n
lim
2
2 A P sin 2 A D I cos A I cos
as close to M ( 0 ) , M ( ) M ( 0 ) , as possible, is given by satisfying the following three conditions d
2
R ( ) 2 A D cos 2 A I cos
2
R ( ) I ( )
(24)
and their squares give the expressions
Proposition 3. Consider the PID control loop given by the plant and controller models (17) and (19) respectively, and by the characteristic quasi-polynomial (20). The MO setting of the controller parameter values P , I , D which make the square magnitude function A
2
I ( ) A A D sin A P cos A I sin , (25)
Consider the control loop arisen from feedback of plant (17) and controller (19) with the characteristic quasi-polynomial (20) and let us take up applying the synthesis according to (9). In converting the mentioned models to the frequency response description the dimensionless operator s s is fixed to the imaginary axis i.e. to s j where represents an angle of the phasor rotation within the dead time corresponding to the usual angular frequency. The characteristic quasi-polynomial H ( s ) can then be considered split into the real and imaginary parts, H ( j ) R ( ) jI ( ) . The application of the optimizing conditions results in the following proposition.
. The real and imaginary parts of H ( j ) are
R ( ) A D cos A P sin A I cos ,
4. APPLYING SIMILARITY NUMBERS IN MAGNITUDE OPTIMUM DESIGN
2
2
as follows
specified by only five parameters , , P , I , D .
A
is given only by the integration gain
0
M ( 0 ) (1 / I )
(20)
M ( )
449
(25)
449
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5. ASSESSMENT OF PID MO SETTING
approximate model and the MO setting is directly assigned by the relationships corresponding to the Figures 2, 3, 4.
Due to dimensionless model (17) the plant is dynamically specified by the similarity numbers and only ( A 2 ) while K is included into the controller parameters. The repeated solution of the set (23), (24), (25) was provided by the Levenberg – Marquardt iteration method for various combinations of and . The needed initial estimates of P , I , D were taken from the analogous solution of the dominant pole assignment in Zítek et al. (2013a). Since the plant dynamics is specified only by the similarity numbers and the controller gains P , I , D are evaluated over the relevant area of , , given by the intervals 0 . 1, 2 and 0.5, 3 . The results of the P , I , D computations are displayed in Fig. 2, 3 and 4 respectively. Each point of these graphs represents a set of all similar control loops with common and . The Figures present the magnitude
Fig. 4 MO derivative gains for given and . 5.1 Dominant poles provided by the modified magnitude optimum setting of the PID control loop. The above presented MO setting of PID provides a strongly damped oscillating response in disturbance rejection. It is worthwhile to pay attention to the frequency and the damping of the natural oscillations which are given by the dominant, i.e. the rightmost roots of the characteristic equation H ( s ) 0 , given in (20). Over the whole above considered area of , by means of the toolbox from Vyhlídal and Zítek (2009) the rightmost parts of the control loop spectra of the quasi-polynomial H ( s ) were computed. In agreement with Uchida et al. (1988) and Fišer et al. (2014) these computations confirm that the dominant group of poles is always formed as a double-pair of roots, Fig. 5 s1,2 1 ( 1 j ),
Fig. 2 MO proportional gains for given and .
s 3,4 2 ( 2 j ) ,
(29)
where 1,2 are the frequency angles of natural oscillations and 1, 2 are the respective damping ratios. The imaginary parts of the poles increase (and the real parts decrease with increasing both and . It is remarkable that the influence of two and results in the line displacements of s1, 2 and
Fig. 3 MO integration gains for given and . optimum settings of PID controllers as surfaces over the area of similarity numbers , . Unimodal character of these surfaces is a common feature and in fact for a particular case of a control loop synthesis it is possible only to assess the swingability and laggardness numbers for the process
Fig. 5 Displacement of the dominant poles of the loop for the whole set of and . 450
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margin based on the modulus optimum criterion. Journal of Process Control, Vol. 23, p. 570 – 584. Fišer, J., Zítek, P., and Kučera, V. (2014). IAE optimization of delayed PID control loops using dimensional analysis approach. In: Proc 6th International Symposium on Communications, Control and Signal Processing (ISCCSP), 2014, 262-265, IEEE, Athens. Kessler, C. (1955). Über die Vorausberechnung optimal abgestimmter Regelkreise. Regelungstechnik, 3(2), p. 40 – 49. Lumbar, S., Vrančić, D., and Strmčnik, S. (2008). Comparative study of decay ratios of disturbancerejection magnitude optimum method for PI controllers. ISA transactions, 47(1), 94-100. Oldenbourg, R. C. and Sartorius, H. (1954). A uniform approach to the optimum adjustment of control loops. Transactions of the ASME, 76(11), 1265-1279. Uchida, K., E. Shimemura, T. Kubo, and N. Abe (1988). The linear-quadratic optimal control approach to feedback control design for systems with delay. Automatica, 24, 773–780. Umland, W.J., Safiuddin, M., (1990). Magnitude and symmetric optimum criterion for the design of linear control systems: what is it and how does it compare with the others. IEEE Transactions on Industry Application, Vol. 26, No. 3, p. 489 – 497. Vrančić, D., Peng, Y., and Strmčnik, S. (1999). A new PID controller tuning method based on multiple integrations. Control Engineering Practice, 7(5), 623-633. Vrančić, D., Strmčnik, S., and Juričić, Đ. (2001). A magnitude optimum multiple integration tuning method for filtered PID controller. Automatica, 37(9), 14731479. Vrančić, D., Strmčnik, S., and Kocijan, J. (2004). Improving disturbance rejection of PI controllers by means of the magnitude optimum method. ISA transactions, 43(1), 7384. Vrančić, D., Strmčnik, S., Kocijan, J., and de Moura Oliveira, P. B. (2010). Improving disturbance rejection of PID controllers by means of the magnitude optimum method. ISA transactions, 49(1), 47-56. Vyhlídal, T. and P. Zítek (2009). Mapping Based Algorithm for Large-Scale Computation of Quasi-Polynomial Zeros. IEEE Transactions on Automatic Control, 54(1), 171 – 177. Whiteley, A. L. (1946). Theory of Servo Systems, with particular reference to Stabilization. Journal of the Institution of Electrical Engineers-Part II: Power Engineering, 93(34), 353-367. Zítek, P., J. Fišer, and Vyhlídal, T. (2013a). Dimensional analysis approach to dominant three-pole placement in delayed PID control loops. Journal of Process Control, 23(8), 1063–1074. Zítek, P., Fišer, J., and Vyhlídal, T. (2013b). Dominant three pole placement in PID control loop with delay. In: Proc. 9th Asian Control Conference (ASCC), 2013, 1-6, IEEE, Istanbul.
s 3 , 4 in Fig. 5. The first one from the pairs is decisive for the
natural frequency of the response
Fig. 6 Comparison of the natural frequency angle of the loop (lower surface) with the ultimate frequency angle of the plant (upper surface). oscillation 1 . The second frequency 2 is much lower and scarcely visible in the response. In fact the pair s 3 , 4 rather increases the damping effects in the response. In comparison with the IAE optimum setting of PID the presented MO design is similar only its damping is higher, (Fišer et al., 2014), (Lumbar et al., 2008). It is to notice that frequency 1 is in a close relation to the ultimate frequency k of the plant, namely it is about eighty per cent of k , see Fig. 6. Very uniform is the damping of the disturbance rejection. Throughout the whole area of and the damping ratio is close to 0.4. 6. CONCLUSIONS The above presented MO synthesis is advantageous with its generic approach due to the totally dimensionless control loop description. This approach has made it possible to apply the MO principle to the meromorphic description of the PID control loop with delay. ACKNOWLEDGEMENT This research was supported by the Technology Agency of the Czech Republic under the Competence Centre Programme, Project TE01020197 “Centre for Applied Cybernetics 3”. REFERENCES Angot, A. (1952). Complements des Mathématiques a l´Usage des Ingenieurs de l´Electrotechnique, Masson et Cie, Paris. Åström, K. J., and Hägglund, T. (1995). PID controllers: theory, design, and tuning. 2nd Ed. (Instrument Society of America), Research Triangle Park, North Carolina. Cvejn, J. (2013). The design of PID controller for nonoscillating time-delay plants with guaranteed stability 451
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Magnitude Magnitude Optimum Optimum Design Design of of PID PID Control Control Loop Loop with with Delay Delay Magnitude Optimum Design of PID Control Loop with Delay 1 1 1,2
Vyhlídal Jaromír Fišer Fišer11,, Pavel 1,2 1,2 Jaromír Pavel Zítek Zítek1111,,, Tomáš Tomáš Vyhlídal Tomáš Vyhlídal Jaromír Fišer 1 1,2 1,, Pavel 1,2 Zítek Pavel Zítek , Tomáš Vyhlídal Jaromír Fišer , Pavel Zítek , Tomáš Vyhlídal Jaromír Fišer 1 11Department of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, Department of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, 11Department Czech Technical University in Prague, Technická Str. 4, CZ 166 07 Czech Republic Department of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, Department of Instrumentation and Control Engineering, Faculty of Mechanical Engineering, Czech Technical University in Prague, Technická Str. 4, CZ 166 07 Czech Republic Czech Technical University in Prague, Technická Str. 4, CZ 166 07 Czech Republic (Tel: +420-22435-3953; e-mail:{ jaromir.fiser, pavel.zitek, tomas.vyhlidal} @fs.cvut.cz) Czech Technical University in Prague, Technická Str. 4, CZ 166 07 Czech Republic Czech Technical University in Prague, Technická Str. 4, CZ 166 07 Czech Republic (Tel: +420-22435-3953; e-mail:{ jaromir.fiser, pavel.zitek, tomas.vyhlidal} @fs.cvut.cz) (Tel: (Tel: +420-22435-3953; +420-22435-3953; e-mail:{ e-mail:{ jaromir.fiser, jaromir.fiser, pavel.zitek, pavel.zitek, tomas.vyhlidal} tomas.vyhlidal} @fs.cvut.cz) @fs.cvut.cz) (Tel: +420-22435-3953; e-mail:{ jaromir.fiser, pavel.zitek, tomas.vyhlidal} @fs.cvut.cz) 2 Institute of Informatics, Robotics and Cybernetics (CIIRC), 2 2Czech Czech Institute Institute of of Informatics, Informatics, Robotics Robotics and and Cybernetics Cybernetics (CIIRC), (CIIRC), 2 2Czech Czech Technical University in Prague Czech of Robotics Cybernetics Czech Institute Institute of Informatics, Informatics, Robotics and and Cybernetics (CIIRC), (CIIRC), Czech Technical University in Prague Czech Technical University in Prague Czech Czech Technical Technical University University in in Prague Prague Abstract: Abstract: The The paper paper deals deals with with aaa magnitude-optimum magnitude-optimum based based synthesis synthesis of of PID PID control control loops loops with with respect respect Abstract: The paper deals with magnitude-optimum based synthesis of PID control loops with respect to the disturbance rejection. To obtain as general as possible formulation of the control synthesis the Abstract: The paper deals with a magnitude-optimum based synthesis of PID control loops with Abstract: The paper deals with aobtain magnitude-optimum based synthesis of PID control loopssynthesis with respect respect to the disturbance rejection. To as general as possible formulation of the control the to the disturbance rejection. To obtain as general as possible formulation of the control synthesis the dimensional analysis is applied resulting in the use of similarity numbers of the so-called swingability to the disturbance rejection. To obtain as general as possible formulation of the control synthesis the to the disturbance rejection. To resulting obtain asin general asofpossible formulation ofthe theso-called control synthesis the dimensional analysis is applied the use similarity numbers of swingability dimensional analysis is applied resulting the use similarity the so-called swingability and laggardness laggardness of the the time delay delay plant.in Just this of approach hasnumbers made it itof possible to work work with the the dimensional analysis is applied resulting in the use of similarity numbers of the so-called swingability dimensional analysis is applied resulting in the use of similarity numbers of the so-called swingability and of time plant. Just this approach has made possible to with and laggardness of the time delay plant. Just this approach has made it possible to work with the transcendental form of the optimized magnitude function. From the meromorphic description of the and laggardness of the time delay plant. Just this approach has made it possible to work with and laggardnessform of the timeoptimized delay plant. Just thisfunction. approach has the made it possible description to work with the transcendental of the magnitude From meromorphic of the transcendental form of the optimized magnitude function. the meromorphic description of control loop loop with with delay the PID setting then then can be be evaluated evaluated by means means of Levenberg-Marquardt iteration transcendental form of the thePID optimized magnitude function. From From the of meromorphic descriptioniteration of the the transcendental form of the optimized magnitude function. From the meromorphic description of the control delay setting can by Levenberg-Marquardt control delay the setting can by Levenberg-Marquardt iteration method.loop Thewith obtained PID settings arethen systematically compared fromof the viewpoint of of the the dominant dominant control loop with delay PID the PID PID setting then can be be evaluated evaluated by means means ofthe Levenberg-Marquardt iteration control loop with delay the PID setting then can be evaluated by means of Levenberg-Marquardt iteration method. The obtained settings are systematically compared from viewpoint method. The obtained PID settings are systematically compared from the viewpoint of the dominant modes of the control loop responses. method. The obtained PID settings are systematically compared from the viewpoint of the dominant method. The obtained PID settings are systematically compared from the viewpoint of the dominant modes of the control loop responses. modes modes of of the the control control loop loop responses. responses. modes the control loop responses. © 2015,ofIFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. Alldominant rights reserved. Keywords: Magnitude optimum criterion, dimensional analysis, disturbance rejection, poles. Keywords: Magnitude optimum criterion, dimensional analysis, disturbance rejection, dominant poles. Keywords: Magnitude optimum criterion, dimensional analysis, disturbance rejection, dominant poles. Keywords: Magnitude optimum criterion, dimensional analysis, disturbance rejection, dominant poles. Keywords: Magnitude optimum criterion, dimensional analysis, disturbance rejection, dominant poles.
1. INTRODUCTION 1. INTRODUCTION 1. INTRODUCTION 1. 1. INTRODUCTION INTRODUCTION The magnitude optimum (MO) method of control loop The magnitude optimum (MO) method of control loop The magnitude optimum (MO) method of control loop synthesis emerged as early as in 1954, introduced by R.C. The magnitude optimum (MO) method of control loop The magnitude optimum (MO) method of control loop synthesis emerged as early as in 1954, introduced by R.C. synthesis emerged as early as in 1954, introduced by R.C. Oldenbourg and H. Sartorius. More specific procedure in synthesis emerged as early as in 1954, introduced by R.C. synthesis emerged as early as in 1954, introduced by R.C. Oldenbourg and H. Sartorius. More specific procedure in Oldenbourg and H. Sartorius. More specific procedure in finding the controller setting fulfilling the requirements of Oldenbourg and H. Sartorius. More specific procedure in Oldenbourg and H. Sartorius. More specific procedure in finding the controller setting fulfilling the requirements of finding the controller setting fulfilling the requirements of this design approach was proposed by C. Kessler (1955). The finding the controller setting fulfilling the requirements of finding the controller setting fulfilling the requirements of this design approach was proposed by C. Kessler (1955). The this design approach was by (1955). original aim was to unify various approaches to optimization this design approach was proposed proposed by C. C. Kessler Kessler (1955). The The this design approach was proposed by C. Kessler (1955). The original aim was to unify various approaches to optimization original aim was to unify various approaches to optimization of the feedback control loops. Classical optimization methods original aim was to unify various approaches to optimization original aim was to unify various approaches to optimization of the feedback control loops. Classical optimization methods of the control loops. optimization methods based on finding the minimum extreme of various integral of the feedback feedback control loops. Classical Classical optimization methods of the feedback control loops. Classical optimization methods based on finding the minimum extreme of various integral based on finding the minimum extreme of various integral criteria are of an indirect character – the result of this design based on finding the minimum extreme of various integral based on finding the minimum extreme of various integral criteria are of an indirect character –– the result of this design criteria are of an indirect character the result of this design is fully subject to the chosen criterion and the obtained criteria are of an indirect character – the result of this design criteria are of an indirect character – the result of this design is fully subject to the chosen criterion and the obtained is fully subject to the chosen criterion and the obtained control loop may not always be the most suitable for practical is fully subject to the chosen criterion and the obtained is fully subject to the chosen criterion and the obtained control loop may not always be the most suitable for practical control loop may not always be the most suitable for practical implementation. For instance the quadratic integral control loop always be most suitable for control loop may may not not always be the the most suitableerror for practical practical implementation. For instance the quadratic error integral implementation. For instance the quadratic error integral (ISE) optimum may lead to insufficiently damped oscillating implementation. For instance the quadratic error integral implementation. For instance the quadratic error integral (ISE) optimum may lead to insufficiently damped oscillating (ISE) optimum may lead to insufficiently damped oscillating control process, while the linear optimum criterion with its (ISE) optimum may lead to insufficiently damped oscillating (ISE) optimum may lead to insufficiently damped oscillating control process, while the linear optimum criterion with its control process, while the linear optimum criterion with its constraint to the aperiodic step response results in overcontrol process, while the linear optimum criterion with its control process, while the linear optimum criterion with its constraint to the aperiodic step response results in overconstraint to the aperiodic step response results in overdamped and rather sluggish control behaviour. On the other constraint to the aperiodic step response results in overconstraint to the aperiodic step response results in overdamped and rather sluggish control behaviour. On the other damped and rather sluggish control behaviour. On the some other hand the immediate approaches consist in prescribing damped and rather control behaviour. On other damped and rather sluggish sluggish control behaviour. On the the some other hand the immediate approaches consist in prescribing hand the immediate approaches consist in prescribing some behaviour properties required to be reached by the design. In hand the immediate approaches consist in prescribing some hand the immediate approaches consist in prescribing some behaviour properties required to be reached by the design. In behaviour properties required to be reached by the design. In the synthesis one can prescribe the frequency and damping of behaviour properties required to be reached by the design. In behaviour properties required to be reached by the design. In the synthesis one can prescribe the frequency and damping of the synthesis one can prescribe the frequency and damping of control step response, the phase margin of the open loop the synthesis one can prescribe the frequency and damping of the synthesis one can prescribe the frequency and damping of the control step response, the phase margin of the open loop the control step the phase margin of the loop frequency or to assign the system the controlresponse step response, response, the phase margin ofdominant the open openpoles loop the control step response, the phase margin of the open loop frequency response or to assign the system dominant poles frequency response or to assign the system dominant poles etc. frequency response or to assign the system dominant poles frequency response or to assign the system dominant poles etc. etc. etc. etc. The The magnitude magnitude optimum optimum approach approach is is related related to to both both these these The magnitude optimum approach is related to both these directions of control loop synthesis. As regards the conditions The magnitude optimum approach is related to both these The magnitude optimum approach is related to both these directions of control loop synthesis. As regards the conditions directions of control loop synthesis. As regards the conditions of zero valued derivatives it represents an extreme seeking directions of control loop synthesis. As regards the conditions directions of control loop synthesis. As regards the conditions of zero zero valued valued derivatives derivatives it it represents represents an an extreme extreme seeking seeking of task but on the other hand directly prescribes aa specific of zero valued derivatives it represents an seeking of zero valued derivatives itit represents an extreme extreme seeking task but on the other hand it directly prescribes specific task but on the other hand it directly prescribes a specific dynamic property of the control loop – the flattest possible task but on the other hand it directly prescribes a specific task but on the other hand it directly prescribes a specific dynamic property of the control loop – the flattest possible dynamic property of the control loop –– the flattest possible shape of the frequency response in the dynamic property of loop flattest possible dynamic property of the the control control loop – the thelowest flattestfrequency possible shape of the frequency response in the lowest frequency shape of the frequency response in the lowest frequency band. Essentially this property is considered to be claimed for shape of the frequency response in the lowest frequency shape of the frequency response in the lowest frequency band. Essentially this property is considered to be claimed for band. Essentially this property is considered to be claimed for the reference tracking frequency response for which the band. Essentially this property is considered to be claimed for band. Essentially this property is considered to be claimed for the reference tracking frequency response for which the the reference tracking frequency response for which the lowest frequency limit value is one. In principle this aim the reference tracking frequency response for which the the reference tracking frequency response for which the lowest frequency limit value is one. In principle this aim lowest frequency limit is In this aim brings to some extent tendency to cancel the closed lowest frequency limitaa value value is one. one. In principle principle this loop aim lowest frequency limit value is one. In principle this aim brings to some extent tendency to cancel the closed loop brings to some extent a tendency to cancel the closed loop poles by the controller parameters, but this kind of control brings to some extent a tendency to cancel the closed loop brings to some extent a tendency to cancel the closed loop poles by poles by the the controller controller parameters, parameters, but but this this kind kind of of control control poles poles by by the the controller controller parameters, parameters, but but this this kind kind of of control control
loop tends to result in over-damped control process, loop tends to result in over-damped control process, loop tends to result in over-damped control process, particularly as regards the load disturbance rejection loop tends to result in over-damped control process, loop tends to result in over-damped control process, particularly as regards the load disturbance rejection particularly as regards the load disturbance rejection (Papadopoulos et al., 2012). As to the reference tracking the particularly as regards the load disturbance rejection particularly as regards the load disturbance rejection (Papadopoulos et et al., 2012). As to the reference tracking the (Papadopoulos al., 2012). As to the reference tracking the MO based synthesis provides relatively fast and non(Papadopoulos et al., 2012). As to the reference tracking the (Papadopoulos et al., 2012). As to the reference tracking the MO based synthesis provides relatively fast and nonMO based synthesis provides relatively fast and nonoscillatory responses well satisfying the demands in most MO based synthesis provides relatively fast and nonMO based synthesis provides relatively fast and nonoscillatory responses well satisfying the demands in most oscillatory responses well satisfying the demands in most applications. As the weak points of the synthesis are oscillatory responses well the demands in oscillatory responses well satisfying satisfying the MO demands in most most applications. As the weak points of the MO synthesis are applications. As the weak points of the MO synthesis are usually mentioned the following. The method is demanding applications. As the weak points of the MO synthesis are applications. As thethe weak points The of the MO is synthesis are usually mentioned following. method demanding usually mentioned the following. The method is demanding for the required assessment of a large number of parameters usually mentioned the following. The method is demanding usually mentioned the following. The method is demanding for the required assessment of a large number of parameters for the required assessment of a large number of parameters which its practical implementation rather A for the required assessment of of parameters for the makes required assessment of aa large large number number of difficult. parameters which makes its practical implementation rather difficult. A which makes its practical implementation rather difficult. A general weak point is the primary concentration on the which makes its practical implementation rather difficult. A which makes its practical implementation rather difficult. A general weak point is the primary concentration on the general weak point is the primary concentration on the reference tracking since the load disturbance compensation is general weak point is the primary concentration on the general weak point is the primary concentration on the reference tracking since the load disturbance compensation is reference tracking since the load disturbance compensation is more significant in most industrial process control. Also the reference tracking since the load disturbance compensation is reference tracking since the load disturbance compensation is more significant in most industrial process control. Also the more significant in most industrial process control. Also the sticking to the rational functions as frequency responses is an more significant in most industrial process control. Also the more significant in most industrial process control. Also the sticking to the rational functions as frequency responses is an sticking to the rational functions as frequency responses is an unnecessary constraint. Somehow more than two decades sticking to rational as frequency responses is sticking to the the rational functions functions asfor frequency responses is an an unnecessary constraint. Somehow for more than two decades unnecessary constraint. Somehow for more than two decades the research of MO based synthesis was attenuated. Only unnecessary constraint. Somehow for more than two decades unnecessary constraint. Somehow for more than two decades the research of MO based synthesis was attenuated. Only the research of MO based synthesis was attenuated. Only towards the end century revival of interest of magnitude the research of MO based was attenuated. Only the research of of MO basedaaa synthesis synthesis was attenuated. Only towards the end of century revival of interest of magnitude towards the end of century revival of interest of magnitude optimum principle may be observed (Umland and Safiuddin, towards the end of century a revival of interest of magnitude towards the end of century a revival of interest of magnitude optimum principle may be observed (Umland and Safiuddin, optimum principle may be observed (Umland and 1990), (Vrančić al., 1999, 2001), where either the Padé optimum principleet may be observed (Umland and Safiuddin, Safiuddin, optimum principle may be observed (Umland and Safiuddin, 1990), (Vrančić et al., 1999, 2001), where either the Padé 1990), (Vrančić et al., 1999, 2001), where either the Padé approximation or Taylor series expansion of time delay is 1990), (Vrančić et al., 1999, 2001), where either the Padé 1990), (Vrančić et al., 1999, 2001), where either the Padé approximation or Taylor series expansion of time delay is approximation or Taylor series expansion of time delay is considered. approximation or Taylor series expansion of time delay is approximation or Taylor series expansion of time delay is considered. considered. considered. considered. Just Just the the aim aim to to overcome overcome the the mentioned mentioned disadvantages disadvantages has has Just the aim to overcome the mentioned disadvantages has revived the interest in further development of the MO based Just the aim to overcome the mentioned has Just the the aiminterest to overcome thedevelopment mentioned disadvantages disadvantages has revived in further of the MO based revived the interest in further development of the MO based synthesis. Several attempts have been worked out in order to revived the interest in further development of the MO based revived the interest in further development of the MO based synthesis. Several attempts have been worked out in order to synthesis. Several attempts have been worked out in order to obtain a better performance in the load disturbance rejection synthesis. Several attempts have been worked out in order to synthesis. Several attempts have been worked out in order to obtain a better performance in the load disturbance rejection obtain a better performance in the load disturbance rejection (Vrančić et al., 2004, 2010). But most of these modifications obtain a better performance in the load disturbance rejection obtain a better performance in the load disturbance rejection (Vrančić et al., 2004, 2010). But most of these modifications (Vrančić et al., 2004, But most of modifications keep on unit value of the frequency (Vrančić etthe al.,requirement 2004, 2010). 2010).of But most of these these modifications (Vrančić et al., 2004, 2010). But most of these modifications keep on the requirement of unit value of the frequency keep on the requirement of unit value of the frequency response function to be optimized (e.g. by means of multiple keep on the requirement of unit value of the frequency keep on the requirement of unit value of the frequency response function to be optimized (e.g. by means of multiple response function to be optimized (e.g. by means of multiple integrations) even though it does not fall in with the system response function to be optimized (e.g. by means of multiple response function to be optimized (e.g. by means of multiple integrations) even though it does not fall in with the system integrations) even though it does not fall in with the system behaviour. Also the keeping on the Padé approximation of integrations) even though it does not fall in with the system integrations) even though it does not fall in with the system behaviour. Also the keeping on the Padé approximation of behaviour. Also the keeping on the Padé approximation of the time delay effect is not necessary with respect to behaviour. Also the keeping on the Padé approximation of behaviour. Also the keeping on necessary the Padé approximation of the time delay effect is not with respect to the time delay effect is not necessary with respect to contemporary potentials of the theory of meromorphic respect to necessary with the time delay effect is not the time delaypotentials effect is of notthe necessary withmeromorphic respect to contemporary theory of contemporary potentials of the theory of meromorphic functions. A modification for non-oscillating plants with contemporary potentials of the theory of meromorphic contemporary potentials offor thenon-oscillating theory of meromorphic functions. A modification plants with functions. A for plants delay and with guaranteed stability margin is proposed by functions. A modification modification for non-oscillating non-oscillating plants with with functions. A modification for non-oscillating plants with delay and with guaranteed stability margin is proposed by delay and with guaranteed stability margin is proposed by Cvejn (2013). delay and with guaranteed stability margin is proposed by delay and with guaranteed stability margin is proposed by Cvejn (2013). Cvejn (2013). Cvejn (2013). (2013). Cvejn
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In the following sections a novel meromorphic modification of the MO design is proposed for optimizing the load disturbance rejection for the time delay plants. To get more generic kind of the results a generalized control loop model based on the dimensional analysis is applied.
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principle of MO, i.e. to the Parseval theorem providing the equality between a square error integral in time domain and the integral of its square Fourier transform magnitude in the frequency domain. Proposition 1. Suppose an asymptotically stable control loop as in Fig. 1 with disturbance frequency response D ( j ) ( j ) and with the disturbance step response e D (t ) for which a convergent Fourier transform E ( j ) exists
2. DISTURBANCE REJECTION MODIFICATION OF MAGNITUDE OPTIMUM DESIGN Unlike the original formulation of MO based design by Whiteley (1946), Oldenbourg and Sartorius (1954) or Kessler (1955), explicitly tied with the reference tracking, our intent is to optimize the load disturbance rejection in the control loop in Fig. 1. The PID controller transfer function R(s) is considered as
E D ( j )
e D ( t ) exp( j t ) dt .
(5)
If for optimizing the load disturbance rejection the integral square error criterion
2
Q e D ( t ) dt ,
(6)
0
in time domain and the square magnitude integral of the frequency response
Fig. 1 The considered PID control loop R(s)
rP s rI rD s
2
s
~ R (s)
,
(1)
0
~
K a 0 exp( s ) a 0 a1 s s
2
,
D (s)
1 R ( s )G ( s )
limit of D ( j ) /( j )
2
2
.
,
(4a)
lim e D ( t ) lim D ( j ) 0
.
(4b)
2
to zero, the closer to
zero is also the square control error in disturbance rejection. The frequency response integrand function in (7) is real and even, containing exclusively even powers of and harmonic functions of multiples of . To simplify further elaboration of the modified magnitude optimization the notation
(3)
lim e D ( t ) lim D ( j ) 0
integral is non-zero. With respect to
The meaning of the equality Q F is clear: The closer the
For this type of transfer function both the step response and the module of frequency response start from zero for t 0 and 0 respectively. Also for t and both these characteristics vanish to zero. So for the step response and the module it holds
t 0
2
square magnitude response D ( j )
s (1 a 1 s a 2 s ) ( rP s rI rD s ) K exp( s )
(7)
d
the properties (4a) and (4b) for t 0 , and t , as well as for 0 , and , both the improper integrals Q and F are convergent. The equality Q F then results just from the well-known Parseval theorem (Angot, 1952), (Oldenbourg and Sartorius, 1955). █
0
j
2
Proof. Although D ( j ) 0 for 0 the corresponding
(2)
s K exp( s )
t
0
D ( j )
in the frequency domain are evaluated then between these criteria the equality Q F holds.
for both the disturbance and the actuating variable. The grounds for this choice of the plant model are discussed in the next Section. The disturbance response of the controlled variable is then given by the closed loop meromorphic transfer function G (s)
F E D ( j ) d
s
where R ( 0 ) rI 0 . With regard to the ability of PID controller to influence no more than three low order terms of the control loop model assume a stable linear plant described by second order model with delay Zítek et al. (2013b) G (s)
2
D ( j ) j
2
D ( j ) D ( j ) j ( j )
2
K M ( )
(8)
let be introduced. Our aim is to achieve this function as flat as possible, i.e. M ( ) M ( 0 ) , over a largest possible bandwidth. Proposition 2. To select controller R(s) such that the flattest possible shape of M ( ) in the lowest frequency bandwidth is achieved the following conditions are to be satisfied
The classical MO criterion is essentially connected with the tracking performance, i.e. with the unit value of the frequency magnitude in the lowest frequency band. But in the area of disturbance rejection this magnitude is zero and due to this we propose to get to the core of the theoretical 447
IFAC TDS 2015 448 June 28-30, 2015. Ann Arbor, MI, USA
lim
d
2i
2
M ( )
d
0
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i 1, 2 ,...,
0,
2i
,
d P ( )
(9)
d
2
(14)
4
M ( ) M ( 0 )
k 1
d k M ( ) . k k! d 0
d
(10)
Due to the even character of M ( ) the odd derivatives in (10) are zero for 0 . Hence in fact the expansion (10) consists of the even terms with k 2 i , i 1, 2 ,... , only and only these terms are relevant in the low frequency band. Therefore only the even derivatives in (10) can influence the shape of M ( ) and the more of them are zero the more flat
(1 9 s )(1 s )
.
(11)
Find the PI controller parameters providing the flattest possible magnitude optimum performance of the loop in its load disturbance compensation. The closed loop disturbance frequency response function in this case is D ( j )
j j (1 9 j )( 1 j ) (1 rP ) j rI
.
2
a 0 /( a1 ) ,
D ( j ) D ( j ) j ( j )
1 P ( )
1 6
4
4
2
2
2
81 18(1 rP ) 100 (1 rP ) 20 rI rI
2
(15)
a1 .
2
(16)
2
.
(17)
The plants with the same and are referred to as dynamically similar. It means that for a pair of such plants whose u ( t ), d ( t ) are identical their responses y ( t ) are identical as well (e.g. the step responses considered in the common relative time t ). The advantage of model (17) is the reduced number of parameters: instead of four parameters in (2) only three numbers , , K determine the set of dynamically similar plants, i.e. the plants with different a 0 , a 1 , , K but with the same , , K .
remain and where
2
. In applying conditions (9) it is apparent that the requirement of the flattest possible shape of M ( ) , i.e. M ( ) M ( 0 ) , is equivalent with the zero valued even derivatives of P ( ) . With regard to only two controller parameters it is possible to satisfy only two equalities from (9). Hence the PI controller setting making M ( ) as close to M ( 0 ) as possible is given by the following two derivatives M ( 0 ) 1 /( rI )
,
y ( t ) y ( t ) y ( t ) K [ u ( t 1) d ( t 1)]
,
(13) where only even powers of
432 (1 rP ) 2400
The dimensionless and are referred to as the swingability and laggardness numbers of the plant respectively. The static gain K as the third parameter of (2) is considered as joined to the controller gains in the control loop model. The time t is replaced by the ratio t t / and the Laplace operator is substituted by s s . Then applying these variables and parameters to model (2) all the considered plants are described by a common dimensionless model
(12)
The square magnitude of the frequency response function M ( ) then acquires the following form M ( )
2
In order to obtain a broader comparison of the controller tunings obtained by means of the synthesis proposed in Section 2 a sufficiently generic model of the control loop is needful. The model given by (1) and (2) is able to describe particular cases of control loop dynamics from a rather wide class of stable plants free of RHP zero effects. A really generic model can be obtained from the application of dimensional analysis and the rules of physical similarity. In Zítek et al. (2013a) a generic dimensionless model of the control loop is introduced by means of the following similarity numbers
█
Example. To start with a simple case suppose a control loop with PI controller only ( rD 0 ) and with a delayless plant transfer function 1
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3. DIMENSIONLESS MODEL OF THE PID CONTROL LOOP
and closer to zero is the frequency response module D ( j )
G (s)
4
By equating their values to zero for 0 the controller setting rP ( 2400 / 432 ) 1 4 . 55 , rI 2 (1 4 . 55 ) 2 / 40 =1.54, is obtained. It is obvious that in case of using the PID controller the zero-valued derivatives could be provided up to the sixth derivative of P ( ) . The example does not contain the time delay, 0 , but in case of 0 it is easy to see that the derivatives and the controller parameters may be evaluated too.
k
in the lowest frequency bandwidth.
2
4
d P ( )
Proof. The square module function may be expanded in the Taylor series about the point 0
2
2430 216(1 rP ) 1200 2 (1 rP ) 40 rI ,
where is equal to the number of the controller adjustable parameters.
2
Also the standard form of the PID controller (1) with the gains r0 , rI , rD is modified in an analogous way by means of introducing the dimensionless parameters
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IFAC TDS 2015 June 28-30, 2015. Ann Arbor, MI, USA
P Kr P ,
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D K rD / ,
I K rI ,
The MO setting of the PID is given by the solution of this set of equations for P , I , D .
(18)
so that the dimensionless controller model is in the form u ( t ) P e ( t ) I e ( t ) D e ( t )
.
Proof. The numerator of M ( ) in (21) is constant A 2 since exp( j ) exp( j ) 1 and also H ( 0 ) A I is constant and therefore for the zero value of the M ( ) derivatives with respect to it is sufficient to achieve the zero value of the
(19)
After closing the control feedback and applying the dimensionless Laplace transform with s s (corresponding to t ) the following characteristic quasi-polynomial of the control loop is obtained 3
2
2
2
2
even derivatives of H ( j ) P ( ) for 0 . The M value for
2
H ( s ) s s s exp( s )[ P s I D s ] ,
2
3
H ( j )
2
2
A
P ( )
2
2
,
2
lim
0
n
H ( j ) d
n
2
2
2
d
n
0,
A 2 A A A 2
2
,
2
2
2
2
2
2
2
2
(26)
2
2
4
2
2
6
4
5
I ( ) 2 A 2 A D sin A 3
2
2
2
2 A ( I A D ) sin 2 A I D sin 2
2
2
2
2
2
2 A P ( A ) cos 2 A I sin A I sin 2
2 P
2
2
2
2
4
2
2
(27)
2
A cos 2 A P ( D I ) sin cos A D sin .
The denominator function of (21) to be differentiated is the sum P ( ) R 2 ( ) I 2 ( ) . The squares (26) and (27) have 25 terms together but in their summations the number of the P ( ) terms is reduced to 17 by the cancellations and due to the identity cos 2 sin 2 1 . The sum can be considered in the form ,
P ( ) P0 ( ) PS ( ) sin PC ( ) cos
(22)
(28)
6
4
,
(23)
2
2
2
/ 3 0 , 2
I ( 4 A A 0.2 A ) 0 .
2
2
4
2
2
2
2
2
2
2
2
5
3
4
4
3
2
2
2
2
2
The second, fourth and sixth derivatives of P ( ) result in rather long and complicated expressions but only a little part of their terms has a non-zero value for 0 . Just these values with 0 result in the left-hand sides of (23), (24) and (25), given by the second, fourth and sixth P ( ) derivatives, respectively. The equations set (23), (24) are nonlinear only the third (25) is linear.
(24)
12 A 4 A A P
4
PC ( ) 2 A D 2 A P 2 A P 2 A I .
2
I
2
P0 ( ) 2 A A D A A P
PS ( ) 2 A D 2 A I 2 A P 2 A ( D I ) ,
D
12 D 24 A 12 A 4 A
2
A I 2 A D I ,
A (1 P 2 P 2 I 2 D I ) 2 I 0
P 2 A 2 A A
2
A D cos ,
(21)
n 2, 4, 6
2
2
2
3
2
2
2
2 A P ( I D ) sin cos A P sin
which result in the set of equations
2 A D 2 A 2 A
4
where
d P ( )
0
4
3
n
lim
2
2 A P sin 2 A D I cos A I cos
as close to M ( 0 ) , M ( ) M ( 0 ) , as possible, is given by satisfying the following three conditions d
2
R ( ) 2 A D cos 2 A I cos
2
R ( ) I ( )
(24)
and their squares give the expressions
Proposition 3. Consider the PID control loop given by the plant and controller models (17) and (19) respectively, and by the characteristic quasi-polynomial (20). The MO setting of the controller parameter values P , I , D which make the square magnitude function A
2
I ( ) A A D sin A P cos A I sin , (25)
Consider the control loop arisen from feedback of plant (17) and controller (19) with the characteristic quasi-polynomial (20) and let us take up applying the synthesis according to (9). In converting the mentioned models to the frequency response description the dimensionless operator s s is fixed to the imaginary axis i.e. to s j where represents an angle of the phasor rotation within the dead time corresponding to the usual angular frequency. The characteristic quasi-polynomial H ( s ) can then be considered split into the real and imaginary parts, H ( j ) R ( ) jI ( ) . The application of the optimizing conditions results in the following proposition.
. The real and imaginary parts of H ( j ) are
R ( ) A D cos A P sin A I cos ,
4. APPLYING SIMILARITY NUMBERS IN MAGNITUDE OPTIMUM DESIGN
2
2
as follows
specified by only five parameters , , P , I , D .
A
is given only by the integration gain
0
M ( 0 ) (1 / I )
(20)
M ( )
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(25)
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5. ASSESSMENT OF PID MO SETTING
approximate model and the MO setting is directly assigned by the relationships corresponding to the Figures 2, 3, 4.
Due to dimensionless model (17) the plant is dynamically specified by the similarity numbers and only ( A 2 ) while K is included into the controller parameters. The repeated solution of the set (23), (24), (25) was provided by the Levenberg – Marquardt iteration method for various combinations of and . The needed initial estimates of P , I , D were taken from the analogous solution of the dominant pole assignment in Zítek et al. (2013a). Since the plant dynamics is specified only by the similarity numbers and the controller gains P , I , D are evaluated over the relevant area of , , given by the intervals 0 . 1, 2 and 0.5, 3 . The results of the P , I , D computations are displayed in Fig. 2, 3 and 4 respectively. Each point of these graphs represents a set of all similar control loops with common and . The Figures present the magnitude
Fig. 4 MO derivative gains for given and . 5.1 Dominant poles provided by the modified magnitude optimum setting of the PID control loop. The above presented MO setting of PID provides a strongly damped oscillating response in disturbance rejection. It is worthwhile to pay attention to the frequency and the damping of the natural oscillations which are given by the dominant, i.e. the rightmost roots of the characteristic equation H ( s ) 0 , given in (20). Over the whole above considered area of , by means of the toolbox from Vyhlídal and Zítek (2009) the rightmost parts of the control loop spectra of the quasi-polynomial H ( s ) were computed. In agreement with Uchida et al. (1988) and Fišer et al. (2014) these computations confirm that the dominant group of poles is always formed as a double-pair of roots, Fig. 5 s1,2 1 ( 1 j ),
Fig. 2 MO proportional gains for given and .
s 3,4 2 ( 2 j ) ,
(29)
where 1,2 are the frequency angles of natural oscillations and 1, 2 are the respective damping ratios. The imaginary parts of the poles increase (and the real parts decrease with increasing both and . It is remarkable that the influence of two and results in the line displacements of s1, 2 and
Fig. 3 MO integration gains for given and . optimum settings of PID controllers as surfaces over the area of similarity numbers , . Unimodal character of these surfaces is a common feature and in fact for a particular case of a control loop synthesis it is possible only to assess the swingability and laggardness numbers for the process
Fig. 5 Displacement of the dominant poles of the loop for the whole set of and . 450
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margin based on the modulus optimum criterion. Journal of Process Control, Vol. 23, p. 570 – 584. Fišer, J., Zítek, P., and Kučera, V. (2014). IAE optimization of delayed PID control loops using dimensional analysis approach. In: Proc 6th International Symposium on Communications, Control and Signal Processing (ISCCSP), 2014, 262-265, IEEE, Athens. Kessler, C. (1955). Über die Vorausberechnung optimal abgestimmter Regelkreise. Regelungstechnik, 3(2), p. 40 – 49. Lumbar, S., Vrančić, D., and Strmčnik, S. (2008). Comparative study of decay ratios of disturbancerejection magnitude optimum method for PI controllers. ISA transactions, 47(1), 94-100. Oldenbourg, R. C. and Sartorius, H. (1954). A uniform approach to the optimum adjustment of control loops. Transactions of the ASME, 76(11), 1265-1279. Uchida, K., E. Shimemura, T. Kubo, and N. Abe (1988). The linear-quadratic optimal control approach to feedback control design for systems with delay. Automatica, 24, 773–780. Umland, W.J., Safiuddin, M., (1990). Magnitude and symmetric optimum criterion for the design of linear control systems: what is it and how does it compare with the others. IEEE Transactions on Industry Application, Vol. 26, No. 3, p. 489 – 497. Vrančić, D., Peng, Y., and Strmčnik, S. (1999). A new PID controller tuning method based on multiple integrations. Control Engineering Practice, 7(5), 623-633. Vrančić, D., Strmčnik, S., and Juričić, Đ. (2001). A magnitude optimum multiple integration tuning method for filtered PID controller. Automatica, 37(9), 14731479. Vrančić, D., Strmčnik, S., and Kocijan, J. (2004). Improving disturbance rejection of PI controllers by means of the magnitude optimum method. ISA transactions, 43(1), 7384. Vrančić, D., Strmčnik, S., Kocijan, J., and de Moura Oliveira, P. B. (2010). Improving disturbance rejection of PID controllers by means of the magnitude optimum method. ISA transactions, 49(1), 47-56. Vyhlídal, T. and P. Zítek (2009). Mapping Based Algorithm for Large-Scale Computation of Quasi-Polynomial Zeros. IEEE Transactions on Automatic Control, 54(1), 171 – 177. Whiteley, A. L. (1946). Theory of Servo Systems, with particular reference to Stabilization. Journal of the Institution of Electrical Engineers-Part II: Power Engineering, 93(34), 353-367. Zítek, P., J. Fišer, and Vyhlídal, T. (2013a). Dimensional analysis approach to dominant three-pole placement in delayed PID control loops. Journal of Process Control, 23(8), 1063–1074. Zítek, P., Fišer, J., and Vyhlídal, T. (2013b). Dominant three pole placement in PID control loop with delay. In: Proc. 9th Asian Control Conference (ASCC), 2013, 1-6, IEEE, Istanbul.
s 3 , 4 in Fig. 5. The first one from the pairs is decisive for the
natural frequency of the response
Fig. 6 Comparison of the natural frequency angle of the loop (lower surface) with the ultimate frequency angle of the plant (upper surface). oscillation 1 . The second frequency 2 is much lower and scarcely visible in the response. In fact the pair s 3 , 4 rather increases the damping effects in the response. In comparison with the IAE optimum setting of PID the presented MO design is similar only its damping is higher, (Fišer et al., 2014), (Lumbar et al., 2008). It is to notice that frequency 1 is in a close relation to the ultimate frequency k of the plant, namely it is about eighty per cent of k , see Fig. 6. Very uniform is the damping of the disturbance rejection. Throughout the whole area of and the damping ratio is close to 0.4. 6. CONCLUSIONS The above presented MO synthesis is advantageous with its generic approach due to the totally dimensionless control loop description. This approach has made it possible to apply the MO principle to the meromorphic description of the PID control loop with delay. ACKNOWLEDGEMENT This research was supported by the Technology Agency of the Czech Republic under the Competence Centre Programme, Project TE01020197 “Centre for Applied Cybernetics 3”. REFERENCES Angot, A. (1952). Complements des Mathématiques a l´Usage des Ingenieurs de l´Electrotechnique, Masson et Cie, Paris. Åström, K. J., and Hägglund, T. (1995). PID controllers: theory, design, and tuning. 2nd Ed. (Instrument Society of America), Research Triangle Park, North Carolina. Cvejn, J. (2013). The design of PID controller for nonoscillating time-delay plants with guaranteed stability 451