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Advanced Tuning for Ziegler-Nichols Plants
Proceedings of 20th The International Federation of Congress Automatic Control Proceedings of the the 20th World World Congress The Federation of Automatic Control Proceedings of the 20th9-14, World Toulouse, France, July 2017 The International International Federation of Congress Automatic Control Available online at www.sciencedirect.com Toulouse, France, July The International of Automatic Control Toulouse, France,Federation July 9-14, 9-14, 2017 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 1805–1810 Advanced Tuning for Ziegler-Nichols Advanced Tuning for Ziegler-Nichols Plants Advanced Tuning for Ziegler-Nichols Plants Plants ∗ ∗ ∗∗
Iv´ an D. D´ıaz-Rodr´ıguez ∗ Sangjin Han ∗ L.H. Keel ∗∗ ∗ Sangjin Han ∗ L.H. Keel ∗∗ ∗ Iv´ a D´ıaz-Rodr´ Iv´ an n D. D. D´ ıaz-Rodr´ ıguez Sangjin Han L.H. Keel ∗∗ S.P.ıguez Bhattacharyya ∗ ∗ ∗ ∗ Iv´ an D. D´ıaz-Rodr´ ıguez Sangjin Han L.H. Keel S.P. S.P. Bhattacharyya Bhattacharyya ∗ S.P. Bhattacharyya ∗ Department of Electrical and Computer Engineering, Texas A&M ∗ ∗ Department of and Computer Engineering, Department of Electrical Electrical and Station, Computer Engineering, Texas A&M A&M University, College TX, 77840 USATexas ∗ Department of Electrical and Computer Engineering, University, College Station, TX, 77840 USA University, College Station, TX, 77840 USATexas A&M (e-mail: ivan diaz 09,sangjin.han,[email protected]). University, College TX,Engineering, 77840 USA Tennessee ∗∗ (e-mail: diaz 09,sangjin.han,[email protected]). (e-mail: ivan diaz 09,sangjin.han,[email protected]). Department ofivan Electrical andStation, Computer ∗∗ ∗∗ (e-mail: ivan diaz 09,sangjin.han,[email protected]). Department of Electrical and Computer Engineering, Department of Electrical and Computer Engineering, Tennessee State University, Nashville, TN, 37209 USA Tennessee ∗∗ Department Electrical and Computer State Nashville, TN, 37209 StateofUniversity, University, Nashville, TN,Engineering, 37209 USA USA Tennessee (e-mail: [email protected]). State University, Nashville, TN, 37209 USA (e-mail: (e-mail: [email protected]). [email protected]). (e-mail: [email protected]). Abstract: In this paper, we describe an advanced tuning approach for the design of PI Abstract: an for of Abstract: In In this this paper, paper, we describe an advanced advanced tuning tuning approach approach for the thefordesign design of PI PI (Proportional-Integral) and we PIDdescribe (Proportional-Integral-Derivative) controllers the ZieglerAbstract: In that this ispaper, we an advanced tuningcontinuous-time approach for LTI thefor design of PI (Proportional-Integral) and PID (Proportional-Integral-Derivative) controllers the (Proportional-Integral) andOder PIDdescribe (Proportional-Integral-Derivative) controllers for(Linear the ZieglerZieglerNichols plants, First Plus Time-Delay (FOPTD) Time (Proportional-Integral) and PID (Proportional-Integral-Derivative) controllers for the ZieglerNichols plants, that is First Oder Plus Time-Delay (FOPTD) continuous-time LTI (Linear Time Nichols plants, that The is First Oder Plus continuous-time LTI Invariant) systems. objective is toTime-Delay provide to (FOPTD) the designer an efficient tool to(Linear design Time PI or Nichols plants, that is First Plus continuous-time LTI Invariant) systems. The is provide to the an tool to design PI or Invariant) systems. The objective is to to provide to (FOPTD) the designer designer an efficient efficientspecifications tool to(Linear designofTime PI or PID controllers where itobjective is Oder possible toTime-Delay select simultaneous performance gain Invariant) systems. The objective is to provide to the designer an efficient tool to design PI or PID controllers where it is possible to select simultaneous performance specifications of gain PID controllers where it is possible to select simultaneous performance specifications of gain margin, phase margin, and gain crossover frequency from a set of achievable performance design PID controllers where and it is possible to select performance ofdesign margin, phase margin, gain crossover frequency aa set of performance margin, phase margin, and gain crossover frequency from set of achievable achievable performance design curves for Ziegler-Nichols plants. To succeed insimultaneous this,from we first construct thespecifications stabilizing set ofgain PI margin, phase margin, andplants. gain crossover frequency a setconstruct of achievable performance curves Ziegler-Nichols To succeed in we first stabilizing set of curves for Ziegler-Nichols plants. To succeed in this, this,from we plant. first construct the stabilizing setdesign of PI PI or PIDfor controllers corresponding to the Ziegler-Nichols Next, wethe generate an achievable curves for Ziegler-Nichols plants. To succeed in this, we first construct the stabilizing set PI or PID controllers corresponding to the Ziegler-Nichols plant. Next, we generate an achievable or PID controllers corresponding to curves the Ziegler-Nichols plant. Next, we generate an achievable performance set displayed as design in the gain and phase margin plane, indexed byofgain or PID controllers corresponding to the Ziegler-Nichols plant. Next, we generate an achievable performance set displayed as design curves in the gain and phase margin plane, indexed by gain performance set displayed as design curves in the gain and phase margin plane, indexed by gain crossover frequencies. Each point in this achievable performance plot represents a prescribed performance set displayed design curves in theobtained gain andbyphase margin plane,aaindexed by gain crossover frequencies. Each point in achievable performance plot prescribed crossoverphase frequencies. Each point in this this achievable performance plot represents prescribed gain margin, margin, andas crossover frequency, a PI or represents PID controller contained in crossover frequencies. Each point in this achievable performance plot represents a prescribed margin, phase margin, and crossover frequency, obtained by a PI or PID controller contained in margin, phase margin, andbycrossover obtained by a PI orperformance PID controller in the stabilizing set. Then, selectingfrequency, a point from the achievable set, contained a uniquegain PI margin, phase margin, and crossover frequency, obtained by a PI or PID controller contained in the stabilizing set. Then, by selecting a point from the achievable performance set, a unique PI thePID stabilizing set.achieving Then, bythese selecting a point from the achievable performance set, a unique PI or controller simultaneous specifications is found from the intersection of an the stabilizing set. Then, by selecting a point from the achievable performance set, a unique PI or PID controller achieving these simultaneous specifications is found from the intersection of an or PID controller achieving these simultaneous specifications is found from the intersection of an ellipse and straight line parametrized from constant magnitude and constant phase loci in the or PIDofand controller achieving these simultaneous specifications found the intersection an ellipse straight line from magnitude and constant phase loci ellipse and straight line parametrized parametrized from constant constant magnitude and from constant phaseapproach. loci in inofthe the space controller gains. We present illustrative examples toisvalidate the proposed ellipse and straight line parametrized from constant magnitude and constant phase loci in the space of controller gains. We present illustrative examples to validate the proposed approach. space of controller gains. We present illustrative examples to validate the proposed approach. space ofIFAC controller gains. We presentofillustrative examples to validate the Ltd. proposed approach. © 2017, (International Federation Automatic Control) Hosting by Elsevier All rights reserved. Keywords: Linear systems, Time-Delay, PI/PID Controllers, Stability limits, Computer aided Keywords: Linear systems, Time-Delay, Time-Delay, PI/PID PI/PID Controllers, Controllers, Stability Stability limits, limits, Computer Computer aided aided Keywords: Linear systems, control system design Keywords: Linear systems, Time-Delay, PI/PID Controllers, Stability limits, Computer aided control system system design control design control system design 1. INTRODUCTION stabilizing set. The PI or PID structure has been used in 1. INTRODUCTION INTRODUCTION stabilizing set. The Thesuch PI or orasPID PID structure has been used in in 1. stabilizing set. PI structure used many applications in Shafai and has Saif been (2015). 1. INTRODUCTION stabilizing set. Thesuch PI oras structure used in many applications applications such asPID in Shafai Shafai and has Saif been (2015). many in and Saif (2015). Many control systems inevitably possess some time delay Unlike classical tuning whichand giveSaif limited number many applications suchmethods as in Shafai (2015). Manytocontrol control systems inevitably inevitably possess some time time delay delay Many systems possess some Unlike classical tuning methods which which give limited number due communication, transmission, transportation, or of Unlike classical tuning methods give limited number sets of stabilizing parameters, we now have all stabilizing Many control systems inevitably possess some time delay due to to effects communication, transmission, transportation, or Unlike classical tuning methods which limited number due communication, transmission, transportation, or of sets sets of of stabilizing parameters, we nowgive have all stabilizing stabilizing inertia (Malek-Zavarei and Jamshidi (1987)). This of stabilizing parameters, we now have all controllers due to the results in (Silva et al (2002); Bhatdue to effects communication, transmission, or of inertia effects (Malek-Zavarei andsuch Jamshidi (1987)). This sets of stabilizing parameters, we now have all stabilizing inertia and Jamshidi (1987)). This controllers due to the results in (Silva et al (2002); Bhatphenomenon is(Malek-Zavarei found in systems astransportation, electrical, chemicontrollerset due to the results in (Silvathe et al (2002); Bhattacharyya al (2009)) who extended Hermite-Biehler inertia effectsis andsuch Jamshidi (1987)). This phenomenon is(Malek-Zavarei found in in systems systems such aslines, electrical, chemicontrollers due to the results in (Silva et al (2002); Bhatphenomenon found as electrical, chemitacharyyafor et al (2009)) who extended the Hermite-Biehler cal, hydraulic, pneumatic, transmission robotics, and theorem tacharyya et al (2009)) who extended the Hermite-Biehler a quasipolynomials. phenomenon ispneumatic, found(Gu in systems such as electrical, chemical, hydraulic, hydraulic, pneumatic, transmission lines, robotics, and tacharyya etaaalquasipolynomials. (2009)) who extended the Hermite-Biehler cal, lines, robotics, and theorem for for quasipolynomials. industrial processes ettransmission al (2003); Leon et al (2008)). theorem cal, hydraulic, pneumatic, lines, et robotics, and theorem for 2. industrial processes (Gu cases ettransmission al is(2003); (2003); Leon al (2008)). (2008)). a quasipolynomials. industrial (Gu et al Leon et al One of theprocesses most studied the Ziegler-Nichols plant, PROBLEM FORMULATION industrial processes (Gu et al (2003); Leon et al (2008)). One of the most studied cases is the Ziegler-Nichols plant, 2. PROBLEM PROBLEM FORMULATION FORMULATION One of the most studied cases is the Ziegler-Nichols plant, 2. FOPTD (Ziegler et al (1942)) because many of the indusOne of the mostare studied cases is the plant, Let us consider 2. PROBLEM FOPTD (Ziegler et approximated al (1942)) (1942)) because many of of thesystem. indusFOPTD (Ziegler et al because many of the indusan FOPTDFORMULATION continuous-time LTI system trial processes byZiegler-Nichols this type FOPTD (Ziegler et approximated al (1942)) many ofmargin thesystem. indusk Let us ustransfer consider an FOPTD FOPTD continuous-time LTIksystem system trial problem processes are approximated byand this type of system. Let consider an continuous-time LTI trial processes by this type of function P (s) := e−Ls where is the The of are achieving exactbecause gain phase be- with T s+1 k −Ls k −Ls Let ustransfer consider an FOPTD continuous-time LTIk system trial processes are approximated byand thisphase type of system. := with transfer function P (s) e where is the The problem problem ofbecause achieving exact gain and phase margin be- with := function P (s) e where k is the the The achieving gain becomes difficultof of exact the non-linearity andmargin solvability T s+1 steady state gain, T is the timeT s+1 constant, and L is k −Ls with transfer function (s) := e where is the the The problem achieving gain phase be- steady comes difficult because of exact the non-linearity andmargin solvability state gain, T the time constant, L comes difficult because of the non-linearity and solvability conditions of ofthe problem (Tan et and al (2012)). The consteady stateThis gain, T is isPis the timeT s+1 constant, and Lk is is the time-delay. system considered with aand controller in comes difficult because of the non-linearity and solvability conditions of the problem (Tan et al (2012)). The consteady state gain, T is the time constant, and L is the time-delay. This system is considered with a controller in conditions of focused the problem (Tan al (2012)). con- atime-delay. This system considered with controller in troller design on gain andetphase margin The has been unity feedback system is configuration as in aFig. 1. conditions of the problem (Tan et al (2012)). The controller design focused on gain and phase margin has been time-delay. This system is considered with a controller in a unity feedback system configuration as in Fig. 1. ˚ troller design focused on gain and phase margin has been studied since (Astr¨ om and H¨ agglund (1984)) resulting in a unity feedback system configuration as in Fig. 1. ˚ troller design focused on gain and phase margin has been studied since ( A str¨ o m and H¨ a gglund (1984)) resulting in a unity feedback system configuration as in Fig. 1. ˚ studied since ( A str¨ o m and H¨ a gglund (1984)) resulting in ˚ different new approaches. For example, see Astr¨om and C(s) P (s) ˚str¨ studied since (approaches. A om and For H¨ agglund (1984)) in ˚ different new approaches. For example, see Aresulting str¨ m and and ˚ different example, A str¨ ooHo m C(s) P H¨ agglundnew (2001); Srivastava and Pandit see (2016); + C(s) P (s) (s) ˚str¨oHo different new approaches. example, see A m and H¨ gglund (2001); Srivastava and Pandit (2016); Ho + C(s) P (s) H¨ aagglund (2001); Srivastava Pandit (2016); and + − Wang (2003). In this paper, For weand present an advanced tuning − H¨ agglund (2001); Srivastava and Pandit (2016); Ho and + − Wang (2003). In this paper,ofwe wePI present an advanced advanced tuning Wang (2003). In paper, present an tuning approach for thethis design and PID controllers for − Wang (2003). In this paper, we present an advanced tuning approach for the design of PI and PID controllers for approach for the design of PI and PID controllers for desired simultaneous gain margin, phase margin, and gain approach for the design of PI and PID controllers for desired simultaneous simultaneous gain margin, margin, phase margin,achievable and gain gain desired gain margin, and crossover frequency selected from aphase constructed 1. FOPTD unity feedback block diagram desired simultaneous gain margin, margin, and gain Fig. crossover frequency selected from constructed achievable crossover frequency from aaphase constructed Gain-Phase margin selected set. The controller gains areachievable retrieved Fig. 1. Fig. 1. FOPTD FOPTD unity unity feedback feedback block block diagram diagram crossover frequency from aconstant constructed Gain-Phase margin selected set. Thesimple controller gainsgain areachievable retrieved the PI unity and PID controllers, Gain-Phase margin set. The controller gains are retrieved Fig. 1. FOPTD feedback block diagram from the stabilizing set by and con- Consider Consider the PI and PID controllers, Gain-Phase margin set. Thesimple controller gains are retrieved the from the the stabilizing set by by simple constant gain and conconfrom stabilizing set constant and KPPI s +and KI PID controllers, KD s2 + KP s + KI stant phase loci conveniently represented asgain an ellipse or Consider the PI and PID and controllers, C (s) := D s22 + KP s + KI . CP I (s) := from stabilizing set line by simple constant andinconK ss s+ K stant the phase loci conveniently represented asgain an ellipse ellipse or Consider P K KD s + K + KII and CP ID (s) := K stant phase loci conveniently as an or P ellipsoid and a straight orrepresented plane superimposed the s P s + KI . C (s) := and CP CP P ID ID (s) := KD s2 + K PI I (s) := KP s s+ KI stant phase loci conveniently represented as an ellipse or ellipsoid and a straight line or plane superimposed in the ss P s + KI . ellipsoid and a straight line or plane superimposed in the s and CP ID (s) := . CP I (s) := ellipsoid and a straight line or plane superimposed in the s s
Copyright © 2017 IFAC 1841 2405-8963 © 2017 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 1841 Copyright © © 2017 IFAC IFAC 1841 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 1841Control. 10.1016/j.ifacol.2017.08.168
Proceedings of the 20th IFAC World Congress 1806 Iván D. Díaz-Rodríguez et al. / IFAC PapersOnLine 50-1 (2017) 1805–1810 Toulouse, France, July 9-14, 2017
The characteristic equations of the closed loop systems are δP I (s) := ke−Ls (KP s + KI ) + (1 + T s)s (1) δP ID (s) := ke−Ls (KD s2 + KP s + KI ) + (1 + T s)s. (2)
(6) Find the root zn from T (9) kKP + cos(z) − z sin(z) = 0 L that is δi (z) = 0. The roots can be found graphically +cos(z) by taking the intersection of kKPsin(z) and TL z. (7) Compute an := a(zn ). (8) If cos(zn ) > 0 go to the next step. If not, n = n + 2 and go to step 6. (9) Determine the lower and upper bounds on KI from max {an } < KI < 0 (10)
The controller design problems to be considered are the following: (1) Construction of the achievable Gain-Phase margin design curves indexed by gain crossover frequencies. (2) Design a PI or PID controller based on desired simultaneous values of gain margin (GM), phase margin (PM), and gain crossover frequency (ωg ), by selecting the intersection point of an ellipse and a straight line superimposed over the stabilizing set.
n:odd
(10) Go to step 5. PID Controller Design for Stable FOPTD Systems. For stable first order systems, we have T > 0, k > 0, and L > 0. The procedure to compute the PID stabilizing set is the following:
This is described next. 3. DESIGN METHODOLOGY The PI or PID Controller design approach developed here for FOPTD can be summarized as follows. A. B. C. D.
Computation of the PI or PID Stabilizing set. Parametrization of a constant gain and phase loci. Construction the gain-phase margin design curves. Selection of simultaneous design specification from the achievable performance set. E. Retrieval of the PI or PID controller gains satisfying the design specifications.
3.1 Computation of the Stabilizing Set Here, we present a summary of the steps to compute the stabilizing sets. For illustration, we consider an unstable system with PI controller and stable system with PID controller (see Silva et al (2002) for more details). PI Controller Design for Unstable FOPTD Systems. For unstable first order systems, we have T < 0, k > 0, and L > 0. The procedure to compute the stabilizing set is the following: (1) For L = 0, calculate the characteristic equation (1). For stability, it is required KP < − k1 , KI < 0. (2) For L > 0, calculate the characteristic equation (1). Considering δ ∗ (s) = eLs δ(s) we have (3) δ ∗ (s) = k(KI + KP s) + s(T s + 1)eLs ∗ (3) Calculate δ (jω) = δr (ω) + jδi (ω), where (4) δr (ω) = kKI − ω sin(Lω) − T ω 2 cos(Lω) (5) δi (ω) = ω kKP + cos(Lω) − T ω sin(Lω) (4) Let z = Lω and calculate the real and imaginary parts of δ ∗ (jω) δr (z) = k KI − a(z) (6) T z kKP + cos(z) − z sin(z) (7) δi (z) = L L z where a(z) = kL sin(z) + TL z cos(z) (5) Pick a value for KP and set n = 1 in the range 1 L2 T (8) α21 + 2 < KP < − kL T k where α1 is the solution of the equation tan(α) = − TL α in the interval (0, π2 ).
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(1) For L = 0, calculate the characteristic equation (2). For stability, it is required 1 T KP > − , KI > 0 and KD > − (11) k k (2) For L > 0, calculate the characteristic equation (2). Considering δ ∗ (s) = eLs δ(s) we have δ ∗ (s) = k(KI + KP s + KD s2 ) + s(T s + 1)eLs (12) (3) Calculate δ ∗ (jω) = δr (ω) + jδi (jω) where δr (ω) = kKI − kKD ω 2 − ω sin(Lω) − T ω 2 cos(Lω) δi (ω) = ω kKP + cos(Lω) − T ω sin(Lω) (4) Let z = Lω and calculate the real and imaginary parts of δ ∗ (jω) T kKD 2 1 z − z sin(z) − 2 z 2 cos(z) δr (z) = kKI − 2 L L L T z kKP + cos(z) − z sin(z) δi (z) = L L (5) Pick a value for KP in the range 1 T 1 α1 sin(α1 ) − cos(α1 ) (13) − < KP < k k L T where α1 is the solution of tan(α) = − T +L α in the interval (0, π). (6) Find the roots z1 and z2 from T (14) kKP + cos(z) − z sin(z) = 0 L that is δi (z) = 0, ∀z > 0. The roots can be found graphically for the following cases: • − k1 < KP < k1 : in this case, take the intersection +cos(z) of the functions kKPsin(z) and TL z. • KP = k1 : in this case, take the intersection of the T functions kKP + cos(z) and L z sin(z). 1 k
T T < KP < kL L α1 sin(α1 ) − cos(α1 ) , in this case take the intersection of the functions kKP +cos(z) and TL z. sin(z) (7) Compute the parameters mn := m(zn ) and bn := b(zn ) for n = 1, 2 where L2 T L sin(zn ) + zn cos(zn ) . m(zn ) = 2 , b(zn ) = − zn kzn L
•
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Iván D. Díaz-Rodríguez et al. / IFAC PapersOnLine 50-1 (2017) 1805–1810
(8) Determine the stabilizing region in the (KI , KD ) space using (a) − k1 < KP < k1 , KD = m1 KI + b1 , (b) KP = k1 , KD = m1 KI + b1 ,(c) k1 < KP <
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We have ∠P (ejωg ) = ∠ e−jLωg P0 (ejωg ) = −Lωg + ∠P0 (jωg ).
So, (24) becomes Φg := π + φ∗g + Lωg − ∠P0 (jωg ). Thus, for the PID case for equation (21) is a straight line in (KI , KD ) plane for a constant value of KP represented in equation (21). For the PI case, from (17) we obtain the ellipse and straight lines corresponding to M = Mg ∗ ∗ 3.2 Constant gain and constant phase loci for PI Controllers and Φ = Φg , giving the design point (KP , KI ). If these intersection points lie in the stabilizing set, denoted by S, Since FOPTD plant has single time delay, we can write the design is feasible, otherwise the specifications have to P (s) = e−Ls P0 (s), where L > 0 is the time-delay and be revised. P0 (s) is a delay-free plant. The frequency response of the plant and controller are e−Ljω P0 (jω), C(jω) respectively 3.4 Computation of the achievable performance Gain-Phase margin design curves where ω ∈ [0, ∞]. Let T kL
T L α1
sin(α1 ) − cos(α1 ) , KD = m1 KI + b1 , KD = m2 K I + b 2 . (9) Go to step 5.
KI2 ω2 −KI Φ := ∠C(jω) = arctan ωKP
M 2 := |C(jω)|2 = KP2 +
(15) (16)
The gain-phase margin design curves represent the actual performance in terms of GM, PM, and ωg for our FOPTD system achievable with a PI or PID controller. The procedure to construct these design curves is the following:
I with the design parameters, where C(jω) = KP (jω)+K jω KP and KI . Equations (15) and (16) can be rewritten as
+ − + (1) Set a test range of φ∗g ∈ [φ− g , φg ] and ωg ∈ [ωg , ωg ]. ∗ (2) For fixed values of φg and ωg , plot an ellipse and a straight line following the description in subsections 3.2 and 3.3. (3) If the intersection point of the ellipse and straight line lies outside of the stabilizing set, then this point is rejected and go to step (2). (4) If the intersection of the ellipse and straight line is contained in the stabilizing set, it represents the design point with the PI or PID controller gains ∗ (KP∗ , KI∗ ) or (KP∗ , KI∗ , KD ) that satisfies the fixed φ∗g and ωg . (5) Given the selected PI or PID controller gains (KP∗ , KI∗ ) ∗ or (KP∗ , KI∗ , KD ), the upper and lower GM of the system are given by K lb K ub GMupper = P∗ and GMlower = P∗ (25) KP KP
(KI )2 (KP )2 + = 1 and KI = cKP (17) a2 b2 where a2 = M 2 , b2 = M 2 ω 2 , and c = ω tan Φ. Clearly (17) is an ellipse and a straight line in KP , KI space. The major and minor axes of the ellipse are given by the square roots of a and b. The slope of the line is c. 3.3 Constant gain and constant phase loci for PID Controllers Now, let 2 KI 2 2 := |C(jω)| = KP + KD ω − M ω KI KD ω − ω −1 Φ := ∠C(jω) = tan KP 2
(18)
where KPub and KPlb are the controller gains at the upper and lower boundary respectively of the stabilizing set following the straight line ray intersecting the ellipse. (6) Go to step 2 and repeat for all values of φ∗g and ωg in the ranges.
(19)
2
P (jω)+KI where C(jω) = KD (jω) +K and KP , KI , and KD jω are the design parameters. From (18) and (19) we have that 2 2 KD ω − KωI K I . (20) = KP2 = M 2 − KD ω − ω tan2 Φ From (20) we can have the following expressions
KI = KD ω 2 ± ω
M 2 tan2 Φ 1 + tan2 Φ
and
KP = ±
M2 (21) 1 + tan2 Φ
Suppose ωg is the prescribed closed-loop gain crossover frequency. Then 1 1 Mg := = −Ljωg . (22) |P (ejωg )| |e ||P0 (jωg )| By the identity ∀x ∈ R, |e−jx | = | cos x − j sin x| = cos2 x + sin2 x = 1, (23)
1 (22) becomes Mg := |P0 (jω . Suppose φ∗g is the desired g )| phase margin in radians, Φg := π + φ∗g − ∠P (ejωg ). (24)
3.5 Selecting an achievable GM, PM, and ωg and retrieving the Controller gains In the last stage of the design process, the designer can select a desired point from the achievable performance Gain-Phase margin set and retrieve the controller gains corresponding to that simultaneous specification of desired GM, PM, and ωg . The controller gain retrieval process is the following. (1) Select desired GM, PM, and ωg from the achievable gain-margin set. (2) For the specified point, construct the ellipse and straight line by using the selected PM and ωg in the constant gain and constant phase loci. (3) Take the intersection of the ellipse and straight line contained in the stabilizing set. This will provide the ∗ gains (KP∗ , KI∗ ) or (KP∗ , KI∗ , KD ).
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Proceedings of the 20th IFAC World Congress 1808 Iván D. Díaz-Rodríguez et al. / IFAC PapersOnLine 50-1 (2017) 1805–1810 Toulouse, France, July 9-14, 2017
(4) The controller that satisfies the prescribed margin K ∗ s+K ∗ specifications is CP I (s) = P s I or CP ID (s) =
4
∗ 2 ∗ KD s +KP s+KI∗ . s
2
4. ILLUSTRATIVE EXAMPLES
0
KP
In this section, we present two numerical examples to illustrate the proposed approach for the design of PI and PID controllers satisfying a desired simultaneous GM, PM, and ωg from a gain-phase achievable performance set.
Computation of the stabilizing set. The characteristic equation is given by δ(s) = (−12s + 1)s + 5(KP s + KI )e−0.5s . (27) By (3), δ ∗ (s) = e0.5s (−12s + 1)s + (KP s + KI )5. For L = 0, we have δ(s) = −12s2 + (5KP + 1)s + 5KI .
(28)
P
-4
I
(K* = -3.2276, K * = -1.3373) P
4.1 Example 1: PI Controller Design for Unstable FOPTD System Let us consider an unstable continuous-time FOPTD system 5 e−0.5s (26) P (s) = −12s + 1 and the PI controller, CP I . We proceed to apply the procedure presented in the methodology.
(Kub = -0.2540, K ub = -0.1052)
-2
I
-6 (Klb = -6.6177, K lb = -2.7419) P
-5
I
0
5
KI
Fig. 2. Stabilizing set in yellow for PI controller design in Example 1, intersection of an ellipse and a straight line (dot in black), and the controller gains (KPlb , KIlb ) and (KPub , KIub ) at the lower and upper boundary points in the stabilizing set (dots in magenta). rad/sec. However, for this value, we get a lower GM of 0.6404. We notice that for a bigger GM from the achievable Gain-Phase margin set, we get lower PM. The blue dots represent the specification points corresponding to a PM of 30o . The designer has the liberty to choose values for GM, PM, and ωg that best suits his design needs.
(29)
− 51 ,
KI < 0. For L > 0 For stability, it is required KP < and by (4) and (5), δ ∗ (jω) = δr (ω) + jδi (jω) (30) where δr (ω) = 5KI − ω sin(0.5ω) + 12ω 2 cos(0.5ω) (31) (32) δi (ω) = ω 5KP + cos(0.5ω) + 12ω sin(0.5ω) . By (8), we can calculate the range fo KP for stability 1 (33) −4.8 α21 + 0.0017 < KP < − 5 Following all the steps we get the stabilizing set in Fig. 2 Construction of the Achievable Gain-Phase margin design curves. For the construction of the achievable GainPhase margin set in this example, the evaluated range of ωg is [0.1, 3] and the range for PM is from 0o to 70o . The calculation of the GM for each case is done by (25). Using the ellipse and straight line intersection points, we can construct the achievable Gain-Phase margin set presented in Fig 3. Selection of simultaneous desired GM, PM, and ωg specifications from the achievable gain-phase margin design curves. In Fig 3, we can see the achievable Gain-Phase margin set of curves indexed by fixed ωg∗ in different colors. Notice that the curves above the 100 GM represent the upper GM and the curves below 100 GM represent the lower GM. We notice that the maximum PM that we can get is 66o for a ωg = 0.4 rad/sec with a value of upper GM of 7.547 and lower GM of 0.2045. Another example of the values of GM and PM that we can get is the point with a PM of 47o with an upper GM of 23.72 and a ωg = 0.1
10
2
w g = 0.1
Ki: 47 Kd: 23.72
10
Ki: 61 Kd: 14.24
g
w g = 0.4 w g = 0.6
1
Ki: 66 Kd: 7.547
Ki: 30 Kd: 2.05
GM
w = 0.2
g
w =1 g
w g = 1.2
Ki: 47 Kd: 0.6404
10 0
w = 0.8
Ki: 61 Kd: 0.3851
w = 1.4 g
w = 1.6 g
w g = 1.8 10
wg = 2
Ki: 30 Kd: 0.0787
-1
Ki: 66 Kd: 0.2045 w g = 2.2 w = 2.4 g
w g = 2.6 10 -2
w g = 2.8 0
10
20
30
40
50
60
70
w =3 g
PM
Fig. 3. Achievable performance in terms of GM, PM, and ωg for PI Controller design in Example 1. The Blue dots represent the intersections of ellipses and straight lines with a PM of 30o. Retrieval of the PI controller gains corresponding to a selected desired point in the achievable performance set. After the selection of simultaneous GM, PM, and ωg from the achievable Gain-Phase margin set, the designer can retrieve the controller gains corresponding to the point. For illustration purposes, let us say that the desired performance values chosen for this example are a PM of 30o , GM = 2.05, and a ωg = 1.4 rad/s from Fig 3. Then, taking these values for the constant gain and constant phase loci presented in the methodology, we can find the intersection of an ellipse and a straight line shown in Fig 1. The controller gains are KP∗ = −3.2276
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and KI∗ = −1.3373. In Fig 4 we can see the Nyquist plot for the controller gains selected. Here, we can see that those controller gains satisfy the desired performance specifications, P M = 30o , GM = 2.05 (6.23 dB). Nyquist Diagram
1.5 1 System: untitled1 Gain Margin (dB): 6.23 At frequency (rad/s): 2.79 Closed loop stable? Yes
Imaginary Axis
0.5 0 -0.5
Fig. 5. Intersection of a cylinder and a plane superimposed in the PID stabilizing set and a PID design point for Example 2.
System: untitled1 Phase Margin (deg): 30 Delay Margin (sec): 0.374 At frequency (rad/s): 1.4 Closed loop stable? Yes
-1 -1.5 -2 -3
-2
-1
0
1
Real Axis
Fig. 4. Nyquist plot for KP∗ = −3.2276 and KI∗ = −1.3373 in the PI controller design in Example 1. 4.2 Example 2: PID Controller Design for Stable FOPTD System Let us consider an stable continuous-time FOPTD system 1 e−2s P (s) = (34) 2s + 1 and the PID controller CP ID (s). We proceed to apply the procedure presented in the methodology. Computation of the stabilizing set. equation (2) is given by
get a cylinder and a plane in the (KP , KI , KD ) 3D space, respectively. The cylinder and the plane, superimposed in the stabilizing set (see Fig. 5) will have two intersection line segments in the (KI , KD ) plane. The specific value where the intersection occurs can be obtained using (21). Equation (21) will give us two values for KP , but only one is contained in the stabilizing set. The intersection line segment in the (KP , KI , KD ) represents the PID controller gains that satisfy the PM and ωg . Evaluating the range of PM and ωg , we can construct the achievable GainPhase margin set represented in 3D in Fig. 6. If we take a fixed value of ωg = 0.2 rad/sec, we can see the achievable performance in 2D in Fig 7. Here we can see that the maximum GM we can get is 8.95 with a PM of 57o .
The characteristic
δP ID (s) = (2s + 1)s + (KD s2 + KP s + KI )e−2s and by (12)
(35)
(36) δ ∗ (s) = e2s (2s + 1)s + (KD s2 + KP s + KI ) For L = 0 we have δ(s) = (KD + 2)s2 + (KP + 1)s + KI (37) For stability, it is required KP > −1, KI > 0, KD > −2 (38) For L > 0 δ ∗ (jω) = δr (ω) + jδi (jω) (39) where δr (ω) = KI − KD ω 2 − ω sin(2ω) − 2ω 2 cos(2ω) (40) (41) δi (ω) = ω KP + cos(2ω) − 2ω sin(2ω) By (13), we can calculate the range fo KP for stability −1 < KP < α1 sin(α1 ) − cos(α1 ) (42) Following all the steps, we get the stabilizing set in Fig. 5 Construction of the Achievable Gain-Phase margin design curves. For the construction of the achievable GainPhase margin set for the PID controller design case, the evaluated range of ωg is [0.1, 1.3] and the range for PM is from 1o to 120o. For the PID case, using the constant gain and constant phase loci equations, (18) and (19) we now
Fig. 6. Achievable performance in terms of GM, PM, and ωg for PID controller design in Example 2. Selection of simultaneous desired GM, PM, and ωg specification from the achievable Gain-Phase margin design curves. In Fig. 6, we can see the achievable Gain-Phase margin set of curves indexed by fixed ωg in different colors. Notice that we can get more GM and PM for lower values of ωg . For example, for ωg = 0.1 rad/sec, the maximum GM that we can get is 20 with a PM of 72o . For ωg = 0.2 rad/sec, the maximum GM is 8.95 with a P M = 57o . For a bigger value of ωg , we get lower values for GM and PM.
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Proceedings of the 20th IFAC World Congress 1810 Iván D. Díaz-Rodríguez et al. / IFAC PapersOnLine 50-1 (2017) 1805–1810 Toulouse, France, July 9-14, 2017
8 7 6
wg: 0.2 PM: 57 GM: 8.95
w g = 0.2, PM = 57, GM = 8.95
5
GM
4 3
2
0
20
40
60
80
100
120
PM
Fig. 7. Achievable gain-phase margin set for ωg = 0.2 rad/sec for PID controller design in Example 2. For example, for ωg = 1.3 rad/sec we get a maximum GM = 1.012 and P M = 19o . The designer has the liberty to choose values for GM, PM, and ωg that best suits his design needs. Retrieval of the PID controller gains corresponding to a selected point in the achievable performance set. After the selection of simultaneous GM, PM, and ωg from the achievable gain-phase margin set, the designer can retrieve the controller gains corresponding to the point. For illustration purposes, let us say that the desired performance values chosen for this example are a PM of 57o , GM of 8.95, and a ωg of 0.2 rad/s (see Fig 6.) Then, taking these values and the constant gain and constant phase loci for PID controllers presented in the methodology, we can find the intersection of the cylinder and the plane in the (KP , KI , KD ) 3D space shown in Fig 5. The controller gains are KP∗ = 0.2188, KI∗ = ∗ 0.2189, and KD = 0.2. In Fig 8 we can see the Nyquist plot for the controller gains selected. Here, we can see that those controller gains satisfy the desired performance specifications, P M = 57o , GM = 8.95 (19 dB). Nyquist Diagram 1
Imaginary Axis
0.5
System: untitled1 Gain Margin (dB): 19 At frequency (rad/s): 0.893 Closed loop stable? Yes
0 System: untitled1 Phase Margin (deg): 57 Delay Margin (sec): 4.98 At frequency (rad/s): 0.2 Closed loop stable? Yes
-0.5
-1
-1
-0.8
-0.6
-0.4
-0.2
0
Real Axis
Fig. 8. Nyquist plot for KP∗ = 0.2188, KI∗ = 0.2189, and ∗ KD = 0.2 in the PID controller design in Example 2. 5. CONCLUSIONS In this paper, we have presented an advanced tuning approach for PI and PID controllers for Ziegler-Nichols
plants, namely FOPTD systems. It was shown how PI and PID controllers can be designed to satisfy simultaneously desired GM, PM, and ωg by selecting the specifications from an achievable Gain-Phase margin set. First, the stabilizing set was computed using recent results for FOPTD. Then, we showed a graphical approach for the parametrization of constant gain and constant phase loci respectively by ellipses and straight lines. We then constructed an achievable Gain-Phase margin set indexed by ωg . After that, given a selected desired specification from the achievable performance set, retrieval of the PI or PID controller was presented as the intersection of an ellipse and a straight line superimposed on the stabilizing set. In the end, two illustrative examples were given to show the proposed approach. For future research, this approach will be extended to multivariable systems. REFERENCES ˚ Astr¨om, Karl Johan and H¨agglund, Tore. Automatic tuning of simple regulators with specifications on phase and amplitude margins. Automatica. 20:5, 645–651, 1984. ˚ Astr¨om, Karl Johan and H¨agglund, Tore. The future of PID control. Control engineering practice. 9:11, 1163– 1175, 2001. Bhattacharyya, S.P., Datta, A., and Keel, L.H. Linear Control Theory: Structure, Robustness, and Optimization. CRC Press, 2009. Gu, Keqin, Chen, Jie and Kharitonov, Vladimir L. Stability of time-delay systems. Springer Science & Business Media, 2003. Ho, Ming-Tzu and Wang, Hsing-Sen. PID controller design with guaranteed gain and phase margins. Asian Journal of Control. 5:3, 374–381, 2003. Leon de la Barra, B.A., Jin, Lihua, Kim, Y.C., and Mossberg, M. Identification of First-Order Time-Delay Systems using Two Different Pulse Inputs Proceedings of the 17th World Congress. Seoul, Korea, July 6-11, 2008. Malek-Zavarei, Manu and Jamshidi, Mohammad. Timedelay systems : analysis, optimization, and applications. North-Holland systems and control series: v. 9, New York, 1987. Shafai, B and Saif, M. Proportional-integral Observer in Robust Control. Fault Detection, and Decentralized Control of Dynamic Systems. Control and Systems Engineering, 13–43, Springer International Publishing, 2015. Silva, Guillermo J and Datta, Aniruddha and Bhattacharyya, Shankar P. New results on the synthesis of PID controllers. IEEE transactions on automatic control. 47:2, 241–252, 2002. Srivastava, Saurabh and Pandit, V.S. A PI/PID controller for time delay systems with desired closed loop time response and guaranteed gain and phase margins. Journal of Process Control. 37, 70–77, 2016. Tan, Kok K and Wang, Qing-Guo, and Hang, Chang C. Advances in PID control. Springer Science & Business Media, 2012. Ziegler, John, G, and Nichols, Nathaniel B. Optimum settings for automatic controllers. Trans. ASME, 64:11, 1942.
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IFAC PapersOnLine 50-1 (2017) 1805–1810 Advanced Tuning for Ziegler-Nichols Advanced Tuning for Ziegler-Nichols Plants Advanced Tuning for Ziegler-Nichols Plants Plants ∗ ∗ ∗∗
Iv´ an D. D´ıaz-Rodr´ıguez ∗ Sangjin Han ∗ L.H. Keel ∗∗ ∗ Sangjin Han ∗ L.H. Keel ∗∗ ∗ Iv´ a D´ıaz-Rodr´ Iv´ an n D. D. D´ ıaz-Rodr´ ıguez Sangjin Han L.H. Keel ∗∗ S.P.ıguez Bhattacharyya ∗ ∗ ∗ ∗ Iv´ an D. D´ıaz-Rodr´ ıguez Sangjin Han L.H. Keel S.P. S.P. Bhattacharyya Bhattacharyya ∗ S.P. Bhattacharyya ∗ Department of Electrical and Computer Engineering, Texas A&M ∗ ∗ Department of and Computer Engineering, Department of Electrical Electrical and Station, Computer Engineering, Texas A&M A&M University, College TX, 77840 USATexas ∗ Department of Electrical and Computer Engineering, University, College Station, TX, 77840 USA University, College Station, TX, 77840 USATexas A&M (e-mail: ivan diaz 09,sangjin.han,[email protected]). University, College TX,Engineering, 77840 USA Tennessee ∗∗ (e-mail: diaz 09,sangjin.han,[email protected]). (e-mail: ivan diaz 09,sangjin.han,[email protected]). Department ofivan Electrical andStation, Computer ∗∗ ∗∗ (e-mail: ivan diaz 09,sangjin.han,[email protected]). Department of Electrical and Computer Engineering, Department of Electrical and Computer Engineering, Tennessee State University, Nashville, TN, 37209 USA Tennessee ∗∗ Department Electrical and Computer State Nashville, TN, 37209 StateofUniversity, University, Nashville, TN,Engineering, 37209 USA USA Tennessee (e-mail: [email protected]). State University, Nashville, TN, 37209 USA (e-mail: (e-mail: [email protected]). [email protected]). (e-mail: [email protected]). Abstract: In this paper, we describe an advanced tuning approach for the design of PI Abstract: an for of Abstract: In In this this paper, paper, we describe an advanced advanced tuning tuning approach approach for the thefordesign design of PI PI (Proportional-Integral) and we PIDdescribe (Proportional-Integral-Derivative) controllers the ZieglerAbstract: In that this ispaper, we an advanced tuningcontinuous-time approach for LTI thefor design of PI (Proportional-Integral) and PID (Proportional-Integral-Derivative) controllers the (Proportional-Integral) andOder PIDdescribe (Proportional-Integral-Derivative) controllers for(Linear the ZieglerZieglerNichols plants, First Plus Time-Delay (FOPTD) Time (Proportional-Integral) and PID (Proportional-Integral-Derivative) controllers for the ZieglerNichols plants, that is First Oder Plus Time-Delay (FOPTD) continuous-time LTI (Linear Time Nichols plants, that The is First Oder Plus continuous-time LTI Invariant) systems. objective is toTime-Delay provide to (FOPTD) the designer an efficient tool to(Linear design Time PI or Nichols plants, that is First Plus continuous-time LTI Invariant) systems. The is provide to the an tool to design PI or Invariant) systems. The objective is to to provide to (FOPTD) the designer designer an efficient efficientspecifications tool to(Linear designofTime PI or PID controllers where itobjective is Oder possible toTime-Delay select simultaneous performance gain Invariant) systems. The objective is to provide to the designer an efficient tool to design PI or PID controllers where it is possible to select simultaneous performance specifications of gain PID controllers where it is possible to select simultaneous performance specifications of gain margin, phase margin, and gain crossover frequency from a set of achievable performance design PID controllers where and it is possible to select performance ofdesign margin, phase margin, gain crossover frequency aa set of performance margin, phase margin, and gain crossover frequency from set of achievable achievable performance design curves for Ziegler-Nichols plants. To succeed insimultaneous this,from we first construct thespecifications stabilizing set ofgain PI margin, phase margin, andplants. gain crossover frequency a setconstruct of achievable performance curves Ziegler-Nichols To succeed in we first stabilizing set of curves for Ziegler-Nichols plants. To succeed in this, this,from we plant. first construct the stabilizing setdesign of PI PI or PIDfor controllers corresponding to the Ziegler-Nichols Next, wethe generate an achievable curves for Ziegler-Nichols plants. To succeed in this, we first construct the stabilizing set PI or PID controllers corresponding to the Ziegler-Nichols plant. Next, we generate an achievable or PID controllers corresponding to curves the Ziegler-Nichols plant. Next, we generate an achievable performance set displayed as design in the gain and phase margin plane, indexed byofgain or PID controllers corresponding to the Ziegler-Nichols plant. Next, we generate an achievable performance set displayed as design curves in the gain and phase margin plane, indexed by gain performance set displayed as design curves in the gain and phase margin plane, indexed by gain crossover frequencies. Each point in this achievable performance plot represents a prescribed performance set displayed design curves in theobtained gain andbyphase margin plane,aaindexed by gain crossover frequencies. Each point in achievable performance plot prescribed crossoverphase frequencies. Each point in this this achievable performance plot represents prescribed gain margin, margin, andas crossover frequency, a PI or represents PID controller contained in crossover frequencies. Each point in this achievable performance plot represents a prescribed margin, phase margin, and crossover frequency, obtained by a PI or PID controller contained in margin, phase margin, andbycrossover obtained by a PI orperformance PID controller in the stabilizing set. Then, selectingfrequency, a point from the achievable set, contained a uniquegain PI margin, phase margin, and crossover frequency, obtained by a PI or PID controller contained in the stabilizing set. Then, by selecting a point from the achievable performance set, a unique PI thePID stabilizing set.achieving Then, bythese selecting a point from the achievable performance set, a unique PI or controller simultaneous specifications is found from the intersection of an the stabilizing set. Then, by selecting a point from the achievable performance set, a unique PI or PID controller achieving these simultaneous specifications is found from the intersection of an or PID controller achieving these simultaneous specifications is found from the intersection of an ellipse and straight line parametrized from constant magnitude and constant phase loci in the or PIDofand controller achieving these simultaneous specifications found the intersection an ellipse straight line from magnitude and constant phase loci ellipse and straight line parametrized parametrized from constant constant magnitude and from constant phaseapproach. loci in inofthe the space controller gains. We present illustrative examples toisvalidate the proposed ellipse and straight line parametrized from constant magnitude and constant phase loci in the space of controller gains. We present illustrative examples to validate the proposed approach. space of controller gains. We present illustrative examples to validate the proposed approach. space ofIFAC controller gains. We presentofillustrative examples to validate the Ltd. proposed approach. © 2017, (International Federation Automatic Control) Hosting by Elsevier All rights reserved. Keywords: Linear systems, Time-Delay, PI/PID Controllers, Stability limits, Computer aided Keywords: Linear systems, Time-Delay, Time-Delay, PI/PID PI/PID Controllers, Controllers, Stability Stability limits, limits, Computer Computer aided aided Keywords: Linear systems, control system design Keywords: Linear systems, Time-Delay, PI/PID Controllers, Stability limits, Computer aided control system system design control design control system design 1. INTRODUCTION stabilizing set. The PI or PID structure has been used in 1. INTRODUCTION INTRODUCTION stabilizing set. The Thesuch PI or orasPID PID structure has been used in in 1. stabilizing set. PI structure used many applications in Shafai and has Saif been (2015). 1. INTRODUCTION stabilizing set. Thesuch PI oras structure used in many applications applications such asPID in Shafai Shafai and has Saif been (2015). many in and Saif (2015). Many control systems inevitably possess some time delay Unlike classical tuning whichand giveSaif limited number many applications suchmethods as in Shafai (2015). Manytocontrol control systems inevitably inevitably possess some time time delay delay Many systems possess some Unlike classical tuning methods which which give limited number due communication, transmission, transportation, or of Unlike classical tuning methods give limited number sets of stabilizing parameters, we now have all stabilizing Many control systems inevitably possess some time delay due to to effects communication, transmission, transportation, or Unlike classical tuning methods which limited number due communication, transmission, transportation, or of sets sets of of stabilizing parameters, we nowgive have all stabilizing stabilizing inertia (Malek-Zavarei and Jamshidi (1987)). This of stabilizing parameters, we now have all controllers due to the results in (Silva et al (2002); Bhatdue to effects communication, transmission, or of inertia effects (Malek-Zavarei andsuch Jamshidi (1987)). This sets of stabilizing parameters, we now have all stabilizing inertia and Jamshidi (1987)). This controllers due to the results in (Silva et al (2002); Bhatphenomenon is(Malek-Zavarei found in systems astransportation, electrical, chemicontrollerset due to the results in (Silvathe et al (2002); Bhattacharyya al (2009)) who extended Hermite-Biehler inertia effectsis andsuch Jamshidi (1987)). This phenomenon is(Malek-Zavarei found in in systems systems such aslines, electrical, chemicontrollers due to the results in (Silva et al (2002); Bhatphenomenon found as electrical, chemitacharyyafor et al (2009)) who extended the Hermite-Biehler cal, hydraulic, pneumatic, transmission robotics, and theorem tacharyya et al (2009)) who extended the Hermite-Biehler a quasipolynomials. phenomenon ispneumatic, found(Gu in systems such as electrical, chemical, hydraulic, hydraulic, pneumatic, transmission lines, robotics, and tacharyya etaaalquasipolynomials. (2009)) who extended the Hermite-Biehler cal, lines, robotics, and theorem for for quasipolynomials. industrial processes ettransmission al (2003); Leon et al (2008)). theorem cal, hydraulic, pneumatic, lines, et robotics, and theorem for 2. industrial processes (Gu cases ettransmission al is(2003); (2003); Leon al (2008)). (2008)). a quasipolynomials. industrial (Gu et al Leon et al One of theprocesses most studied the Ziegler-Nichols plant, PROBLEM FORMULATION industrial processes (Gu et al (2003); Leon et al (2008)). One of the most studied cases is the Ziegler-Nichols plant, 2. PROBLEM PROBLEM FORMULATION FORMULATION One of the most studied cases is the Ziegler-Nichols plant, 2. FOPTD (Ziegler et al (1942)) because many of the indusOne of the mostare studied cases is the plant, Let us consider 2. PROBLEM FOPTD (Ziegler et approximated al (1942)) (1942)) because many of of thesystem. indusFOPTD (Ziegler et al because many of the indusan FOPTDFORMULATION continuous-time LTI system trial processes byZiegler-Nichols this type FOPTD (Ziegler et approximated al (1942)) many ofmargin thesystem. indusk Let us ustransfer consider an FOPTD FOPTD continuous-time LTIksystem system trial problem processes are approximated byand this type of system. Let consider an continuous-time LTI trial processes by this type of function P (s) := e−Ls where is the The of are achieving exactbecause gain phase be- with T s+1 k −Ls k −Ls Let ustransfer consider an FOPTD continuous-time LTIk system trial processes are approximated byand thisphase type of system. := with transfer function P (s) e where is the The problem problem ofbecause achieving exact gain and phase margin be- with := function P (s) e where k is the the The achieving gain becomes difficultof of exact the non-linearity andmargin solvability T s+1 steady state gain, T is the timeT s+1 constant, and L is k −Ls with transfer function (s) := e where is the the The problem achieving gain phase be- steady comes difficult because of exact the non-linearity andmargin solvability state gain, T the time constant, L comes difficult because of the non-linearity and solvability conditions of ofthe problem (Tan et and al (2012)). The consteady stateThis gain, T is isPis the timeT s+1 constant, and Lk is is the time-delay. system considered with aand controller in comes difficult because of the non-linearity and solvability conditions of the problem (Tan et al (2012)). The consteady state gain, T is the time constant, and L is the time-delay. This system is considered with a controller in conditions of focused the problem (Tan al (2012)). con- atime-delay. This system considered with controller in troller design on gain andetphase margin The has been unity feedback system is configuration as in aFig. 1. conditions of the problem (Tan et al (2012)). The controller design focused on gain and phase margin has been time-delay. This system is considered with a controller in a unity feedback system configuration as in Fig. 1. ˚ troller design focused on gain and phase margin has been studied since (Astr¨ om and H¨ agglund (1984)) resulting in a unity feedback system configuration as in Fig. 1. ˚ troller design focused on gain and phase margin has been studied since ( A str¨ o m and H¨ a gglund (1984)) resulting in a unity feedback system configuration as in Fig. 1. ˚ studied since ( A str¨ o m and H¨ a gglund (1984)) resulting in ˚ different new approaches. For example, see Astr¨om and C(s) P (s) ˚str¨ studied since (approaches. A om and For H¨ agglund (1984)) in ˚ different new approaches. For example, see Aresulting str¨ m and and ˚ different example, A str¨ ooHo m C(s) P H¨ agglundnew (2001); Srivastava and Pandit see (2016); + C(s) P (s) (s) ˚str¨oHo different new approaches. example, see A m and H¨ gglund (2001); Srivastava and Pandit (2016); Ho + C(s) P (s) H¨ aagglund (2001); Srivastava Pandit (2016); and + − Wang (2003). In this paper, For weand present an advanced tuning − H¨ agglund (2001); Srivastava and Pandit (2016); Ho and + − Wang (2003). In this paper,ofwe wePI present an advanced advanced tuning Wang (2003). In paper, present an tuning approach for thethis design and PID controllers for − Wang (2003). In this paper, we present an advanced tuning approach for the design of PI and PID controllers for approach for the design of PI and PID controllers for desired simultaneous gain margin, phase margin, and gain approach for the design of PI and PID controllers for desired simultaneous simultaneous gain margin, margin, phase margin,achievable and gain gain desired gain margin, and crossover frequency selected from aphase constructed 1. FOPTD unity feedback block diagram desired simultaneous gain margin, margin, and gain Fig. crossover frequency selected from constructed achievable crossover frequency from aaphase constructed Gain-Phase margin selected set. The controller gains areachievable retrieved Fig. 1. Fig. 1. FOPTD FOPTD unity unity feedback feedback block block diagram diagram crossover frequency from aconstant constructed Gain-Phase margin selected set. Thesimple controller gainsgain areachievable retrieved the PI unity and PID controllers, Gain-Phase margin set. The controller gains are retrieved Fig. 1. FOPTD feedback block diagram from the stabilizing set by and con- Consider Consider the PI and PID controllers, Gain-Phase margin set. Thesimple controller gains are retrieved the from the the stabilizing set by by simple constant gain and conconfrom stabilizing set constant and KPPI s +and KI PID controllers, KD s2 + KP s + KI stant phase loci conveniently represented asgain an ellipse or Consider the PI and PID and controllers, C (s) := D s22 + KP s + KI . CP I (s) := from stabilizing set line by simple constant andinconK ss s+ K stant the phase loci conveniently represented asgain an ellipse ellipse or Consider P K KD s + K + KII and CP ID (s) := K stant phase loci conveniently as an or P ellipsoid and a straight orrepresented plane superimposed the s P s + KI . C (s) := and CP CP P ID ID (s) := KD s2 + K PI I (s) := KP s s+ KI stant phase loci conveniently represented as an ellipse or ellipsoid and a straight line or plane superimposed in the ss P s + KI . ellipsoid and a straight line or plane superimposed in the s and CP ID (s) := . CP I (s) := ellipsoid and a straight line or plane superimposed in the s s
Copyright © 2017 IFAC 1841 2405-8963 © 2017 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright 1841 Copyright © © 2017 IFAC IFAC 1841 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 1841Control. 10.1016/j.ifacol.2017.08.168
Proceedings of the 20th IFAC World Congress 1806 Iván D. Díaz-Rodríguez et al. / IFAC PapersOnLine 50-1 (2017) 1805–1810 Toulouse, France, July 9-14, 2017
The characteristic equations of the closed loop systems are δP I (s) := ke−Ls (KP s + KI ) + (1 + T s)s (1) δP ID (s) := ke−Ls (KD s2 + KP s + KI ) + (1 + T s)s. (2)
(6) Find the root zn from T (9) kKP + cos(z) − z sin(z) = 0 L that is δi (z) = 0. The roots can be found graphically +cos(z) by taking the intersection of kKPsin(z) and TL z. (7) Compute an := a(zn ). (8) If cos(zn ) > 0 go to the next step. If not, n = n + 2 and go to step 6. (9) Determine the lower and upper bounds on KI from max {an } < KI < 0 (10)
The controller design problems to be considered are the following: (1) Construction of the achievable Gain-Phase margin design curves indexed by gain crossover frequencies. (2) Design a PI or PID controller based on desired simultaneous values of gain margin (GM), phase margin (PM), and gain crossover frequency (ωg ), by selecting the intersection point of an ellipse and a straight line superimposed over the stabilizing set.
n:odd
(10) Go to step 5. PID Controller Design for Stable FOPTD Systems. For stable first order systems, we have T > 0, k > 0, and L > 0. The procedure to compute the PID stabilizing set is the following:
This is described next. 3. DESIGN METHODOLOGY The PI or PID Controller design approach developed here for FOPTD can be summarized as follows. A. B. C. D.
Computation of the PI or PID Stabilizing set. Parametrization of a constant gain and phase loci. Construction the gain-phase margin design curves. Selection of simultaneous design specification from the achievable performance set. E. Retrieval of the PI or PID controller gains satisfying the design specifications.
3.1 Computation of the Stabilizing Set Here, we present a summary of the steps to compute the stabilizing sets. For illustration, we consider an unstable system with PI controller and stable system with PID controller (see Silva et al (2002) for more details). PI Controller Design for Unstable FOPTD Systems. For unstable first order systems, we have T < 0, k > 0, and L > 0. The procedure to compute the stabilizing set is the following: (1) For L = 0, calculate the characteristic equation (1). For stability, it is required KP < − k1 , KI < 0. (2) For L > 0, calculate the characteristic equation (1). Considering δ ∗ (s) = eLs δ(s) we have (3) δ ∗ (s) = k(KI + KP s) + s(T s + 1)eLs ∗ (3) Calculate δ (jω) = δr (ω) + jδi (ω), where (4) δr (ω) = kKI − ω sin(Lω) − T ω 2 cos(Lω) (5) δi (ω) = ω kKP + cos(Lω) − T ω sin(Lω) (4) Let z = Lω and calculate the real and imaginary parts of δ ∗ (jω) δr (z) = k KI − a(z) (6) T z kKP + cos(z) − z sin(z) (7) δi (z) = L L z where a(z) = kL sin(z) + TL z cos(z) (5) Pick a value for KP and set n = 1 in the range 1 L2 T (8) α21 + 2 < KP < − kL T k where α1 is the solution of the equation tan(α) = − TL α in the interval (0, π2 ).
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(1) For L = 0, calculate the characteristic equation (2). For stability, it is required 1 T KP > − , KI > 0 and KD > − (11) k k (2) For L > 0, calculate the characteristic equation (2). Considering δ ∗ (s) = eLs δ(s) we have δ ∗ (s) = k(KI + KP s + KD s2 ) + s(T s + 1)eLs (12) (3) Calculate δ ∗ (jω) = δr (ω) + jδi (jω) where δr (ω) = kKI − kKD ω 2 − ω sin(Lω) − T ω 2 cos(Lω) δi (ω) = ω kKP + cos(Lω) − T ω sin(Lω) (4) Let z = Lω and calculate the real and imaginary parts of δ ∗ (jω) T kKD 2 1 z − z sin(z) − 2 z 2 cos(z) δr (z) = kKI − 2 L L L T z kKP + cos(z) − z sin(z) δi (z) = L L (5) Pick a value for KP in the range 1 T 1 α1 sin(α1 ) − cos(α1 ) (13) − < KP < k k L T where α1 is the solution of tan(α) = − T +L α in the interval (0, π). (6) Find the roots z1 and z2 from T (14) kKP + cos(z) − z sin(z) = 0 L that is δi (z) = 0, ∀z > 0. The roots can be found graphically for the following cases: • − k1 < KP < k1 : in this case, take the intersection +cos(z) of the functions kKPsin(z) and TL z. • KP = k1 : in this case, take the intersection of the T functions kKP + cos(z) and L z sin(z). 1 k
T T < KP < kL L α1 sin(α1 ) − cos(α1 ) , in this case take the intersection of the functions kKP +cos(z) and TL z. sin(z) (7) Compute the parameters mn := m(zn ) and bn := b(zn ) for n = 1, 2 where L2 T L sin(zn ) + zn cos(zn ) . m(zn ) = 2 , b(zn ) = − zn kzn L
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(8) Determine the stabilizing region in the (KI , KD ) space using (a) − k1 < KP < k1 , KD = m1 KI + b1 , (b) KP = k1 , KD = m1 KI + b1 ,(c) k1 < KP <
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We have ∠P (ejωg ) = ∠ e−jLωg P0 (ejωg ) = −Lωg + ∠P0 (jωg ).
So, (24) becomes Φg := π + φ∗g + Lωg − ∠P0 (jωg ). Thus, for the PID case for equation (21) is a straight line in (KI , KD ) plane for a constant value of KP represented in equation (21). For the PI case, from (17) we obtain the ellipse and straight lines corresponding to M = Mg ∗ ∗ 3.2 Constant gain and constant phase loci for PI Controllers and Φ = Φg , giving the design point (KP , KI ). If these intersection points lie in the stabilizing set, denoted by S, Since FOPTD plant has single time delay, we can write the design is feasible, otherwise the specifications have to P (s) = e−Ls P0 (s), where L > 0 is the time-delay and be revised. P0 (s) is a delay-free plant. The frequency response of the plant and controller are e−Ljω P0 (jω), C(jω) respectively 3.4 Computation of the achievable performance Gain-Phase margin design curves where ω ∈ [0, ∞]. Let T kL
T L α1
sin(α1 ) − cos(α1 ) , KD = m1 KI + b1 , KD = m2 K I + b 2 . (9) Go to step 5.
KI2 ω2 −KI Φ := ∠C(jω) = arctan ωKP
M 2 := |C(jω)|2 = KP2 +
(15) (16)
The gain-phase margin design curves represent the actual performance in terms of GM, PM, and ωg for our FOPTD system achievable with a PI or PID controller. The procedure to construct these design curves is the following:
I with the design parameters, where C(jω) = KP (jω)+K jω KP and KI . Equations (15) and (16) can be rewritten as
+ − + (1) Set a test range of φ∗g ∈ [φ− g , φg ] and ωg ∈ [ωg , ωg ]. ∗ (2) For fixed values of φg and ωg , plot an ellipse and a straight line following the description in subsections 3.2 and 3.3. (3) If the intersection point of the ellipse and straight line lies outside of the stabilizing set, then this point is rejected and go to step (2). (4) If the intersection of the ellipse and straight line is contained in the stabilizing set, it represents the design point with the PI or PID controller gains ∗ (KP∗ , KI∗ ) or (KP∗ , KI∗ , KD ) that satisfies the fixed φ∗g and ωg . (5) Given the selected PI or PID controller gains (KP∗ , KI∗ ) ∗ or (KP∗ , KI∗ , KD ), the upper and lower GM of the system are given by K lb K ub GMupper = P∗ and GMlower = P∗ (25) KP KP
(KI )2 (KP )2 + = 1 and KI = cKP (17) a2 b2 where a2 = M 2 , b2 = M 2 ω 2 , and c = ω tan Φ. Clearly (17) is an ellipse and a straight line in KP , KI space. The major and minor axes of the ellipse are given by the square roots of a and b. The slope of the line is c. 3.3 Constant gain and constant phase loci for PID Controllers Now, let 2 KI 2 2 := |C(jω)| = KP + KD ω − M ω KI KD ω − ω −1 Φ := ∠C(jω) = tan KP 2
(18)
where KPub and KPlb are the controller gains at the upper and lower boundary respectively of the stabilizing set following the straight line ray intersecting the ellipse. (6) Go to step 2 and repeat for all values of φ∗g and ωg in the ranges.
(19)
2
P (jω)+KI where C(jω) = KD (jω) +K and KP , KI , and KD jω are the design parameters. From (18) and (19) we have that 2 2 KD ω − KωI K I . (20) = KP2 = M 2 − KD ω − ω tan2 Φ From (20) we can have the following expressions
KI = KD ω 2 ± ω
M 2 tan2 Φ 1 + tan2 Φ
and
KP = ±
M2 (21) 1 + tan2 Φ
Suppose ωg is the prescribed closed-loop gain crossover frequency. Then 1 1 Mg := = −Ljωg . (22) |P (ejωg )| |e ||P0 (jωg )| By the identity ∀x ∈ R, |e−jx | = | cos x − j sin x| = cos2 x + sin2 x = 1, (23)
1 (22) becomes Mg := |P0 (jω . Suppose φ∗g is the desired g )| phase margin in radians, Φg := π + φ∗g − ∠P (ejωg ). (24)
3.5 Selecting an achievable GM, PM, and ωg and retrieving the Controller gains In the last stage of the design process, the designer can select a desired point from the achievable performance Gain-Phase margin set and retrieve the controller gains corresponding to that simultaneous specification of desired GM, PM, and ωg . The controller gain retrieval process is the following. (1) Select desired GM, PM, and ωg from the achievable gain-margin set. (2) For the specified point, construct the ellipse and straight line by using the selected PM and ωg in the constant gain and constant phase loci. (3) Take the intersection of the ellipse and straight line contained in the stabilizing set. This will provide the ∗ gains (KP∗ , KI∗ ) or (KP∗ , KI∗ , KD ).
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(4) The controller that satisfies the prescribed margin K ∗ s+K ∗ specifications is CP I (s) = P s I or CP ID (s) =
4
∗ 2 ∗ KD s +KP s+KI∗ . s
2
4. ILLUSTRATIVE EXAMPLES
0
KP
In this section, we present two numerical examples to illustrate the proposed approach for the design of PI and PID controllers satisfying a desired simultaneous GM, PM, and ωg from a gain-phase achievable performance set.
Computation of the stabilizing set. The characteristic equation is given by δ(s) = (−12s + 1)s + 5(KP s + KI )e−0.5s . (27) By (3), δ ∗ (s) = e0.5s (−12s + 1)s + (KP s + KI )5. For L = 0, we have δ(s) = −12s2 + (5KP + 1)s + 5KI .
(28)
P
-4
I
(K* = -3.2276, K * = -1.3373) P
4.1 Example 1: PI Controller Design for Unstable FOPTD System Let us consider an unstable continuous-time FOPTD system 5 e−0.5s (26) P (s) = −12s + 1 and the PI controller, CP I . We proceed to apply the procedure presented in the methodology.
(Kub = -0.2540, K ub = -0.1052)
-2
I
-6 (Klb = -6.6177, K lb = -2.7419) P
-5
I
0
5
KI
Fig. 2. Stabilizing set in yellow for PI controller design in Example 1, intersection of an ellipse and a straight line (dot in black), and the controller gains (KPlb , KIlb ) and (KPub , KIub ) at the lower and upper boundary points in the stabilizing set (dots in magenta). rad/sec. However, for this value, we get a lower GM of 0.6404. We notice that for a bigger GM from the achievable Gain-Phase margin set, we get lower PM. The blue dots represent the specification points corresponding to a PM of 30o . The designer has the liberty to choose values for GM, PM, and ωg that best suits his design needs.
(29)
− 51 ,
KI < 0. For L > 0 For stability, it is required KP < and by (4) and (5), δ ∗ (jω) = δr (ω) + jδi (jω) (30) where δr (ω) = 5KI − ω sin(0.5ω) + 12ω 2 cos(0.5ω) (31) (32) δi (ω) = ω 5KP + cos(0.5ω) + 12ω sin(0.5ω) . By (8), we can calculate the range fo KP for stability 1 (33) −4.8 α21 + 0.0017 < KP < − 5 Following all the steps we get the stabilizing set in Fig. 2 Construction of the Achievable Gain-Phase margin design curves. For the construction of the achievable GainPhase margin set in this example, the evaluated range of ωg is [0.1, 3] and the range for PM is from 0o to 70o . The calculation of the GM for each case is done by (25). Using the ellipse and straight line intersection points, we can construct the achievable Gain-Phase margin set presented in Fig 3. Selection of simultaneous desired GM, PM, and ωg specifications from the achievable gain-phase margin design curves. In Fig 3, we can see the achievable Gain-Phase margin set of curves indexed by fixed ωg∗ in different colors. Notice that the curves above the 100 GM represent the upper GM and the curves below 100 GM represent the lower GM. We notice that the maximum PM that we can get is 66o for a ωg = 0.4 rad/sec with a value of upper GM of 7.547 and lower GM of 0.2045. Another example of the values of GM and PM that we can get is the point with a PM of 47o with an upper GM of 23.72 and a ωg = 0.1
10
2
w g = 0.1
Ki: 47 Kd: 23.72
10
Ki: 61 Kd: 14.24
g
w g = 0.4 w g = 0.6
1
Ki: 66 Kd: 7.547
Ki: 30 Kd: 2.05
GM
w = 0.2
g
w =1 g
w g = 1.2
Ki: 47 Kd: 0.6404
10 0
w = 0.8
Ki: 61 Kd: 0.3851
w = 1.4 g
w = 1.6 g
w g = 1.8 10
wg = 2
Ki: 30 Kd: 0.0787
-1
Ki: 66 Kd: 0.2045 w g = 2.2 w = 2.4 g
w g = 2.6 10 -2
w g = 2.8 0
10
20
30
40
50
60
70
w =3 g
PM
Fig. 3. Achievable performance in terms of GM, PM, and ωg for PI Controller design in Example 1. The Blue dots represent the intersections of ellipses and straight lines with a PM of 30o. Retrieval of the PI controller gains corresponding to a selected desired point in the achievable performance set. After the selection of simultaneous GM, PM, and ωg from the achievable Gain-Phase margin set, the designer can retrieve the controller gains corresponding to the point. For illustration purposes, let us say that the desired performance values chosen for this example are a PM of 30o , GM = 2.05, and a ωg = 1.4 rad/s from Fig 3. Then, taking these values for the constant gain and constant phase loci presented in the methodology, we can find the intersection of an ellipse and a straight line shown in Fig 1. The controller gains are KP∗ = −3.2276
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and KI∗ = −1.3373. In Fig 4 we can see the Nyquist plot for the controller gains selected. Here, we can see that those controller gains satisfy the desired performance specifications, P M = 30o , GM = 2.05 (6.23 dB). Nyquist Diagram
1.5 1 System: untitled1 Gain Margin (dB): 6.23 At frequency (rad/s): 2.79 Closed loop stable? Yes
Imaginary Axis
0.5 0 -0.5
Fig. 5. Intersection of a cylinder and a plane superimposed in the PID stabilizing set and a PID design point for Example 2.
System: untitled1 Phase Margin (deg): 30 Delay Margin (sec): 0.374 At frequency (rad/s): 1.4 Closed loop stable? Yes
-1 -1.5 -2 -3
-2
-1
0
1
Real Axis
Fig. 4. Nyquist plot for KP∗ = −3.2276 and KI∗ = −1.3373 in the PI controller design in Example 1. 4.2 Example 2: PID Controller Design for Stable FOPTD System Let us consider an stable continuous-time FOPTD system 1 e−2s P (s) = (34) 2s + 1 and the PID controller CP ID (s). We proceed to apply the procedure presented in the methodology. Computation of the stabilizing set. equation (2) is given by
get a cylinder and a plane in the (KP , KI , KD ) 3D space, respectively. The cylinder and the plane, superimposed in the stabilizing set (see Fig. 5) will have two intersection line segments in the (KI , KD ) plane. The specific value where the intersection occurs can be obtained using (21). Equation (21) will give us two values for KP , but only one is contained in the stabilizing set. The intersection line segment in the (KP , KI , KD ) represents the PID controller gains that satisfy the PM and ωg . Evaluating the range of PM and ωg , we can construct the achievable GainPhase margin set represented in 3D in Fig. 6. If we take a fixed value of ωg = 0.2 rad/sec, we can see the achievable performance in 2D in Fig 7. Here we can see that the maximum GM we can get is 8.95 with a PM of 57o .
The characteristic
δP ID (s) = (2s + 1)s + (KD s2 + KP s + KI )e−2s and by (12)
(35)
(36) δ ∗ (s) = e2s (2s + 1)s + (KD s2 + KP s + KI ) For L = 0 we have δ(s) = (KD + 2)s2 + (KP + 1)s + KI (37) For stability, it is required KP > −1, KI > 0, KD > −2 (38) For L > 0 δ ∗ (jω) = δr (ω) + jδi (jω) (39) where δr (ω) = KI − KD ω 2 − ω sin(2ω) − 2ω 2 cos(2ω) (40) (41) δi (ω) = ω KP + cos(2ω) − 2ω sin(2ω) By (13), we can calculate the range fo KP for stability −1 < KP < α1 sin(α1 ) − cos(α1 ) (42) Following all the steps, we get the stabilizing set in Fig. 5 Construction of the Achievable Gain-Phase margin design curves. For the construction of the achievable GainPhase margin set for the PID controller design case, the evaluated range of ωg is [0.1, 1.3] and the range for PM is from 1o to 120o. For the PID case, using the constant gain and constant phase loci equations, (18) and (19) we now
Fig. 6. Achievable performance in terms of GM, PM, and ωg for PID controller design in Example 2. Selection of simultaneous desired GM, PM, and ωg specification from the achievable Gain-Phase margin design curves. In Fig. 6, we can see the achievable Gain-Phase margin set of curves indexed by fixed ωg in different colors. Notice that we can get more GM and PM for lower values of ωg . For example, for ωg = 0.1 rad/sec, the maximum GM that we can get is 20 with a PM of 72o . For ωg = 0.2 rad/sec, the maximum GM is 8.95 with a P M = 57o . For a bigger value of ωg , we get lower values for GM and PM.
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8 7 6
wg: 0.2 PM: 57 GM: 8.95
w g = 0.2, PM = 57, GM = 8.95
5
GM
4 3
2
0
20
40
60
80
100
120
PM
Fig. 7. Achievable gain-phase margin set for ωg = 0.2 rad/sec for PID controller design in Example 2. For example, for ωg = 1.3 rad/sec we get a maximum GM = 1.012 and P M = 19o . The designer has the liberty to choose values for GM, PM, and ωg that best suits his design needs. Retrieval of the PID controller gains corresponding to a selected point in the achievable performance set. After the selection of simultaneous GM, PM, and ωg from the achievable gain-phase margin set, the designer can retrieve the controller gains corresponding to the point. For illustration purposes, let us say that the desired performance values chosen for this example are a PM of 57o , GM of 8.95, and a ωg of 0.2 rad/s (see Fig 6.) Then, taking these values and the constant gain and constant phase loci for PID controllers presented in the methodology, we can find the intersection of the cylinder and the plane in the (KP , KI , KD ) 3D space shown in Fig 5. The controller gains are KP∗ = 0.2188, KI∗ = ∗ 0.2189, and KD = 0.2. In Fig 8 we can see the Nyquist plot for the controller gains selected. Here, we can see that those controller gains satisfy the desired performance specifications, P M = 57o , GM = 8.95 (19 dB). Nyquist Diagram 1
Imaginary Axis
0.5
System: untitled1 Gain Margin (dB): 19 At frequency (rad/s): 0.893 Closed loop stable? Yes
0 System: untitled1 Phase Margin (deg): 57 Delay Margin (sec): 4.98 At frequency (rad/s): 0.2 Closed loop stable? Yes
-0.5
-1
-1
-0.8
-0.6
-0.4
-0.2
0
Real Axis
Fig. 8. Nyquist plot for KP∗ = 0.2188, KI∗ = 0.2189, and ∗ KD = 0.2 in the PID controller design in Example 2. 5. CONCLUSIONS In this paper, we have presented an advanced tuning approach for PI and PID controllers for Ziegler-Nichols
plants, namely FOPTD systems. It was shown how PI and PID controllers can be designed to satisfy simultaneously desired GM, PM, and ωg by selecting the specifications from an achievable Gain-Phase margin set. First, the stabilizing set was computed using recent results for FOPTD. Then, we showed a graphical approach for the parametrization of constant gain and constant phase loci respectively by ellipses and straight lines. We then constructed an achievable Gain-Phase margin set indexed by ωg . After that, given a selected desired specification from the achievable performance set, retrieval of the PI or PID controller was presented as the intersection of an ellipse and a straight line superimposed on the stabilizing set. In the end, two illustrative examples were given to show the proposed approach. For future research, this approach will be extended to multivariable systems. REFERENCES ˚ Astr¨om, Karl Johan and H¨agglund, Tore. Automatic tuning of simple regulators with specifications on phase and amplitude margins. Automatica. 20:5, 645–651, 1984. ˚ Astr¨om, Karl Johan and H¨agglund, Tore. The future of PID control. Control engineering practice. 9:11, 1163– 1175, 2001. Bhattacharyya, S.P., Datta, A., and Keel, L.H. Linear Control Theory: Structure, Robustness, and Optimization. CRC Press, 2009. Gu, Keqin, Chen, Jie and Kharitonov, Vladimir L. Stability of time-delay systems. Springer Science & Business Media, 2003. Ho, Ming-Tzu and Wang, Hsing-Sen. PID controller design with guaranteed gain and phase margins. Asian Journal of Control. 5:3, 374–381, 2003. Leon de la Barra, B.A., Jin, Lihua, Kim, Y.C., and Mossberg, M. Identification of First-Order Time-Delay Systems using Two Different Pulse Inputs Proceedings of the 17th World Congress. Seoul, Korea, July 6-11, 2008. Malek-Zavarei, Manu and Jamshidi, Mohammad. Timedelay systems : analysis, optimization, and applications. North-Holland systems and control series: v. 9, New York, 1987. Shafai, B and Saif, M. Proportional-integral Observer in Robust Control. Fault Detection, and Decentralized Control of Dynamic Systems. Control and Systems Engineering, 13–43, Springer International Publishing, 2015. Silva, Guillermo J and Datta, Aniruddha and Bhattacharyya, Shankar P. New results on the synthesis of PID controllers. IEEE transactions on automatic control. 47:2, 241–252, 2002. Srivastava, Saurabh and Pandit, V.S. A PI/PID controller for time delay systems with desired closed loop time response and guaranteed gain and phase margins. Journal of Process Control. 37, 70–77, 2016. Tan, Kok K and Wang, Qing-Guo, and Hang, Chang C. Advances in PID control. Springer Science & Business Media, 2012. Ziegler, John, G, and Nichols, Nathaniel B. Optimum settings for automatic controllers. Trans. ASME, 64:11, 1942.
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