o.dyer PID

PI and PID controller tuning rules for time delay processes: a summary

Technical Report AOD-00-01, Edition 1

A. O’Dwyer,School of Control Systems and Electrical Engineering,

Dublin Institute of Technology, Kevin St., Dublin 8, Ireland.15 May 2000

Phone: 353-1-4024875Fax: 353-1-4024992

e-mail: [email protected]

Abstract: The ability of proportional integral (PI) and proportional integral derivative (PID) controllers tocompensate many practical industrial processes has led to their wide acceptance in industrial applications. Therequirement to choose either two or three controller parameters is perhaps most easily done using tuning rules. Asummary of tuning rules for the PI and PID control of single input, single output (SISO) processes with timedelay are provided in this report. Inevitably, this report is a work in progress and will be added to and extendedregularly.

Keywords: PI, PID, tuning rules, time delay.

1. Introduction

The ability of PI and PID controllers to compensate most practical industrial processes has led to theirwide acceptance in industrial applications. Koivo and Tanttu [1], for example, suggest that there are perhaps 5-10% of control loops that cannot be controlled by SISO PI or PID controllers; in particular, these controllersperform well for processes with benign dynamics and modest performance requirements [2, 3]. It has been statedthat 98% of control loops in the pulp and paper industries are controlled by SISO PI controllers [4] and that, inprocess control applications, more than 95% of the controllers are of PID type [3]. The PI or PID controllerimplementation has been recommended for the control of processes of low to medium order, with small timedelays, when parameter setting must be done using tuning rules and when controller synthesis is performedeither once or more often [5]. However, Ender [6] states that, in his testing of thousands of control loops inhundreds of plants, it has been found that more than 30% of installed controllers are operating in manual modeand 65% of loops operating in automatic mode produce less variance in manual than in automatic (i.e. theautomatic controllers are poorly tuned); this is rather sobering, considering the wealth of information available inthe literature for determining controller parameters automatically. It is true that this information is scatteredthroughout papers and books; the purpose of this paper is to bring together in summary form the tuning rules forPI and PID controllers that have been developed to compensate SISO processes with time delay. Tuning rules forthe variations that have been proposed in the ‘ideal’ PI and PID controller structure are included. Considerablevariations in the ideal PID controller structure, in particular, are encountered; these variations are explored inmore detail in Section 2.

2. PID controller structures

The ideal continuous time domain PID controller for a SISO process is expressed in the Laplace domainas follows:

U s G s E sc( ) ( ) ( )= (1)

with G s KTs

T sc ci

d( ) ( )= + +11

(2)

and with Kc = proportional gain, Ti = integral time constant and Td = derivative time constant. If Ti = ∞and Td = 0 (i.e. P control), then it is clear that the closed loop measured value, y, will always be less than the

desired value, r (for processes without an integrator term, as a positive error is necessary to keep the measuredvalue constant, and less than the desired value). The introduction of integral action facilitates the achievement ofequality between the measured value and the desired value, as a constant error produces an increasing controlleroutput. The introduction of derivative action means that changes in the desired value may be anticipated, and

thus an appropriate correction may be added prior to the actual change. Thus, in simplified terms, the PIDcontroller allows contributions from present controller inputs, past controller inputs and future controller inputs.

Many tuning rules have been defined for the ideal PI and PID structures. Tuning rules have also beendefined for other PI and PID structures, as detailed in Section 4.

3. Process modelling

Processes with time delay may be modelled in a variety of ways. The modelling strategy used willinfluence the value of the model parameters, which will in turn affect the controller values determined from thetuning rules. The modelling strategy used in association with each tuning rule, as described in the originalpapers, is indicated in the tables. Of course, it is possible to use the tuning rules proposed by the authors with adifferent modelling strategy than that proposed by the authors; applications where this occurs are not indicated(to date). The modelling strategies are referenced as indicated. The full details of these modelling strategies areprovided in Appendix 2.

A. First order lag plus delay (FOLPD) model ( G sK e

sTmm

s

m

m

( ) =+

− τ

1):

Method 1: Parameters obtained using the tangent and point method (Ziegler and Nichols [8], Hazebroek andVan den Waerden [9]); Appendix 2.

Method 2: Km , τm assumed known; Tm estimated from the open loop step response (Wolfe [12]); Appendix

2.Method 3: Parameters obtained using an alternative tangent and point method (Murrill [13]); Appendix 2.Method 4: Parameters obtained using the method of moments (Astrom and Hagglund [3]); Appendix 2.Method 5: Parameters obtained from the closed loop transient response to a step input under proportional

control (Sain and Ozgen [94]); Appendix 2.Method 6: Km , Tm , τm assumed known.

Method 7: Parameters obtained using a least squares method in the time domain (Cheng and Hung [95]);Appendix 2.

Method 8: Parameters obtained in the frequency domain from the ultimate gain, phase and frequencydetermined using a relay in series with the closed loop system in a master feedback loop. Themodel gain is obtained by the ratio of the integrals (over one period) of the process output to thecontroller output. The delay and time constant are obtained from the frequency domain data(Hwang [160]).

Method 9: Parameters obtained from the closed loop transient response to a step input under proportionalcontrol (Hwang [2]); Appendix 2.

Method 10: Parameters obtained from two points estimated on process frequency response using a relay anda relay in series with a delay (Tan et al. [39]); Appendix 2.

Method 11: Tm and τm are determined from the ultimate gain and period estimated using a relay in series

with the process in closed loop; Km assumed known (Hang and Cao [112]); Appendix 2.

Method 12: Parameters are estimated using a tangent and point method (Davydov et al. [31]); Appendix 2.Method 13: Parameters estimated from the open loop step response and its first time derivative (Tsang and

Rad [109]); Appendix 2.Method 14: Tm and τm estimated from Ku , Tu determined using Ziegler-Nichols ultimate cycle method;

Km estimated from the process step response (Hang et al. [35]); Appendix 2.

Method 15: Tm and τm estimated from Ku , Tu determined using a relay autotuning method; Km estimated

from the process step response (Hang et al. [35]); Appendix 2.Method 16: )j(G 0135p ω , 0135

ω and Km are determined from an experiment using a relay in series with the

process in closed loop; estimates for Tm and τm are subsequently calculated. (Voda and

Landau [40]); Appendix 2.Method 17: Parameter estimates back-calculated from discrete time identification method (Ferretti et al.

[161]); Appendix 2.* Method 18: Parameter estimates calculated from process reaction curve using numerical integration

procedures (Nishikawa et al. [162]).* Method 19: Parameter estimates determined graphically from a known higher order process (McMillan [58]

… also McMillan (1983), pp. 34-40.

* Method 20: Km estimated from the open loop step response. T90% and τm estimated from the closed loop

step response under proportional control (Astrom and Hagglund [93]?)Method 21: Parameters estimated from linear regression equations in the time domain (Bi et al. [46]);

Appendix 2.Method 22: Tm and τm estimated from relay autotuning method (Lee and Sung [163]); Km estimated

from the closed loop process step response under proportional control (Chun et al. [57]);Appendix 2.

* Method 23: Parameters are estimated from a step response autotuning experiment – Honeywell UDC 6000controller (Astrom et al. [30]).Method 24: Parameters are estimated from the closed loop step response when process is in series with a

PID controller (Morilla et al. [104a]); Appendix 2.Method 25: mτ and mT obtained from an open loop step test as follows: )tt(4.1T %33%67m −= ,

m%67m T1.1t −=τ . mK assumed known (Chen and Yang [23a]).

Method 26: mτ and mT obtained from an open loop step test as follows: )tt(245.1T %33%70m −= ,

%70%33m t498.0t498.1 −=τ . mK assumed known (Miluse et al. [27b]).

* Method 27: Data at the ultimate period is deduced from an open loop impulse response (Pi-Mira et al.[97a]).

B. Non-model specific

Method 1: Parameters umu ,K,K ω are estimated from data obtained using a relay in series with the process

in closed loop and from the process step response (Kristiansson and Lennartsson [157]). need to check how the other methods define these parameters –

C. Integral plus time delay (IPD) model ( G sK e

smm

s m

( ) =− τ

)

Method 1: τm assumed known; Km determined from the slope at start of the open loop step response

(Ziegler and Nichols [8]); Appendix 2.Method 2: Km , τm assumed known.

Method 3: Parameters estimated from the ultimate gain and frequency values determined from an experimentusing a relay in series with the process in closed loop (Tyreus and Luyben [75]); Appendix 2.

Method 4: Parameters are estimated from the servo or regulator closed loop transient response, under PIcontrol (Rotach [77]); Appendix 2.

Method 5: Parameters are estimated from the servo closed loop transient response under proportionalcontrol (Srividya and Chidambaram [80]); Appendix 2.

Method 6: uK and uT are estimated from estimates of the ultimate and crossover frequencies. The ultimate

frequency estimate is obtained by placing an amplitude dependent gain in series with theprocess in closed loop; the crossover frequency estimate is obtained by also using an amplitudedependent gain (Pecharroman and Pagola [165]); Appendix 2.

D. First order lag plus integral plus time delay (FOLIPD) model (( )

G sK e

s sTmm

s

m

m

( ) =+

− τ

1)

* Method 1: Method of moments (Astrom and Hagglund [3]).Method 2: Km , Tm , τm assumed known.

Method 3: Parameters estimated from the open loop step response and its first and second time derivatives(Tsang and Rad [109]); Appendix 2.

Method 4: uK and uT are estimated from estimates of the ultimate and crossover frequencies (Pecharroman

and Pagola [165]) – as in Method 6, IPD model.

E. Second order system plus time delay (SOSPD) model ( G sm ( ) =K e

T s T sm

s

m m m

m−

+ +

τ

ξ12 2

12 1,

( )( )K e

T s T sm

s

m m

m−

+ +

τ

1 11 2

)

Method 1: Km , Tm1 , Tm2 , τm or Km , Tm1 , ξ m , τm assumed known.

Method 2: Parameters estimated using a two-stage identification procedure involving (a) placing a relay inseries with the process in closed loop and (b) placing a proportional controller in series with theprocess in closed loop (Sung et al. [139]); Appendix 2.

* Method 3: Parameters obtained in the frequency domain from the ultimate gain, phase and frequencydetermined using a relay in series with the closed loop system in a master feedback loop. Themodel gain is obtained by the ratio of the integrals (over one period) of the process output to thecontroller output. The other parameters are obtained from the frequency domain data (Hwang[160]).

Method 4: Tm and τm estimated from Ku , Tu determined using a relay autotuning method; Km estimated

from the process step response (Hang et al. [35]); Appendix 2.* Method 5: Parameter estimates back-calculated from discrete time identification method (Ferretti et al.

[161]).Method 6: Parameteres estimated from the underdamped or overdamped transient response in open loop to a

step input (Jahanmiri and Fallahi [149]); Appendix 2.* Method 7: Parameters estimated from a least squares time domain method (Lopez et al. [84]).Method 8: Parameters estimated from data obtained when the process phase lag is 090− and 0180− ,

respectively (Wang et al. [143]); Appendix 2.* Method 9: Parameter estimates back-calculated from discrete time identification method (Wang and

Clements [147]).Method 10: mK , 1mT and mτ are determined from the open loop time domain Ziegler-Nichols response

(Shinskey [16], page 151); 2mT assumed known.

Method 11: Parameters estimated from two points determined on process frequency response using a relayand a relay in series with a delay (Tan et al. [39]); Appendix 2.

* Method 12: Parameter estimated back-calculated from discrete time identification method (Lopez et al. [84]).* Method 13: Parameters estimated from a step response autotuning experiment – Honeywell UDC 6000

controller (Astrom et al. [30]).Method 14: 2m1m TT = . mτ and 1mT obtained from an open loop step test as follows:

)tt(794.0T %33%701m −= , %70%33m t937.0t937.1 −=τ . mK assumed known (Miluse et al.

[27b]).Method 15: uK and uT are estimated from estimates of the ultimate and crossover frequencies (Pecharroman

and Pagola [165]) – as in Method 6, IPD model.

F. Integral squared plus time delay ( PDI 2 ) model ( G sm ( ) =2

sm

seK mτ−

)

Method 1: Km , Tm , τm assumed known.

G. Second order system (repeated pole) plus integral plus time delay (SOSIPD) model ( G sm ( ) =( )2

m

sm

sT1s

eK m

+

τ−

)

Method 1: uK and uT are estimated from estimates of the ultimate and crossover frequencies (Pecharroman and

Pagola [165]) – as in Method 6, IPD model.Method 2: Km , Tm , τm assumed known.

H. Third order system plus time delay (TOLPD) model ( G sm ( ) =( )( )( )

K esT sT sT

ms

m m m

m−

+ + +

τ

1 1 11 2 3

).

Method 1: Km , Tm1 , Tm2 , Tm3 , τm known.

I. Unstable first order lag plus time delay model ( G sm ( ) =K e

sTm

s

m

m−

−

τ

1)

Method 1: Km , Tm , τm known.

Method 2: The model parameters are obtained by least squares fitting from the open loop frequencyresponse of the unstable process; this is done by determining the closed loop magnitude andphase values of the (stable) closed loop system and using the Nichols chart to determine theopen loop response (Huang and Lin [154], Deshpande [164]).

J. Unstable second order system plus time delay model ( G sm ( ) =( )( )

K esT sT

ms

m m

m−

− +

τ

1 11 2

)

Method 1: Km , Tm1 , Tm2 , τm known.

Method 2: The model parameters are obtained by least squares fitting from the open loop frequencyresponse of the unstable process; this is done by determining the closed loop magnitude andphase values of the (stable) closed loop system and using the Nichols chart to determine theopen loop response (Huang and Lin [154], Deshpande [164]).

K. Second order system plus time delay model with a positive zero ( G sm ( ) =( )

( )( )K sT e

sT sTm m

s

m m

m1

1 13

1 2

−

+ +

− τ

)


L. Second order system plus time delay model with a negative zero ( G sm ( ) =( )

( )( )K sT e

sT sTm m

s

m m

m1

1 13

1 2

+

+ +

− τ

)


M. Fifth order system plus delay model (( )

G sK b s b s b s b s b s e

a s a s a s a s a sm

m ss m

( )( )

=+ + + + +

+ + + + +

−1

11 2

23

34

4 5

1 22

33

44

55

τ

)

Method 1: Km , b1 , b 2 , b 3 , b 4 , b5 , a1 , a2 , a3 , a4 , a5 , τm known.

N. General model with a repeated pole ( G sK e

sTmm

s

mn

m

( )( )

=+

− τ

1)

* Method 1: Strejc’s method

O. General stable non-oscillating model with a time delay

P. Delay model ( )e)s(G msm

τ−=

Note: * means that the procedure has not been fully described to date.

4. Organisation of the report

The tuning rules are organised in tabular form, as is indicated in the list of tables below. Within each table, thetuning rules are classified further; the main subdivisions made are as follows:(i) Tuning rules based on a measured step response (also called process reaction curve methods).(ii) Tuning rules based on minimising an appropriate performance criterion, either for optimum regulator or

optimum servo action.(iii) Tuning rules that gives a specified closed loop response (direct synthesis tuning rules). Such rules may be

defined by specifying the desired poles of the closed loop response, for instance, though more generally,the desired closed loop transfer function may be specified. The definition may be expanded to covertechniques that allow the achievement of a specified gain margin and/or phase margin.

(iv) Robust tuning rules, with an explicit robust stability and robust performance criterion built in to the designprocess.

(v) Tuning rules based on recording appropriate parameters at the ultimate frequency (also called ultimatecycling methods).

(vi) Other tuning rules, such as tuning rules that depend on the proportional gain required to achieve a quarterdecay ratio or magnitude and frequency information at a particular phase lag.

Some tuning rules could be considered to belong to more than one subdivision, so the subdivisions cannot beconsidered to be mutually exclusive; nevertheless, they provide a convenient way to classify the rules. Tuningrules for the variations that have been proposed in the ‘ideal’ PI and PID controller structure are included in theappropriate table. In all cases, one column in the tables summarise the conditions under which the tuning rulesare designed to operate, if appropriate ( Y s( ) = closed loop system output, R s( ) = closed loop system input).

Tables 1-3: PI tuning rules – FOLPD model - G sK e

sTmm

s

m

m

( ) =+

− τ

1

Table 1: Ideal controller G s KTsc c

i

( ) = +

1 1

. Eighty-five such tuning rules are defined; the references are

(a) Process reaction methods: Ziegler and Nichols [8], Hazebroek and Van der Waerden [9], Astromand Hagglund [3], Chien et al. [10], Cohen and Coon [11], Wolfe [12], Murrill [13] – page 356,McMillan [14] – page 25, St. Clair [15] – page 22 and Shinskey [15a]. Twelve tuning rules aredefined.

(b) Performance index minimisation (regulator tuning): Minimum IAE - Murrill [13] – pages 358-363,Shinskey [16] – page 123, ** Shinskey [17], Huang et al. [18], Yu [19]. Minimum ISE - Hazebroekand Van der Waerden [9], Murrill [13]– pages 358-363, Zhuang and Atherton [20], Yu [19].Minimum ITAE - Murrill [13] – pages 358-363, Yu [19]. Minimum ISTSE - Zhuang and Atherton[20]. Minimum ISTES – Zhuang and Atherton [20]. Thirteen tuning rules are defined.

(c) Performance index minimisation (servo tuning): Minimum IAE – Rovira et al. [21], Huang et al. [18].Minimum ISE - Zhuang and Atherton [20], han and Lehman [22]. Minimum ITAE - Rovira et al. [21].Minimum ISTSE - Zhuang and Atherton [20]. Minimum ISTES – Zhuang and Atherton [20]. Seventuning rules are defined.

(d) Direct synthesis: Haalman [23], Chen and Yang [23a], Pemberton [24], Smith and Corripio [25], Smithet al. [26], Hang et al. [27], Miluse et al. [27a], Gorecki et al. [28], Chiu et al. [29], Astrom et al. [30],Davydov et al. [31], Schneider [32], McAnany [33], Leva et al. [34], Khan and Lehman [22], Hang etal. [35, 36], Ho et al. [37], Ho et al [104], Tan et al. [39], Voda and Landau [40], Friman and Waller[41], Smith [42], Cox et al. [43], Cluett and Wang [44], Abbas [45], Bi et al. [46], Wang and Shao[47]. Thirty-one tuning rules are defined.

(e) Robust: Brambilla et al. [48], Rivera et al. [49], Chien [50], Thomasson [51], Fruehauf et al. [52],Chen et al. [53], Ogawa [54], Lee et al. [55], Isaksson and Graebe [56], Chun et al. [57]. Ten tuningrules are defined.

(f) Ultimate cycle: McMillan [58], Shinskey [59] – page 167, **Shinskey [17], Shinskey [16] – page148, Hwang [60], Hang et al. [65], Zhuang and Atherton [20], Hwang and Fang [61]. Twelve tuningrules are defined.

Table 2: Controller G s K bT sc c

i

( ) = +

1 Two direct synthesis tuning rules are defined by Astrom and Hagglund

[3] - page 205-208.

Table 3: Two degree of freedom controller: )s(RK)s(EsT

11K)s(U ci

c α−

+= . One performance index

minimisation tuning rule is defined by Taguchi and Araki [61a].

Tables 4-7: PI tuning rules - non-model specific

Table 4: Controller G s KTsc c

i

( ) = +

1 1 . Nineteen such tuning rules are defined; the references are:

(a) Ultimate cycle: Ziegler and Nichols [8], Hwang and Chang [62], ** Hang et al. [36], McMillan [14] –page 90, Pessen [63], Astrom and Hagglund [3] – page 142, Parr [64] – page 191, Yu [122] – page11. Seven tuning rules are defined.

(b) Other tuning rules: Parr [64] – page 191, McMillan [14] – pages 42-43, Parr [64] – page 192,Hagglund and Astrom [66], Leva [67], Astrom [68], Calcev and Gorez [69], Cox et al. [70]. Eighttuning rules are defined.

(c) Direct synthesis: Vrancic et al. [71], Vrancic [72], Friman and Waller [41], Kristiansson andLennartson [158a]. Four tuning rules are defined.

Table 5: Controller G s K bT sc c

i

( ) = +

1. One direct synthesis tuning rule is defined by Astrom and Hagglund

[3] – page 215.

Table 6: Controller ( )U s KT s

E s K b R sci

c( ) ( ) ( )= +

+ −1 1 1 . One direct synthesis tuning rule is defined by

Vrancic [72].

Table 7: Controller U s K Y sKT s

E scc

i

( ) ( ) ( )= − . One direct synthesis tuning rule is defined by Chien et al. [74].

Tables 8-11: PI tuning rules – IPD model G sK e

smm

s m

( ) =− τ

Table 8: Controller G s KTsc c

i

( ) = +

1 1

. Twenty such tuning rules are defined; the references are:

(a) Process reaction methods: Ziegler and Nichols [8], Wolfe [12], Tyreus and Luyben [75], Astrom andHagglund [3] – page 138. Four tuning rules are defined.

(b) Regulator tuning – performance index minimisation: Minimum ISE – Hazebroek and Van derWaerden [9]. Minimum IAE - Shinskey [59] – page 74. Minimum ITAE - Poulin and Pomerleau [82].Four tuning rules are defined.

(c) Ultimate cycle: Tyreus and Luyben [75], ** Shinskey [17]. Two tuning rules are defined.(d) Robust: Fruehauf et al. [52], Chien [50], Ogawa [54]. Three tuning rules are defined.(e) Direct synthesis: Wang and Cluett [76], Cluett and Wang [44], Rotach [77], Poulin and Pomerleau

[78], Kookos et al. [38]. Five tuning rules are defined.(f) Other methods: Penner [79], Srividya and Chidambaram [80]. Two tuning rules are defined.

Table 9: Controller G s KTs T sc c

i f

( ) = +

+1 1 1

1. One robust tuning rule is defined by Tan et al. [81].

Table 10: Controller U s K Y sKT s

E scc

i

( ) ( ) ( )= − . One direct synthesis tuning rule is defined by Chien et al. [74].


11K)s(U ci

c α−

+= . Two performance index

minimisation - servo/regulator tuning rules are defined by Taguchi and Araki [61a] and Pecharromanand Pagola [134b].

Tables 12-14: PI tuning rules – FOLIPD model ( )

G sK e

s sTmm

s

m

m

( ) =+

− τ

1

Table 12: Ideal controllerG s KTsc c

i

( ) = +

1 1

. Six such tuning rules are defined; the references are:

(a) Ultimate cycle: McMillan [58]. One tuning rule is defined.(b) Regulator tuning – minimum performance index: Minimum IAE – Shinskey [59] – page 75. Shinskey

[59] – page 158. Minimum ITAE - Poulin and Pomerleau [82]. Four tuning rules are defined.(c) Direct synthesis - Poulin and Pomerleau [78]. One tuning rule is defined.

Table 13: ControllerG s K bT sc c

i

( ) = +

1. One direct synthesis tuning rule is defined by Astrom and Hagglund

[3] – pages 210-212.


11K)s(U ci

c α−


minimisation tuning rules are defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134b].

Tables 15-16: PI tuning rules – SOSPD model K e

T s T sm

s

m m m

m−

+ +

τ

ξ12 2

12 1 or

( )( )K e

T s T sm

s

m m

m−

+ +

τ

1 11 2


i

( ) = +

1 1

. Ten tuning rules are defined; the references are:

(a) Robust: Brambilla et al. [48]. One tuning rule is defined.(b) Direct synthesis: Tan et al. [39]. One tuning rule is defined.(c) Regulator tuning – minimum performance index: Minimum IAE - Shinskey [59] – page 158, **

Shinskey [17], Huang et al. [18], Minimum ISE – McAvoy and Johnson [83], Minimum ITAE –Lopez et al. [84]. Five tuning rules are defined.

(d) Servo tuning – minimum performance index: Minimum IAE - Huang et al. [18]. One tuning rule isdefined.

(e) Ultimate cycle: Hwang [60], ** Shinskey [17]. Two tuning rules are defined.


11K)s(U ci

c α−

+= . Three performance index

minimisation tuning rules are defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134a],[134b].

Table 17: PI tuning rules – SOSIPD model (repeated pole) G sm ( ) =( )2

m

sm

sT1s

eK m

+

τ−


11K)s(U ci

c α−


minimisation tuning rules are defined by Taguchi and Araki [61a] and Pecharroman and Pagola [134b].

Tables 18-19: PI tuning rules – third order lag plus delay (TOLPD) model ( )( )( )

K esT sT sT

ms

m m m

m−

+ + +

τ

1 1 11 2 3


i

( ) = +

1 1

. One *** tuning rule is defined. The reference is Hougen [85].


11K)s(U ci

c α−

+= . One performance index

minimisation tuning rule is defined by Taguchi and Araki [61a].

Table 20: PI tuning rules - unstable FOLPD model K e

sTm

s

m

m−

−

τ

1

Table 20: Ideal controllerG s KTsc c

i

( ) = +

1 1

. Six tuning rules are defined; the references are:

(a) Direct synthesis: De Paor and O’Malley [86], Venkatashankar and Chidambaram [87], Chidambaram[88], Ho and Xu [90]. Four tuning rules are defined.

(b) Robust: Rotstein and Lewin [89]. One tuning rule is defined.(c) Ultimate cycle: Luyben [91]. One tuning rule is defined.

Table 21: PI tuning rules - unstable SOSPD model ( )( )

K esT sT

ms

m m

m−

− +

τ

1 11 2


i

( ) = +

1 1

. Three tuning rules are defined; the references are:

(a) Ultimate cycle: McMillan [58]. One tuning rule is defined.(b) Minimum performance index – regulator tuning: Minimum ITAE – Poulin and Pomerleau [82]. Two

tuning rules are defined.

Table 22: PI tuning rules – delay model mse τ−


i

( ) = +

1 1

. Two tuning rules are defined; the references are:

(a) Direct synthesis: Hansen [91a].(b) Minimum performance index – regulator tuning: Shinskey [57].

Tables 23-40: PID tuning rules - FOLPD model K esT

ms

m

m−

+

τ

1

Table 23: Ideal controller G s KT s

T sc ci

d( ) = + +

1 1

. Fifty-seven tuning rules are defined; the references are:

(a) Process reaction: Ziegler and Nichols [8], Astrom and Hagglund [3] – page 139, Parr [64] – page194, Chien et al. [10], Murrill [13]- page 356, Cohen and Coon [11], Astrom and Hagglund [93]-pages 120-126, Sain and Ozgen [94]. Eight tuning rules are defined.

(b) Minimum performance index – regulator tuning: Minimum IAE – Murrill [13] – pages 358-363,Cheng and Hung [95]. Minimum ISE - Murrill [13] – pages 358-363, Zhuang and Atherton [20].Minimum ITAE - Murrill [13] – pages 358-363. Minimum ISTSE - Zhuang and Atherton [20].Minimum ISTES - Zhuang and Atherton [20]. Minimum error - step load change - Gerry [96]. Eighttuning rules are defined.

(c) Minimum performance index – servo tuning: Minimum IAE - Rovira et al. [21], Wang et al. [97].Minimum ISE - Wang et al. [97], Zhuang and Atherton [20]. Minimum ITAE - Rovira et al. [21],Cheng and Hung [95], Wang et al. [97]. Minimum ISTSE – Zhuang and Atherton [20]. MinimumISTES – Zhuang and Atherton [20]. Nine tuning rules are defined.

(d) Ultimate cycle: Pessen [63], Zhuang and Atherton [20], Pi-Mira et al. [97a], Hwang [60], Hwang andFang [61], McMillan [58], Astrom and Hagglund [98], Li et al. [99], Tan et al. [39], Friman andWaller [41]. Fourteen tuning rules are defined.

(e) Direct synthesis: Gorecki et al. [28], Smith and Corripio [25], Suyama [100], Juang and Wang [101],Cluett and Wang [44], Zhuang and Atherton [20], Abbas [45], Camacho et al.[102, Ho et al. [103],Ho et al [104], Morilla et al. [104a]. Fourteen tuning rules are defined.

(f) Robust: Brambilla et al. [48], Rivera et al. [49], Fruehauf et al. [52], Lee et al. [55]. Four tuning rules.

Table 24: Ideal controller with first order filter G s KT s

T sT sc c

id

f

( ) = + +

+1 1 1

1. Three robust tuning rules are

defined by ** Morari and Zafiriou [105], Horn et al. [106] and Tan et al. [81].

Table 25: Ideal controller with second order filter G s KT s

T s b sa s a sc c

id( ) = + +

++ +

1 1 11

1

1 22

. One robust tuning

rule is defined by Horn et al. [106].

Table 26: Ideal controller with set-point weighting G s K bT s

T sc ci

d( ) = + +

1. One direct synthesis tuning rule

is defined by Astrom and Hagglund [3] – pages 208-210.Table 27. Ideal controller with first order filter and set-point weighting:

−

++

+

++= )s(Y

sT1sT4.01)s(R

1sT1sT

sT11K)s(U

r

r

fd

ic . One direct synthesis tuning rule is defined

by Normey-Rico et al. [106a].

Table 28: Classical controller

NT

s1

sT1sT

11K)s(Gd

d

icc

+

+

+= . Twenty tuning rules are defined; the references are:

(a) Process reaction: Hang et al. [36] – page 76, Witt and Waggoner [107], St. Clair [15] – page 21,Shinskey [15a]. Five tuning rules are defined.

(b) Minimum performance index – regulator tuning: Minimum IAE - Kaya and Scheib [108], Witt andWaggoner [107]. Minimum ISE - Kaya and Scheib [108]. Minimum ITAE - Kaya and Scheib [108],Witt and Waggoner [107]. Five tuning rules are defined.

(c) Minimum performance index – servo tuning: Minimum IAE - Kaya and Scheib [108], Witt andWaggoner [107]. Minimum ISE - Kaya and Scheib [108]. Minimum ITAE - Kaya and Scheib [108],Witt and Waggoner [107]. Five tuning rules are defined.

(d) Direct synthesis: Tsang and Rad [109], Tsang et al. [111]. Two tuning rules are defined.(e) Robust: Chien [50]. One tuning rule is defined.

(f) Ultimate cycle: Shinskey [59] – page 167, Shinskey [16] – page 143. Two tuning rules are defined.

Table 29: Non-interacting controller

+−

+= )s(Y

NsT1

sT)s(E

sT1

1K)s(Ud

d

ic . Two tuning rules are defined; the

references are:(a) Minimum performance index – regulator tuning: Minimum IAE - Huang et al. [18].(b) Minimum performance index – servo tuning: Minimum IAE - Huang et al. [18].

Table 30: Non-interacting controller U s KT s

E s T ssTN

Y sci

d

d( ) ( ) ( )= +

−

+1 1

1. Five tuning rules are defined; the

references are:(a) Minimum performance index – servo tuning: Minimum ISE - Zhuang and Atherton [20], Minimum

ISTSE - Zhuang and Atherton [20]. Minimum ISTES - Zhuang and Atherton [20]. Three tuningrules are defined.

(b) Ultimate cycle: Zhuang and Atherton [20], Shinskey [16] – page 148. Two tuning rules are defined.

Table 31: Non-interacting controller )s(Y

NsT

1

sT)s(E

sT1K)s(U

d

d

ic

+−

+= . Six tuning rules are defined; the

references are:(a) Minimum performance index – regulator tuning: Minimum IAE – Kaya and Scheib [108]. Minimum

ISE – Kaya and Scheib [108]. Minimum ITAE – Kaya and Scheib [108].(b) Minimum performance index – servo tuning: Minimum IAE – Kaya and Scheib [108]. Minimum ISE –

Kaya and Scheib [108]. Minimum ITAE – Kaya and Scheib [108].Table 32: Non-interacting controller with setpoint weighting:

( )U s K bT s

E s K T sT s N

Y s K b Y sci

c d

dc( ) ( ) ( ) ( )= +

−

++ −1

11 . Three ultimate cycle tuning rules are

defined by Hang and Astrom [111], Hang et al. [65] and Hang and Cao [112].

Table 33: Industrial controller U s KT s

R sT sT sN

Y sci

d

d( ) ( ) ( )= +

−

+

+

11 1

1. Six tuning rules are defined: the

reference are:(a) Minimum performance index – regulator tuning: Minimum IAE - Kaya and Scheib [108]. Minimum

ISE - Kaya and Scheib [108]. Minimum ITAE - Kaya and Scheib [108]. Three tuning rules aredefined.

(b) Minimum performance index – servo tuning: Minimum IAE - Kaya and Scheib [108]. Minimum ISE -Kaya and Scheib [108]. Minimum ITAE - Kaya and Scheib [108]. Three tuning rules are defined.

Table 34: Series controller ( )G s KT s

sTc ci

d( ) = +

+1 1 1 . Three tuning rules are defined; the references are:

(a) ******: Astrom and Hagglund [3] – page 246.(b) Ultimate cycle: Pessen [63].(c) Direct synthesis: Tsang et al. [110].

Table 35: Series controller with filtered derivative G s KT s

sTsTN

c ci

d

d( ) = +

+

+

11

11

. One robust tuning rule is

defined by Chien [50].

Table 36: Controller with filtered derivative G s KT s

T s

s TN

c ci

d

d( ) = + +

+

11

1. Three tuning rules are defined; the

references are:

(a) Robust: Chien [50], Gong et al. [113]. Two tuning rules are defined.(b) Direct synthesis: Davydov et al. [31]. One tuning rule is defined.

Table 37: Alternative non-interacting controller 1 - U s KT s

E s K T sY sci

c d( ) ( ) ( )= +

−1 1

. Six ultimate cycle

tuning rules are defined; the references are: Shinskey [59] – page 167, ** Shinskey [17], Shinskey [16] –page 143, VanDoren [114].

Table 38: Alternative filtered derivative controller - [ ]

G s KT s

s s

sc c

i

m m

m

( ) . .

.= +

+ +

+

1 1 1 05 0 0833

1 01

2 2

2

τ τ

τ. One direct

synthesis tuning rule is defined by Tsang et al. [110].

Table 39: I-PD controller ( )U sKT s

E s K T s Y sc

ic d( ) ( ) ( )= − +1 . Two direct synthesis tuning rules are defined by Chien

et al. [74] and Argelaguet et al. [114a].

Table 40: Two degree of freedom controller: )s(Rs

NT1

sTK)s(E

sNT1

sTsT

11K)s(U

d

dc

d

d

ic

+

β+α−

+++= . One

performance index minimisation tuning rule is defined by Taguchi and Araki [61a].

Tables 41-48: PID tuning rules - non-model specific

Table 41: Ideal controllerG s KT s

T sc ci

d( ) = + +

1 1

. Twenty five tuning rules are defined; the references are

(a) Ultimate cycle: Ziegler and Nichols [8], Blickley [115], Parr [64] – pages 190-191, De Paor [116],Corripio [117] – page 27, Mantz andTacconi [118], Astrom and Hagglund [3] – page 142, Astromand Hagglund [93], Atkinson and Davey [119], ** Perry and Chilton [120], Yu [122] – page 11, Luoet al. [121], McMillan [14] – page 90, McAvoy and Johnson [83], Karaboga and Kalinli [123], Hangand Astrom [124], Astrom et al. [30], St. Clair [15] - page 17, Shin et al. [125]. Nineteen tuning rulesare defined.

(b) Other tuning: Harriott [126], Parr [64] – pages 191, 193, McMillan [14] - page 43, Calcev and Gorez[69], Zhang et al. [127], Garcia and Castelo [127a]. Six tuning rules are defined.


T sT sN

c ci

d

d( ) = + +

+

11

1. Eight tuning rules are defined; the

references are:(a) Direct synthesis: Vrancic [72], Vrancic [73], Lennartson and Kristiansson [157], Kristiansson and

Lennartson [158], Kristiansson and Lennartson [158a]. Six tuning rules are defined.(b) Other tuning: Leva [67], Astrom [68]. Two tuning rules are defined.

Table 43: Ideal controller with set-point weighting:

( ) ( ) ( )U s K F R(s Y sTs

FR(s Y s T s F R(s Y sc pi

i d d( ) ) ( ) ) ( ) ) ( )= − + − + −1

. One ultimate cycle tuning rule is

defined by Mantz and Tacconi [118].

Table 42: Ideal controller with proportional weighting G s K bT s

T sc ci

d( ) = + +

1. One direct synthesis tuning

rule is defined by Astrom and Hagglund [3] – page 217.


E s K T ssTN

Y sci

c d

d( ) ( ) ( )= +

−

+1 1

1. One ultimate cycle tuning rule is

defined by Fu et al. [128].

Table 45: Series controller ( )U s KT s

sTci

d( ) = +

+1 1 1 . Three ultimate cycle tuning rules are defined by Pessen

[131], Pessen [129] and Grabbe et al. [130].

Table 46: Series controller with filtered derivative U s KT s

sTsTN

ci

d

d( ) = +

+

+

11

11

. One ultimate cycle tuning

rule is defined by Hang et al. [36] - page 58.

Table 47: Classical controller U s KT s

sT

sTN

ci

d

d( ) = +

+

+1 1 1

1. One ultimate cycle tuning rule is defined by Corripio

[117].


E s K T sY sci

c d( ) ( ) ( )= +

−1 1

. One ultimate cycle tuning rule is

defined by VanDoren [114].

Tables 49-58: PID tuning rules - IPD model K e

sm

s m− τ


T sc ci

d( ) = + +

1 1

. Five tuning rules are defined; the references are:

(a) Process reaction: Ford [132], Astrom and Hagglund [3] – page 139. Two tuning rules are defined.(b) Direct synthesis: Wang and Cluett [76], Cluett and Wang [44], Rotach [77]. Three tuning rules are

defined.Table 50: Ideal controller with first order filter, set-point weighting and output feedback:

)s(YK)s(YsT1

sT4.01)s(R1sT

1sTsT

11K)s(U 0r

r

fd

ic −

−

++

+

++= . One direct synthesis tuning rule

has been defined by Normey-Rico et al. [106a].

Table 51: Ideal controller with filtered derivative G s KTs

T ssTN

c ci

d

d( ) = + +

+

11

1. One robust tuning rule has

been defined by Chien [50].

Table 52: Series controller with filtered derivative G s KTs

T sT sN

c ci

d

d( ) = +

+

+

11

11

. One robust tuning rule has


Table 53: Classical controller G s KT s

T sT sN

c ci

d

d( ) = +

+

+

11 1

1. Five tuning rules have been defined; the references

are:(a) Ultimate cycle: Luyben [133], Belanger and Luyben [134]. Two tuning rules have been defined.(b) Robust: Chien [50]. One tuning rule has been defined.(c) Performance index minimisation – regulator tuning: ** Minimum IAE - Shinskey [17], Shinskey [59]

– page 74. Two tuning rules have been defined.

Table 54: Alternative non-interacting controller 1: U s KT s

E s K T sY sci

c d( ) ( ) ( )= +

−1 1

. Two performance index

minimisation rules – minimum IAE regulator tuning have been defined by Shinskey [59] – page 74 and** Shinskey [17].

Table 55: I-PD controller ( )U sKT s

E s K T s Y sc

ic d( ) ( ) ( )= − +1 . One direct synthesis tuning rule has been defined by

Chien et al. [74].

Table 56: Controller )s(sYTK)s(R)1b(K)s(E)sT

11(K)s(U dcc

ic −−++= . One direct synthesis tuning rule has

been defined by Hansen [91a].


NT1

sTK)s(E

sNT1

sTsT

11K)s(U

d

dc

d

d

ic

+

β+α−

+++= . Two

minimum performance index – servo/regulator tuning have been defined by Taguchi and Araki [61a] andPecharroman and Pagola [134a].

Tables 58-67: PID tuning rules - FOLIPD model ( )K e

s sTm

s

m

m−

+

τ

1


T sc ci

d( ) = + +

1 1

. One ultimate cycle tuning rule has been defined by Millan

[58].

Table 59: Ideal controller with filter G s KT s

T sT sc c

id

f

( ) = + +

+1 1 1

1. Three robust tuning rules have been

defined by Tan et al. [81], Zhang et al. [135] and Tan et al. [136].

Table 60: Ideal controller with set-point weighting G s K bT s

T sc ci

d( ) = + +

1. One direct synthesis tuning rule

has been defined by Astrom and Hagglund [3] - pages 212-213.


T sT sN

c ci

d

d( ) = +

+

+

11 1

1. Five tuning rules have been defined; the references

are as follows:(a) Robust: Chien [50]. One tuning rule is defined.(b) Minimum performance index – regulator tuning: Minimum IAE – Shinskey [59] – page 75, Shinskey

[59] – pages 158-159, Minimum ITAE - Poulin and Pomerleau [82], [92]. Four tuning rules aredefined.

Table 62: Series controller with derivative filtering G s KTs

T sT sN

c ci

d

d( ) = +

+

+

11

11




E s K T sY sci

c d( ) ( ) ( )= +

−1

1. Two minimum

performance index (minimum IAE) – regulator tuning rules have been defined by Shinskey [59] – page75, page 159.

Table 64: Ideal controller with filtered derivative: G s KT s

T s

s TN

c ci

d

d( ) = + +

+

11

1. One robust tuning rule has



( ) [ ] [ ]U s K F R s Y sKT s

F R s Y s K T s F R s Y sc pc

ii c d d( ) ( ) ( ) ( ) ( ) ( ) ( )= − + − + − . One ultimate cycle tuning rule

has been defined by Oubrahim and Leonard [138].

Table 66: Alternative classical controller: G s KT sT sN

T sT sN

c ci

d

d

d( ) =

+

+

+

+

1

1

1

1. One direct synthesis tuning rule has

been defined by Tsang and Rad [109].

Table 67: Two degree of freedom controller:

)s(Rs

NT1

sTK)s(E

sNT1

sTsT

11K)s(U

d

dc

d

d

ic

+

β+α−

+++= . Two minimum performance index –

servo/regulator tuning have been defined by Taguchi and Araki [61a] and Pecharroman and Pagola[134a].

Tables 68-79: PID tuning rules - SOSPD model ( )( )

K esT sT

ms

m m

m−

+ +

τ

1 11 2

or K eT s T s

ms

m m m

m−

+ +

τ

ξ12 2

12 1


T sc ci

d( ) = + +

1 1

. Twenty seven tuning rules have been defined; the

references are:(a) Minimum performance index – servo tuning: Minimum ITAE – Sung et al. [139]. One tuning rule is

defined.(b) Minimum performance index – regulator tuning: Minimum ITAE – Sung et al. [139], Lopez et al.

[84]. One tuning rule is defined.(c) Ultimate cycle: Hwang [60], Shinskey [16] – page 151. Three tuning rules are defined.(d) Direct synthesis: Hang et al. [35], Ho et al. [140], Ho et al. [141], Ho et al. [142], Wang et al. [143],

Leva et al. [34], Wang and Shao [144], Pemberton [145], Pemberton [24], Suyama [100], Smith et al.[146], Chiu et al. [29], Wang and Clemens [147], Gorez and Klan [147a], Miluse et al. [27a], Miluse etal. [27b], Seki et al. [147b], Landau and Voda [148]. Nineteen tuning rules are defined.

(e) Robust: Brambilla et al. [48], Chen et al. [53], Lee et al. [55]. Three tuning rules are defined.

Table 69: Filtered controller G s KT s

T sT sc c

id

f

( ) = + +

+1 1 1

1. One robust tuning rule has been defined by Hang

et al. [35].

Table 70: Filtered controller G s KT s

T s b sa sc c

id( ) = + +

++

1 1 11

1

1

. One robust tuning rule has been defined by

Jahanmiri and Fallahi [149].


T sT sN

c ci

d

d( ) = +

+

+

11 1

1. Seven tuning rules have been defined; the

references are:(a) Minimum performance index – regulator tuning: Minimum IAE - Shinskey [59] – page 159, **

Shinskey [59], ** Shinskey [17], ** Shinskey [17]. Minimum ISE – McAvoy and Johnson [83]. Fivetuning rules are defined.

(b) Direct synthesis: Astrom et al. [30], Smith et al. [26]. Two tuning rules are defined.

Table 72: Alternative classical controller G s KT s

NT sT sc c

i

d

d

( ) = +

++

1 1 1

1. One ***** tuning rule has been

defined by Hougen [85].


E s K T sY sci

c d( ) ( ) ( )= +

−1 1

. Three minimum

performance index (minimum IAE) – regulator tuning rules have been defined by Shinskey [59] – page158, ** Shinskey [17], ** Shinskey [17].


T sc ci

d( ) = +

+1 1 1 . One minimum performance index - regulator tuning rule

has been defined by Haalman [23].

Table 75: Non-interacting controller

+−

+= )s(Y

NsT1

sT)s(E

sT1

1K)s(Ud

d

ic . Two tuning rules have been

defined. The references are:(a) Minimum performance index – regulator tuning: Minimum IAE - Huang et al. [18].(b) Minimum performance index – servo tuning: Minimum IAE - Huang et al. [18].


( ) [ ] [ ]U s K F R s Y sKT s


ii c d d( ) ( ) ( ) ( ) ( ) ( ) ( )= − + − + − . One ultimate cycle tuning rule

has been defined by Oubrahim and Leonard [138].

Table 77: Non-interacting controller [ ] ( )U s K bTs

R s Y s c T s Y sci

d( ) ( ) ( ) ( )= +

− − +1

. One direct synthesis tuning

rule has been defined by Hansen [150].

Table 78: Ideal controller with filtered derivative G s KTs

T ssTN

c ci

d

d( ) = + +

+

11

1. Two tuning rules are defined;

the references are:(a) Direct synthesis: Hang et al. [151].(b) Robust: Hang et al. [151].


NT1

sTK)s(E

sNT1

sTsT

11K)s(U

d

dc

d

d

ic

+

β+α−

+++= . Three

minimum performance index – servo/regulator tuning rules have been defined by Taguchi and Araki[61a] and Pecharroman and Pagola [134a], [134b].

Table 80: PID tuning rules - PDI 2 model G sm ( ) =2

sm

seK mτ−

Table 80: Controller )s(sYTK)s(R)1b(K)s(E)sT

11(K)s(U dcc

ic −−++= . One direct synthesis tuning rule has

been defined by Hansen [91a].

Table 81: PID tuning rules – SOSIPD model (repeated pole) G sm ( ) =( )2

m

sm

sT1s

eK m

+

τ−


NT1

sTK)s(E

sNT1

sTsT

11K)s(U

d

dc

d

d

ic

+

β+α−

+++= . Two

minimum performance index – servo/regulator tuning rules have been defined by Taguchi and Araki[61a] and Pecharroman and Pagola [134a].

Tables 82-84: PID tuning rules - SOSPD model with a positive zero ( )

( )( )K sT e

sT sTm m

s

m m

m1

1 13

1 2

−

+ +

− τ


T sT sN

c ci

d

d( ) = + +

+

11

1. One robust tuning rule has been



T sT sN

c ci

d

d( ) = +

+

+

11 1

1. One robust tuning rule has been defined by Chien

[50].


T sT sN

c ci

d

d( ) = +

+

+

11

11



Tables 85-88: PID tuning rules - SOSPD model with a negative zero ( )

( )( )K sT e

sT sTm m

s

m m

m1

1 13

1 2

+

+ +

− τ


T sc ci

d( ) = + +

1 1

. One minimum performance index tuning rule has been

defined by Wang et al. [97].

Table 86: Ideal controller with filtered derivative G s KT s

T sT sN

c ci

d

d( ) = + +

+

11

1. One robust tuning rule has been



T sT sN

c ci

d

d( ) = +

+

+

11 1

1. One robust tuning rule has been defined by Chien

[50].


T sT sN

c ci

d

d( ) = +

+

+

11

11



Table 89-90: PID tuning rules - TOLPD model ( )( )( )

K esT sT sT

ms

m m m

m−

+ + +

τ

1 1 11 2 3


T sc ci

d( ) = + +

1 1

. Two minimum performance index tuning rules have been

defined by Polonyi [153].


NT1

sTK)s(E

sNT1

sTsT

11K)s(U

d

dc

d

d

ic

+

β+α−

+++= .One

minimum performance index tuning rule has been defined by Taguchi and Araki [61a].

Tables 91-93: PID tuning rules - unstable FOLPD model K e

sTm

s

m

m−

−

τ

1


T sc ci

d( ) = + +

1 1

. Three direct synthesis tuning rules are defined by De Paor

and O’Malley [86], Chidambaram [88] and Valentine and Chidambaram [154].


E s K T ssTN

Y sci

c d

d( ) ( ) ( )= +

−

+1 1

1. Two tuning rules have been

defined; the references are:(a) Minimum performance index – servo tuning: Minimum IAE - Huang and Lin [155](b) Minimum performance index – servo tuning: Minimum IAE - Huang and Lin [155]


T sT sN

c ci

d

d( ) = +

+

+

11 1

1. One performance index minimisation – regulator

tuning rule has been defined by Shinskey [16]– page 381.

Tables 94-97: PID tuning rules - unstable SOSPD model ( )( )

G sK e

sT sTmm

s

m m

m

( ) =− +

− τ

1 11 2


T sc ci

d( ) = + +

1 1

. Two tuning rules have been defined; the references are

(a) Ultimate cycle: McMillan [58](b) Robust: Rotstein and Lewin [89].


T sTN

sc c

i

d

d( ) = +

+

+

11 1

1. Two minimum performance index tuning rules

(regulator - minimum ITAE) have been defined by Poulin and Pomerleau [82], [92].


T sc ci

d( ) = +

+1 1 1 . One direct synthesis tuning rule has been defined by

Ho and Xu [90].


E s K T ssTN

Y sci

c d

d( ) ( ) ( )= +

−

+1 1

1. Two tuning rules have been

defined; the references are(a) Minimum performance index – servo tuning: Minimum IAE - Huang and Lin [155](b) Minimum performance index – regulator tuning: Minimum IAE - Huang and Lin [155]

Table 98: PID tuning rules – general model with a repeated pole G sK e

sTmm

s

mn

m

( )( )

=+

− τ

1


T sc ci

d( ) = + +

1 1

. One direct synthesis tuning rule has been defined by

Skoczowski and Tarasiejski [156]

Table 99: PID tuning rules – general stable non-oscillating model with a time delay


T sc ci

d( ) = + +

1 1

. One direct synthesis tuning rule has been defined by Gorez

and Klan [147a].

Tables 100-101: PID tuning rules – fifth order model with delay

( )G s

K b s b s b s b s b s e


m ss m

( )( )

=+ + + + +

+ + + + +

−1

11 2

23

34

4 5

1 22

33

44

55

τ


T sc ci

d( ) = + +

1 1

. One direct synthesis tuning rule is defined by Vrancic et

al. [159].

Table 101: Controller with filtered derivative G s KTs

T sT sN

c ci

d

d( ) = + +

+

11

1. One direct synthesis tuning rule is

defined by Vrancic et al. [159].

** some more information needed.

The number of tuning rules in each table is included in the data. Servo and regulator tuning rules are countedseparately; otherwise, rules in which different tuning parameters are provided for a number of variations inprocess parameters or desired response parameters (such as desired gain margin, phase margin or closed loopresponse time constant) are counted as one tuning rule. Tabular summaries are provided below.

Table A: Model structure and tuning rules – a summary for PI controllers

ModelProcessreaction

MinimisePerformanc

e index

DirectSynthesis

Ultimatecycle

Robusttuning Other rules Total

StableFOLPD

12 21 33 10 12 0 88 (53%)

Non-modelspecific

0 0 7 7 0 8 23 (14%)

IPD 4 6 6 2 4 2 24 (14%)FOLIPD 0 6 2 1 0 0 9 (5%)SOSPD 0 9 1 2 1 0 13 (7%)SOSIPD 0 2 0 0 0 0 2 (1%)TOLPD 0 1 0 0 0 1 2 (1%)

UnstableFOLPD

0 0 4 1 1 0 6 (4%)

UnstableSOSPD

0 2 0 1 0 0 3 (2%)

Delaymodel

0 1 1 0 0 0 2 (1%)

TOTAL 16 48 54 24 18 11 171

Table B: Model structure and tuning rules – a summary for PID controllers


MinimisePerformanc

e index

DirectSynthesis

Ultimatecycle


StableFOLPD

13 45 24 28 12 1 123 (44%)

Non-modelspecific

0 0 7 27 0 8 42 (15%)

IPD 2 6 6 2 3 0 19 (7%)FOLIPD 0 8 2 2 6 0 18 (6%)SOSPD 0 16 23 4 6 1 50 (17%)

PDI 2 0 0 1 0 0 0 1 (0%)

SOSIPD 0 2 0 0 0 0 2 (1%)SOSPD –pos. zero

0 0 0 0 3 0 3 (1%)

SOSPD –neg. zero

0 1 0 0 3 0 4 (1%)

TOLPD 0 3 0 0 0 0 3 (1%)UnstableFOLPD

0 3 3 0 0 0 6 (2%)

UnstableSOSPD

0 4 1 1 1 0 7 (2%)

Higherorder

0 0 4 0 0 0 4 (1%)

TOTAL 15 88 71 64 34 10 282

Table C: Model structure and tuning rules – a summary for PI/PID controllers


MinimisePerformanc

e index

DirectSynthesis

Ultimatecycle


StableFOLPD

25 66 57 38 24 1 211 (47%)

Non-modelspecific

0 0 14 34 0 16 65 (15%)

IPD 6 12 12 4 7 2 43 (9%)FOLIPD 0 14 4 3 6 0 27 (6%)SOSPD 0 25 24 6 7 1 63 (14%)

PDI 2 0 0 1 0 0 0 1 (0%)

SOSIPD 0 4 0 0 0 0 4 (1%)SOSPD –pos. zero

0 0 0 0 3 0 3 (1%)

SOSPD –neg. zero

0 1 0 0 3 0 4 (1%)

TOLPD 0 4 0 0 0 1 5 (1%)UnstableFOLPD

0 3 7 0 1 0 12 (3%)

UnstableSOSPD

0 6 1 2 1 0 10 (2%)

Delaymodel

0 1 1 0 0 0 2 (0%)

Higherorder

0 0 4 0 0 0 4 (1%)

TOTAL 31 136 125 88 52 21 453

Table D: PI controller structure and tuning rules – a summary

Controller structureStable

FOLPDNon-model

specificIPD FOLIP

DSOSPD Other Total

G s KTsc c

i

( ) = +

1 1

85 19 20 6 10 12 152(92%)

G s K bT sc c

i

( ) = +

12 1 0 1 2 0 6

(4%)

)s(RK)s(EsT

11K)s(U ci

c α−

+= 1 1 2 2 1 3 10

(4%)

U s K Y sKT s

E scc

i

( ) ( ) ( )= − 0 1 1 0 0 0 2(1%)

G s KTs T sc c

i f

( ) = +

+1 1 1

1 0 0 1 0 0 0 1(0%)

Total 88 21 23 8 12 15 167

Table E: PID controller structure and tuning rules – a summary


FOLPDNon-model

specificIPD FOLIP

DSOSPD Other Total

G s KT s

T sc ci

d( ) = + +

1 1

57 25 5 1 27 11126

(45%)

G s KT s

T s

s TN

c ci

d

d( ) = + +

+

11

13 8 1 1 2 3 18

(6%)

G s KT s

T sT sc c

id

f

( ) = + +

+1 1 1

1 3 0 0 3 1 07

(3%)

G s KT s

T s b sa sc c

id( ) = + +

++

1 1 11

1

10 0 0 0 1 0

1(0%)

G s KT s

T s b sa s a sc c

id( ) = + +

++ +

1 1 11

1

1 22 3 0 0 0 0 0

3(1%)

G s K bT s

T sc ci

d( ) = + +

11 1 0 1 0 0

3(1%)

Subtotal 67 34 6 6 31 14 158(56%)

NT

s1

sT1sT

11K)s(Gd

d

icc

+

+

+= 20 1 5 5 7 5 43

(15%)

G s KT sT sN

T sT sN

c ci

d

d

d( ) =

+

+

+

+

1

1

1

10 0 0 1 0 0

1(0%)

G s KT s

NT sT sc c

i

d

d

( ) = +

++

1 1 1

1 0 0 0 0 1 01

(0%)

( )G s KT s

sTc ci

d( ) = +

+1 1 1 3 3 0 0 1 1

8(3%)

G s KT s

sTsTN

c ci

d

d( ) = +

+

+

11

11

1 1 1 1 0 26

(2%)

[ ]G s K

T ss s

sc c

i

m m

m

( ). .

.= +

+ +

+

1

1 1 05 0 0833

1 01

2 2

2

τ τ

τ1 0 0 0 0 0

1(0%)

Subtotal 25 5 6 7 9 8 60(22%)

U s KT s

E s T ssTN

Y sci

d

d( ) ( ) ( )= +

−

+1 1

15 0 0 0 0 0 5

(2%)

+−

+= )s(Y

NsT1

sT)s(E

sT1

1K)s(Ud

d

ic 2 0 0 0 4 0 6

(2%)


FOLPDNon-model

specificIPD FOLIP

DSOSPD Other Total

)s(Y

NsT

1

sT)s(E

sT1K)s(U

d

d

ic

+−

+= 6 0 0 0 0 0 6

(2%)

U s KT s

E s K T ssTN

Y sci

c d

d( ) ( ) ( )= +

−

+1 1

10 1 0 0 1 0 6

(2%)

( )U s K bTs

E sK T s

T s NY s K b Y sc

i

c d

dc( ) ( ) ( ) ( )= +

−

++ −

11

13 0 0 0 0 0

3(1%)

U s KT s

R sT sT sN

Y sci

d

d( ) ( ) ( )= +

−

+

+

11 1

1

6 0 0 0 0 0 6(2%)

U s KT s

E s K T sY sci

c d( ) ( ) ( )= +

−1 1

6 1 2 2 3 014

(5%)

[ ] ( )U s K bTs

R s Y s c T s Y sci

d( ) ( ) ( ) ( )= +

− − +

10 0 0 0 1 0

1(0%)

( )U sKT s

E s K T s Y sc

ic d( ) ( ) ( )= − +1

2 0 1 0 0 03

(1%)

)s(Rs

NT

1

sTK)s(E

sNT

1

sTsT

11K)s(U

d

dc

d

d

ic

+

β+α−

+++= 1 0 1 1 3 3 9

(3%)

( ) ( ) ( )U s K F R(s Y sT s

F R(s Y s T s F R(s Y sc pi

i d d( ) ) ( ) ) ( ) ) ( )= − + − + −1 0 1 0 1 1 0 3(1%)

)s(sYTK)s(R)1b(K)s(E)sT

11(K)s(U dcc

ic −−++= 0 0 1 0 0 1 2

(1%)

−

++

+

++= )s(Y

sT1sT4.01

)s(R1sT

1sT

sT1

1K)s(Ur

r

fd

ic

1 0 0 0 0 0 1(0%)

)s(YK)s(YsT1

sT4.01)s(R1sT

1sTsT

11K)s(U 0r

r

fd

ic −

−

++

+

++= 0 0 1 0 0 0 1

(0%)Subtotal 32 3 6 4 8 8 61

(22%)

Total 124 42 18 17 49 30 280

3. Tuning rules for PI and PID controllers

Table 1: PI tuning rules - FOLPD model - G sK e

sTmm

s

m

m

( ) =+

− τ

1– ideal controller G s K

Tsc ci

( ) = +

1 1

. 84 tuning

rules

Rule Kc Ti Comment

Process reactionZiegler and Nichols [8]

Model: Method 1 .

09. TK

m

m mτ 3 33. τmQuarter decay ratio.

τm

mT≤ 1

ατ

TK

m

m m

βτm

τm mT α β τm mT α β τm mT α β

0.2 0.68 7.14 1.1 0.90 1.49 2.0 1.20 1.000.3 0.70 4.76 1.2 0.93 1.41 2.2 1.28 0.950.4 0.72 3.70 1.3 0.96 1.32 2.4 1.36 0.910.5 0.74 3.03 1.4 0.99 1.25 2.6 1.45 0.880.6 0.76 2.50 1.5 1.02 1.19 2.8 1.53 0.850.7 0.79 2.17 1.6 1.06 1.14 3.0 1.62 0.830.8 0.81 1.92 1.7 1.09 1.10 3.2 1.71 0.810.9 0.84 1.75 1.8 1.13 1.06 3.4 1.81 0.801.0 0.87 1.61 1.9 1.17 1.03

Hazebroek and Van derWaerden [9]

Model: Method 1

ατ

= +0 5 01. .m

mTβ

ττ

=−m

m mT16 1 2. .τm

mT> 35.

Astrom and Hagglund [3] –page 138

Model: Not relevant

063. TK

m

m mτ 3 2. τm

Ultimate cycle Ziegler-Nichols equivalent

0 6. TK

m

m mτ 4τm

0% overshoot -

011 10. .< <τm

mTChien et al. [10] - regulator

Model: Method 107. TK

m

m mτ2 33.

τm

mK20% overshoot -

011 10. .< <τm

mT

Astrom and Hagglund [3] –regulator – page 150

Model: Method 1

07. TK

m

m mτ 2 3. τm 20% overshoot

035. TK

m

m mτ 117. Tm

0% overshoot -

011 10. .< <τm

mTChien et al. [10] - servo

Model: Method 1 0 6. TK

m

m mτ Tm

20% overshoot-

011 10. .< <τm

mT

Cohen and Coon [11]process reaction

Model: Method 1.

1 0 9 0083K

T

m

m

m

. .τ

+

TT T

T

m

m

m

m

m

m

m

333 031

1 2 22

2

. .

.

τ τ

τ

+

+

Quarter decay ratio

Rule Kc Ti Comment

ατ

TK

m

m m

βτm

τm mT α β τm mT α β

0.2 4.4 3.23 1.0 0.78 1.28

Two constraints criterion -Wolfe [12]

Model: Method 2.0.5 1.8 2.27 5.0 0.30 0.53

Decay ratio = 0.4; minimumerror integral (regulator

mode).

Two constraints criterion -Murrill [13] – page 356

Model: Method 3

09280 946

..

KT

m

m

mτ

TT

m m

m1078

0 583

.

.τ

Quarter decay ratio;minimum error integral

(servo mode).

01 10. .≤ ≤τm

mT

McMillan [14] – page 25Model: Method 3

K m

3τm Time delay dominant

processes

St. Clair [15] – page 22Model: Method 3

0333. TK

m

m mτTm Tm

mτ≤ 30.

Shinskey [15a]Model: Method 1

mm

m

KT667.0

τm78.3 τ

167.0T

m

m =τ

Regulator tuning Performance index minimisationMinimum IAE - Murrill [13] –

pages 358-363Model: Method 3

09840 986

..

KT

m

m

mτ

TT

m m

m0608

0 707

.

.τ

01 10. .≤ ≤

τm

mT

100. T Km m mτ 30. τm τm mT = 0 2.

104. T Km m mτ 2 25. τm τm mT = 0 5.

111. T Km m mτ 145. τm τm mT = 1

Minimum IAE - Shinskey[16] – page 123

Model: Method 6139. T Km m mτ τm τm mT = 2

0 95. T Km m mτ 34. τm τm mT = 01.Minimum IAE - Shinskey[17] – page 38

Model: Method 60 95. T Km m mτ 2 9. τm τm mT = 0 2.

Minimum IAE – Huang etal. [18]

Model: Method 6Kc

( )1 1 Ti( )1 01 1. ≤ ≤

τm

mT

0 6850214 0 346 1256

.. . .

KTT Tm

L

m

T m

m

m

m

− −τ

τ T TT T

m L

m

Tm

m

m

m

0 214

1977 055 1123

.

. . .

−τ

τTT T

L

m

m

m

≤ +2 641 016. .τ ;

τ m

mT≤ 0 35.

Minimum IAE – Yu [19]

(Load model = K e

sTL

s

L

L−

+

τ

1)

Model: Method 60874

0099 0159 1041.

. . .

KTT Tm

L

m

Tm

m

m

m

− + −τ

τ T TT T

m L

m

Tm

m

m

m

0 415

4 515 0 067 0 876

.

. . .

− +τ

τ 2 641 0 16 1. .τm

m

L

mTTT

+ ≤ ≤ ;

τ m

mT≤ 0 35.

1 KK T T T T

ecm

m

m

m

m

m

m

m

m

Tm

m( ). . .

. . . . . .10 9077 0 063 0 5961

1 6 4884 4 6198 0 8196 52132 7 2712 0 7241= + +

−

−

−

− −τ τ τ τ

τ

T TT T T T T Ti m

m

m

m

m

m

m

m

m

m

m

m

m

( ) . . . . . . .12 3 4 5 6

00064 3 9574 64789 9 4348 10 7619 7 5146 2 2236= + −

+

−

+

−

τ τ τ τ τ τ

Rule Kc Ti Comment

08710015 0 384 1055

.. . .

KTT Tm

L

m

Tm

m

m

m

− + −τ

τ T TT T

m L

m

Tm

m

m

m

0 444

0 217 0 213 0867

.

. . .

− −τ

τ 1 3< ≤TT

L

m

; τ m

mT≤ 0 35.Minimum IAE – Yu [19]

(Load model = K e

sTL

s

L

L−

+

τ

1)

(continued)Model: Method 6

05130 218 1451

.. .

KTT Tm

L

m

Tm

m

m

m

−τ

τ T TT T

m L

m

Tm

m

m

m

0 670

0 003 0 084 0 56

.

. . .

− −τ

ττ m

mT> 035.

TK T

m

m m

m

mττ074 0 3. .+

143. Tm τm mT < 0 2.

ατ

TK

m

m m

βτm

τm mT α β τm mT α β τm mT α β

0.2 0.80 7.14 0.7 0.96 2.44 2.0 1.46 1.180.3 0.83 5.00 1.0 1.07 1.85 3.0 1.89 0.95

Minimum ISE - Hazebroekand Van der Waerden [9]

Model: Method 1

0.5 0.89 3.23 1.5 1.26 1.41 5.0 2.75 0.81Minimum ISE - Murrill [13]

– pages 358-363Model: Method 3

13050 959

..

KT

m

m

mτ

TT

m m

m0492

0 739

.

.τ

01 10. .≤ ≤

τm

mT

12790 945

..

KT

m

m

mτ

TT

m m

m0535

0 586

.

.τ

01 10. .≤ ≤

τm

mTMinimum ISE – Zhuang andAtherton [20]

Model: Method 61346

0 675.

.

KT

m

m

mτ

TT

m m

m0552

0 438

.

.τ

11 2 0. .≤ ≤

τm

mT

0 9210 181 0 205 1 214

.. . .

KTT Tm

L

m

Tm

m

m

m

− −τ

τ T TT T

m L

m

Tm

m

m

m

0 430

0 954 0 49 0 639

.

. . .

−τ

τTT T

L

m

m

m≤ +2 310 0 077. .

τ ;

τ m

mT≤ 0 35.

11570 045 0 344 1 014

.. . .

KTT Tm

L

m

Tm

m

m

m

− + −τ

τ T TT T

m L

m

Tm

m

m

m

0 359

2 532 0 292 0 899

.

. . .

− −τ

τ 2 310 0 077 1. .τ m

m

L

mTTT

+ ≤ ≤ ;

τ m

mT≤ 0 35.

1070 065 0 234 1047

.. . .

KTT Tm

L

m

Tm

m

m

m

− + −τ

τ T TT T

m L

m

Tm

m

m

m

0 347

1112 0 094 0 898

.

. . .

− −τ

τ 1 3< ≤TT

L

m

; τ m

mT≤ 0 35.

Minimum ISE – Yu [19]

(Load model = K e

sTL

s

L

L−

+

τ

1)

Model: Method 6

12890 04 0 067 0889

.. . .

KTT Tm

L

m

Tm

m

m

m

+ −τ

τ T TT T

m L

m

Tm

m

m

m

0596

0 372 0 44 0 46

.

. . .

−τ

ττ m

mT> 035.

Minimum ITAE - Murrill[13] – pages 358-363

Model: Method 3

08590 977

..

KT

m

m

mτ

TT

m m

m0674

0 680

.

.τ

01 10. .≤ ≤

τm

mT

05980 272 0 254 1341

.. . .

KTT Tm

L

m

Tm

m

m

m

− −τ

τ T TT T

m L

m

Tm

m

m

m

0805

0 304 0 112 0 196

.

. . .

−τ

τTT T

L

m

m

m≤ +2 385 0 112. .

τ ;

τ m

mT≤ 0 35.

0 7350 011 1945 1 055

.. . .

KTT Tm

L

m

Tm

m

m

m

− − −τ

τ T TT T

m L

m

Tm

m

m

m

0 425

5 809 0 241 0 901

.

. . .

− +τ

τ 2 385 0112 1. .τ m

m

L

mTTT

+ ≤ ≤ ;

τ m

mT≤ 0 35.

Minimum ITAE – Yu [19]

(Load model = K e

sTL

s

L

L−

+

τ

1)

Model: Method 6

0 7870 084 0154 1 042

.. . .

KTT Tm

L

m

Tm

m

m

m

+ −τ

τ T TT T

m L

m

Tm

m

m

m

0 431

0148 0 365 0 901

.

. . .

− −τ

τ 1 3< ≤TT

L

m

; τ m

mT≤ 0 35.

Rule Kc Ti Comment

Minimum ITAE – Yu [19]

(Load model = K e

sTL

s

L

L−

+

τ

1)


08780172 0 057 0 909

.. . .

KTT Tm

L

m

Tm

m

m

m

− −τ

τ T TT T

m L

m

Tm

m

m

m

0 794

0 228 0257 0 489

.

. . .

−τ

ττ m

mT> 035.

10150 957

..

KT

m

m

mτ

TT

m m

m0667

0 552

.

.τ

01 10. .≤ ≤

τm

mTMinimum ISTSE - Zhuangand Atherton [20]Model: Method 6 1065

0 673.

.

KT

m

m

mτ

TT

m m

m0687

0 427

.

.τ

11 2 0. .≤ ≤

τm

mT

10210 953

..

KT

m

m

mτ

TT

m m

m0629

0 546

.

.τ

01 10. .≤ ≤

τm

mTMinimum ISTES – Zhuangand Atherton [20]

Model: Method 61076

0 648.

.

KT

m

m

mτ

TT

m m

m0650

0 442

.

.τ

11 2 0. .≤ ≤

τm

mT

Servo tuning Performance index minimisationMinimum IAE – Rovira et al.

[21]Model: Method 3

07580 861

..

KT

m

m

mτ

T

T

m

m

m

1020 0323. .− τ 01 10. .≤ ≤τm

mT

Minimum IAE - Huang et al.[18]

Model: Method 6Kc

( )2 2 Ti( )2 01 1. ≤ ≤

τm

mT

09800.892

.K

T

m

m

mτ

T

T

m

m

m

0690 0155. .− τ 01 10. .≤ ≤τm

mTMinimum ISE - Zhuang andAtherton [20]

Model: Method 6 10720 560

..

KT

m

m

mτ

T

T

m

m

m

0648 0114. .−τ 11 2 0. .≤ ≤

τm

mT

0 7388 03185. .τm m

m

mTTK

+

K K

T

c m

m m m

0 5291 000030822

. .τ τ

−

0 01 0 2. .≤ ≤τm

mTMinimum ISE – Khan andLehman [22]

Model: Method 6 0 808 0 511 0 255. . .τ τm m m m

m

mT T

TK

+ −

K K

T T

c m

m m m m m m

0 095 0 846 0 3812

. . .

τ τ τ τ+ −

0 2 20. ≤ ≤τm

mT

Minimum ITAE – Rovira etal. [21]

Model: Method 3

05860 916

..

KT

m

m

mτ

T

T

m

m

m

1030 0165. .−τ 01 10. .≤ ≤

τm

mT

2 KK T T T T

ecm

m

m

m

m

m

m

m

m

Tm

m( ). . .

. . . . . .21 0169 3 5959 3 6843

1 130454 9 0916 0 3053 11075 2 2927 4 8259= − − +

+

−

+

−τ τ τ τ

τ

T TT T T T T Ti m

m

m

m

m

m

m

m

m

m

m

m

m

( ) . . . . . . .2

2 3 4 5 6

0 9771 0 2492 34651 7 4538 82567 4 7536 11496= − +

−

+

−

+

τ τ τ τ τ τ

Rule Kc Ti Comment

07120 921

..

KT

m

m

mτ

T

T

m

m

m

0968 0 247. .−τ 01 10. .≤ ≤

τm

mTMinimum ISTSE - Zhuangand Atherton [20]

Model: Method 6 07860 559

..

KT

m

m

mτ

T

T

m

m

m

0883 0158. .−τ 11 2 0. .≤ ≤

τm

mT

05690 951

..

KT

m

m

mτ

T

T

m

m

m

1023 0179. .− τ 01 10. .≤ ≤τm

mTMinimum ISTES – Zhuangand Atherton [20]Model: Method 6

06280.583

.K

T

m

m

mτ

T

T

m

m

m

1007 0167. .−τ 11 2 0. .≤ ≤

τm

mT

Direct synthesisHaalman [23]

Model: Method 62

3T

Km

m mτ Tm

Closed loop sensitivityM s = 19. . (Astrom and

Hagglund [3])Chen and Yang [23a]Model: Method 25

mm

m

KT7.0τ Tm

26.1Ms = ; 24.2A m = ;0

m 50=φ

Minimum IAE – regulator -Pemberton [24], Smith andCorripio [25] – page 343-

346. Model: Method 6

TK

m

m mτTm 01 0 5. .≤ ≤

τm

mT

Minimum IAE – servo -Smith and Corripio [25] –page 343-346. Model:

Method 6

35

TK

m

m mτ Tm01 05. .≤ ≤

τm

mT

5% overshoot – servo –Smith et al. [26] – deduced

from graph.Model: Method not stated

052. TK

m

m mτ

Tm0 04 14. .≤ ≤

τm

mT

1% overshoot – servo –Smith et al. [26] – deduced

from graphModel: Method not stated

044T. m

m mK τ

Tm0 04 14. .≤ ≤

τm

mT

5% overshoot - servo -Smith and Corripio [25] –page 343-346. Model:

Method 6

TK

m

m m2 τ Tm

5% overshoot - servo -Hang et al. [27]

Model: Method 1

1325

TK

m

m mτ Tm

mm

m

KT368.0

τ Tm

Closed loop responseovershoot = 0%

(Model: Method 26 – Miluseet al. [27b])

mm

m

KT514.0

τ Tm


Miluse et al. [27a]

Model: Method not stated

mm

m

KT581.0

τ Tm


Rule Kc Ti Comment

mm

m

KT641.0

τ Tm


mm

m

KT696.0

τ Tm


mm

m

KT748.0

τ Tm


mm

m

KT801.0

τ Tm


mm

m

KT853.0

τ Tm


mm

m

KT906.0

τ Tm


mm

m

KT957.0

τ Tm


Miluse et al. [27a] -continued

Model: Method not stated

mm

m

KT008.1

τ Tm


Kc( 3) Ti

( )3

Pole is real and has max.attainable multiplicityRegulator – Gorecki et al.

[28](considered as 2 rules)

Model: Method 6 Kc( )4 3 Ti

( )4

Low freq. part of magnitudeBode diagram is flat

Chiu et al. [29]Model: Method 6 ( )

λλτ

TK

m

m m1+ Tm

λ variable; suggestedvalues: 0.2, 0.4, 0.6, 1.0.

Astrom et al. [30]

Model: Method 22

3

13

KTm

m

m

+

τ Tm Honeywell UDC 6000controller

Davydov et al. [31]

Model: Method 12

1

1905 0 826KTm

m

m

. .τ

+

0153 0 362. .τm

mmT

T+

Closed loop responsedamping factor =

0.9; 02 1. ≤ ≤τm mT .

3 KK

TT

ecm

m

m

m

m

T Tm

m

m

m( )32

22

222

22

1

2

= +

−

+

− −

ττ

τ τ

TT

T T T T T

i m

m

m

m

m

m

m

m

m

m

m

m

m

( )3

2

2 3 2 2

12

32 2

22

22

=

+

+

+

+

− +

+

τ

τ

τ τ τ τ τ

KK

T T T

T Tc

m

m

m

m

m

m

m

m

m

m

m

( )4

2 3

2

11 3 6 6

4 1 3 3

=+ +

+

+ +

τ τ τ

τ τ

, T

T T T

T Ti m

m

m

m

m

m

m

m

m

m

m

( )4

2 3

2

1 3 6 6

3 1 2 2

=+ +

+

+ +

ττ τ τ

τ τ

Rule Kc Ti Comment

0 368.T

Km

m mτ Tm

Closed loop responsedamping factor = 1Schneider [32]

Model: Method 6 0 403.T

Km

m mτ Tm

Closed loop responsedamping factor = 0.6

McAnany [33]

Model: Method 5

( )( )

1 44 0 72 0 43 2 14

12 0 36 22

. . . .

. .

T T

K

m m m m

m m m

+ − −

+ +

τ τ

τ τ( )K

Tm m m

m m

556 2

4 128 2 4

2.

. .

+ +

+ −

τ τ

τ

Closed loop time constant =167. Tm .

Leva et al. [34]

Model: Method 16

ω ωω

cn m

m

cn m

cn i

TK

TT

11

2 2

2 2

++

( )tan tanφπ

τ ω ω

ω

m m cn cn m

cn

T− + +

−

21

ω

φτ

πφ

τcn

mm

mm

m

T

=

− − −

−2822

1. tan

TK T T

m

m m m

m

m

0 3852 0 7230 404 2

. ..

ττ

+ −

K K

T T

c m

m m m m

0 4104 0 00024 0 5252 2

. . .τ τ

− −

0 01 0 2. .≤ ≤τm

mTKhan and Lehman [22]

Model: Method 6TK T T

m

m m m m m

0 404 0 256 01275. . .τ τ

+ −

K K

T T

c m

m m m m m m

0 719 0 0808 0 3242

. . .τ τ τ τ

+ −

0 2 20. ≤ ≤τm

mT

1048. TK

m

m mτTm Gain Margin = 1.5

Phase Margin = 30 0

07854. TK

m

m mτTm Gain Margin = 2

Phase Margin = 450

0524. TK

m


Phase Margin = 600

0 393. TK

m


Phase Margin = 67 50.

Hang et al. [35, 36]

Model: Method 11Or

Model: Method 14

0314T. m

m mK τTm Gain Margin = 5

Phase Margin = 720

ωp m

m m

T

K A

1

24 1

2

ωω τ

πpp m

mT− +

( )( )

ωφ π

τp

m m m m

m m

A A A

A=

+ −

−

0 5 1

12

.

Given A m , ISE is minimised when φτ

m mm

m

AT

= − +688884 34 3534 91606. . . for servo

tuning (Ho et al [104]).

Gain and phase margin – Hoet al. [37]

(considered as 2 rules)

Model: Method 6

Given A m , ISE is minimised when ( )φ τm m m mA T= 459848 0.2677 0. 2755. for regulator

tuning (Ho et al [104]).

Tan et al. [39]

Model: Method 10

( )( )

β ω β ω

β ω

φ φ

φ

T T

A T

i m

m i

1

1

2

2

+

+

[ ]1

1βω β ω βτ ω φφ φ φtan tan− − −− Tm m

ω ωφ < u

βτ

= <08 05. , .m

mT;

βτ

= >0 5 05. , .m

mT

1

35135 0. ( )G jp ω

4 6

135 0

.ω

τm

mT≤ 01.

1

2 828135 0. ( )G jp ω

4

135 0ω01 015. .< ≤

τm

mT

Symmetrical optimumprinciple - Voda and Landau

[40]

Model: Method not relevant

1

4 6 0 6135 0. ( ) .G j Kp mω − [ ]

115 0 75

2 3 0 3

135

135 135

0

0 0

. ( ) .

. ( ) .

G j K

G j K

p m

p m

ω

ω ω

+

− 0 15 1. < ≤τm

mT

Rule Kc Ti Comment

Friman and Waller [41]Model: Method 6

0 2333

135 0

.

( )G jp ω

1

135 0ωτm mT> 2 . Gain margin = 3;

Phase margin = 450

Voda and Landau [40]

Model: Method 6T

Km

m m2 τTm

Phase margin = 600 ;

0 25 1. ≤ ≤τm

mT

Smith [42]Model: Method not

specified

0 35.Km

042. τm

6 dB gain margin - dominantdelay process

Kc( )5 Ti

( )5 τm

mT≤ 1Modulus optimum principle

- Cox et al. [43]4

Model: Method 17

0 5. TK

m

m mτ Tm

τm

mT> 1

0 019952 0 20042. .ττ

m m

m m

TK

+ 0 099508 0 999560 99747 8742510 5

. .. . .

ττ

τm m

m mm

TT

+− −

Closed loop time constant =4τm

0 05548 0 33639. .ττ

m m

m m

TK

+ 016440 0 995580 98607 15032 10 4

. .. . .

ττ

τm m

m mm

TT

+− −


0 092654 0 43620. .ττ

m m

m m

TK

+ 0 20926 0985180 96515 4 255010 3

. .. . .

ττ

τm m

m mm

TT

++ −

Closed loop time constant =133. τm

012786 051235. .ττ

m m

m m

TK

+ 0 24145 0 967510 93566 2 298810 2

. .. . .

ττ

τm m

m mm

TT

++ −

Closed loop time constant =τm

016051 0 57109. .ττ

m m

m m

TK

+ 0 26502 0 94291089868 6 935510 2

. .. . .

ττ

τm m

m mm

TT

++ −


Cluett and Wang [44]

Model: Method 6

019067 0 61593. .ττ

m m

m m

TK

+ 0 28242 0 912310 85491 015937. .. .

ττ

τm m

m mm

TT

++


Abbas [45]

Model: Method 6

0148 0186

497 0 464

1

0.590

. .

(0. . )

.045

+

−

−τm

m

m

T

K V

Tm m+ 0 5. τV = fractional overshoot,

0 0 2≤ ≤V .

01 5 0. .≤ ≤τm

mT

Bi et al. [46]Model: Method 20

05064. TK

m

m mτ Tm

4 KK

T T TT Tc

m

m m m m m m

m m m m m

( ) . . ..

53 2 2 3

2 2 3

0 5 05 01670667

=+ + +

+ +

τ τ τ

τ τ τ, T

T T TT Ti

m m m m m m

m m m m

( ) . ..

53 2 2 3

2 2

05 01670 5

=+ + +

+ +

τ τ τ

τ τ

Rule Kc Ti Comment

Wang and Shao [47]

Model: Method 6

)a5(cK 5 )a5(

iTλ = inverse of the maximumof the absolute real part of

the open loop transferfunction; ]5.2,5.1[=λ

Robust

( )TK

m m

m m

++05. τ

λ τ Tm m+ 0 5. τBrambilla et al. [48] -

Model: Method 6 Closed loop response has less than 5% overshoot with no model uncertainty:

λ = 1 , 01 1. ≤ ≤τm

mT; λ

τ= −1 05 10. log m

mT, 1 10< ≤

τm

mT

TK

m

mλ Tm λ τ≥ 17. m , λ > 01. Tm .Rivera et al. [49]

Model: Method 6 22T

Km m

m

+ τλ Tm m+ 0 5. τ λ τ≥ 17. m , λ > 01. Tm .

Chien [50]

Model: Method 6

TK

m

m m( )τ λ+ Tm

λ = Tm [50];

λ τ> +Tm m , τm mT<<(Thomasson [51])

Thomasson [51]

Model: Method not defined

ττ λ

m

m mK2 ( )+ 05. τm

τm mT>> ; λ = desired

closed loop time constant

59

TKm

m mτ 5τm

τm

mT< 0 33.Fruehauf et al. [52]

Model: Method 1 TKm

m m2τ Tm

τm

mT≥ 0 33.

mm

m

KT50.0

τ Tm

14.3A m = , 0m 4.61=φ ,

00.1Ms =

mm

m

KT61.0

τ Tm

58.2A m = , 0m 0.55=φ ,

10.1Ms =

mm

m

KT67.0

τ Tm

34.2A m = , 0m 6.51=φ ,

20.1Ms =

mm

m

KT70.0

τ Tm

24.2A m = , 0m 0.50=φ ,

26.1M s =

Chen et al. [53]

Model: Method 6

mm

m

KT72.0

τ Tm

18.2A m = , 0m 7.48=φ ,

30.1Ms =

5

ω−ω

ωλ=

0

0

0 90902

901

)a5(c

1)(f

)(f1

K ,

( )[ ])Ttancos(T)Ttansin()}T1{T(

T1

K)(f m90

1m90

2m90m90

1m90m

2m

290m5.12

m2

90

m901 000000

0

0 ω−τω−ω−ω−τω−τω++ω+

=ω −−

[ ]2m90m90

1m90m

2m

290m2

m2

90902 T)Ttancot()}T1{T(

T1

1)(f 0000

0

0 ω+ω−τω−τω++ω+

−=ω −

[ ])Ttansin(])T1(T[)Ttancos(T

)Ttansin()T21()Ttancos()T1(TT

m901

m90m2

m2

90m2

90m901

m902

m3

90

m901

m902

m2

90m901

m90m2

m2

90m90)a5(i

0000000

0000000

ω−τω−τω++ω+ω−τω−ω−

ω−τω−ω++ω−τω−τω++ω=

−−

−−

Rule Kc Ti Comment

mm

m

KT76.0

τ Tm

07.2A m = , 0m 5.46=φ ,

40.1Ms =

Chen et al. [53]- continued

Model: Method 6

mm

m

KT80.0

τ Tm

96.1A m = , 0m 1.44=φ ,

50.1Ms =

αK m

βTm

τm mT α β τm mT α β

0.5 0.9 1.3 2.0 0.45 2.01.0 0.6 1.6 10.0 0.4 7.0

20% uncertainty in theprocess parameters

0.5 0.7 1.3 2.0 0.4 2.21.0 0.47 1.7 10.0 0.35 7.5


0.5 0.47 1.3 2.0 0.32 2.41.0 0.36 1.8 10.0 0.3 8.5


0.5 0.4 1.3 2.0 0.3 2.4

Ogawa [54] – deduced fromgraphs

Model: Method 6

1.0 0.33 1.8 10.0 0.29 9.060% uncertainty in the

process parameters

Lee et al. [55]

Model: Method 6( )

TK

i

m mλ τ+ ( )Tm

m

m

++

τλ τ

2

2

Desired closed loop

response = ( )

es

ms−

+

τ

λ 1,

λ τ= 0 333. m

Isaksson and Graebe [56]Model: Method 6

TK

m m

m

+ 0 25. τλ

Tm m+ 025. τ Tm m> τ

Chun et al. [57]

Model: Method 21

( )( )

T

Km m

m m

τ λ λ

τ λ

+ −

+

2 2

2

( )T

Tm m

m m

τ λ λ

τ

+ −

+

2 2

λ = 0 4. Tm

Ultimate cycle

McMillan [58]

Model: Method 1 orMethod 18

1881 1

10 65

..K

T

TT

m

m

m m

m m

τ

τ+

+

166 10

..65

ττm

m

m m

TT

++

Tuning rules developedfrom K Tu u,

Regulator – minimum IAE –Shinskey [59] – page 167.

Model: Method notspecified

KT

u

u

m

305 0 35. .−τ

TT T

uu

m

u

m

0 87 0 855 0 1722

. . .− +

τ τ

055. K u 0 78. Tu τ m mT = 0 2.Regulator – minimum IAE –Shinskey [17] – page 121.


0 48. Ku uT47.0 τm mT = 1

0 5848. Ku 0 81. Tu τ m mT = 0 2.

0 5405. K u 0 66. Tu τ m mT = 0 5.

0 4762. K u 0 47. Tu τm mT = 1

Regulator – minimum IAE –Shinskey [16] – page 148

Model: Method 60 4608. K u 0 37. Tu τ m mT = 2

Regulator – nearly minimumIAE, ISE, ITAE – Hwang

[60]Model: Method 8

( )1 1− µ K u ,

( )µ

ω τ ω τ1

2 2114 1 0 482 0068

1=

− +

+

. . .u m u m

u m u mK K K K

K Kc u uµ ω2 ,

( )µ

ω τ ω τ2

2 20 0694 1 2 1 0 367

1=

− + −

+

. . .u m u m

u m u mK K K K

Decay ratio = 0.15 -

01 2 0. .≤ ≤τm

mT

Rule Kc Ti Comment

( )1 1− µ K u ,

( )µ

ω τ ω τ1

2 2109 1 0 497 0 0724

1=

− +

+

. . .u m u m

u m u mK K K K

K Kc u uµ ω2 ,

( )µ

ω τ ω τ2

2 20054 1 2 54 0457

1=

− + −

+

. . .u m u m

u m u mK K K K

Decay ratio = 0.2 -

01 2 0. .≤ ≤τm

mTRegulator – nearly minimum

IAE, ISE, ITAE – Hwang[60]


( )1 1− µ K u ,

( )µ

ω τ ω τ1

2 2103 1 051 00759

1=

− +

+

. . .u m u m

u m u mK K K K

K Kc u uµ ω2 ,

( )µ

ω τ ω τ2

2 200386 1 326 0 6

1=

− + −

+

. . .u m u m

u m u mK K K K


01 2 0. .≤ ≤τm

mT

( )1 1− µ K u ,

( )µ

ω τ ω τ1

2 2107 1 0 466 00667

1=

− +

+

. . .u m u m

u m u mK K K K

K Kc u uµ ω2 ,

( )µ

ω τ ω τ2

2 200328 1 221 0 338

1=

− + −

+

. . .u m u m

u m u mK K K K

Decay ratio = 0.1, r = 0.5 -

01 2 0. .≤ ≤τm

mT

( )1 1− µ K u ,

( )µ

ω τ ω τ1

2 21111 0467 00657

1=

− +

+

. . .u m u m

u m u mK K K K

K Kc u uµ ω2 ,

( )µ

ω τ ω τ2

2 200477 1 207 0333

1=

− + −

+

. . .u m u m

u m u mK K K K

Decay ratio = 0.1, r = 0.75 -

01 2 0. .≤ ≤τm

mT

Servo – small IAE – Hwang[60]

Model: Method 8

r = parameter related to theposition of the dominant real

pole.( )1 1− µ K u ,

( )µ

ω τ ω τ1

2 2114 1 0466 00647

1=

− +

+

. . .u m u m

u m u mK K K K

K Kc u uµ ω2 ,

( )µ

ω τ ω τ2

2 200609 1 197 0323

1=

− + −

+

. . .u m u m

u m u mK K K K

Decay ratio = 0.1, r = 1.0 -

01 2 0. .≤ ≤τm

mT

Hang et al. [65]


56

12 211 13

37 4

15 1411 13

37 4

++

−

++

−

τ

τ

τ

τ

m

m

m

m

m

m

m

m

u

T

T

T

T

Ku

m

m

m

m

T14

T37

13T

11

154

2.0

+

−τ

+τ

016 096. .≤ <τm

mT;

Servo response: 10%overshoot, 3% undershoot

Servo – minimum ISTSE –Zhuang and Atherton [20]

Model: Method not relevant0 361. K u ( ) uum T1KK935.1083.0 + 01 2 0. .≤ ≤

τm

mT

Regulator – minimum ISTSE– Zhuang and Atherton [20]Model: Method not relevant

1892 02443249 2 097. .. .

K KK K

Km u

m uu

++

0 706 0 2270 7229 12736

. .. .

K KK K

Km u

m uu

−+

01 2 0. .≤ ≤

τm

mT

Servo – nearly minimum IAEand ITAE – Hwang and

Fang [61]Model: Method 9

0 438 0110 0 03762

. . .− +

τ τm

m

m

muT T

K( / )

. . .

K K

T T

c u u

m

m

m

m

ω

τ τ0 0388 0108 0 0154

2

+ −

01 2 0. .≤ ≤τm

mT;

decay ratio = 0.03

Regulator – nearly minimumIAE and ITAE – Hwang and


0 515 0 0521 0 02542

. . .− +

τ τm

m

m

muT T

K( / )

. . .

K K

T T

c u u

m

m

m

m

ω

τ τ0 0877 0 0918 0 0141

2

+ −

01 2 0. .≤ ≤τm

mT;

decay ratio = 0.12

SimultaneousServo/regulator – Hwang

and Fang [61]Model: Method 9

0 46 0 0835 0 03052

. . .− +

τ τm

m

m

muT T

K( / )

. . .

K K

T T

c u u

m

m

m

m

ω

τ τ0 0644 0 0759 0 0111

2

+ −

01 2 0. .≤ ≤τm

mT


sTmm

s

m

m

( ) =+

− τ

1– Controller G s K b

T sc ci

( ) = +

1. 2 tuning rules

Rule Kc Ti Comment

Direct synthesis (Maximum sensitivity)

029 2 7 3 7 2

. . .e TK

m

m m

− +τ τ

τ,

( )τ τ τ= +m m mT

8 9 6 6 3 0 2

. . .τ τ τme− + or

0 79 14 2 4 2

. . .T em− +τ τ

b e= +0 81 0 73 1 9 2

. . .τ τ

Ms = 1.4 , 014 55. .≤ ≤τm

mT

Astrom and Hagglund [3] -dominant pole design –

page 205-208

Model: Method 3 or 4078 4 1 5 7 2

. . .e TK

m

m m

− +τ τ

τ8 9 6 6 3 0 2

. . .τ τ τme− + or

0 79 14 2 4 2

. . .T em− +τ τ

b e= −0 44 0 78 0 45 2

. . .τ τ ;

Ms = 2.0, 014 55. .≤ ≤τm

mT

Astrom and Hagglund [3] -modified Ziegler-Nichols –

page 208Model: Method 3 or 4

04. TK

m

m mτ0 7. Tm b = 0.5; 01 2. ≤ ≤

τm

mT


sTmm

s

m

m

( ) =+

− τ

1– Two degree of freedom controller:

)s(RK)s(EsT

11K)s(U ci

c α−

+= . 1 tuning rule

Rule Kc Ti Comment

Minimum servo/regulator Performance index minimisation

−τ

+002434.0

T

7382.01098.0

K1

m

mm

)b5(iT 6Taguchi and Araki [61a]

Model: ideal process2

m

m

m

m

T06568.0

T4242.06830.0

τ+τ−=α

0.1Tm

m ≤τ

.

Overshoot (servo step)%20≤ ; settling time

≤ settling time of tuningrules of Chien et al. [10]

6

τ+

τ−τ+=3

m

m

2

m

m

m

mm

)b5(i T

205.1T

058.3T

171.306216.0TT

Table 4: PI tuning rules - non-model specific – controller G s KTsc c

i

( ) = +

1 1 . 20 tuning rules

Rule Kc Ti Comment

Ultimate cycle

Ziegler and Nichols [8] 0 45. Ku 0 83. Tu Quarter decay ratio

Hwang and Chang [62]0 45. Ku 1 522 5 22

1 1p T. .−

p T1 1, = decay rate, period

measured underproportional control when

K Kc u= 05.

** Hang et al. [36] 0 25. Ku 0 2546. Tu dominant time delay process

McMillan [14] – page 90 0 3571. K u Tu

Pessen [63] 0 25. Ku 0 042. K Tu u dominant time delay process

0 4698. K u 0 4373. Tu Gain margin = 2, phasemargin = 200

01988. K u 0 0882. Tu Gain margin = 2.44, phasemargin = 610


0 2015. Ku 01537. Tu Gain margin = 3.45, phasemargin = 46 0

Parr [64] – page 191 05. Ku 043. Tu Quarter decay ratio

Yu [122] – page 11 0 33. Ku 2Tu

Other tuning rulesParr [64] – page 191 0667 25%. K T25% Quarter decay ratio

042 25%. K T25% ‘Fast’ tuningMcMillan [14] – pages 42-43 0 33 25%. K T25% ‘Slow’ tuning

Parr [64] – page 1920333.

( )G jp uω 2Tu Bang-bang oscillation test

0 5

1350

.

( )G jp ω

4

135 0ωAlfa-Laval Automation

ECA400 controllerHagglund and Astrom [66]

0 25

1350

.

( )G jp ω

16

135 0

.ω

Alfa-Laval AutomationECA400 controller - process

has a long delay

Rule Kc Ti Comment

Leva [67]( )

( )tan .

( ) tan .

φ φ π

ω φ φ π

ω

ω

m p

p m pG j

− −

+ − −

0 5

1 052

( )tan .φ φ π

ωωm p− − 05 φ φ πωm p> + 05. ,

φ ωp = process phase at

frequency ω

Astrom [68]sin

( )

φ

ωm

pG j 90 0

tanφω

m

90 0

Calcev and Gorez [69]1

2 2 G jp u( )ω

1ω u

φm = 450 ,‘small’ τm

φm = 150 , ‘large’ τm

Cox et al. [70] 020. sinVTAu mφ 016. tanTu mφ V = relay amplitude, A = limit

cycle amplitude.

Direct synthesis

Vrancic et al. [71]05 3

1 2 3

. AA A K Am−

1 AA

3

2

A m m≥ ≥2 600, φ

Vrancic [72]05 3

1 2 3

. AA A K Am− 333. τm

Modified Ziegler-Nicholsprocess reaction method

Friman and Waller [41]0 4830

150 0

.

( )G jp ω

37321

150 0

.ω

Gain margin = 2,Phase margin = 300

2mu

mu

KK72.1KK18.1 −

umu

mu

)17.0KK33.0(72.1KK18.1

ω−− 5.0KK1.0 mu ≤≤

2mu

mu

KK36.0KK50.0 −

umu

mu

)17.0KK33.0(36.0KK50.0

ω−− 5.0KK mu >

2m135

m135

KK

160KK20

0

0 −

um135

m135

)175.0KK315.0(

160KK20

0

0

ω−

− 1.0KK mu < ;

1.0KK m135 0 ≤

Kristiansson andLennartson [158a]

2m135

m135

KK

6.13KK4.5

0

0 −

um135

m135

)2.3KK32.1(

6.13KK4.5

0

0

ω−

− 1.0KK mu < ;

1.0KK m135 0 >

1 ( )A y1 1= ∞ , ( )A y2 2= ∞ , ( )A y3 3= ∞

Alternatively, if the process model is G s Kb s b s b sa s a s a s

em ms m( ) =

+ + ++ + +

−11

1 22

33

1 23

33

τ , then

( )A K a bm m1 1 1= − + τ , ( )A K b a A a bm m m2 2 2 1 1 1205= − + − +τ τ. ,

( )A K a b A a A a b bm m m m3 3 3 2 1 1 2 2 12 305 0167= − + − + − +τ τ τ. .

Table 5: PI tuning rules - non-model specific – controller G s K bT sc c

i

( ) = +

1. 1 tuning rule

Rule Kc Ti Comment


0 053 2 9 2 6 2

. . .K euκ κ− ,

κ = 1 K Km u0 90 4 4 2 7 2

. . .T eu− +κ κ

b e= − +11 0 0061 18 2

. . .κ κ ;0 < < ∞K Km u .

maximum sensitivity Ms

=1.4

Astrom and Hagglund [3] -Ms = 1.4 – page 215

013 1 9 13 2

. . .K euκ κ− 0 90 4 4 2 7 2

. . .T eu− +κ κ

b e= −048 0 40 017 2

. . .κ κ ;0 < < ∞K Km u .

Ms = 2.0

Table 6: PI tuning rules - non-model specific – controller ( )U s KT s

E s K b R sci

c( ) ( ) ( )= +

+ −1 1 1 . 1 tuning rule

Rule Kc Ti Comment

Direct synthesis

Vrancic [72] 2 Kc( )6 ( )

A

KK

K Kbm

c

c m

1

6

6 221

2 21+ + −

( )

( )[ ]b = 0 5 08. , . - good servo

and regulator response

2 ( ) ( ) ( )

( )( )K

A A K A K A A A b A K A A K A A

b K A A K A Ac

m m m m

m m

( )6 1 2 3 3 1 22 2

32

3 13

1 2

2 23 1

31 2

1 2

1 2=

− − − − − + −

− + −, K A A Am 3 1 2 0− <

( ) ( ) ( )( )( )

KA A K A K A A A b A K A A K A A

b K A A K A Ac

m m m m

m m

( )6 1 2 3 3 1 22 2

32

3 13

1 2

2 23 1

31 2

1 2

1 2=

− + − − − + −

− + −, K A A Am 3 1 2 0− >

y t Ky

udm

t

1

0

( )( )

= −

∫

ττ

∆ , ( )y t A y d

t

2 1 1

0

( ) ( )= −∫ τ τ , ( )y t A y dt

3 2 2

0

( ) ( )= −∫ τ τ

Table 7: PI tuning rules - non-model specific – controller U s K Y sKT s

E scc

i

( ) ( ) ( )= − . 1 tuning rule

Rule Kc Ti Comment

Direct synthesis

Chien et al. [74] ( )− + +

+ +

T T T T

K T TCL CL m m m

m CL CL m m

2

2 2

1414

1414

.

.

τ

τ τ

− + +

+

T T T T

TCL CL m m m

m m

2 1414. τ

τ

Underdamped systemresponse - ξ = 0 707. .

τm mT> 0 2.

Table 8: PI tuning rules - IPD model G sK e

smm

s m

( ) =− τ

- controller G s KTsc c

i

( ) = +

1 1

. 21 tuning rules

Rule Kc Ti Comment

Process reactionZiegler and Nichols [8]

Model: Method 1.0 9.

Km mτ 333. τm

Quarter decay ratio

0 6.Km mτ 2 78. τm

Decay ratio = 0.4; minimumerror integral (regulator

mode).Two constraints method –

Wolfe [12]

Model: Method 1 087.Km mτ

4 35. τm

Decay ratio is as small aspossible; minimum error

integral (regulator mode).

Tyreus and Luyben [75]

Model: Method 2 or 30 487.Km mτ

8 75. τm

Maximum closed loop logmodulus = 2dB ; closed loop

time constant = τm 10


Model: Not relevant

0 63.Km mτ 3 2. τm

Ultimate cycle Ziegler-Nichols equivalent

Regulator tuning Performance index minimisationMinimum ISE – Hazebroekand Van der Waerden [9]

Model: Method 1

15.K m mτ 556. τm

Shinskey [59] – minimumIAE regulator – page 74.


09259.Km mτ

4τm

Poulin and Pomerleau [82] –minimum ITAE (process

output step loaddisturbance)

Model: Method 2

05264.K m mτ

4 5804. τm

Poulin and Pomerleau [82] –minimum ITAE (process

input step load disturbance)Model: Method 2

05327.K m mτ 38853. τm

Ultimate cycleTyreus and Luyben [75]

Model: Method 2 or 3 .0 31. Ku 2 2. Tu

Maximum closed loop logmodulus = 2dB ; closed loop

time constant = τm 10

Regulator – minimum IAE –Shinskey [17] – page 121.

Model: method notspecified

061. Ku Tu

RobustFruehauf et al. [52]Model: Method 5

05.Km mτ 5τm

Rule Kc Ti Comment

[ ]1 2

2Km

m

m

λ τ

λ τ

+

+

2λ τ+ m

λ τ=

1Km

m, [50];

λ τ> +m mT (Thomasson

[51])

λ Overshoot TS PP RT15. τm 58% 6τm 17. Km

mτ7Km

mτ

2 5. τm 35% 11τm 2 0. K m

mτ16K m

mτ

35. τm 26% 16τm 2 2. K m

mτ23Km

mτ

Chien [50]

Model: Method 2

45. τm 22% 20τm 25. Km

mτ30K m

mτ

Zhang et al. [135]

[ ]λ τ τ= 15 4 5. , .m m

- values deduced fromgraphs

0 45.K m mτ 11τm


0 39.

K m mτ 12τm


0 34.K m mτ 13τm


0 30.Km mτ 14τm


Ogawa [54] – deduced fromgraph

Model: Method 5

0 27.K m mτ 15τm


Rule Kc Ti Comment

Direct synthesisατK m m

βτm

Closedlooptime

const.

Damp.Factor

ξ

Gainmargin

A m

Phasemargin

φm

[deg.]

α βClosed

looptime

const.

Damp.Factor

ξ

Gainmargin

A m

Phasemargin

φm

[deg.]

α β

τm 0.707 1.3 11 0.9056 2.6096 τm 1.0 1.3 14 0.8859 3.212

2τm 0.707 2.5 33 0.5501 4.0116 2τm 1.0 2.3 37 0.6109 5.2005

3τm 0.707 3.6 42 0.3950 5.4136 3τm 1.0 3.0 46 0.4662 7.1890

4τm 0.707 4.7 47 0.3081 6.8156 4τm 1.0 4.0 52 0.3770 9.1775

5τm 0.707 5.9 50 0.2526 8.2176 5τm 1.0 4.8 56 0.3164 11.166

6τm 0.707 7.1 52 0.2140 9.6196 6τm 1.0 5.6 59 0.2726 13.155

7τm 0.707 8.2 54 0.1856 11.022 7τm 1.0 6.3 61 0.2394 15.143

8τm 0.707 9.2 55 0.1639 12.424 8τm 1.0 7.2 62 0.2135 17.132

9τm 0.707 10.4 56 0.1467 13.826 9τm 1.0 8.0 64 0.1926 19.120

10τm 0.707 11.5 57 0.1328 15.228 10τm 1.0 8.7 65 0.1754 21.109

11τm 0.707 12.7 58 0.1213 16.630 11τm 1.0 9.6 66 0.1611 23.097

12τm 0.707 13.8 59 0.1117 18.032 12τm 1.0 10.4 67 0.1489 25.086

13τm 0.707 14.9 59 0.1034 19.434 13τm 1.0 11.2 67 0.1384 27.074

14τm 0.707 16.0 60 0.0963 20.836 14τm 1.0 12.0 68 0.1293 29.063

15τm 0.707 17.0 60 0.0901 22.238 15τm 1.0 12.7 68 0.1213 31.051

Wang and Cluett [76] –deduced from graph

Model: Method 2

16τm 0.707 18.2 60 0.0847 23.640 16τm 1.0 13.6 69 0.1143 33.040

0 9588.Km mτ 30425. τm

Closed loop time constant =τm

0 6232.Km mτ 52586. τm


0 4668.Km mτ 7 2291. τm


0 3752.Km mτ 91925. τm


0 3144.Km mτ 111637. τm



Model: Method 2

0 2709.Km mτ 131416. τm


Rotach [77]

Model: Method 4

0 75.K m mτ 241. τm

Damping factor foroscillations to a disturbance

input = 0.75.Poulin and Pomerleau [78]

Model: Method 2 034. K u or 2 13.

K Tm u104. Tu

Maximum sensitivity= 5 dB

Rule Kc Ti Comment

ωp

m mA K ( )1

05ω π ω τp p m. −( )

( )ω

φ π

τp

m m m m

m m

A A A

A=

+ −

−

0 5 1

12

.

0 942.K m mτ

4510. τm Gain Margin = 1.5Phase Margin = 22 50.

0 698.K m mτ

4 098. τm Gain Margin = 2Phase Margin = 30 0

0 491.K m mτ

6942. τm Gain Margin = 3Phase Margin = 450

Gain and phase margin –Kookos et al. [38]Model: Method 2

Representative results

0384.K m mτ

18 710. τm Gain Margin = 4Phase Margin = 600

Other methods0 58.

K m mτ 10τm

Maximum closed loop gain =1.26Penner [79]

Model: Method 20 8.

K m mτ 59. τm

Maximum closed loop gain =2.0

Srividya and Chidambaram[80]

Model: Method 5

0 67075.K m mτ 36547. τm


smm

s m

( ) =− τ

- controller G s KTs T sc c

i f

( ) = +

+1 1 1

1. 1 tuning rule

Rule Kc Ti Comment

Robust

Tan et al. [81]Model: Method 2

0463 0 277. .λτ+

K m m

τλ

m

0 238 0123. .+Tf

m=+

τλ5750 0590. .

,

λ = 0 5.


smm

s m

( ) =− τ

- controller U s K Y sKT s

E scc

i

( ) ( ) ( )= − . 1 tuning rule

Rule Kc Ti Comment

Direct synthesisChien et al. [74]

Model: Method 1(

1414

14142

.

.

T

K T TCL m

m CL CL m m

+

+ +

τ

τ τ 1414. TCL m+ τUnderdamped systemresponse - ξ = 0 707. .

τm mT≤ 0 2.


smm

s m

( ) =− τ

- Two degree of freedom controller:

)s(RK)s(EsT

11K)s(U ci

c α−

+= . 1 tuning rule

Rule Kc Ti Comment

Servo/regulator tuning Performance index minimisation

Taguchi and Araki [61a]

Model: ideal processmmK

7662.0τ

6810.0=α

m091.4 τ

0.1Tm

m ≤τ

.



uK049.0 uT826.2 506.0=α , 0c 164−=φ

uK066.0 uT402.2 512.0=α , 0c 160−=φ

uK099.0 uT962.1 522.0=α , 0c 155−=φ

uK129.0 uT716.1 532.0=α , 0c 150−=φ

uK159.0 uT506.1 544.0=α , 0c 145−=φ

uK189.0 uT392.1 555.0=α , 0c 140−=φ

uK218.0 uT279.1 564.0=α , 0c 135−=φ

uK250.0 uT216.1 573.0=α , 0c 130−=φ

uK286.0 uT127.1 578.0=α , 0c 125−=φ

uK330.0 uT114.1 579.0=α , 0c 120−=φ

Minimum ITAE -Pecharroman and Pagola

[134b]

1K m =

Model: Method 6

uK351.0 uT093.1 577.0=α , 0c 118−=φ

Table 12: PI tuning rules – FOLIPD model ( )

G sK e

s sTmm

s

m

m

( ) =+

− τ

1 - controllerG s K

Tsc ci

( ) = +

1 1

. 6 tuning rules

Rule Kc Ti Comment

Ultimate cycle

McMillan [58]

Model: Method not relevant1477 1

12 0.65

2

.K

T

Tm

m

m m

m

τ

τ+

332 10.65

. ττm

m

m

T+

Tuning rules developed

from K Tu u,

Regulator tuning Minimum performance indexMinimum IAE – Shinskey

[59] – page 75.Model: Method not

specified( )0 556.

K Tm m mτ +( )3 7. τm mT+

Minimum IAE – Shinskey[59] – page 158

Model: Open loop methodnot specified

0 952.(T )Km m m+ τ

( )4 Tm m+ τ

( ) ( )b

K TT

a Tm m m

m

m mτ τ+ +

+2

2 1 ( )a Tm mτ +

τm mT a b τm mT a b τm mT a b

0.2 5.0728 0.5231 1.0 4.7839 0.5249 1.8 4.6837 0.52560.4 4.9688 0.5237 1.2 4.7565 0.5250 2.0 4.6669 0.52570.6 4.8983 0.5241 1.4 4.7293 0.52520.8 4.8218 0.5245 1.6 4.7107 0.5254

0.2 3.9465 0.5320 1.0 4.0397 0.5311 1.8 4.0218 0.53130.4 3.9981 0.5315 1.2 4.0337 0.5312 2.0 4.0099 0.53140.6 4.0397 0.5311 1.4 4.0278 0.5312

Minimum ITAE – Poulin andPomerleau [82] – deduced

from graph

Model: Method 2

Output step loaddisturbance

(2 tuning rules)

Input step load disturbance

0.8 4.0397 0.5311 1.6 4.0278 0.5312Direct synthesis

Poulin and Pomerleau [78]

Model: Method 2

034. K u or 2 13.

K Tm u104. Tu

Maximum sensitivity= 5 dB

Table 13: PI tuning rules – FOLIPD model ( )

G sK e

s sTmm

s

m

m

( ) =+

− τ

1 - controllerG s K b

T sc ci

( ) = +

1. 1 tuning rule

Rule Kc Ti Comment

Direct synthesis

( )041 0 23 0 019 2

. . .eK Tm m m

− +

+

τ τ

τ,


57 17 0 69 2

. . .τ τ τme −

b e= −0 33 2 5 1 9 2

. . .τ τ .

Ms = 1.4; 014 55. .≤ ≤τm

mTAstrom and Hagglund [3] -maximum sensitivity – pages

210-212

Model: Method 1. ( )081 11 0 76 2

. . .eK Tm m m

− +

+

τ τ

τ 3 4 0.28 0.0089 2

. τ τ τme −

b e= − +078 1 9 1 2 2

. . .τ τ .

Ms = 2.0; 014 55. .≤ ≤τm

mT

Table 14: PI tuning rules - FOLIPD model ( )

G sK e

s sTmm

s

m

m

( ) =+

− τ

1 - Two degree of freedom controller:

)s(RK)s(EsT

11K)s(U ci

c α−

+= . 1 tuning rule.

Rule Kc Ti Comment

Servo/regulator tuning Minimum performance index

+τ

+001723.0

T

2839.01787.0

K1

m

mm

2

m

m

m

m

T2591.0

T794.3296.4

τ+

τ+


Model: ideal process

m

m

T01877.06551.0

τ+=α

0.1Tm

m ≤τ

.



uK049.0 uT826.2 506.0=α , 0c 164−=φ

uK066.0 uT402.2 512.0=α , 0c 160−=φ

uK099.0 uT962.1 522.0=α , 0c 155−=φ

uK129.0 uT716.1 532.0=α , 0c 150−=φ

uK159.0 uT506.1 544.0=α , 0c 145−=φ

uK189.0 uT392.1 555.0=α , 0c 140−=φ

uK218.0 uT279.1 564.0=α , 0c 135−=φ

uK250.0 uT216.1 573.0=α , 0c 130−=φ

uK286.0 uT127.1 578.0=α , 0c 125−=φ

uK330.0 uT114.1 579.0=α , 0c 120−=φ


[134b]

1K m = ; 1Tm =

Model: Method 4

uK351.0 uT093.1 577.0=α , 0c 118−=φ

Table 15: PI tuning rules - SOSPD model K e

T s T sm

s

m m m

m−

+ +

τ

ξ12 2

12 1or

( )( )K e

T s T sm

s

m m

m−

+ +

τ

1 11 2


i

( ) = +

1 1

. 11 tuning rules.

Rule Kc Ti Comment

Robust

( )T T

Km m m

m m

1 2 052 1

+ ++

. ττ λ

T Tm m m1 2 05+ + . τλ varies graphically with

τm m mT T( )1 2+ -

01 101 2. ( )≤ + ≤τm m mT Tτm m mT T( )1 2+ λ τm m mT T( )1 2+ λ τm m mT T( )1 2+ λ

0.1 3.0 1.0 0.6 10.0 0.20.2 1.8 2.0 0.4

Brambilla et al. [48]

Model: Method 1

0.5 1.0 5.0 0.2Direct synthesis

Gain and phase margin - Tanet al. [39] – repeated pole

Model: Method 11

( )( )

β ω β ω

β ω

φ φ

φ

T T

A T

i m

m i

1

1

2

2

+

+

[ ]1

2 1βω β ω βτ ω φφ φ φtan tan− − −− Tm m m

ω ωφ < u

βτ

= <0 8 0 33. , .m

mT;

βτ

= >0 5 0 33. , .m

mT

Regulator tuning Minimum performance indexMinimum IAE - Shinskey

[59] – page 158Model: Open loop method

not specified

( )

100

50 55 1

1

2

1

2

T

K T e

m

m m m

T

Tm

m mτ τ+ + −

−+

( )τ τm

TTem

m m0 5 35 13 1

2. .+ −

−+

αKm

βτm

ξ mτm mT 1 α β ξ m

τm mT 1 α β ξ mτm mT 1 α β

1 0.5 0.8 1.82 4 0.5 4.3 3.45 7 0.5 7.8 3.851 4.0 5.7 12.5 4 4.0 27.1 6.67 7 4.0 51.2 5.88

Minimum ISE – McAvoyand Johnson [83] – deduced

from graph

Model: Method 11 10.0 13.6 25.0

αKm

βTm

ξ mτm mT 1 α β ξ m

τm mT 1 α β ξ mτm mT 1 α β

0.5 0.1 3.0 2.86 1 0.1 7.0 2.00 4 0.1 40.0 0.830.5 1.0 0.2 0.83 1 1.0 0.95 2.22 4 1.0 6.0 3.33

Minimum ITAE – Lopez etal. [84] – deduced from

graph

Model: Method 70.5 10.0 0.3 4.0 1 10.0 0.35 5.00 4 4.0 0.75 10.0

0 77 1. TK

m

m mτ ( )283 2. τm mT+τm

mT 1

02= . ,TT

m

m

2

1

01= .

070 1. TK

m

m mτ( )2 65 2. τm mT+ τm

mT 1

02= . ,TT

m

m

2

1

0 2= .

080 1. TK

m

m mτ( )2 29 2. τm mT+ τm

mT 1

02= . ,TT

m

m

2

1

05= .

Minimum IAE – Shinskey[17] – page 48. Model:method not specified.

080 1. TK

m

m mτ( )167 2. τm mT+ τm

mT 1

02= . ,TT

m

m

2

1

10= .

Rule Kc Ti Comment


Model: Method 1

Kc( )31 1 Ti

( )31 0 12

1

< ≤TT

m

m

; 01 11

. ≤ ≤τm

mT

1 KK T

TT

T

T T Tcm

m

m

m

m

m m

m

m

m

m

m

( ). .

. . . . . .31

1

2

1

2

12

1

0 9077

1

0 0631

6 4884 4 6198 3491 253143 0 8196 52132= + − − +

−

− −τ τ τ τ

+ −

−

+

+

+

1 7 2712 180448 5 3263 139108 0 49371

0 5961

2

1

0 7204

2

1

1 0049

2

1

1 005

2

K TTT

TT

TT

T

m

m

m

m

m

m

m

m

m

m

m

. . . . .. . . .

ττ

+

+

+

−

1 191783 12 2494 8 4355 17 67812

1 1

0 8529

2

1 1

0 5613

1

2

1

0 557

1

2

1

11818

KTT T

TT T T

TT T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

. . . .. . . .

τ τ τ τ

+ − − +

10 7241 2 2525 549591

2

1

2

12

Ke e e

m

T

T

T

T

Tm

m

m

m

m m

m. . .τ τ

T TT

TT T

T

T

TT Ti m

m

m

m

m

m

m

m m

m

m

m

m

m

( ) . . . . . . .311

1

2

1 1

2

2

12

2

1

2

1

3

0 0064 3 9574 4 4087 6 4789 128702 15083 9 4348= + + −

− −

+

τ τ τ τ

+

+

−

−

T TT T T

TT

TT Tm

m

m

m

m

m

m

m

m

m

m

m

m1

2

1 1

2

1

2

1

2

2

1

2

1

4

17 0736 15 9816 3909 10 7619. . . .τ τ τ

+ −

−

−

+

T TT T T

TT T

TT

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

2

1 1

3

1

2

2

1

2

1

2

1

3

2

1

4

10 684 22 3194 6 6602 6 8122. . . .τ τ τ

+

+

+

+

TT

TT T

TT T T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

1

5

2

1 1

4

2

1

2

1

3

1

2

2

1

3

75146 28724 114666 111207. . . .τ τ τ τ

+ −

−

−

−

+

TT

TT

TT T

TT T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

1

2

1

4

2

1

5

1

6

2

1 1

5

2

1

6

12174 4 3675 2 2236 0112 10308. . . . .τ τ τ

+ −

−

−

+

TT

TT T

TT T

TT T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

1

4

2

1

2

1

3

2

1

3

1

2

2

1

4

1

2

1

5

1 9136 34994 15777 11408. . . .τ τ τ τ

Rule Kc Ti Comment


Model: Method 1

Kc( )32 2 Ti

( )32

0 4 1. ≤ ≤ξm ;

0 05 11

. ≤ ≤τm

mT

2 KK T T T Tc

m

m

mm m

m

m

m

m

m

m

( ). .

. . . . . .32

1 1 1

1 4439

1

014561

10 4183 209497 55175 265149 42 7745 105069= − − − − +

+

τξ ξ

τ τ τ

+

+ − − +

−1 15 4103 34 3236 17 8860 54 0584 22 4263

1

0 31573 7057 4 5359 19593

1

0 0541

K T Tm

m

mm m m m

m

m

. . . . ..

. . ..

τξ ξ ξ ξ

τ

+

+

− +

1 2 7497 50 2197 171968 102931

4 742618288

1 1

2 7227 1

K T T TT

mm

m

mm

m

m

m

mm m

m

m

. . . ..

. .ξτ

ξτ τ

ξ ξτ

+ − + −

1 167667 14 5737 7 30251 1

Ke e e

m

T Tm

m mm

m

m. . .τ

ξξ

τ

T TT T T Ti m

m

mm

m

mm

m

mm

m

m

( ) . . . . . . .321

1 1

2

1

2

1

3

11447 45128 75 2486 110807 12 282 345 3228 1919539= + − −

− + +

τξ

τξ

τξ

τ

+

− − −

TT T Tm m

m

m

m

mm m

m

m1

1

2

1

2 2

1

4

359 3345 158 7611 770 2897 153633. . . .ξ τ τ ξ ξ τ

+ −

−

+ +

TT T Tm m

m

mm

m

m

m

mm m1

1

32

1

2

1

3 4412 5409 414 7786 4850976 864 5195. . . .ξ τ ξ τ τ ξ ξ

+

+

+

+

TT T T Tm

m

mm

m

mm

m

m

m

mm1

1

5

1

42

1

3

1

2355 4366 222 2865 275166 2052493. . . .τ ξ τ ξ τ τ ξ

+ −

− −

−

+

TT T Tm

m

mm m

m

mm

m

mm1

1

4 5

1

6

1

56479 5627 4731346 6 547 432822 99 8717. . . . .τ ξ ξ τ ξ τ ξ

+ −

−

−

+

TT T T Tm

m

mm

m

mm

m

mm

m

mm1

1

42

1

33

1

24

1

5735666 56 4418 37 497 160 7714. . . .τ ξ τ ξ τ ξ τ ξ

Rule Kc Ti Comment

Servo tuning Minimum performance indexMinimum IAE - Huang et al.

[18]Model: Method 1

Kc( )33 3 Ti

( )33 0 12

1

< ≤TT

m

m

; 01 11

. ≤ ≤τm

mT

3 KK T

TT

T

T T Tcm

m

m

m

m

m m

m

m

m

m

m

( ). .

. . . . . .33

1

2

1

2

12

1

10169

1

359591

130454 9 0916 2 6647 9 162 0 3053 11075= − − + + +

+

−τ τ τ τ

+ −

−

−

+

−

1 2 2927 310306 13 0155 9 6899 0 64181

3 6843

2

1

0 8476

2

1

2 6083

2

1

2 9049

2

K TTT

TT

TT

T

m

m

m

m

m

m

m

m

m

m

m

. . . . .. . . .

ττ

+

−

+

−1 189643 39 7340 281552

1 1

0 2016

2

1 1

13293

1

2

1

0801

KTT T

TT T T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

. . .. . .

τ τ τ

+ −

+ + +

12 0067 4 8259 2 1137 84511

1

2

1

3 956

1

2

1

2

12

K TTT

e e em

m

m

m

m

T

T

T

T

Tm

m

m

m

m m

m. . . ..

ττ τ

T TT

TT T

T

T

TT Ti m

m

m

m

m

m

m

m m

m

m

m

m

m

( ) . . . . . . .331

1

2

1 1

2

2

12

2

1

2

1

3

0 9771 0 2492 0 8753 34651 38516 7 5106 7 4538= − + +

− +

−

τ τ τ τ

+

−

−

+

TTT T T

TT

TT Tm

m

m

m

m

m

m

m

m

m

m

m

m1

2

1 1

2

1

2

1

22

1

3

1

4

116768 10 9909 161461 82567. . . .τ τ τ

+ −

+

+

+

T TT T T

TT T

TT

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

2

1 1

3

1

2

2

1

2

1

2

1

3

2

1

4

181011 6 2208 219893 158538. . . .τ τ τ

+ −

+

−

−

TT

TT T

TT T T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

1

52

1 1

42

1

2

1

3

1

22

1

3

4 7536 14 5405 2 2691 8 387. . . .τ τ τ τ

+ −

−

+

−

+

TT

TT

TT T

TT T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

1

2

1

4

2

1

5

1

6

2

1 1

5

2

1

6

16 651 71990 11496 4 728 11395. . . . .τ τ τ

+

+

+

+

TT

TT T

TT T

TT T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

1

4

2

1

2

1

3

2

1

3

1

2

2

1

4

1

2

1

5

0 6385 10885 31615 4 5398. . . .τ τ τ τ

Rule Kc Ti Comment


Model: Method 1

Kc( )34 4 Ti

( )34

0 4 1. ≤ ≤ξm ;

0 05 11

. ≤ ≤τm

mT

4 KK T T T Tc

m

m

mm m

m

m

m

mm

m

m

( ) . . . . . .34

1 1 1

3

1

21

10 95 18845 3 4123 4 5954 17002 21324= − − − + −

−

τξ ξ

τ τξ

τ

+ −

− +

+

+

1 14 4149 0 7683 7 5142 3 7291 534442

1

3

1

0 421

1

0 1984

1

18033

K T T T Tmm

m

mm

m

m

m

m

m

m

. . . . .. . .

ξτ

ξτ τ τ

+ − − + +

−

− −1 0 0819 3603 71163 3206 7 848019 5419 10749 11006

1

0 6753

1

0 1642

K T Tmm m m m

m

mm

m

m

. . . . .. . .. .

ξ ξ ξ ξτ

ξτ

+

+ + + −

1 113222 2 4239 34137 10251 055931 9948

1

11 1

K Te e e T

mm

m

m

T Tm

m

m

m

m m

m m

m. . . . ..ξ τ ξτ

τξ

ξ τ

T TT T T Ti m

m

mm

m

mm

m

mm

m

m

( ) . . . . . . .341

1 1

2

1

2

1

3

2 4866 233234 53662 656053 29 0062 24 1648 83 6796= − + +

+ − −

τ ξ τ ξ τ ξ τ

+ −

+ + +

TT T Tm m

m

m

m

mm m

m

m1

1

2

1

2 3

1

4

1359699 431477 519749 86 0228. . . .ξτ τ

ξ ξτ

+

+

− −

TT T Tm m

m

mm

m

m

m

mm m1

1

32

1

2

1

3 4704553 1534877 1250112 685893. . . .ξ τ ξ τ τ ξ ξ

+ −

+

−

+

TT T T Tm

m

mm

m

mm

m

m

m

mm1

1

5

1

42

1

3

1

2362 7517 27 6178 152 7422 20 8705. . . .τ ξ τ ξ τ τ ξ

+

+ +

+

−

TT T Tm

m

mm m

m

mm

m

mm1

1

4 5

1

6

1

5654 0012 58 7376 131193 202645 232064. . . . .τ ξ ξ τ ξ τ ξ

+ −

+

−

+

TT T T Tm

m

mm

m

mm

m

mm

m

mm1

1

42

1

33

1

24

1

5616742 136 2439 954092 204168. . . .τ ξ τ ξ τ ξ τ ξ

Rule Kc Ti Comment

Ultimate cycle

Kc( )7 Ti

( )7

Decay ratio = 0.15 - ε < 2 4. ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

Kc( )8 Ti

( )8


2 4 3. ≤ <ε , 0 2 2 01

. .≤ ≤τ m

mT,

0 6 4 2. .≤ ≤ξm

Regulator - nearly minimumIAE, ISE, ITAE – Hwang

[60]

Model: Method 3

Kc( )9 5 Ti

( )9


3 20≤ <ε , 0 2 2 01

. .≤ ≤τ m

mT,

0 6 4 2. .≤ ≤ξm

5 εξ τ τ

τ ω=

+ +6 42

12

12

12

T T K KT

m m m m H m m

m m H

, KK

TT

Hm m

mm

m m m= − −

92 18 182

2

12 1

2

ττ ξ τ

,

ωτ ξ τH

H m

mm m m H m m

K K

TT K K

= +

+ +

12

3 612 1

2

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

7

2

10 674 1 0447 00607

1= −

− +

+

ω τ ω τ,

( )TK K K

Ki

c H m

H m H m H m

( ) ( )

. . .7

2 2

1

0 0607 1 105 0233=

+

+ −ω ω τ ω τ

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

8

2

10 778 1 0 467 0 0609

1= −

− +

+

ω τ ω τ,

( )TK K K

Ki

c H m

H m H m H m

( ) ( )

. . .8

2 2

1

0 0309 1 284 0532=

+

+ −ω ω τ ω τ

( ) [ ]( )K

K K K KKc

H m H mH

H m

( ). . . .

92

1131 0519 1 103 0514

1= −

− +

+

ω τ ε ε,

[ ]( )( )TK K K

ic H m

H m

( ) ( ). . ln . .

92

10 0603 1 0 929 1 2 01 12

=+

+ + −ω τ ε ε

Rule Kc Ti Comment

Kc( )10 6 Ti

( )10 Decay ratio = 0.15 - ε > 20 ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

Kc( )11 Ti

( )11

Decay ratio = 0.2 - ε < 2 4. ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

Kc( )12 Ti

( )12

Decay ratio = 0.2 -

2 4 3. ≤ <ε , 0 2 2 01

. .≤ ≤τ m

mT,

0 6 4 2. .≤ ≤ξm

Kc( )13 Ti

( )13

Decay ratio = 0.2 -

3 20≤ <ε , 0 2 2 01

. .≤ ≤τ m

mT,

0 6 4 2. .≤ ≤ξm

Kc( )14 Ti

( )14

Decay ratio = 0.2 - ε > 20 ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

Kc( )15 Ti

( )15

Decay ratio = 0.25 - ε < 2 4. ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

Regulator – nearly minimumIAE, ISE, ITAE - Hwang [60]

Model: Method 3

Kc( )16 Ti

( )16


2 4 3. ≤ <ε , 0 2 2 01

. .≤ ≤τ m

mT,

0 6 4 2. .≤ ≤ξm

6 ( )[ ]

( )KK K K K

Kc

H m H m

H m H mH

( ). . .

10

2

1114 1 0 482 0 068

1= −

− +

+

ω τ ω τ,

( )TK K K

Ki

c H m

H m H m H m

( ) ( )

. . .10

2 2

1

0 0694 1 21 0 367=

+

− + −ω ω τ ω τ

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

11

2

10622 1 0 435 0 052

1= −

− +

+

ω τ ω τ,

( )TK K K

Ki

c H m

H m H m H m

( ) ( )

. . .11

2 2

1

00697 1 0 752 0145=

+

+ −ω ω τ ω τ

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

12

2

10 724 1 0 469 00609

1= −

− +

+

ω τ ω τ,

( )TK K K

Ki

c H m

H m H m H m

( ) ( )

. . .12

2 2

1

0 0405 1 193 0 363=

+

+ −ω ω τ ω τ

( ) [ ]( )K

K K K KKc

H m H mH

H m

( ). . . .

132

1126 0 506 1 107 0 616

1= −

− +

+

ω τ ε ε,

[ ]( )( )T

K K Ki

c H m

H m

( ) ( )

. . ln . .13

2

1

0 0661 1 0 824 1 171 117=

+

+ + −ω τ ε ε

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

14

2

1109 1 0497 0 0724

1= −

− +

+

ω τ ω τ,

( )TK K K

Ki

c H m

H m H m H m

( ) ( )

. . .14

2 2

1

0 054 1 2 54 0 457=

+

− + −ω ω τ ω τ

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

15

2

10584 1 0 439 00514

1= −

− +

+

ω τ ω τ, ( )T

K K K

Ki

c H m

H m H m H m

( ) ( )

. . .15

2 2

1

0 0714 1 0 685 0 131=

+

+ −ω ω τ ω τ

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

16

2

10 675 1 0 472 0061

1= −

− +

+

ω τ ω τ,

( )TK K K

Ki

c H m

H m H m H m

( ) ( )

. . .16

2 2

1

0 0484 1 143 0273=

+

+ −ω ω τ ω τ

Rule Kc Ti Comment

Kc( )17 7 Ti

( )17


3 20≤ <ε , 0 2 2 01

. .≤ ≤τ m

mT,

0 6 4 2. .≤ ≤ξm

Regulator - nearly minimumIAE, ISE, ITAE - Hwang [60]

Model: Method 3

Kc( )18 Ti

( )18

Decay ratio = 0.25 - ε > 20 ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

Kc( )19 Ti

( )19

Decay ratio = 0.1 - ε < 2 4. ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

ξτ τ

mm

m

m

mT T≤ + +

0 776 0 0568 018

1 1

2

. . .

Kc( )20 Ti

( )20

Decay ratio = 0.1 - 2 4 3. ≤ <ε ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

ξτ τ

mm

m

m

mT T≤ + +

0 776 0 0568 018

1 1

2

. . .

Kc( )21 Ti

( )21

Decay ratio = 0.1 - 3 20≤ <ε ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

ξτ τ

mm

m

m

mT T≤ + +

0 776 0 0568 018

1 1

2

. . .

Servo - nearly minimum IAE,ISE, ITAE - Hwang [60]

Model: Method 3

Kc( )22 Ti

( )22

Decay ratio = 0.1 - ε > 20 ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

ξτ τ

mm

m

m

mT T≤ + +

0 776 0 0568 018

1 1

2

. . .

7 ( ) [ ]

( )K

K K K KKc

H m H mH

H m

( ). . . .

172

112 0495 1 11 0 698

1= −

− +

+

ω τε ε

,[ ]( )( )T

K K Ki

c H m

H m

( ) ( ). . ln . .

172

10 0702 1 0 734 1 148 11

=+

+ + −ω τ ε ε

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

18

2

1103 1 0 51 0 0759

1= −

− +

+

ω τ ω τ,

( )TK K K

Ki

c H m

H m H m H m

( ) ( )

. . .18

2 2

1

0 0386 1 326 0 6=

+

− + −ω ω τ ω τ

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

19

2

10 822 1 0 549 0112

1= −

− +

+

ω τ ω τ,

( )TK K K

Ki

c H m

H m H m H m

( ) ( )

. . .19

2 2

1

0 0142 1 696 177=

+

+ −ω ω τ ω τ

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

20

2

10786 1 0 441 0 0569

1= −

− +

+

ω τ ω τ,

( )TK K K

Ki

c H m

H m H m H m

( ) ( )

. . .20

2 2

1

00172 1 4 62 0 823=

+

+ −ω ω τ ω τ

( ) [ ]( )K

K K K KKc

H m H mH

H m

( ). . . .

212

1128 0542 1 0 986 0558

1= −

− +

+

ω τ ε ε,

[ ]( )( )T

K K Ki

c H m

H m

( ) ( )

. . ln . .21

2

1

0 0476 1 0 996 1 213 113=

+

+ + −ω τ ε ε

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

22

2

1114 1 0 466 00647

1= −

− +

+

ω τ ω τ, ( )T

K K K

Ki

c H m

H m H m H m

( ) ( )

. . .22

2 2

1

0 0609 1 1 97 0 323=

+

− + −ω ω τ ω τ

Rule Kc Ti Comment

Kc( )23 8 Ti

( )23

Decay ratio = 0.1 - ε < 2 4. ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

ξτ τ

> + +

0889 0 496 0 26

1 1

2

. . .m

m

m

mT T

Kc( )24 Ti

( )24

Decay ratio = 0.1 - 2 4 3. ≤ <ε ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

ξτ τ

> + +

0889 0 496 0 26

1 1

2

. . .m

m

m

mT T

Kc( )25 Ti

( )25

Decay ratio = 0.1 - 3 20≤ <ε ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

ξτ τ

> + +

0889 0 496 0 26

1 1

2

. . .m

m

m

mT T

Kc(26) Ti

(26)

Decay ratio = 0.1 - ε > 20 ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

ξτ τ

> + +

0889 0 496 0 26

1 1

2

. . .m

m

m

mT T


Model: Method 3

Kc( )27 Ti

( )27

Decay ratio = 0.1 - ε < 2 4. ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

ξτ τ

mm

m

m

mT T> + +

0 776 0 0568 018

1 1

2

. . .

ξτ τ

mm

m

m

mT T≤ + +

0 889 0 496 0 26

1 1

2

. . .

8 ( )[ ]

( )KK K K K

Kc

H m H m

H m H mH

( ). . .

23

2

10 794 1 0541 0 126

1= −

− +

+

ω τ ω τ, ( )T

K K K

Ki

c H m

H m H m H m

( ) ( )

. . .23

2 2

1

0 0078 1 8 38 197=

+

+ −ω ω τ ω τ

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

24

2

10 738 1 0 415 0 0575

1= −

− +

+

ω τ ω τ, ( )T

K K K

Ki

c H m

H m H m H m

( ) ( )

. . .24

2 2

1

0 0124 1 4 05 0 63=

+

+ −ω ω τ ω τ

( ) [ ]( )K

K K K KKc

H m H mH

H m

( ). . . .

252

1115 0 564 1 0 959 0 773

1= −

− +

+

ω τ ε ε,

[ ]( )( )T

K K Ki

c H m

H m

( ) ( )

. . ln . .25

2

1

0 0355 1 0 947 1 19 107=

+

+ + −ω τ ε ε

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

26

2

1107 1 0 466 0 0667

1= −

− +

+

ω τ ω τ, ( )T

K K K

Ki

c H m

H m H m H m

( ) ( )

. . .26

2 2

1

0 0328 1 2 21 0338=

+

− + −ω ω τ ω τ

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

27

2

10 789 1 0 527 0 11

1= −

− +

+

ω τ ω τ, ( )T

K K K

Ki

c H m

H m H m H m

( ) ( )

. . .27

2 2

1

0 009 1 9 7 2 4=

+

+ −ω ω τ ω τ

Rule Kc Ti Comment

Kc(28) 9 Ti

(28)

Decay ratio = 0.1 - 2 4 3. ≤ <ε ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

ξτ τ

mm

m

m

mT T> + +

0 776 0 0568 018

1 1

2

. . .

ξτ τ

mm

m

m

mT T≤ + +

0 889 0 496 0 26

1 1

2

. . .

Kc( )29 Ti

( )29

Decay ratio = 0.1 - 3 20≤ <ε ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

ξτ τ

mm

m

m

mT T> + +

0 776 0 0568 018

1 1

2

. . .

ξτ τ

mm

m

m

mT T≤ + +

0 889 0 496 0 26

1 1

2

. . .


Model: Method 3

Kc( 30) Ti

(30)

Decay ratio = 0.1 - ε > 20 ,

0 2 2 01

. .≤ ≤τ m

mT, 0 6 4 2. .≤ ≤ξm

ξτ τ

mm

m

m

mT T> + +

0 776 0 0568 018

1 1

2

. . .

ξτ τ

mm

m

m

mT T≤ + +

0 889 0 496 0 26

1 1

2

. . .

Regulator - minimum IAE -Shinskey [17] – page 121.


048. K u 083. Tu τm

mT 1

02= . , TT

m

m

2

1

0 2= .

9 ( )[ ]

( )KK K K K

Kc

H m H m

H m H mH

(. . .

28)

2

10 76 1 0426 0 0551

1= −

− +

+

ω τ ω τ, ( )T

K K K

Ki

c H m

H m H m H m

( ) ( )

. . .28

2 2

1

0 0153 1 4 37 0 743=

+

+ −ω ω τ ω τ

( ) [ ]( )K

K K K KKc

H m H mH

H m

( ). . . .

292

1122 0 55 1 0 978 0 659

1= −

− +

+

ω τ ε ε,

[ ]( )( )T

K K Ki

c H m

H m

( ) ( )

. . ln . .29

2

1

0 0421 1 0 969 1 2 02 111=

+

+ + −ω τ ε ε

( )[ ]( )K

K K K KKc

H m H m

H m H mH

( ). . .

30

2

1111 1 0 467 0 0657

1= −

− +

+

ω τ ω τ, ( )T

K K K

Ki

c H m

H m H m H m

( ) ( )

. . .30

2 2

1

0 0477 1 2 07 0 333=

+

− + −ω ω τ ω τ

Table 16: PI tuning rules - SOSPD model -K e

T s T sm

s

m m m

m−

+ +

τ

ξ12 2

12 1

- Two degree of freedom controller: )s(RK)s(EsT

11K)s(U ci

c α−

+= . 3 tuning rules.

Rule Kc Ti Comment


+τ

+0003414.0

T

5613.03717.0

K1

m

mm

)a30(iT 10

3

m

m2

m

m

m

m

T1201.0

T3087.0

T5056.06438.0

τ−

τ+

τ−=α

0.1Tm

m ≤τ

; 1m =ξ



+τ

+2

m

mm ]06041.0T

[

05627.01000.0

K1 )b30(

iT



4

m

m

3

m

m

2

m

m

m

m

T182.3

T317.9

T575.7

T4439.06178.0

τ−

τ+

τ−

τ−=α

0.1Tm

m ≤τ

; 5.0m =ξ




[134a]

Model: Method 15

uK1713.0 uT0059.14002.0=α ,

0c 65.139−=φ1K m = ; 1Tm = ; 1m =ξ

uK147.0 uT150.1 411.0=α , 0c 146−=φ

uK170.0 uT013.1 401.0=α , 0c 140−=φ

uK195.0 uT880.0 386.0=α , 0c 133−=φ

uK210.0 uT720.0 342.0=α , 0c 125−=φ

uK234.0 uT672.0 345.0=α , 0c 115−=φ

uK249.0 uT610.0 323.0=α , 0c 105−=φ

uK262.0 uT568.0 308.0=α , 0c 94−=φ

uK274.0 uT545.0 291.0=α , 0c 84−=φ

uK280.0 uT512.0 281.0=α , 0c 73−=φ

uK291.0 uT503.0 270.0=α , 0c 63−=φ

uK297.0 uT483.0 260.0=α , 0c 52−=φ

uK303.0 uT462.0 246.0=α , 0c 41−=φ


[134b]

1K m = ; 1Tm = ; 1m =ξ

Model: Method 15

uK307.0 uT431.0 229.0=α , 0c 30−=φ

10

τ−

τ+τ−=3

m

m

2

m

m

m

mm

)a30(i T

5524.0T

081.1T

3692.0069.2TT

τ+

τ−

τ+

τ−=

4

m

m

3

m

m

2

m

m

m

mm

)b30(i T

567.8T

85.25T

04.30T

39.16340.4TT

Rule Kc Ti Comment

uK317.0 uT386.0 171.0=α , 0c 19−=φ

uK324.0 uT302.0 004.0=α , 0c 10−=φ


[134b] - continued

uK320.0 uT223.0 204.0−=α , 0c 6−=φ

Table 17: PI tuning rules - SOSIPD model (repeated pole) -2

1m

sm

)sT1(seK m

+

τ−


11K)s(U ci

c α−

+= . 1 tuning rule.

Rule Kc Ti Comment


+τ

+7640.0

T

3840.007368.0

K1

m

mmm

m

T029.4549.8

τ+



m

m

T006606.06691.0

τ+=α

0.1Tm

m ≤τ

;



uK049.0 uT826.2 506.0=α , 0c 164−=φ

uK066.0 uT402.2 512.0=α , 0c 160−=φ

uK099.0 uT962.1 522.0=α , 0c 155−=φ

uK129.0 uT716.1 532.0=α , 0c 150−=φ

uK159.0 uT506.1 544.0=α , 0c 145−=φ

uK189.0 uT392.1 555.0=α , 0c 140−=φ

uK218.0 uT279.1 564.0=α , 0c 135−=φ

uK250.0 uT216.1 573.0=α , 0c 130−=φ

uK286.0 uT127.1 578.0=α , 0c 125−=φ

uK330.0 uT114.1 579.0=α , 0c 120−=φ


[134b]

1K m = ; 1Tm =

Model: Method 1

uK351.0 uT093.1 577.0=α , 0c 118−=φ

Table 19: PI tuning rules - TOLPD model ( )( )( )

K esT sT sT

ms

m m m

m−

+ + +

τ

1 1 11 2 3


i

( ) = +

1 1

. 1

tuning rule

Rule Kc Ti Comment

0 7 1

0 333.

.

KT

m

m

mτ

( )15 0 08

1 2 3. .τm m m mT T T+

τm

mT 1

0 04> . ; T T Tm m m1 2 3≥ ≥Hougen [85]

Model: Method 1

( )1

207 081

0 333

1 2 3

1 2 3

0 333K

T T T T

T T Tm

m

m

m m m

m m m

. ..

.τ

+

+ +

( )15 0 081 2 3. .τm m m mT T T+

τm

m1T≤ 0 04. ; T T Tm m m1 2 3≥ ≥

Table 20: PI tuning rules - TOLPD model (repeated pole) -3

1m

sm

)sT1(eK m

+

τ−


11K)s(U ci

c α−

+= . 1 tuning rule.

Rule Kc Ti Comment


+τ

+5009.0

T

7399.02713.0

K1

m

mm

)c30(iT 1Taguchi and Araki [61a]

Model: ideal process2

m

m

m

m

T05159.0

T2648.04908.0

τ+τ−=α

0.1Tm

m ≤τ

;



1

τ+τ−=2

m

m

m

mm

)c30(i T

1354.0T

003899.0759.2TT

Table 21: PI tuning rules - unstable FOLPD model K e

sTm

s

m

m−

−

τ

1 - controllerG s K

Tsc ci

( ) = +

1 1

. 6 tuning rules

Rule Kc Ti Comment

Direct synthesis

De Paor and O’Malley [86]

Model: Method 1

Kc( )35

( )

1

10 5T

TTm

m m

m mm

−

ττ

φtan .

gain margin = 2; τm

mT< 1

( )φτ

ττ τm

m m

m mm m m m

T

TT T=

−− −−tan 1 1

1

Venkatashankar andChidambaram [87]Model: Method 1

Kc( )36 ( )25 Tm m− τ τm

mT< 0 67.

Chidambaram [88]Model: Method 1

1 1 0 26K

T

m

m

m

+

.

τ 25 27Tm m− ττm

mT< 06.

Ho and Xu [90]

Model: Method 1

ωp m

m m

T

A K1

15712. ω ω τp p mmT

− −

( )( )

ωφ

τp

m m m m

m m

A A A

A=

+ −

−

157 1

12

.,

τm

mT< 0 62.

Robust

TTK

mm

m

λλ

λ

+

2

2

λ λTm

+

2 λ obtained graphically –

sample values below

Km uncertainty = 50% τm mT = 0.2 λ = [ . , . ]0 6 1 9T Tm m

τm mT = 0.2 λ = [ . , . ]0 5 45T Tm m

τm mT = 0.4 λ = [ . , . ]15 4 5T Tm m

Rotstein and Lewin [89]

Model: Method 1

Km uncertainty = 30%

τm mT = 0.6 λ = [ . , . ]39 41T Tm m

Ultimate cycle

Luyben [91]Model: Method not relevant

031. Ku 2 2. Tu

Maximum closed loop logmodulus = 2 dB; closed

loop time constant =316. τm

2

2 ( ) ( )KTK

T TT

TT Tc

m

mm m m m

m m

m mm m m m

( cos sin35) 11

1= − +−

−

τ ττ

ττ τ

( )( )

( )( )

( )K

KT

TT

T T

T

Tc

m

m

m m mm m

m m m

m m m

mm m

( ) ..36

2

2

2 2 2

2 2

22

2

1 0 98 10 04 25

1

1625

= +−

−

+−

−

+ −τ τβ τ

β τ

τ τ

βτ

τ: β = 1373. ,

τm

mT< 025.

β = 0 953. , 0 25 0 67. .≤ <τm

mT

Table 22: PI tuning rules - unstable SOSPD model ( )( )

K esT sT

ms

m m

m−

− +

τ

1 11 2


i

( ) = +

1 1

. 3

tuning rules.

Rule Kc Ti Comment

Ultimate cycleMcMillan [58]

Model: Method not relevantKc

( )37 3 Ti( )37 Tuning rules developed

from K Tu u,

Regulator tuning Minimum performance index

( )[ ]( )

bTaT

T

K aT T

mm

m m

m m m m

12

2

22

1 2

14

4

++

− +

τ

τ

( )( )

4

41 2

1 2

T T

aT Tm m m

m m m

τ

τ

+

− +

τm mT a b τm mT a b τm mT a b

0.05 0.9479 2.3546 0.25 1.4905 2.5992 0.45 2.0658 2.90040.10 1.0799 2.4111 0.30 1.6163 2.6612 0.50 2.2080 2.98260.15 1.2013 2.4646 0.35 1.7650 2.73680.20 1.3485 2.5318 0.40 1.9139 2.8161

0.05 1.1075 2.4230 0.25 1.5698 2.6381 0.45 2.1022 2.92100.10 1.2013 2.4646 0.30 1.6943 2.7007 0.50 2.2379 3.00030.15 1.3132 2.5154 0.35 1.8161 2.7637

Minimum ITAE – Poulin andPomerleau [82] – deduced

from graph

Model: Method 1


(considered as 2 tuningrules)

Input step load disturbance0.20 1.4384 2.5742 0.40 1.9658 2.8445

3 ( )

( )( )

KK

T T

T T T T

T T T

cm

m m

m m m m m

m m m m m

( ).

.37 1 22

1 2 1 2

1 2 1

0 65

2

1477 1

1

=

++

− −

τ

τ τ

, ( )

( )( )T

T T T T

T T Ti mm m m m

m m m m m

( )

.

.37 1 2 1 2

1 2 1

0 65

332 1= ++

− −

τ

τ τ

( )K T A

K

T T T T

Tci

m

m m m m

i

( ) max max max max

max

38 12

22 6

12

22 4 2

2 21=

+ + +

+

ω ω ω

ω

Table 23: PI tuning rules – delay model mse τ− - controller G s KTsc c

i

( ) = +

1 1

. 2 tuning rules

Rule Kc Ti Comment

Direct synthesisHansen [91a]


0.2 m3.0 τ

Regulator tuning Minimum performance indexShinskey [57] – minimumIAE – regulator - page 67.


mK4.0 m5.0 τ

Table 25: PID tuning rules - FOLPD model K esT

ms

m

m−

+

τ

1 - ideal controller G s K

T sT sc c

id( ) = + +

1 1

. 56.

Rule Kc Ti Td Comment

Process reactionZiegler and Nichols

[8]Model: Method 1

[12. TK

m

m mτ,

2TK

m

m mτ] 2τm 05. τm Quarter decay ratio

Astrom andHagglund [3] – page

139Model: Method 6

0 94T. m

m mK τ 2τm 05. τm Ultimate cycle Ziegler-Nichols equivalent

Parr [64] – page 194Model: Method 1

125. TK

m

m mτ 2 5. τm 0 4. τm

095. TK

m

m mτ2 38. τm 0 42. τm

0% overshoot;

011 1. < <τm

mTChien et al. [10] –

regulator


m

m mτ2τm 0 42. τm

20% overshoot;

011 1. < <τm

mT

0 6. TK

m

m mτTm 0 5. τm

0% overshoot;

011 1. < <τm

mTChien et al. [10] -

servo


m

m mτ1 36. Tm 0 47. τm

20% overshoot;

011 1. < <τm

mT

Three constraintsmethod - Murrill [13]-

page 356

Model: Method 3

13700.950

.K

T

m

m

mτ

TT

m m

m1351

0 738

.

.τ

0 365

0.950

. TTm

m

m

τ

Quarter decay ratio;minimum integral error

(servo mode);K K T

Tc m d

m

= 05. ;

01 1. ≤ ≤τm

mT

Cohen and Coon [11]

Model: Method 1

1135 025

KT

m

m

m

. .τ

+

T

T T

T

m

m

m

m

m

m

m

2 5 0 46

1 0 61

2

. .

.

τ τ

τ

+

+

037

1 0 2

.

.

ττm

m

mT+

Quarter decay ratio

Astrom andHagglund [93]-pages 120-126

Model: Method 19

3Km

T90% 05. τm

T90% = 90% closed

loop step responsetime. Leeds and

Northrup Electromax V.

Sain and Ozgen [94]Model: Method 5

10 6939 01814

KT

m

m

m

. .τ

+

0 8647 0 226

0 8647

. .

.

TT

m m

m

m

+

+

τ

τ

0 0565

0 8647 0 226

.

. .

TT

m

m

mτ+

Regulator tuning Minimum performance indexMinimum IAE –

Murrill [13] – pages358-363

Model: Method 3

14350 921

..

KT

m

m

mτ

TT

m m

m0878

0.749

.τ

0 482

1 137

..

TTm

m

m

τ

01 1. < ≤

τm

mT


Modified minimumIAE - Cheng and

Hung [95]Model: Method 7

30 921

KT

m

m

mτ

.T

Tm m

m0878

0.749

.τ

0 482

1 137

..

TTm

m

m

τ

Minimum ISE -Murrill [13] – pages

358-363Model: Method 3

14950 945

..

KT

m

m

mτ

TT

m m

m1101

0.771

.τ

0 56

1 006

..

TTm

m

m

τ

01 1. < ≤

τm

mT

14730 970

..

KT

m

m

mτ

TT

m m

m1115

0.753

.τ

0 55

0 948

..

TTm

m

m

τ

01 10. .≤ ≤

τm

mTMinimum ISE -Zhuang andAtherton [20]

Model: Method 61524

0 735.

.

KT

m

m

mτ

TT

m m

m1130

0.641

.τ

0 552

0851

..

TTm

m

m

τ

11 2 0. .≤ ≤

τm

mT

Minimum ITAE -Murrill [13] – pages

358-363Model: Method 3

13570 947

..

KT

m

m

mτ

TT

m m

m0842

0.738

.τ

0 381

0 995

..

TTm

m

m

τ

01 1. < ≤

τm

mT

14680 970

..

KT

m

m

mτ

TT

m m

m0942

0.725

.τ

0 443

0 939

..

TTm

m

m

τ

01 10. .≤ ≤

τm

mTMinimum ISTSE -

Zhuang andAtherton [20]

Model: Method 61515

0 730.

.

KT

m

m

mτ

TT

m m

m0957

0.598

.τ

0 444

0 847

..

TTm

m

m

τ

11 2 0. .≤ ≤

τm

mT

15310 960

..

KT

m

m

mτ

TT

m m

m0971

0.746

.τ

0 413

0 933

..

TTm

m

m

τ

01 10. .≤ ≤

τm

mTMinimum ISTES -


Model: Method 61592

0 705.

.

KT

m

m

mτ

TT

m m

m0957

0.597

.τ

0 414

0 850

..

TTm

m

m

τ

11 2 0. .≤ ≤

τm

mT

Minimum error - stepload change - Gerry

[96]Model: Method 6

03.K m

0 5. τm Tm

τm

mT> 5

Servo tuning Minimum performance indexMinimum IAE -Rovira et al. [21]Model: Method 3

10860 869

..

KT

m

m

mτ

T

T

m

m

m

0740 013. .−τ 0 348

0 914

..

TTm

m

m

τ

01 1. < ≤

τm

mT

Minimum IAE -Wang et al. [97]

Model: Method 6

( )

( )

0 76450 6032

05..

.+

+

+

ττ

τm m

m m

m m m

TT

K TTm m+ 0 5. τ

0 505

..

TT

m m

m m

ττ+

0 05 6. < <τm

mT

Minimum ISE - Wanget al. [97]

Model: Method 6

( )

( )

0 91550 7524

0 5..

.+

+

+

ττ

τm m

m m

m m m

TT

K TTm m+ 0 5. τ 0 5

05.

.T

Tm m

m m

ττ+

0 05 6. < <τm

mT

10480 897

..

KT

m

m

mτ

T

T

m

m

m

1195 0 368. .−τ 0 489

0 888

..

TTm

m

m

τ

01 10. .≤ ≤

τm

mTMinimum ISE -Zhuang andAtherton [20]

Model: Method 61154

0 567.

.

KT

m

m

mτ

T

T

m

m

m

1047 0 220. .−τ 0 490

0 708

..

TTm

m

m

τ

11 2 0. .≤ ≤

τm

mT


Minimum ITAE -Rovira et al. [21]Model: Method 3

09650 85

..

KT

m

m

mτ

T

T

m

m

m

0796 01465. .−τ 0 308

0 929

..

TTm

m

m

τ

01 1. ≤ ≤

τm

mT

Modified minimumITAE - Cheng and

Hung [95]Model: Method 7

120.855

.K

T

m

m

mτ

T

T

m

m

m

0796 0147. .−τ 0 308

0 929

..

TTm

m

m

τ

Damping factor ofclosed loop system =

0.707.

Minimum ITAE –Wang et al. [97]

Model: Method 6

( )

( )

0 730305307

0 5..

.+

+

+

ττ

τm m

m m

m m m

TT

K TTm m+ 0 5. τ 0 5

05.

.T

Tm m

m m

ττ+

0 05 6. < <τm

mT

10420 897

..

KT

m

m

mτ

T

T

m

m

m

0987 0238. .−τ 0 385

0 906

..

TTm

m

m

τ

01 10. .≤ ≤

τm

mTMinimum ISTSE –Zhuang andAtherton [20]

Model: Method 61142

0579.

.

KT

m

m

mτ

T

T

m

m

m

0919 0172. .−τ 0 384

0 839

..

TTm

m

m

τ

11 2 0. .≤ ≤

τm

mT

09680 904

..

KT

m

m

mτ

T

T

m

m

m

0977 0 253. .−τ 0 316

0892

..

TTm

m

m

τ

01 10. .≤ ≤

τm

mTMinimum ISTES –Zhuang andAtherton [20]

Model: Method 61061

0 583.

.

KT

m

m

mτ

T

T

m

m

m

0892 0165. .−τ 0 315

0 832

..

TTm

m

m

τ

11 2 0. .≤ ≤

τm

mT

Ultimate cycleRegulator – minimum

IAE – Pessen [63]Model: Method 6

0 7. K u 0 4. Tu 0149. Tu01 10. .≤ ≤

τm

mT

Servo – minimumISTSE – Zhuang and

Atherton [20]Model: Method not

relevant

0 509. Ku ( )0 051 3 302 1. . K K Tm u u+ 0125. Tu 01 2 0. .≤ ≤τm

mT

Servo – minimumISTSE – Pi-Mira et

al. [97a]Model: Method 27

uK604.0 ( )um T1KK972.404.0 + uT130.0

Regulator - minimumISTSE - Zhuang and


relevant

4 434 0 966512 1734. .. .

K KK K

Km u

m uu

−+

1751 0 6123776 1388. .. .

K KK K

Tm u

m uu

−+

0144. Tu 01 2 0. .≤ ≤τm

mT

Rule Kc Ti Td Comment1

Kc( )38 Ti

( )38 0471. KK

u

m uω

Model: Method 8Decay ratio = 0.15 -

0 1 2 0. .≤ ≤τ m

mT; ε < 2 4.

Kc( )39 Ti

( )39 0471. KK

u

m uω


0 1 2 0. .≤ ≤τ m

mT;

2 4 3. ≤ <ε

Kc( )40 Ti

( )40 0471. KK

u

m uω


0 1 2 0. .≤ ≤τ m

mT;

3 20≤ <ε

Kc( )41 Ti

( )41 0471. KK

u

m uω


0 1 2 0. .≤ ≤τ m

mT; ε ≥ 20

Kc( )42 Ti

( )42 0471. KK

u

m uω

Decay ratio = 0.2 -

0 1 2 0. .≤ ≤τ m

mT; ε < 2 4.

Regulator - nearlyminimum IAE, ISE,ITAE - Hwang [60]

Model: Method 8

Kc( )43 Ti

( )43 0471. KK

u

m uω

Decay ratio = 0.2 -

0 1 2 0. .≤ ≤τ m

mT;

2 4 3. ≤ <ε2


1 KK

T K K K T T K K T K KH

m m

m m m m c u m

u

m m m m m u c

u

m m m c u m

u

= − + + + + − +

−

92 18 18

18849 324

49324

7162

0 471 1081

1681

1775812

2 4 2 2 3 3 2 2 2 2

2ττ τ τ

ωτ τ τ

ωτ τ τ

ω. . .

ωτ τ τ

ω

HH m

m m H m m c u m

u

K K

T K K K K=

+

+ −

1

3 60 942

3

2 . , ε

ω τ ωω τ

=+ −2 1884

0471T K K K K

K Km u H m m u u c

c u H m

..

[ ]( )K K

K K Kc HH m H m

m H m

( ). . .

382 20 674 1 0 447 0 0607

1= −

− +

+

ω τ ω τ,

( )( )

TK K K

Kic H m

H m H m H m

( )( )

. . .38

38

2 2

1

0 0607 1 105 0233=

+

+ −ω ω τ ω τ

[ ]( )K K

K K Kc HH m H m

m H m

( ). . .

392 20 778 1 0 467 0 0609

1= −

− +

+

ω τ ω τ,

( )( )

TK K K

Kic H m

H m H m H m

( )( )

. . .39

39

2 2

1

0 0309 1 2 84 0532=

+

+ −ω ω τ ω τ

2 ( ) [ ]( )

K KK K Kc H

m H m

H m

( ). . . .

402131 0519 1 103 0 514

1= −

− +

+

ω τ ε ε,

( )[ ]( )( )T

K K K

Ki

c H m

H m H m

( )( )

. . ln . .40

40

2

1

00603 1 0 929 1 2 01 12=

+

+ + −ω ω τ ε ε

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

412 2114 1 0 482 0 068

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .41

41

2 2

1

0 0694 1 21 0 367=

+

− + −ω ω τ ω τ

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

422 20622 1 0 435 0 052

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .42

42

2 2

1

00697 1 0752 0 145=

+

+ −ω ω τ ω τ

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

432 20 724 1 0469 00609

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .43

43

2 2

1

0 0405 1 193 0 363=

+

+ −ω ω τ ω τ

3

Kc( )44 Ti

( )44 0471. KK

u

m uω

Decay ratio = 0.2 -

0 1 2 0. .≤ ≤τ m

mT;

3 20≤ <ε

Kc( )45 Ti

( )450471. KK

u

m uωDecay ratio = 0.2 -

0 1 2 0. .≤ ≤τ m

mT; ε ≥ 20

Kc( )46 Ti

( )460471. KK

u


0 1 2 0. .≤ ≤τ m

mT; ε < 2 4.

Kc( )47 Ti

( )47 0471. KK

u

m uω


0 1 2 0. .≤ ≤τ m

mT;

2 4 3. ≤ <ε

Kc( )48 Ti

( )48 0471. KK

u

m uω


0 1 2 0. .≤ ≤τ m

mT;

3 20≤ <ε

Regulator – nearlyminimum IAE, ISE,ITAE - Hwang [60]

(continued)

Model: Method 8

Kc( )49 Ti

( )490471. KK

u


0 1 2 0. .≤ ≤τ m

mT; ε ≥ 20

Servo - nearly min.IAE, ISE, ITAE -

Hwang [60]

Model: Method 8

Kc( )50 Ti

( )500471. KK

u


01 2 0. .≤ ≤τm

mT; ε < 2 4.

3 ( ) [ ]

( )K K

K K Kc Hm H m

H m

( ). . . .

442126 0 506 1 107 0616

1= −

− +

+

ω τ ε ε,

( )[ ]( )( )T

K K K

Ki

c H m

H m H m

( )( )

. . ln . .44

44

2

1

00661 1 0 824 1 171 117=

+

+ + −ω ω τ ε ε

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

452 2109 1 0 497 0 0724

1= −

− +

+

ω τ ω τ,

( )( )T

K K K

Ki

c H m

H m H m H m

((

. . .45)

45)

2 2

1

0054 1 254 0 457=

+

− + −ω ω τ ω τ

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

462 20 584 1 0439 00514

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .46

46

2 2

1

0 0714 1 0 685 0131=

+

+ −ω ω τ ω τ

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

472 20675 1 0 472 0061

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .47

47

2 2

1

00484 1 143 0 273=

+

+ −ω ω τ ω τ

( ) [ ]( )

K KK K Kc H

m H m

H m

( ). . . .

48212 0 495 1 11 0 698

1= −

− +

+

ω τ ε ε,

( )[ ]( )( )T

K K K

Ki

c H m

H m H m

( )( )

. . ln . .48

48

2

1

0 0702 1 0 734 1 148 11=

+

+ + −ω ω τ ε ε

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

492 2103 1 051 0 0759

1= −

− +

+

ω τ ω τ,

( )( )T

K K K

Ki

c H m

H m H m H m

( )( )

. . .49

49

2 2

1

0 0386 1 3 26 0 6=

+

− + −ω ω τ ω τ

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

502 20 822 1 0 549 0112

1= −

− +

+

ω τ ω τ,

( )( )T

K K K

Ki

c H m

H m H m H m

( )( )

. . .50

50

2 2

1

0 0142 1 6 96 177=

+

+ −ω ω τ ω τ

Rule Kc Ti Td Comment4

Kc( )51 Ti

( )51 0471. KK

u

m uω

Decay ratio = 0.1 -

0 1 2 0. .≤ ≤τ m

mT;

2 4 3. ≤ <ε

Kc( )52 Ti

( )52 0471. KK

u

m uω

Decay ratio = 0.1 -

01 2 0. .≤ ≤τm

mT;

3 20≤ <ε

Servo - nearly min.IAE, ISE, ITAE -

Hwang [60](continued)

Model: Method 8

Kc( )53 Ti

( )530471. KK

u


01 2 0. .≤ ≤τm

mT; ε ≥ 20

Servo – nearlyminimum IAE, ITAE– Hwang and Fang

[61]Model: Method 9

c cT

cT

Km

m

m

mu1 2 3

2

+ +

τ τ

c c1 20 537 0 0165= = −. , .

c 3 0 00173= .

K

K c cT

cT

c

u um

m

m

m

ωτ τ

4 5 6

2

+ +

c c4 500503 0163= =. , .

c 6 0 0389= − .

c cT

cT

KK

m

m

m

m

u

u c7 8 9

2

+ +

τ τω

c c7 80 350 00344= = −. , .

c 9 0 00644= .

01 2 0. .≤ ≤τm

mT-

decay ratio = 0.03

Regulator – nearlyminimum IAE, ITAE– Hwang and Fang

[61]Model: Method 9

c cT

cT

Km

m

m

m1 2 3

2

+ +

τ τ

c c1 20802 0154= = −. , .

c 3 0 0460= .

K

K c cT

cT

c

u um

m

m

m

ωτ τ

4 5 6

2

+ +

c c4 50190 0 0532= =. , .

c 6 0 00509= − .

c cT

cT

KK

m

m

m

m

u

u c7 8 9

2

+ +

τ τω

c c7 80 421 0 00915= =. , .

c 9 000152= − .

01 2 0. .≤ ≤τm

mT-

decay ratio = 0.12

SimultaneousServo/regulator -

nearly minimum IAE,ITAE - Hwang and


c cT

cT

Km

m

m

mu1 2 3

2

+ +

τ τ

c c1 20 713 0176= = −. , .

c 3 0 0513= .

K

K c cT

cT

c

u um

m

m

m

ωτ τ

4 5 6

2

+ +

c c4 50149 0 0556= =. , .

c 6 0 00566= − .

c cT

cT

KK

m

m

m

m

u

u c7 8 9

2

+ +

τ τω

c c7 80 371 00274= = −. , .

c9 0 00557= .

01 2 0. .≤ ≤τm

mT

McMillan [58]

Model: Method 1 orMethod 18

1 415 1

10 65

..K

T

T

T

m

m

m m

m m

τ

τ+

+

ττm

m

m m

TT

10 65

++

.

0 25 10 65

..

ττm

m

m m

TT

++

Tuning rulesdeveloped from K Tu u,

KA

u

marbitrary

TT

u

i4 2πSpecify gain margin

A mAstrom andHagglund [98]

Model: Method notrelevant

Ku mcosφ αTd Tu

m mtan tanφα

φ

π

+ +4

4

2 Specify phase marginφm ; α = 4 (Astrom et

al. [30])

4 [ ]

( )K K

K K Kc HH m H m

m H m

( ). . .

512 20786 1 0441 00569

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .51

51

2 2

1

0 0172 1 4 62 0 823=

+

+ −ω ω τ ω τ

( ) [ ]( )

K KK K Kc H

m H m

H m

( ). . . .

522128 0 542 1 0 986 0 558

1= −

− +

+

ω τ ε ε,

( )[ ]( )( )T

K K K

Ki

c H m

H m H m

( )( )

. . ln . .52

52

2

1

0 0476 1 0 996 1 2 131 113=

+

+ + −ω ω τ ε ε

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

532 2114 1 0 466 0 0647

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .53

53

2 2

1

0 0609 1 197 0 323=

+

− + −ω ω τ ω τ


Kc( )54 5 εTd

T4

42

πη η

η+ +

φm Amη φm Am

η φm Amη φm Am

η

150 2 2.8 30 0 1.67 3.8 450 1.25 4.6 60 0 1.11 5.420 0 1.67 3.2 350 1.43 4.0 500 1.25 4.9 650 1.11 5.5250 1.67 3.5 40 0 1.43 4.2 550 1.25 5.2

Li et al. [99]


4hAA mπ 03183. T 00796. T simplified algorithm

KA

u

mmcosφ αTd Tu

m mtan tanφα

φ

π

+ +4

4

2 Am = 2 , φm = 450 ;

α chosen arbitrarilyTan et al. [39]

Model: Method 10 K

A m

φ ( )rK

K r K

u

u u

φ φ

φ φ

ω ω

ω ω

2 2

2 2 2 2

−

−

12ωφ Ti

Arbitrary A m , φm at

ωφ ;

r K Ku= +01 09. . ( )φ

( )0 25.

G jp uω05774.

ω u

01443.ω u

τm

mT< 0 25. , Am = 2 ,

φm = 600

Friman and Waller[41]


( )04830

150 0

.

G jp ω

37321

150 0

.ω

09330

150 0

.ω

0 25 2 0. .≤ ≤τm

mT

Am = 2 , φm = 450

5 ( )

K h

A A b bT

T Tc

m

m i d

( ) cos54

2 2

42

=− + −

φ

π π,

( )( )

ηφ

=− −

+ −

tan m b A b

b A b

2 2

2 21 with ±h = amplitude of relay, ±b =

deadband of relay, A, T = limit cycle amplitude and period, respectively.


Direct synthesis

( )Kc55 Ti

( )55 Td( )55

Pole is real and hasmaximum attainable

multiplicity; τm

mT< 2

Regulator - Goreckiet al. [28]

Model: Method 6

(2 tuning rules)( )Kc56 6 Ti

( )56 Td( )56

Low frequency part ofmagnitude Bodediagram is flat.

Regulator - minimumIAE - Smith and

Corripio [25] – page343-346

Model: Method 6

TK

m

m mτ Tm 0 5. τ m01 15. .≤ ≤

τm

mT

Servo – minimumIAE – Smith and

Corripio [25] – page343-346

Model: Method 6

56

TK

m

m mτ Tm 0 5. τ m 01 15. .≤ ≤τm

mT

Servo – 5%overshoot – Smithand Corripio [25] –

page 343-346Model: Method 6

TK

m

m m2 τ Tm 0 5. τ m

6 ( )KK

TT T T T

ecm

m

m

m

m

m

m

m

m

m

m

T Tm

m

m

m552 2

32

322

62

32

92 2

2

= + +

− −

−

+

− −

ττ τ τ τ

τ τ

TT T T T

T T T T T T

i m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

(

.

55)

2 2

2 2 2 3

62

32

92 2

21 32

32

36 4 5 62 2

=

+

+

− − −

+ +

+

− − −

−

τ

τ τ τ τ

τ τ τ τ τ τ,

TT

T T T T

d m

m

m

m

m

m

m

m

m

m

m

( .55)

2

2 205

32

1

62

32

92 2

=

+

−

+

+

− − −

τ

τ

τ τ τ τ

KK

TTc

m m

i

m

m

( )

( )

56

56

1 1

2 1 2

=

+

−

ττ

, T

T T T T

T T Ti m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

( )56

2 3 4

2

7 42 135 240 180

15 2 1 1 3 6

=+ +

+

+

+

+ +

ττ τ τ τ

τ τ τ

T

T T T T

T T T Td m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

( )56

2 3 4

2 3 4

1 7 27 60 60

7 42 135 240 180

=+ +

+

+

+ +

+

+

ττ τ τ τ

τ τ τ τ

Suyama [100]Model: Method 6

10 7236 0 2236

KT

m

m

m

. .τ

+

Tm m+ 0 309. τ

2 2367 236 2 236

.. .

TT

m m

m m

ττ+


Juang and Wang[101]

Model: Method 6

ατ τ

ατ

+ +

+

m

m

m

m

mm

m

T T

KT

052

2

.T

T T

T

m

m

m

m

m

m

m

ατ τ

ατ

+ +

+

052

. 05 05

05

2

2

. .

.

τα

τα

τ

ατ

ατ τ

m

mm

m

m

m

m

m

m

m

m

m

m

TT

T T

T T T

+ −

+

+ +

Closed loop timeconstant = αTm ,

0 1< <α

0 019952 0 20042. .ττ

m m

m m

TK

+ 0 099508 0 999560 99747 8 7425 10 5

. .. . .

ττ

τm m

m mm

TT

+− −

− ++

0 0069905 0 0294800 029773 0 29907. .. .

ττ

τm m

m mm

TT

Closed loop timeconstant = 4τm

0 055548 0 33639. .ττ

m m

m m

TK

+ 0 16440 0 995580 98607 15032 10 4

. .. . .

ττ

τm m

m mm

TT

+− −

− ++

0 016651 0 0936410 093905 0 56867

. .. .

ττ

τm m

m mm

TT


0 092654 0 43620. .ττ

m m

m m

TK

+ 0 20926 0 985180 96515 4255010 3

. .. .

ττ

τm m

m mm

TT

++ −

− ++

0 024442 0 176690 17150 080740. .. .

ττ

τm m

m mm

TT

Closed loop timeconstant = 133. τm

012786 0 51235. .ττ

m m

m m

TK

+ 0 24145 0 967510 93566 2 298810 2

. .. . .

ττ

τm m

m mm

TT

++ −

− ++

0 030407 0 274800 25285 1 0132. .. .

ττ

τm m

m mm

TT

Closed loop timeconstant = τm

016051 057109. .ττ

m m

m m

TK

+ 0 26502 0 942910 89868 6 935510 2

. .. . .

ττ

τm m

m mm

TT

++ −

− ++

0 035204 0 388230 33303 11849. .. .

ττ

τm m

m mm

TT


Cluett and Wang[44]

Model: Method 6

019067 0 61593. .ττ

m m

m m

TK

+ 0 28242 0 912310 85491 0 15937. .. .

ττ

τm m

m mm

TT

++

− ++

0 039589 0 519410 40950 13228. .. .

ττ

τm m

m mm

TT


Gain and phasemargin - Zhuang and


relevant

( )mK u mcos ,φm e K Km u= − −0 614 1 0 233 0 347. ( . ).

( )φ mK Ke m u= − −338 1 0 970 0 45. . .

( ) ( )α

φα

φ

ω

tan tanm m

u

+ +4

2

2

,

α = +0 413 3 302 1. ( . )K Km u

( ) ( )tan tanφα

φ

ω

m m

u

+ +4

2

2 Gain margin = 2, phasemargin = 600

01 2 0. .≤ ≤τm

mT

Abbas [45]

Model: Method 6

0177 0 348

0531 0 359

1 002

0 713

. .

( . . )

.

.

+

−

−τm

m

m

T

K V

Tm m+ 05. τ TT

m m

m m

ττ2 +

V = fractionalovershoot0 0 2≤ ≤V .

01 5 0. .≤ ≤τm

mT

Camacho et al.[102]Model: Method 6

1K

TTm

m m

m m

+ ττ

4TT

m m

m m

ττ+

TT

m m

m m

ττ+

18578 0.0821

0.9087

0.9471.K A Tm

m

m

m

m

φ τ

− ( )Ti57 7 0 4899 0.1457

0.0845

1.

.0264T

A Tm m

m

m

m

φ τ

[ ]Am ∈ 2 5, ,

[ ]φm ∈ 30 600 0, ,

01 10. .≤ ≤τm

mT.

Servo – minimum ISE- Ho et al. [103]

Model: Method 6

Given A m , ISE is minimised when φτ τ

mm

m

m

m

m mA T A T= + + −29 7985

621189 403182 76 2833.

. . .

(Ho et al [104])

10722 0.116

0.8432

0.908.K A Tm

m

m

m

m

φ τ− −

12497 1

0.2099

0.3678. .0082T

A Tm m

m

m

m

φ τ

0 4763 0.328

0.0961

1.

.0317T

A Tm m

m

m

m

φ τ−

[ ]Am ∈ 2 5, ,

[ ]φm ∈ 30 600 0, ,

01 10. .≤ ≤τm

mT

Regulator - minimumISE – Ho et al. [103]

Model: Method 6

Given A m , ISE is minimised when ( )φ τm m m mA T= 465489 0.2035 0.3693. (Ho et al [104])


7 ( )[ ]

( )TT A T

A Tim m m m m

m m m m

( ). . . .

. . .57

0 0211 1 03289 64572 251914

1 0 0625 08079 0 347=

+ + +

+ − +

φ τ

φ τ

Morilla et al. [104a]Model: Method 24

)a57(cK 8

α−+

τ

411m

)a57(iTα ]5.0,2.0[;1.0 0 =δ=α

RobustRobust - Brambilla et

al. [48]

Model: Method 6

1 05K

T

m

m m

m

+

. τλτ

Tm m+ 0 5. τ TT

m m

m m

ττ2 +

01 10. ≤ ≤τm

mT and no

model uncertainty -λ ≈ 0 35.

Rivera et al. [49]

Model: Method 6

1 0505K

T

m

m m

m

++

..

τλ τ Tm m+ 0 5. τ

TT

m m

m m

ττ2 +

λ > 01. Tm ,

λ τ≥ 0 8. m .

59

TKm

m mτ 5τm ≤ 0 5. τ m

τm

mT< 033.Fruehauf et al. [52]

Model: Method 1 TKm

m m2τ Tm ≤ 0 5. τ m

τm

mT≥ 0 33.

( )T

Ki

m mλ τ+ ( )Tm

m

m

++

τλ τ

2

2 ( )τ

λ ττm

m

m

iT

2

21

3+−

λ

τ= m

3Lee et al. [55]

Model: Method 6

( )T

Ki

m m2λ τ α+ −

Tm + −α

λ ατ τλ τ α

2 2052+ −

+ −m m

m

.T

T

m

m m

m

i

α

τ ατ

λ τ α−

−

+ −

3 2

6 22

-

λ ατ τλ τ α

2 2052+ −

+ −m m

m

.

Two degrees offreedom controller;

α = −Tm

TT

emm

Tm

m12

−

−λτ

;

Desired response =e

s

ms−

+

τ

λ1

8 ( )]T2[K

TK

)a57(im0n0m

0n)a57(

i)a57(c

−τω+δ

ω= ,

( )( ) 2)a57(

i)a57(

imm

2)a57(i

)a57(imm

2m

20m0

0nTTT

TTTTT

α−−τ

α−−τ+δ+δ−=ω

with

[ ]

2

e

0

ablog2

1

1

π+

=δ , =ab

desired closed loop response decay ratio


ms

m

m−

+

τ

1 - ideal controller with first order filter

G s KT s

T sT sc c

id

f

( ) = + +

+1 1 1

1. 3 tuning rules.


Robust

** Morari andZafiriou [105]

1 05K

T

m

m m

m

++

. τλ τ Tm m+ 0 5. τ T

Tm m

m m

ττ2 +

( )TTF

m

m m

=+

λττ2

,

λ τ> 0 25. m ,

λ > 0 2. Tm .

Horn et al. [106]

Model: Method 6 ( )2

2T

Km m

m m

++

τλ τ

Tm m+ 0 5. τ ττ

m m

m m

TT+

Tf = ( )

λτλ τ

m

m2 +;

λ τ> m , λ < Tm .

H∞ optimal – Tan et

al. [81]

Model: Method 6

0 265 0 30705

. ..

λτ

++

KT

m

m

m

Tm m+ 0 5. τ ττ

m m

m m

TT+ 2

Tfm=+

τλ5314 0 951. .

λ = 2 - ‘fast’response

λ = 1 - ‘robust’ tuningλ = 15. - recommended


ms

m

m−

+

τ

1 - ideal controller with second order filter

G s KT s

T s b sa s a sc c

id( ) = + +

++ +

1 1 11

1

1 22

. 1 tuning rule.


Robust

Horn et al. [106]

Model: Method 6

( )2

2 2 1

Tb K

m m

m m

++ −

τλ τ

,

( )[ ]( )

bT

Tm m m m

m m1

2 2

2=

+ −

+

λ τ τ τ λ

τ λ

+ ( )[ ]

( )2 2

2

λ λ

τ λ

Tm

m

−

+

Tm m+ 05. τ ττ

m m

m m

TT+ 2

Filter

( )

1

12 2

2 2

12

1

12

2

+

++ +

+ −+

b s

bb

s a sm m

m

λτ λ τλ τ

( )a

bm

m2

2

12 2=

+ −

λ τ

λ τ; λ τ> m ,

λ < Tm .


ms

m

m−

+

τ

1 - ideal controller with set-point weighting

G s K bT s

T sc ci

d( ) = + +

1. 1 tuning rule.



38 8 4 7 3 2

. . .e TK

m

m m

− +τ τ

τ,


52 2 5 14 2

. . .τ τ τme− − or

0 46 2 8 2 1 2

. . .T emτ τ−

0 89 0 37 4 1 2

. . .τ τ τme− − or

0 077 5 0 4 8 2

. . .T emτ τ−

b e= +0 40 0.18 2 8 2

. .τ τ ;M s = 1 4. ;

014 55. .≤ ≤τm

mT

Astrom andHagglund [3] –pages 208-210

Model: Method 3

8 4 9 6 9 8 2

. . .e TK

m

m m

− +τ τ

τ32 15 0 93 2

. . .τ τ τme− − or

0 28 3 8 1 2

. . .6T emτ τ−

0 86 1 0.44 2

. .9τ τ τme− − or

0 076 3 4 11 2

. . .T emτ τ−

b e= +0 22 0.65 0.051 2

. τ τ ;Ms = 2 0. ;

014 55. .≤ ≤τm

mT


ms

m

m−

+

τ

1 - ideal controller with first order filter and set-point

weighting

−

++

+

++= )s(Y

sT1sT4.01)s(R

1sT1sT

sT11K)s(U

r

r

fd

ic . 1 tuning rule.


Direct synthesisNormey-Rico et al.

[106a]Model: Method not

specified

( )mm

mm

KT2375.0

τ+τ mm 5.0T τ+

mm

mm

T2T

τ+τ mf 13.0T τ=

mr 5.0T τ=


ms

m

m−

+

τ

1 - classical controller G s K

T sT sT sN

c ci

d

d( ) = +

+

+

11 1

1. 20.


Process reactionHang et al. [36] –

page 76Model: Method 1

083. TK

m

m mτ 15. τm 0 25. τm

Foxboro EXACTcontroller ‘pretune’;

N=10Witt and Waggoner

[107]Model: Method 1 [

0 6. TK

m

m mτ,

TK

m

m mτ]

τm τm

Equivalent to Zieglerand Nichols [8];

[ ]N = 10 20,

Witt and Waggoner[107]

Model: Method 2Kc

( )58 1 Ti( )58 Td

( )58

Equivalent to Cohenand Coon [11];

N = [10,20]T

Km

m mτ 5τm 0 5. τm ‘aggressive’ tuning;St. Clair [15] – page21

Model: Method 1 05. TK

m

m mτ 5τm 0 5. τm ‘conservative’ tuning;

Shinskey [15a]Model: Method 1

mm

m

KT889.0

τ m75.1 τ m70.0 τ 167.0Tm

m =τ

Regulator tuning Minimum performance indexMinimum IAE - Kaya

and Scheib [108]Model: Method 3

0980890 76167

..

KT

m

m

mτ

TT

m m

m091032

1

.

.05221τ

0 59974

0 89819

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10

Minimum IAE – Wittand Waggoner [107]

Model: Method 1Kc

( )59 2 Ti( )59 Td

( )590 1 0 258. .< <

τm

mT; [ ]N = 10 20,

1 K

T TT T

Kc

m

m

m

m

m

m

m

m

m

( )

. . . . .58

2

1350 0 25 0 7425 0 0150 0 0625

2=

+ ± + +

τ ττ τ

T T

TT T

im

m

m

m

m

m

m

( )

. . . . .

58

2

1350 0 25 0 7425 0 0150 0 0625

=

+ + +

ττ τ

m

T T

TT T

dm

m

m

m

m

m

m

( )

. . . . .

58

2

1350 025 07425 0 0150 0 0625

=

+ ± + +

ττ τ

2 KK T Tc

m

m

m

m

m

( ).

..59

0.921 18860718

1 1 1693=

± −

− −τ τ

,

TT

T

T

i

mm

m

m

m

( )

.

.

.

.

59

1 137

1 886

0 964

1 1 1693

=

−

τ

τm

, TT

T

T

d

mm

m

m

m

( )

.

.

.

.

59

1 137

1 886

0 964

1 1 1693

=

+ ±

τ

τ


Minimum ISE - Kayaand Scheib [108]Model: Method 3

1119070 89711

..

KT

m

m

mτ

TT

m m

m07987

0.9548

.τ

0 54766

0 87798

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10

Minimum ITAE -Kaya and Scheib

[108]Model: Method 3

0779021

..06401

KT

m

m

mτ

TT

m m

m114311

0.70949

.τ

0 57137

1

..03826

TTm

m

m

τ

0 1< ≤

τm

mT; N=10

Minimum ITAE -Witt and Waggoner


Kc( )60 Ti

( )60 3 Td( )60

0 1 0 379. .< <τm

mT; [ ]N = 10 20,

Servo tuning Minimum performance indexMinimum IAE - Kaya


065104432

..

KT

m

m

mτ

T

T

m

m

m

09895 0 09539. .+τ 0 50814

1 08433

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10

Minimum IAE - Wittand Waggoner [107]

Model: Method 1Kc

( )61 Ti( )61 Td

( )610 1 1. ≤ ≤

τm

mT; [ ]N = 10 20,


0719591 03092

..

KT

m

m

mτ

T

T

m

m

m

112666 018145. .−τ 0 54568

0 86411

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10



1127620 80368

..

KT

m

m

mτ

T

T

m

m

m

099783 0 02860. .+τ 0 42844

10081

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10

3 KK T Tc

m

m

m

m

m

( ).733

. .600.947 1

0 679 1 1 1283=

± −

−τ τ

TT

T

T

i

mm

m

m

m

( )

.733

.

.

60

0.995

1

0 762

1 1 1283

=

−

τ

τm

,

TT

T

T

d

mm

m

m

m

( )

.733

.

.

60

0.995

1

0 762

1 1 1283

=

+ ±

τ

τ.

KK T T Tc

m

m

m

m

m

m

m

( ). .

.. . .61

0 869 0 9141086

1 1 1392 074 013=

± −

−

−τ τ τ

,

TT

T

T T

i

mm

m

m

m

m

m

( )

.

. . .

61

0.914

0.869

0 696

1 1 1392 074 013

=

−

−

τ

τ τm

, TT

T

T T

d

mm

m

m

m

m

m

( )

.

. . .

61

0.914

0.869

0 696

1 1 1392 0 74 013

=

± −

−

τ

τ τ


Minimum ITAE -Witt and Waggoner


Kc( )62 4 Ti

( )62 Td( )62

0 1 1. ≤ ≤τm

mT; [ ]N = 10 20,

Direct synthesisTsang and Rad [109]Model: Method 13

0809. TK

m

m mτTm 05. τm Overshoot = 16%; N=5

aTK

m

m mτ Tm 025. τm N = 2.5

a ξ a ξ a ξ a ξ a ξ a ξ1.681

80.0 1.161

00.2 0.859

40.4 0.669

30.6 0.542

90.8 0.456

91.0

Tsang et al. [111]

Model: Method 6

1.3829

0.1 0.9916

0.3 0.7542

0.5 0.6000

0.7 0.4957

0.9

Robust

105KT

m

m

mλ τ+

. Tm 05. τm [ ]λ τ= m mT, ; N=10Chien [50]

Model: Method 6 1 0505K m

m

m

..τ

λ τ+

05. τm Tm [ ]λ τ= m mT, ; N=10

Ultimate cycleMinimum IAE

regulator – Shinskey[59] – page 167.


um

mm

T32.03K−τ

τT T

uu

m

015 0 05. .τ

−

014. Tu

095. T Km m mτ 1 43. τm 052. τmτm mT = 02.

095. T Km m mτ 117. τm 048. τmτm mT = 05.

114. T Km m mτ 1 03. τm 0 40. τmτm mT = 1

Minimum IAEregulator - Shinskey

[16] – page 143.Model: Method 6 139. T Km m mτ 077. τm 0 35. τm

τm mT = 2

4 KK T T Tc

m

m

m

m

m

m

m

( ). .

.. . .62

0 85 0 9290 965

1 1 1232 0 796 01465=

± −

−

−τ τ τ

,

TT

T

T T

i

mm

m

m

m

m

m

( )

.

. . .

62

0.929

0.85

0616

1 1 1232 0796 01465

=

−

−

τ

τ τm

, TT

T

T T

d

mm

m

m

m

m

m

( )

.

. . .

62

0.929

0.85

0616

1 1 1232 0 796 01465

=

± −

−

τ

τ τ


ms

m

m−

+

τ

1 - non-interacting controller

+−

+= )s(Y

NsT1

sT)s(E

sT1

1K)s(Ud

d

ic . 2 tuning rules.


Regulator tuning Minimum performance indexMinimum IAE –Huang et al. [18]Model: Method 6

Kc( )63 5 Ti

( )63 Td( )63 01 1

1

. ≤ ≤τm

mT; N =10

Servo tuning Minimum performance indexMinimum IAE -

Huang et al. [18]Model: Method 6

Kc( )64 6 Ti

( )64 Td( )64 01 1. ≤ ≤

τm

mT; N =10

5 KK T T T T

ecm

m

m

m

m

m

m

m

m

Tm

m(.

. . . . . .63)0.8058 0.6642 2 1482

1 01098 8 6290 11863 231098 20 3519 191463= − +

+

+

−

−τ τ τ τ

τ

T TT T T T Ti m

m

m

m

m

m

m

m

m

m

m

( . . . . . .63)2 3 4 5

00145 2 0555 4 4805 7 7916 7 0263 2 4311= − + −

+

−

+

τ τ τ τ τ

TT

K T T T T T Tdm

c

m

m

m

m

m

m

m

m

m

m

m

m

(( . . . . . . .63)63)

2 3 4 5 6

00206 0 9385 2 3820 7 2774 111018 8 0849 2 274= − + −

+

−

+

−

τ τ τ τ τ τ

6 KK T T T T

ecm

m

m

m

m

m

m

m

m

Tm

m( ).6405

. . . . . .640.0865 0.4062 2

1 7 0636 66 6512 261928 7 3453 336578 57 937= + +

+

+

−

−τ τ τ τ

τ

T TT T T T Ti m

m

m

m

m

m

m

m

m

m

m

( ) . . . . . .642 3 4 5

09923 0 2819 14510 2504 18759 05862= + −

+

−

+

τ τ τ τ τ

TT

K T T T T T Tdm

c

m

m

m

m

m

m

m

m

m

m

m

m

( )( ) . . . . . . .64

64

2 3 4 5 6

0 0075 0 3449 0 0793 08089 10884 0 352 0 0471= + −

+

−

+

+

τ τ τ τ τ τ


ms

m

m−

+

τ


)s(Y

NsT

1

sT)s(E

sT11K)s(U

d

d

ic

+−



Servo tuning Minimum performance index

12600 887

..

KT

m

m

mτ

T

T

m

m

m

0701 0147. .−τ 0 375

0 886

..

TTm

m

m

τ

01 10. .≤ ≤

τm

mT; N=10Minimum ISE -


Model: Method 61295

0 619.

.

KT

m

m

mτ

T

T

m

m

m

0661 0110. .−τ 0 378

0 756

..

TTm

m

m

τ

11 2 0. .≤ ≤

τm

mT; N=10

10530 930

..

KT

m

m

mτ

T

T

m

m

m

0736 0126. .−τ 0 349

0 907

..

TTm

m

m

τ

01 10. .≤ ≤

τm

mT; N=10Minimum ISTSE -


Model: Method 61120

0 625.

.

KT

m

m

mτ

T

T

m

m

m

0720 0114. .−τ 0 350

0811

..

TTm

m

m

τ

11 2 0. .≤ ≤

τm

mT; N=10

09420 933

..

KT

m

m

mτ

T

T

m

m

m

0770 0130. .−τ 0 308

0897

..

TTm

m

m

τ

01 10. .≤ ≤

τm

mT; N=10Minimum ISTES -


Model: Method 61001

0 624.

.

KT

m

m

mτ

T

T

m

m

m

0754 0116. .−τ 0 308

0813

..

TTm

m

m

τ

11 2 0. .≤ ≤

τm

mT; N=10

Ultimate cycleServo – minimum

ISTSE – Zhuang andAtherton [20]


4 437 15878 024 1435. .. .

K KK K

Km u

m uu

−−

( )0 037 589 1. . K K Tm u u+ 0112. Tu 01 2 0. .≤ ≤τm

mT; N=10

0 5556. Ku 039. Tu 014. Tuτm mT = 02.

0 4926. Ku 0 34. Tu 014. Tuτm mT = 05.

05051. K u 0 33. Tu 013. Tuτm mT = 1

Regulator - minimumIAE - Shinskey [16] –

page 148.Model: Method 6

0 4608. Ku 0 28. Tu 013. Tuτm mT = 2


ms

m

m−

+

τ


)s(Y

NsT

1

sT)s(E

sT1K)s(U

d

d

ic

+−





1315090 8826

..

KT

m

m

mτ

TT

m m

m12587

1

.

.3756τ

0 5655

0 4576

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10


134660 9308

..

KT

m

m

mτ

TT

m m

m16585

1

.

.25738τ

0 79715

0 41941

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10



131760 7937

..

KT

m

m

mτ

TT

m m

m112499

1

.

.42603τ

0 49547

0 41932

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10



1130310 81314

..

KT

m

m

mτ

T

T

m

m

m

57527 57241. .−τ 0 32175

017707

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10


1262390 8388

..

KT

m

m

mτ

T

T

m

m

m

60356 6 0191. .−τ 0 47617

0 24572

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10



0983840 49851

..

KT

m

m

mτ

T

T

m

m

m

2 71348 2 29778. .−τ 0 21443

0 16768

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10


ms

m

m−

+

τ

1 - non-interacting controller with set-point weighting

( )U s K bT s

E s K T ssTN

Y s K b Y sci

c d

dc( ) ( ) ( ) ( )= +

−

++ −1

11 . 3 tuning rules.


Ultimate cycle

0 6. Ku 0 5. Tu 0125. Tu

τm

mT< 0 3. .

( )b xT

m

m= − +2 01

166.

. τ,

x = overshoot.

0 6. Ku 0 5. Tu 0125. Tu

0 3 0 6. .≤ <τm

mT.

b xT

m

m

= +2τ

0 6. Ku0 5 15 083. . . .−

τm

muT

T 0125. Tu

0 6 0 8. .≤ <τm

mT.

bT

m

m

= −16.τ

0 6. Ku0 5 15 083. . . .−

τm

muT

T 0125. Tu08 10. .≤ <

τm

mT; b=0.8

Hang and Astrom[111]

N=10

Model: Method 1

0 6. Ku 0 335. Tu 0125. Tu10. <

τm

mT; b=0.8

0 6. Ku

bKK

=−+

1515

'

' ,

10% overshoot - servo

0 5. Tu

bK

=+36

27 5 ' ,

20% overshoot - servo

0125. Tu

K ' =

[ ][ ]2

11 1337 4

ττm m

m m

TT

+−

016 0 57. .≤ <τm

mT;

N=10

Hang et al. [65]

Model: Method 1

0 6. Ku

b K= +

817

49

1' ,

20% overshoot,10% undershoot;

servo

0 222. 'K Tu

K ' =

[ ][ ]2

11 1337 4

ττm m

m m

TT

+−

0125. Tu0 57 0 96. .≤ <

τm

mT;

N=10

Hang and Cao [112]Model: Method 11 0 6. Ku

053 0 22. .−

τm

muT

T 053 0 224

. .−

τm

m

u

TT 01 0 5. .≤ <

τm

mT; N=10


ms

m

m−

+

τ

1 - industrial controller

U s KT s

R sT sT sN

Y sci

d

d( ) ( ) ( )= +

−

+

+

11 1

1. 6 tuning rules.




0910 7938

..

KT

m

m

mτ

TT

m m

m101495

1

.

.00403τ

0 5414

0 7848

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10


1114708992

..

KT

m

m

mτ

TT

m m

m09324

0.8753

.τ

0 56508

0 91107

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10



070580 8872

..

KT

m

m

mτ

TT

m m

m103326

0.99138

.τ

0 60006

0 971

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10



0816991004

..

KT

m

m

mτ

T

T

m

m

m

109112 0 22387. .−τ 0 44278

0 97186

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10


114270 9365

..

KT

m

m

mτ

T

T

m

m

m

099223 0 35269. .−τ 0 35308

0 78088

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10



083260 7607

..

KT

m

m

mτ

T

T

m

m

m

100268 0 00854. .+τ 0 44243

111499

..

TTm

m

m

τ

0 1< ≤

τm

mT; N=10


ms

m

m−

+

τ

1 - series controller ( )G s K

T ssTc c

id( ) = +

+1 1 1 . 3 tuning

rules.


Autotuning **Astrom and

Hagglund [3] – page246

Model: Not specified

56

TK

m

m mτ15. τm

0 25. τm Foxboro EXACTcontroller

Ultimate cyclePessen [63]

Model: Method 6035. Ku 0 25.

Tu

0 25.Tu

01 1. ≤ ≤τm

mT

Direct synthesisaT

Km

m mτ Tm 025. τm


40.0 1.269

00.2 0.949

20.4 0.748

20.6 0.617

00.8 0.541

31.0

Tsang et al. [110]

Model: Method 13

1.5039

0.1 1.0894

0.3 0.8378

0.5 0.6756

0.7 0.5709

0.9

Table 35: PID tuning rules – FOLPD model K esT

ms

m

m−

+

τ

1 - series controller with filtered derivative

G s KT s

sTsTN

c ci

d

d( ) = +

+

+

11

11

. 1 tuning rule.


Robust

( )T

Km

m mλ τ+ 05. Tm 05. τm [ ]λ τ= m mT, ; N=10Chien [50]

Model: Method 6( )0 5

05.

.τ

λ τm

m mK + 05. τm Tm [ ]λ τ= m mT, ; N=10


ms

m

m−

+

τ

1 - controller with filtered derivative

G s KT s

T s

s TN

c ci

d

d( ) = + +

+

11

1. 3 tuning rules.


RobustChien [50]

Model: Method 6( )

TK

m m

m m

++0505..τ

λ τ Tm m+ 05. τTm m

m m

ττ2T + [ ]λ τ= m mT, ; N=10

( )T

Km m

m m

++0 38661 0009..

τλ τ Tm m+ 0 3866. τ

038660 3866

..

TT

m m

m m

ττ+ N = [3,10]Gong et al. [113]

Model: Method 6 ( )λ

ττ

τ=+ +

− +01388 01247 00482

03866 1 01495. . .

. ( ) .N T N

N T Nm m

m mm

Direct synthesis1

1552 0078KTm

m

m

. .τ

+

0186 0532. .τm

mmT

T+

0 25 0186 0532. . .

τm

mmT

T+


0.9; 02 1. ≤ ≤τm mT ;N Km=

Davydov et al. [31]

Model: Method 12

1

1209 0103KTm

m

m

. .τ

+

0 382 0 338. .τ m

mmT

T+

0 4 0 382 0 338. . .

τm

mmT

T+


0.9; 02 1. ≤ ≤τm mT ;N Km=


ms

m

m−

+

τ

1 - alternative non-interacting controller 1 -

U s KT s

E s K T sY sci

c d( ) ( ) ( )= +

−1 1

. 6 tuning rules.


Ultimate cycleK

Tu

u

m

373 069. .−τ

01252

.Tu

mτ 012. Tu

Tu

mτ< 2 7.


page 167.Model: method not

specified

KT

u

u

m

2 62 0 35. .−τ

01252

.Tu

mτ 012. Tu

Tu

mτ≥ 2 7.

Minimum IAE -Shinskey [17] – page117. Model: Method

6

132. T Km m mτ 180. τm 0 44. τmτm mT = 01.

132. T Km m mτ 1 77. τm 0 41. τmτm mT = 02.

135. T Km m mτ 1 43. τm 0 41. τmτm mT = 05.

149. T Km m mτ 117. τm 0 37. τmτm mT = 1

Minimum IAE -Shinskey [16] – page

143.Model: Method 6 182. T Km m mτ 0 92. τm 0 32. τm

τm mT = 2

0 7692. Ku 0 48. Tu 011. Tuτm mT = 02.

0 6993. Ku 0 42. Tu 012. Tuτm mT = 05.

0 6623. Ku 0 38. Tu 012T. uτm mT = 1


page 148.Model: Method 6

0 6024. K u 0 34. Tu 012. Tuτm mT = 2

Regulator – minimumIAE - Shinskey [17] –

page 121. Model:method not specified

0 7576. Ku 0 48. Tu 011. Tuτm mT = 02.

Process reaction –VanDoren [114]

Model: Method 1

15. TK

m

m mτ 25. τm 04. τm


ms

m

m−

+

τ

1 - Alternative filtered derivative controller -

[ ]G s K

T ss s

sc c

i

m m

m

( ) . .

.= +

+ +

+

1 1 1 05 0 0833

1 01

2 2

2

τ τ

τ. 1 tuning rule.


Direct synthesisaT

Km

m mτ Tm 025. Tm


20.0 1.329

30.2 1.028

00.4 0.841

10.6 0.695

30.8 0.552

71.0

Tsang et al. [110]

Model: Method 13

1.5520

0.1 1.1595

0.3 0.9246

0.5 0.7680

0.7 0.6219

0.9


ms

m

m−

+

τ

1 - I-PD controller ( )U s

KT s

E s K T s Y sc

ic d( ) ( ) ( )= − +1 . 2 tuning

rules.



Model: Method 6Kc

a( )64 1 Tia( )64 Td

a( )64


τm mT≤ 0 2.

Minimum ISE –Argelaguet et al.[114a]. Model:

Method not definedmm

mm

K2T2

ττ+ mm 5.0T τ+

mm

mm

T2T

τ+τ First order Pade

approximation for mτ

1 ( )KT T T T

K T Tc

a CL m m m m CL

m CL CL m m

( ) . .

. .64

2 2

2 2

1 414 0 25

0 707 0 25=

+ + −

+ +

τ τ

τ τ, T

T T T TTi

a CL m m m m CL

m m

( ) . ..

642 21414 0 25

05=

+ + −+

τ ττ

TT T T T

T T T Tda m CL m m m m CL

m m m CL m CL

( ) . . .. .

642 2

2 20707 0 25 05

025 1414=

+ −+ + −

τ τ ττ τ

Table 40: PID tuning rules - FOLPD model - G sK e

sTmm

s

m

m

( ) =+

− τ

1– Two degree of freedom controller:

)s(Rs

NT1

sTK)s(E

sNT1

sTsT

11K)s(U

d

dc

d

d

ic

+

β+α−

+++= . 1 tuning rule.


Servo/regulatortuning

Minimum performance index

Taguchi and Araki[61a]


)b64(cK 2 )b64(

iT )b64(dT

0.1Tm

m ≤τ

;


≤ settling time oftuning rules of Chien

et al. [10]

2

−τ

+=001582.0

T

224.11415.0

K1

K

m

mc

)b64(c ,

τ+

τ−τ+=3

m

m

2

m

m

m

mm

)b64(i T

4824.0T

452.1T

200.201353.0TT

τ−τ+=2

m

m

m

mm

)b64(d T

04943.0T

4119.00002783.0TT , 2

m

m

m

m

T03966.0

T2786.06656.0

τ+

τ−=α ,

2

m

m

m

m

T03936.0

T2054.06816.0

τ+

τ−=β

Table 41: PID tuning rules - non-model specific – ideal controllerG s KT s

T sc ci

d( ) = + +

1 1

. 25 tuning rules.


Ultimate cycleZiegler and Nichols

[8][ 0 6. Ku , Ku ] 0 5. Tu 0125. Tu Quarter decay ratio

Blickley [115] 0 5. K u Tu [ ]0125 0167. , .T Tu uQuarter decay ratio

0 5. K u Tu 0 2. Tu Overshoot to servoresponse ≈ 20%Parr [64] – pages

190-191 05. Ku 034. Tu 008. Tu Quarter decay ratio

De Paor [116] 0 866. Ku 0 5. Tu 0125. Tu phase margin = 300

Corripio [117] – page27

0 75. Ku 0 63. Tu 01. Tu Quarter decay ratio

Mantz andTacconi[118]

0 906. K u 0 5. Tu 0125. Tu phase margin = 250

0 4698. K u 0 4546. Tu 01136. Tu Gain margin = 2, phasemargin = 20 0

01988. K u 12308. Tu 0 3077. Tu

Gain margin = 2.44,phase margin = 610


142

0 2015. K u 0 7878. Tu 01970. Tu

Gain margin = 3.45,phase margin = 460

Astrom andHagglund [93] 0 35. Ku 0 77. Tu 0 19. Tu

Gain margin ≥ 2 , phasemargin ≥ 450

Atkinson and Davey[119]

0 25. Ku 0 75. Tu 0 25. Tu 20% overshoot - servoresponse

033. Ku 05. Tu 033. Tu ‘Some’ overshoot** Perry and Chilton[120] 02. K u 05. Tu 033. Tu ‘No’ overshoot

033. Ku 05. Tu 0125. Tu ‘Some’ overshootYu [122] – page 11

02. K u 05. Tu 0125. Tu ‘No’ overshoot

Luo et al. [121] 0 48. Ku 0 5. Tu 0125. Tu

McMillan [14] –page 90

0 5. K u 0 5. Tu 0125. Tu

McAvoy andJohnson [83]

054. Ku Tu 0 2. Tu

Karaboga and Kalinli[123] [ ]032 0 6. , .K Ku u

04267 15. , .KT

KT

u

u

u

u

[ ]008 015. , .K T K Tu u u u

Hang and Astrom[124]

Ku msinφ Tu m

m

( cos )sin

1 − φπ φ

Tu m

m

( cos )sin

14

− φπ φ

φm = phase margin

Astrom et al. [30] Ku mcos φ Tum m

41 2

πφ φtan tan+ +

Tum m

161 2

πφ φtan tan+ +

‘small’ time delay

05. Ku 12. Tu 0125. Tu ‘aggressive’ tuningSt. Clair [15] -page 17 0 25. Ku 12. Tu 0125. Tu ‘conservative’ tuning


Pole placement - Shinet al. [125]

Kc( )65 1 Ti

( )65 αTi( )65 Typical α : 0.1

Typical ξ : [0.3,0.7]

Other rules Harriott [126] –pages 179-180

K25% 0167 25%. T 0 667 25%. T Quarter decay ratio

K25% 0 67 25%. T 017 25%. T Quarter decay ratioParr [64] – pages

191, 19305.( )G jp uω Tu 0 25. Tu

083 25%. K 0 5 25%. T 01 25%. T ‘Fast’ tuningMcMillan [14] -Page 43 0 67 25%. K 0 5 25%. T 01 25%. T ‘Slow’ tuning

Calcev and Gorez[69]

1

2 2 G jp u( )ω

1ω u

Ti

4φm = 450 ,‘small’ τm

φm = 150 , ‘large’ τm

Zhang et al. [127] Kc( )66 2 Ti

( )66 Td( )66

1 K

a TT

a TT

b ac

u iu i

ii

( )

( )( )

( )( )

65

165

65 2 165

165 2 2

1 1=

−

− −

− −

ρ

αωω

αωω

ρ

,

( ) ( ) ( )

( )T

a a a a a b a b

a bi

uu

u

( )652 1 1 2

21 2 1

1 2

1

1 2 1

4

2=

− + − + + +

+

αρ ω α ω ρω ω

αρ ω α ω

( )aK

G jp21

11

= ∠cos ( )ω , ( )

ρω ω

ωξ

ξ=

− −u

u

121

, ( )bK

G jp21

11

= ∠sin ( )ω , aK u

11

= −

K1 1,ω = modified ultimate gain and corresponding angular frequency

2

( ) ( )K

A

d

c

m p m

( )

tan sin

66

22 2 2

22

1

116

=

+ −−

+φ φπ ε

φ

, d,ε = relay amp. and deadband, A = limit cycle amp.

Crossing point of the Nyquist curve and relay with hysteresis is outside the unit circle:

( )Ti

u

c

uu

cm p

( )

tan

66

2

1=

−

− −

ωω

ω β ωω

φ φ, ( )10. tan+ −

ωω

φ φu

cm p ≤ ≤β ( )1 2. tan+ −

ωω

φ φu

cm p

( ) ( )Td

u c m p

u c

662 2=

− −

−

βω ω φ φ

ω ω

tan, ωc = frequency when the open loop gain equals unity.

Crossing point of the Nyquist curve and relay with hysteresis is within the unit circle:

( )Ti

u

c

uu

cp m

( )

tan

66

2

1=

−

+ −

ωω

ω β ωω

φ φ,

ω ωω

u r

r

−− 0 4. ≤ ≤β

ω ωω

u r

r

−+ 0 4.

( ) ( )Td

u c m p

u c

662 2=

− −

−

βω ω φ φ

ω ω

tan, ω r = oscillation frequency when a pure relay is switched into closed loop.


Garcia and Castelo[127a]

)a66(cK 3 )a66(

iT )a66(dT

3 ( )

)j(G

)j(G180cosK

1p

1pm0

)a66(c ω

ω∠−φ+= ,

( )( ))j(G180cos

]1)j(G180[sin2T

1pm0

1

1pm0

)a66(i ω∠−φ+ω

+ω∠−φ+= ,

( )( ))j(G180cos2

1)j(G180sinT

1pm0

1

1pm0

)a66(d ω∠−φ+ω

+ω∠−φ+= , 1ω = oscillation frequency when a sine function is placed in series

with the process in closed loop; u1 ω<ω .

Table 42: PID tuning rules - non-model specific – controller with filtered derivative

G s KT s

T sT sN

c ci

d

d( ) = + +

+

11

1. 8 tuning rules.


Direct synthesis

( )T

A Ti

i2 1 −A

A T A TNd

d

3

267

1

67 2

− −( )( ) Td

( )67 4 N <10

( )A

A A A K T Am d

3

1 2 3 122 − −

AA T Ad

3

2 1−A A A AA A A

3 4 2 5

32

1 5

−−

N ≥10

Vrancic [72]

05

1

. TA K T

i

m i−A A A A

A2 2

21 3

1

4

2

− − χ

χχTi

N = 10[ ]χ = 02 0 25. , .

Vrancic [73] Kca( )67

A

A T AT

NKd

a da

m

3

267

1

67 2

− −( )( ) Td

a( )67 8 20≤ ≤N

4 ( ) ( )( )

( )T

A A A A A AN

A A A A A A A

NA A A

d( )67

32

5 1 32

5 1

2

3 2 5 5 2 4 3

3 2 5

4

2=

− − + − − − −

−

y t Ky

udm

t

1

0

( )( )

= −

∫

ττ

∆ , ( )y t A y d

t

2 1 1

0

( ) ( )= −∫ τ τ , ( )y t A y dt

3 2 2

0

( ) ( )= −∫ τ τ , ( )[ ]y t A y dt

4 3 3

0

( ) = −∫ τ τ ,

( )[ ]y t A y dt

5 4 4

0

( ) = −∫ τ τ , ( )A y1 1= ∞ , ( )A y2 2= ∞ , ( )A y3 3= ∞ , ( )A y4 4= ∞ , ( )A y5 5= ∞

Alternatively, if the process model is G s Kb s b s b s b s b sa s a s a s a s a s

em ms m( ) =

+ + + + ++ + + + +

−11

1 22

33

44

55

1 22

33

44

55

τ , then

( )A K a bm m1 1 1= − + τ , ( )A K b a A a bm m m2 2 2 1 1 1205= − + − +τ τ. ,

( )A K a b A a A a b bm m m m3 3 3 2 1 1 2 2 12 305 0167= − + − + − +τ τ τ. . ,

( )A K b a A a A a A a b b bm m m m m4 4 4 3 1 2 2 1 3 3 22

13 405 0167 0 042= − + − + − + + +τ τ τ τ. . . ,

( )A K a b A a A a A a A a b b b bm m m m m m5 5 5 4 1 3 2 2 3 1 4 4 32

23

14 505 0167 0 042 0 008= − + − + − + − + − +τ τ τ τ τ. . . .

K A

A A A A T AT

NA A

ca

da d

a( )

( )( )[ ]

67 3

1 2 0 367

12

67 2

0 12

=

− − −

Tda( )67 is obtained from a solution of the following equation:

[ ] [ ] [ ] ( )[ ] ( )A AN

TA A

NT

A A A AN

T A A A T A A A Ada

da

da

da0 3

367 4

1 32

67 30 5 2 3 67 2

32

1 567

2 5 3 4 0( ) ( ) ( ) ( )+ −−

+ − + − =


Direct synthesisLennartson and

Kristiansson [157]Model: Method 1

Kc( )111 5 Ti

( )111 0 4 111. ( )TiK Ku m ≤ 0 6.

Kc( )111a Ti

( )111a 0 4 111a. ( )TiK Ku m > 10

ω u u mK K ≤ 0 4.

Kc( )111b Ti

( )111b 0 4 111b. ( )TiK Ku m > 10

ω u u mK K ≥ 0 4.

Kcc( )111 6 Ti

( )111c 0 4 111c. ( )TiK Ku m > 10

ω u u mK K ≤ 0 4.

Kc( )111d Ti

( )111d 0 4 111d. ( )TiK Ku m > 10

ω u u mK K ≥ 0 4.

Kristiansson andLennartson [157]

Model: Method 1

Kc( )111e Ti

( )111e 0 4 111e. ( )Ti

K Ku m > 167.ω u u mK K > 0 45.

N = 2.5

5 [ ]

K K K K K K K

K K K K K Kc u m

u m u m

u m u m u m

( )

.111

2 2

2 2

12 35 30

25 12 35 30= − +

+ − +, T

Ki

c

u u u

( )( )

. . . .111

111

3 20 053 0 47 014 011=

− + − +ω ω ω,

NK K K K K Km u m u m u

= − +

2 512

35 302 2

.

[ ]K K K K K K KK K K K K Kc

au m

u m u m

u m u m u m

( )1112 2

2 2

12 35 303 12 35 30

= − ++ − +

, TK

ia c

a

u u u

( )( )

. . . .111

111

3 20525 0 473 0143 0113=

− + − +ω ω ω,


= − +

3 12 35 302 2

[ ]K K K K K K KK K K K K Kc u m

u m u m

u m u m u m

( )111b2 2

2 2

12 35 303 12 35 30

= − ++ − +

, TK

ic

u u u

( )( )

. . . .111b

111b

3 20185 1052 0854 0 309=

− + − +ω ω ω,


= − +

3 12 35 302 2

6 [ ]

K K K K K K K

K K K K K Kc

cu m

u m u m

u m u m u m

( )

.111

2 2

2 2

12 35 30

25 12 35 30= − +

+ − +, T

Ki

c cc

u u u

( )( )

. . . .111

111

3 20525 0 473 0143 0113=

− + − +ω ω ω,


= − +

2 512

35 302 2

.

[ ]K K K K K K K

K K K K K Kc u m

u m u m

u m u m u m

( )

.111d

2 2

2 2

12 35 30

25 12 35 30= − +

+ − +, T

Ki

c

u u u

( )( )

. . . .111d

111d

3 20185 1052 0 854 0309=

− + − +ω ω ω,


= − +

2 512

35 302 2

.

KK K K Kc

e

m u m u

( ) . ..111

2 2

7 71 914314= − + , T

Ki

e ce

u u u

( )( )

. . . .111

111

3 20 63 0 39 015 0 0082=

− + + +ω ω ω


Kristiansson andLennartson [158]Model: Method 1

Kc( )111f 7 Ti

( )111f 0 4 111f. ( )Ti

)g111(cK )g111(

iT )g111(dT

)g111(fT given below;

5.0KK1.0 mu ≤≤Kristiansson andLennartson [158a]


)h111(cK 8 )h111(

iT )h111(dT

)h111(fT given below;

5.0KK mu >

7 [ ]K K K K K KK K K K K Kc

u m u m u

m u u m u m

( )

.111f

3 2 2

3 3 2 2

12 35 3025 12 35 30

= − ++ − +

, [ ]T K K KK K K Ki

c m u

u m u m u

( )( )

. .111f

111f 3 2

2 2095 2 14=

− +ω,


= − +

2 512

35 302 2

.

−

+−

++−

++−

+−= 1

6.1KK3.2KK1.1

)KK37.01)(KK1320(6.1

)KK37.01)(KK1320(KK

)6.1KK3.2KK1.1(K

um2

u2

m

umum2

umumu2

m

2mu

2m

2u)g111(

c

−

+−

++−++−ω

+−= 1

6.1KK3.2KK1.1

)KK37.01)(KK1320(6.1)KK37.01)(KK1320(

)6.1KK3.2KK1.1(KKT

um2

u2

m

umum2

umumu

mu2

m2

uum)g111(i

−−

+−

++−+−

++−

++−ω+−

= 11

6.1KK3.2KK1.1

)KK37.01)(KK1320(6.1)6.1KK3.2KK1.1(

)KK37.01()KK1320(

)KK37.01)(KK1320()6.1KK3.2KK1.1(KK

T

um2

u2

m

umum

2um

2u

2m

2um

2um

2umumu

mu2

m2

uum)g111(d

2umumu

mu2

m2

uum)g111(f )KK37.01)(KK1320(

)6.1KK3.2KK1.1(KKT

++−ω

+−=

8

−

+−

+

+ω

+−= 1

6.1KK3.2KK1.1

)KK37.01(KK8.4

)KK37.01(KK3

)6.1KK3.2KK1.1(K

um2

u2

m

umum2

um2

u2

mu

2mu

2m

2u)h111(

c

−

+−

++ω

+−= 1

6.1KK3.2KK1.1

)KK37.01(KK8.4)KK37.01(3

)6.1KK3.2KK1.1(T

um2

u2

m

umum2

umu

mu2

m2

u)h111(i

−−

+−

++−

+

+ω+−

= 11

6.1KK3.2KK1.1

)KK37.01(KK8.4)6.1KK3.2KK1.1(

)KK37.01(KK3

)KK37.01(3)6.1KK3.2KK1.1(

T

um2

u2

m

umum

2um

2u

2m

2um

2u

2m

2umu

mu2

m2

u)h111(d

2umu

mu2

m2

u)h111(f )KK37.01(3

6.1KK3.2KK1.1T

+ω

+−=


Kristiansson andLennartson [158a]

(continued)Model: Method not

specified

)i111(cK 9 )i111(

iT )i111(dT

)i111(fT given below;

1.0KK mu <

Other

( )ω

ω ω ω

T

G j T T T

i

p i d i( ) 1 2 2 2 2− +

( )

( )

tan .

tan .

φ φ π

ωπ

βφ φ π

ω

ω

m

m

− −

+ − −

0 5

12

0 5

2πβω

β = 10φ φ πω > −m

Leva [67]

( )ω

ω ω ω

T

G j T T T

i

p i d i( ) 1 2 2 2 2− +

αTd( )68 10 Td

( )68 6 10≤ ≤αφ φ πω < −m

Astrom [68] Ku mcosφ [ ]2 1 2tan tanφ φ

ω

m m

u

+ + Ti

4Parameters determined

at φ m = 30 45 600 0 0, ,

9

−

+−

+

+

+−= 1

)KK7.36(

)KK3.08.1(8.20

)KK3.08.1(K13

)KK7.36(K

0

0

0

0

135m

135m

2135mm

2m135)i111(

c

−

+−

+

+ω

+−= 1

)KK7.36(

)KK3.08.1(8.20

)KK3.08.1(13

KK)KK7.36(T

0

0

00

00

135m

135m

2135m135

m135m135)i111(i

−

−+−

++−

+

+ω

+−= 1

1)KK7.36(

)KK3.08.1(8.20)KK7.36(

)KK3.08.1(169

)KK3.08.1(13

KK)KK7.36(T

0

0

0

0

00

00

135m

135m

2135m

2135m

2135m135

135m135m)i111(d ,

2135m135

135m135m)i111(f )KK3.08.1(13

KK)KK7.36(T

00

00

+ω

+−=

10 ( )

( )Td

m

m

( )tan .

tan .68

2 2 2 2

2

4 05

2 0 5=

− + + − −

− −

αω α ω αω φ φ π

αω φ φ πω

ω

Table 43: PID tuning rules - non-model specific – ideal controller with set-point weighting

( ) ( ) ( )U s K F R(s Y sTs

FR(s Y s T s F R(s Y sc pi

i d d( ) ) ( ) ) ( ) ) ( )= − + − + −1

. 1 tuning rule.


Ultimate cycleMantz and Tacconi

[118]0 6. K u

Fp = 017.

0 5. Tu

Fi = 10125. Tu

Fd = 0 654.Quarter decay ratio

Table 44: PID tuning rules - non-model specific – ideal controller with proportional weighting

G s K bT s

T sc ci

d( ) = + +

1. 1 tuning rule.


Direct synthesis

( )033 0.31 2

. e K u− −κ κ ,

κ = 1 K Km u( )076 1 0.36 2

. .6e Tu− −κ κ ( )017 0.46 2 1 2

. .e Tu− −κ κ

b e= − +058 1 3 5 2

. .3 .κ κ

0 < < ∞K Km u

maximum Ms = 1.4Astrom and

Hagglund [3] – page217

( )072 1 1 2

. .6 .2e K u− +κ κ ( )059 1 0.38 2

. .3e Tu− +κ κ ( )015 1 0.56 2

. .4e Tu− +κ κ

b e= −025 0.56 0.12 2

. κ κ

0 < < ∞K Km u

maximum Ms = 2.0

Table 45: PID tuning rules - non-model specific – non-interacting controller

U s KT s

E s K T ssTN

Y sci

c d

d( ) ( ) ( )= +

−

+1 1

1. 1 tuning rule.


Ultimate cycleFu et al. [128] 0 5. K u 0 34. Tu 0 08. Tu

Table 46: PID tuning rules - non-model specific – series controller ( )U s KT s

sTci

d( ) = +

+1 1 1 . 3 tuning rules.


Ultimate cycle

025. Ku 033. Tu

05.Tu

‘optimum’ servoresponsePessen [131]

0 2. K u 0 25. Tu

05.Tu

‘optimum’ regulatorresponse - step

changes0 2. K u 0 5. Tu 0 33. Tu No overshoot; close to

optimum regulatorPessen [129]0 33. Ku 0 33. Tu 0 5. Tu ‘Some’ overshoot

Grabbe et al. [130] 025. Ku 033. Tu 0 5. Tu

Table 47: PID tuning rules - non-model specific – series controller with filtered derivative

U s KT s

sTsTN

ci

d

d( ) = +

+

+

11

11

. 1 tuning rule.


Ultimate cycleHang et al. [36]

- page 580 35. Ku 113. Tu 0 20. Tu

Table 48: PID tuning rules - non-model specific – classical controller U s KT s

sT

sTN

ci

d

d( ) = +

+

+1 1 1

1. 1 tuning rule.


Ultimate cycleCorripio [117] – page

270 6. K u 05. Tu 0125. Tu 10 20≤ ≤N

Table 49: PID tuning rules - non-model specific – non-interacting controller

U s KT s

E s K T sY sci

c d( ) ( ) ( )= +

−1 1

. 1 tuning rule.


Ultimate cycle

VanDoren [114] 0 75. K u 0 625. Tu 01. Tu

Table 50: PID tuning rules - IPD model K e

sm

s m− τ

- ideal controller G s KT s

T sc ci

d( ) = + +

1 1

. 5 tuning rules.


Direct synthesisατK m m

βτm χτm

Closed looptime

constant

DampingFactor, ξ

Gain marginA m

Phase marginφm [degrees] α β χ

τm 0.707 2.0 32 0.9056 2.6096 0.3209

2τm 0.707 3.1 40 0.5501 4.0116 0.2205

3τm 0.707 4.4 46 0.3950 5.4136 0.1681

4τm 0.707 5.5 49 0.3081 6.8156 0.1357

5τm 0.707 6.7 52 0.2526 8.2176 0.1139

6τm 0.707 7.8 54 0.2140 9.6196 0.0980

7τm 0.707 8.9 55 0.1856 11.0216 0.0861

8τm 0.707 10.0 56 0.1639 12.4236 0.0767

9τm 0.707 11.2 57 0.1467 13.8256 0.0692

10τm 0.707 12.2 58 0.1328 15.2276 0.0630

11τm 0.707 13.4 59 0.1213 16.6296 0.0579

12τm 0.707 14.5 59 0.1117 18.0316 0.0535

13τm 0.707 15.6 59 0.1034 19.4336 0.0497

14τm 0.707 16.7 60 0.0963 20.8356 0.0464

15τm 0.707 17.8 60 0.0901 22.2376 0.0436

16τm 0.707 19.0 60 0.0847 23.6396 0.0410

τm 1.0 2.0 37 0.8859 3.2120 0.3541

2τm 1.0 2.9 46 0.6109 5.2005 0.2612

3τm 1.0 3.8 52 0.4662 7.1890 0.2069

4τm 1.0 4.6 56 0.3770 9.1775 0.1713

5τm 1.0 5.5 58 0.3164 11.1660 0.1462

6τm 1.0 6.4 61 0.2726 13.1545 0.1275

7τm 1.0 7.1 62 0.2394 15.1430 0.1311

8τm 1.0 8.0 64 0.2135 17.1315 0.1015

9τm 1.0 8.7 65 0.1926 19.1200 0.0921

10τm 1.0 9.5 66 0.1754 21.1085 0.0843

11τm 1.0 10.4 67 0.1611 23.0970 0.0777

12τm 1.0 11.1 67 0.1489 25.0855 0.0721

13τm 1.0 12.0 68 0.1384 27.0740 0.0672

14τm 1.0 12.8 68 0.1293 29.0625 0.0630

15τm 1.0 13.4 69 0.1213 31.0510 0.0592

Wang and Cluett [76]– deduced from

graph

Model: Method 2

16τm 1.0 14.4 69 0.1143 33.0395 0.0559


0 9588.Km mτ 30425. τm 0 3912. τm

Closed loop timeconstant = τm

0 6232.Km mτ 52586. τm 0 2632. τm


0 4668.Km mτ 7 2291. τm 0 2058. τm


0 3752.Km mτ 91925. τm 01702. τm


0 3144.Km mτ 111637. τm 01453. τm



Model: Method 2

0 2709.Km mτ 131416. τm 01269. τm


Rotach [77]

Model: Method 4

121.K m mτ

160. τm 0 48. τm

Damping factor foroscillations to a

disturbance input =0.75.

Process reactionFord [132]

Model: Method 21 48.

K m mτ 2τm0 37. τm Decay ratio: 2.7:1


139Model: Method not

relevant

094.Km mτ 2τm

0 5. τm Ultimate cycle Ziegler-Nichols equivalent


sm

s m− τ

- Ideal controller with first order filter, set-point weighting and

output feedback )s(YK)s(YsT1

sT4.01)s(R1sT

1sTsT

11K)s(U 0r

r

fd

ic −

−

++

+

++= . 1 tuning rule.


Direct synthesisNormey-Rico et al.

[106a]


mmK563.0

τ m5.1 τ m667.0 τ

mf 13.0T τ=

mm0 K2

1K

τ=

mr 75.0T τ=


sm

s m− τ

- ideal controller with filtered derivative

G s KTs

T ssTN

c ci

d

d( ) = + +

+

11

1. 1 tuning rule.


RobustChien [50]

Model: Method 2( )

205K m mλ τ+ . 2λ τ+ m

( )τ λ τλ τ

m m

m

++025

2. λ =

1Km

; N=10


sm

s m− τ

- series controller with filtered derivative

G s KTs

T sT sN

c ci

d

d( ) = +

+

+

11

11

. 1 tuning rule.


Robust

[ ]1 2 0 5

05 2K m

m

m

λ τ

λ τ

+

+

.

.2 05λ τ+ . m 05. τm

λ τ=

1K m

m, ; N=10Chien [50]

Model: Method 2

[ ]1 05

05 2K m

m

m

.

.

τ

λ τ+

05τm 2 05λ τ+ . m

λ τ=

1K m

m, ; N=10


sm

s m− τ

- classical controller G s KT s

T sT sN

c ci

d

d( ) = +

+

+

11 1

1.

5 tuning rules.


Ultimate cycleLuyben [133]

Model: Method 2 0 46. Ku 2 2. Tu 016. Tu

maximum closed looplog modulus of +2dB ;

N=10 Belanger andLuyben [134]

Model: Method 2311. K u 2 2. Tu 364T. u N=10

Robust

[ ]1 05

0 5 2Km

m

m

.

.

τ

λ τ+

0 5. τm 2 05λ τ+ . m

λ τ=

1K m

m, ; N=10Chien [50]

Model: Method 2

[ ]1 2 0 5

05 2K m

m

m

λ τ

λ τ

+

+

.

.2 0 5λ τ+ . m 0 5. τm

λ τ=

1K m

m, ; N=10

Regulator tuning Minimum performance indexMinimum IAE -

Shinskey [17] – page121. Model: Method

not specified056. Ku 0 39. Tu 015. Tu

Minimum IAE -Shinskey [17] – page117. Model: Method

1

0 93.Km mτ

157. τm 056. τm

09259.Km mτ 160. τm 0 58. τm N=10

Minimum IAE –Shinskey [59] – page

74Model: Method not

specified

09259.Km mτ 1 48. τm 0 63. τm N=20


sm

s m− τ

- Alternative non-interacting controller 1 -

U s KT s

E s K T sY sci

c d( ) ( ) ( )= +

−1 1

. 2 tuning rules.



Shinskey [59] – page74.


12821.K m mτ

19. τm 046. τm

Minimum IAE –Shinskey [17] – page121. Model: Method

not specified0 77. K u 0 48. Tu 015. τm

Minimum IAE –Shinskey [17] – page117. Model: Method

1

128.K m mτ

1 90. τm 0 48. τm


sm

s m− τ

- I-PD controller ( )U sKT s

E s K T s Y sc

ic d( ) ( ) ( )= − +1 .

1 tuning rule.



Model: Method 1Kc

a( )68 1 1414. TCL m+ τ0 25 0 707

1 414

2. ..

τ ττ

m CL m

CL m

TT

++


τm mT≤ 0 2.

1

( )KT

K T Tc

a CL m

m CL CL m m

( ) .

. .68

2 2

1414

0 707 0 25

+

+ +

τ

τ τ

Table 57: PID tuning rules - IPD model G sK e

smm

s m

( ) =− τ

- controller


11(K)s(U dcc

ic −−++= . 1 tuning rule.




mmK938.0 τ m7.2 τ m313.0 τ b = 0.167


sm

s m− τ


)s(Rs

NT1

sTK)s(E

sNT1

sTsT

11K)s(U

d

dc

d

d

ic

+

β+α−

+++= . 2 tuning rules.






τmm

253.1K1

6642.0=α

m388.2 τ

6797.0=β

m4137.0 τ

0.1Tm

m ≤τ

;



et al. [10]

uK672.1 uT366.0 uT136.0601.0=α , 1=β ,

N = 10, 0c 164−=φ

uK236.1 uT427.0 uT149.0607.0=α , 1=β ,

N = 10, 0c 160−=φ

uK994.0 uT486.0 uT155.0610.0=α , 1=β ,

N = 10, 0c 155−=φ

uK842.0 uT538.0 uT154.0616.0=α , 1=β ,

N = 10, 0c 150−=φ

uK752.0 uT567.0 uT157.0605.0=α , 1=β ,

N = 10, 0c 145−=φ

uK679.0 uT610.0 uT149.0610.0=α , 1=β ,

N = 10, 0c 140−=φ

uK635.0 uT637.0 uT142.0612.0=α , 1=β ,

N = 10, 0c 135−=φ

uK590.0 uT669.0 uT133.0610.0=α , 1=β ,

N = 10, 0c 130−=φ

uK551.0 uT690.0 uT114.0616.0=α , 1=β ,

N = 10, 0c 125−=φ

uK520.0 uT776.0 uT087.0609.0=α , 1=β ,

N = 10, 0c 120−=φ

Minimum ITAE -Pecharroman and

Pagola [134a]

=φc phase

corresponding to thecrossover frequency;

1K m =

Model: Method 6

uK509.0 uT810.0 uT068.0611.0=α , 1=β ,

N = 10, 0c 118−=φ

Table 59: PID tuning rules - FOLIPD model ( )K e

s sTm

s

m

m−

+

τ

1- ideal controller G s K

T sT sc c

id( ) = + +

1 1

.

1 tuning rule.




Kc( )69 1 ( )Ti

69 Td( )69 Tuning rules

developed from K Tu u,

1 KK

T

Tc

m

m

m m

m

( ) .692 0.65

2

1111 1

1

=

+

τ

τ

, ( )T Ti m

m

m

690.65

2 1= +

τ

τ, ( )T T

d mm

m

690.65

05 1= +

. τ

τ


s sTm

s

m

m−

+

τ

1- ideal controller with filter

G s KT s

T sT sc c

id

f

( ) = + +

+1 1 1

1. 3 tuning rules.


Robust

Tan et al. [81]

Model: Method 2

[. .

]0 463 0277

2

λτ+

Km m

([ . . ] )0238 0123λ τ+ +Tm m

Tmm++

τλ0 238 0123. . ( )

TT

m m

m m

τλ τ0238 0123. .+ +

Tfm=+

τλ5750 0590. .

λ = 0 5. - performanceλ = 01. - robustnessλ = 0 25. - acceptable

( )3

3 32 2

λ τ

λ λτ τ

+ +

+ +

T

Km m

m m m

3λ τ+ +Tm m ( )3

3

λ τ

λ τ

+

+ +m m

m m

T

T

Tfm m

=+ +

λ

λ λτ τ

3

2 23 3

15 4 5. .τ λ τm m≤ ≤

Zhang et al. [135]

Model: Method 2λ τ= 15. m ….Overshoot = 58%, Settling time = 6τm

λ τ= 25. m ….Overshoot = 35%, Settling time = m11τλ τ= 35. m ….Overshoot = 26%, Settling time = m16τλ τ= 45. m ….Overshoot = 22%, Settling time = m20τ

Obtained from graph

0 03371

012252

..

T

K Tm

m m

m

mτ

τ+

Tm m+ 81633. τ

TT

m m

m m

ττ01225. + Tf m= 05549. τ

0 07541

0 18632

..

TK T

m

m m

m

mττ

+

Tm m+ 53677. τ

TT

m m

m m

ττ01863. + Tf m= 0 4482. τ

Tan et al. [136]

Model: Method 2

013441

0 25232

..

TK T

m

m m

m

mττ

+

Tm m+ 3 9635. τ

TT

m m

m m

ττ0 2523. + Tf m= 0 2863. τ


s sTm

s

m

m−

+

τ

1- ideal controller with set-point weighting

G s K bT s

T sc ci

d( ) = + +

1. 1 tuning rule.


Direct synthesis

( )56 8 8 6 8 2

. . .eK Tm m m

− +

+

τ τ

τ,


11 6 7 4 4 2

. . .τ τ τme − 17 6 4 2 0 2

. . .τ τ τme− + Maximum Ms = 1.4

b e= −012 6 9 6 6 2

. . .τ τAstrom andHagglund [3] - pages

212-213

Model: Method 1 orMethod 2. ( )

86 7 1 54 2

. . .eK Tm m m

− +

+

τ τ

τ10 3 3 2 33 2

. . .τ τ τme − 0 38 0 056 0 60 2

. . .τ τ τme − Maximum Ms = 2.0

b e= − +056 2 2 12 2

. . .τ τ


s sTm

s

m

m−

+

τ

1- classical controller G s K

T sT sT sN

c ci

d

d( ) = +

+

+

11 1

1.

5 tuning rules.


Robust

[ ]1

2KT

m

m

mλ τ+

Tm 2λ τ+ m [ ]λ τ= m mT, ; N=10Chien [50]

Model: Method 2

[ ]1 2

2K m

m

m

λ τ

λ τ

+

+

2λ τ+ m Tm [ ]λ τ= m mT, ; N=10

Regulator tuning Minimum performance indexMinimum IAE –



0 78.( )K Tm m mτ +

1 38. ( )τm mT+ 0 66. ( )τm mT+ τm mT= ; N=10

( )100

108 122 0 03K m mTm

mτ

τ. .− 157 1 12 1. .τ τ

m

T

em

m+ −

−

0 56 0 75. .τm mT+Tm

mτ> 05.Minimum IAE -

Shinskey [59] –pages 158-159

Model: Open loopmethod not specified ( )

100

108 1 0 4K m mT m

mτ

τ+ . 157 1 12 1. .τ τ

m

T

em

m+ −

−

0 56 075. .τm mT+Tm

mτ≤ 05.

( ) ( )b

K T

T

a Tm m m

m

m mτ τ+ +

+2

21 ( )a Tm mτ + Tm ( )0 2≤ ≤

τm

dT N;

01 033. .TTN

Tmd

m≤ ≤

Minimum ITAE -Poulin and

Pomerleau [82], [92] –deduced from graph

Model: Method 2( )τm dT N a b ( )τm dT N a b

0.2 5.0728 0.5231 1.2 4.7565 0.52500.4 4.9688 0.5237 1.4 4.7293 0.52520.6 4.8983 0.5241 1.6 4.7107 0.52540.8 4.8218 0.5245 1.8 4.6837 0.5256


1.0 4.7839 0.5249 2.0 4.6669 0.5257

0.2 3.9465 0.5320 1.2 4.0397 0.53120.4 3.9981 0.5315 1.4 4.0278 0.53120.6 4.0397 0.5311 1.6 4.0278 0.53120.8 4.0397 0.5311 1.8 4.0218 0.5313

Input step loaddisturbance

1.0 4.0397 0.5311 2.0 4.0099 0.5314


s sTm

s

m

m−

+

τ

1- series controller with derivative filtering

G s KTs

T sT sN

c ci

d

d( ) = +

+

+

11

11

. 1 tuning rule.


Robust

[ ]1 2

2Km

m

m

λ τ

λ τ

+

+

2λ τ+ m Tm

[ ]λ τ= m mT, ; N=10

(Chien and Fruehauf[137])

Chien [50]

Model: Method 2

[ ]1

2KT

m

m

mλ τ+

Tm 2λ τ+ m [ ]λ τ= m mT, ; N=10


s sTm

s

m

m−

+

τ

1- alternative non-interacting controller 1

U s KT s

E s K T sY sci

c d( ) ( ) ( )= +

−1

1. 2 tuning rules.





118.( )K Tm m mτ +

1 38. ( )τm mT+ 0 55. ( )τm mT+

Minimum IAE -Shinskey [59] – page

159.Model: Open loopmethod not defined

128

1 024 0142

.

( . . )KT T

m mm

m

m

m

ττ τ

+ −

19 1 0 75 1. . [ ]τ τm

T

em

m+ −

−

0 48 0 7. .τm mT+


s sTm

s

m

m−

+

τ

1- ideal controller with filtered derivative

G s KT s

T s

s TN

c ci

d

d( ) = + +

+

11

1. 1 tuning rule.


RobustChien [50]

Model: Method 2( )

22

λ τ

λ τ

+ +

+m m

m m

T

K 2λ τ+ +Tm m

( )T

Tm m

m m

2

2

λ τ

λ τ

+

+ + λ = Tm ; N = 10


s sTm

s

m

m−

+

τ

1- ideal controller with set-point weighting

( ) [ ] [ ]U s K F R s Y sKT s


ii c d d( ) ( ) ( ) ( ) ( ) ( ) ( )= − + − + − . 1 tuning rule.


Ultimate cycleOubrahim andLeonard [138]


06. Ku

Fp = 01.

05. Tu

Fi = 10125. Tu

Fd = 0 01.0 05 0 8. .< <

τm

mT;

20% overshoot


s sTm

s

m

m−

+

τ

1- Alternative classical controller

G s KT sT sN

T sT sN

c ci

d

d

d( ) =

+

+

+

+

1

1

1

1. 1 tuning rule.


Direct synthesisTsang and Rad [109]

Model: Method 3

0 809.K m mτ Tm 0 5. τm

Maximum overshoot =16%; N = 8.33

Table 68: PID tuning rules – FOLIPD model ( )K e

s sTm

s

m

m−

+

τ

1- Two degree of freedom controller:

)s(Rs

NT1

sTK)s(E

sNT1

sTsT

11K)s(U

d

dc

d

d

ic

+

β+α−







)a69(cK 2 )a69(

iT )a69(dT

0.1Tm

m ≤τ

;



et al. [10]

uK672.1 uT366.0 uT136.0601.0=α , 1=β ,

N = 10, 0c 164−=φ

uK236.1 uT427.0 uT149.0607.0=α , 1=β ,

N = 10, 0c 160−=φ

uK994.0 uT486.0 uT155.0610.0=α , 1=β ,

N = 10, 0c 155−=φ

uK842.0 uT538.0 uT154.0616.0=α , 1=β ,

N = 10, 0c 150−=φ

uK752.0 uT567.0 uT157.0605.0=α , 1=β ,

N = 10, 0c 145−=φ

uK679.0 uT610.0 uT149.0610.0=α , 1=β ,

N = 10, 0c 140−=φ

uK635.0 uT637.0 uT142.0612.0=α , 1=β ,

N = 10, 0c 135−=φ

uK590.0 uT669.0 uT133.0610.0=α , 1=β ,

N = 10, 0c 130−=φ


Pagola [134a]

=φc phase


1Km = ; 1Tm = ;

8.005.0 m <τ< .

Model: Method 4

uK551.0 uT690.0 uT114.0616.0=α , 1=β ,

N = 10, 0c 125−=φ

2

−τ

+=2

m

mc

)a69(c

]01308.0T

[

5184.07608.0

K1

K ,

τ−τ+=2

m

m

m

mm

)a69(i T

5517.0T

997.303330.0TT

τ+

τ−τ+=3

m

m

2

m

m

m

mm

)a69(d T

6878.0T

774.1T

058.203432.0TT ,

6647.0=α , 2

m

m

m

m

T03330.0

T1277.08653.0

τ+

τ−=β


uK520.0 uT776.0 uT087.0609.0=α , 1=β ,

N = 10, 0c 120−=φ


Pagola [134a] -continued

uK509.0 uT810.0 uT068.0611.0=α , 1=β ,

N = 10, 0c 118−=φ

Table 69: PID tuning rules - SOSPD model ( )( )

K esT sT

ms

m m

m−

+ +

τ

1 11 2

or K eT s T s

ms

m m m

m−

+ +

τ

ξ12 2

12 1 - ideal controller

G s KT s

T sc ci

d( ) = + +

1 1

. 27 tuning rules.


Servo tuning Minimum performance indexMinimum ITAE –Sung et al. [139]Model: Method 2

Kc( )70 Ti

( )70 Td( )70 0 05 2

1

. < ≤τm

mT

Regulator tuning Minimum performance indexMinimum ITAE –Sung et al. [139]Model: Method 2

Kc( )71 Ti

( )71 Td( )71 0 05 2

1

. < ≤τm

mT

1

1 KK Tc

m

m

mm

( ) . . .70

1

0.9831

0 04 0333 0 949= − + +

−τ

ξ , ξm ≤ 0 9. or

KK T Tc

m

m

m

m

mm

( ) . . .70

1 1

0.8321 0 544 0308 1408= − + +

−τ τ ξ , ξm > 0 9. .

T TTi m

m

mm

( ) . .701

1

2 055 0 072= +

τ ξ , τm

mT 1

1≤ or T TTi m

m

mm

( ) . .701

1

1768 0 329= +

τ ξ , τm

mT 1

1>

T T

e T

dm

Tm

m

m

mm

( )

.090

. .

70 1

0.870 1

1

1 055 16831

1.060=

−

+

−

τξ

τ

KK T Tc

m

m

m

m

mm

( ).001

. . .71

1

2

1

0.7661 067 0297 2189= − +

+

− −τ τ ξ ,

τm

mT 1

0 9< . or

KK T Tc

m

m

m

m

mm

( ) . . . .71

1

2

1

0.7661 0365 0 260 14 2189= − + −

+

−τ τ ξ ,

τm

mT 1

0 9≥ .

T TTi m

m

m

( ) . .711

1

0.520

2 212 03=

−

τ,

τm

mT 1

0 4< . or

T TT

eTi m

m

m

T m

m

m

m

m( ) { . . . . . . }711

1

20.15 0.33

1

2

0 975 0 910 1845 1 525 088 281= − + −

+ −

− −

−+τ τ

ξτ

, τm

mT 1

0 4≥ .

T T

e TT

dm

T m

m

m

m

m

m

m

( )

.

. . . .

71 1

0.15 0.9391

1171

1

0.530

1 145 0 969 19 15761

1.121

=

−

+

− +

−

− +

−−

ξ

τ

ττ


ξ m τm m1T Kc Ti Td

0.5 0.125Km

05 1. Tm 0 25 1. Tm

0.5 1.007.K m

1 3 1. Tm 12 1. Tm

0.5 10.00 35.Km

5 1Tm 1 0 1. Tm

1.0 0.125Km

0 5 1. Tm 0 2 1. Tm

1.0 1.018.K m

1 7 1. Tm 0 7 1. Tm

Minimum ITAE –Lopez et al. [84]

- taken from plots

Model: Method 12

4.0 1.09 0.K m

2 1Tm 0 45 1. Tm

Representative results

Ultimate cycle

Kc( )72 2 Ti

( )72 ( )1 45 11

1162

0

. .+−

K K

Ku m

m uω ε

( )1 0 612 0103 2 2− +. .ω τ ω τu m u m


0 2 2 01

. .≤ ≤τm

mT and

0 6 4 2. .≤ ≤ξ m ; ε < 2 4.

Regulator – nearlyminimum IAE, ISE,ITAE – Hwang [60]

Model: Method 3Kc

( 73) Ti(73) ( )1 45 1

1116

20

. .+−

K K

Ku m

m uω ε

( )1 0 612 0103 2 2− +. .ω τ ω τu m u m


0 2 2 01

. .≤ ≤τm

mT and

0 6 4 2. .≤ ≤ξ m ;2 4 3. ≤ <ε

2 KK

TT K T K

Hm m

mm

m m m m d c m

u

= − − +

92 18 9

492

2

12

τ

τ ξ τ τω

+

92 324

4981 9

781

109

1081

49

8812 1

44 2 2 2 2

12 3

1 13 3

1 12 2 2 2

2KT

T T T TK T K

T T K K T

m mm

m m m m m m m m m m m mc d m

m m m m m m c m d m

uττ τ ξ τ ξ τ τ ξ τ τ ξ τ τ

ω+ + − + + − +

+

−

( )

ωξ τ τ τ

ω

HH m

mm m m H m m c m d m

u

K K

TT K K K K T

=+

+ + −

12

3 62

312

2 , εξ τ τ

τ ω012

12

12

6 42

=+ +T T K K

Tm m m m u m m

m m u

εξ τ τ τ

τ τ ω=

+ + −+

6 4 42

12

12

212

T T K K K K TK K T T

m m m m H m m m m c d

m c d m m m H( )

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

722 20674 1 0 447 0 0607

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .72

72

2 2

1

00607 1 105 0 233=

+

+ −ω ω τ ω τ

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

732 20 778 1 0 467 0 0609

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .73

73

2 2

1

0 0309 1 2 84 0 532=

+

+ −ω ω τ ω τ


Kc( )74 3 Ti

( )74 ( )1 45 11

1162

0

. .+−

K K

Ku m

m uω ε

( )1 0 612 0103 2 2− +. .ω τ ω τu m u m


0 2 2 01

. .≤ ≤τm

mT and

0 6 4 2. .≤ ≤ξ m ;3 20≤ <ε

Kc( )75 Ti

( )75 ( )1 45 11

1162

0

. .+−

K K

Ku m

m uω ε

( )1 0 612 0103 2 2− +. .ω τ ω τu m u m


0 2 2 01

. .≤ ≤τm

mT and

0 6 4 2. .≤ ≤ξ m ; ε ≥ 20

Kc( )76 Ti

( )76 ( )1 45 11

1162

0

. .+−

K K

Ku m

m uω ε

( )1 0 612 0103 2 2− +. .ω τ ω τu m u m

Decay ratio = 0.2 -

0 2 2 01

. .≤ ≤τm

mT and

0 6 4 2. .≤ ≤ξ m ; ε < 2 4.

Kc( )77 Ti

( )77 ( )1 45 11

1162

0

. .+−

K K

Ku m

m uω ε

( )1 0 612 0103 2 2− +. .ω τ ω τu m u m

Decay ratio = 0.2 -

0 2 2 01

. .≤ ≤τm

mT and

0 6 4 2. .≤ ≤ξ m ;2 4 3. ≤ <ε

Kc( )78 Ti

( )78 ( )1 45 11

1162

0

. .+−

K K

Ku m

m uω ε

( )1 0 612 0103 2 2− +. .ω τ ω τu m u m

Decay ratio = 0.2 -

0 2 2 01

. .≤ ≤τm

mT and

0 6 4 2. .≤ ≤ξ m ;3 20≤ <ε

Regulator – nearlyminimum IAE, ISE,

ITAE – Hwang [60] –continued

Model: Method 3

Kc( )79 Ti

( )79 ( )1 45 11

1162

0

. .+−

K K

Ku m

m uω ε

( )1 0 612 0103 2 2− +. .ω τ ω τu m u m

Decay ratio = 0.2 -

0 2 2 01

. .≤ ≤τm

mT and

0 6 4 2. .≤ ≤ξ m ; ε ≥ 20

3 ( ) [ ]

( )K K

K K Kc Hm H m

H m

( ). . . .

742131 0519 1 103 0514

1= −

− +

+

ω τ ε ε,

( )[ ]( )( )T

K K K

Ki

c H m

H m H m

( )( )

. . ln . .74

74

2

1

0 0603 1 0 929 1 2 01 12=

+

+ + −ω ω τ ε ε

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

752 2114 1 0 482 0 068

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .75

75

2 2

1

0 0694 1 21 0 367=

+

− + −ω ω τ ω τ

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

762 20 622 1 0 435 0 052

1= −

− +

+

ω τ ω τ,

( )( )

TK K K

Ki

c H m

H m H m H m

( )( )

. . .76

76

2 2

1

0 0697 1 0 752 0145=

+

+ −ω ω τ ω τ

[ ]( )K K

K K Kc HH m H m

m H m

( ). . .

772 20 724 1 0 469 0 0609

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .77

77

2 2

1

00405 1 193 0363=

+

+ −ω ω τ ω τ

( ) [ ]( )

K KK K Kc H

m H m

H m

( ). . . .

782126 0 506 1 1 07 0 616

1= −

− +

+

ω τ ε ε,

( )[ ]( )( )T

K K K

Ki

c H m

H m H m

( )( )

. . ln . .78

78

2

1

0 0661 1 0 824 1 171 117=

+

+ + −ω ω τ ε ε

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

792 2109 1 0497 00724

1= −

− +

+

ω τ ω τ,

( )( )T

K K K

Ki

c H m

H m H m H m

( )( )

. . .79

79

2 2

1

0 054 1 2 54 0457=

+

− + −ω ω τ ω τ


Kc( )80 4 Ti

( )80

( )1 45 11

1162

0

. .+−

K K

Ku m

m uω ε

( )1 0 612 0103 2 2− +. .ω τ ω τu m u m


0 2 2 01

. .≤ ≤τm

mT and

0 6 4 2. .≤ ≤ξ m ; ε < 2 4.

Kc( )81 Ti

( )81

( )1 45 11

1162

0

. .+−

K K

Ku m

m uω ε

( )1 0 612 0103 2 2− +. .ω τ ω τu m u m


0 2 2 01

. .≤ ≤τm

mT and

0 6 4 2. .≤ ≤ξ m ;2 4 3. ≤ <ε

Kc( )82 Ti

( )82 ( )1 45 11

1162

0

. .+−

K K

Ku m

m uω ε

( )1 0 612 0103 2 2− +. .ω τ ω τu m u m


0 2 2 01

. .≤ ≤τm

mT and

0 6 4 2. .≤ ≤ξ m ;3 20≤ <ε

Regulator – nearlyminimum IAE, ISE,

ITAE – Hwang [60] –continued

Model: Method 3

Kc(83) Ti

(83) ( )1 45 11

1162

0

. .+−

K K

Ku m

m uω ε

( )1 0 612 0103 2 2− +. .ω τ ω τu m u m


0 2 2 01

. .≤ ≤τm

mT and

0 6 4 2. .≤ ≤ξ m ; ε ≥ 20

Kc( )84 Ti

( )84 0471. KK

u

m uωξ

τ τ≤ + +

0613 0613 0117

1 1

2

. . .m

m

m

mT TServo– nearly

minimum IAE, ISE,ITAE – Hwang [60].In all, decay ratio =

0.1 with

0 2 2 01

. .≤ ≤τm

mT and

0 6 4 2. .≤ ≤ξ m

Model: Method 3

Kc(85) Ti

(85) 0471. KK

u

m uωξ

τ τ≤ + +

0613 0613 0117

1 1

2

. . .m

m

m

mT T

4 [ ]

( )K K

K K Kc HH m H m

m H m

( ). . .

802 20 584 1 0 439 0 0514

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .80

80

2 2

1

0 0714 1 0 685 0 131=

+

+ −ω ω τ ω τ

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

812 20675 1 0 472 0061

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .81

81

2 2

1

00484 1 143 0 273=

+

+ −ω ω τ ω τ

( ) [ ]( )

K KK K Kc H

m H m

H m

( ). . . .

82212 0 495 1 11 0 698

1= −

− +

+

ω τ ε ε,

( )[ ]( )( )T

K K K

Ki

c H m

H m H m

( )( )

. . ln . .82

82

2

1

0 0702 1 0 734 1 148 11=

+

+ + −ω ω τ ε ε

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

832 2103 1 0 51 0 0759

1= −

− +

+

ω τ ω τ,

( )( )T

K K K

Ki

c H m

H m H m H m

( )( )

. . .83

83

2 2

1

0 0386 1 3 26 0 6=

+

− + −ω ω τ ω τ

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

842 20 822 1 0 549 0112

1= −

− +

+

ω τ ω τ,

( )( )T

K K K

Ki

c H m

H m H m H m

( )( )

. . .84

84

2 2

1

0 0142 1 6 96 177=

+

+ −ω ω τ ω τ

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

852 20 786 1 0 441 0 0569

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .85

85

2 2

1

0 0172 1 4 62 0 823=

+

+ −ω ω τ ω τ


Kc( )86 5 Ti

( )860471. KK

u

m uωξ

τ τ≤ + +

0613 0613 0117

1 1

2

. . .m

m

m

mT TServo– nearly

minimum IAE, ISE,ITAE – Hwang [60]

(continued)Kc

( )87 Ti( )87

0471. KK

u

m uω ξτ τ

≤ + +

0613 0613 0117

1 1

2

. . .m

m

m

mT T

Kc( )88 Ti

( )880471. KK

u

m uωξ

τ τ> + −

0649 058 0005

1 1

2

. . .m

m

m

mT T

Kc( )89 Ti

( )890471. KK

u

m uω ξτ τ

> + −

0649 058 0005

1 1

2

. . .m

m

m

mT T

Kc( )90 Ti

( )900471. KK

u

m uωξ

τ τ> + −

0649 058 0005

1 1

2

. . .m

m

m

mT T

Kc( )91 Ti

( )910471. KK

u

m uωξ

τ τ> + −

0649 058 0005

1 1

2

. . .m

m

m

mT T

Kc( )92 Ti

( )920471. KK

u

m uωξ

τ τ≤ + −

0649 058 0005

1 1

2

. . .m

m

m

mT T,

ξτ τ

> + +

0613 0613 0117

1 1

2

. . .m

m

m

mT T

Servo – nearlyminimum IAE, ISE,ITAE – Hwang [60]

6

Model: Method 3

Kc( )93 Ti

( )930471. KK

u

m uωξ

τ τ≤ + −

0649 058 0005

1 1

2

. . .m

m

m

mT T,

ξτ τ

> + +

0613 0613 0117

1 1

2

. . .m

m

m

mT T

5 ( ) [ ]

( )K K

K K Kc Hm H m

H m

( ). . . .

862128 0 542 1 0986 0558

1= −

− +

+

ω τ ε ε,

( )[ ]( )( )T

K K K

Ki

c H m

H m H m

( )( )

. . ln . .86

86

2

1

0 0476 1 0 996 1 2 13 113=

+

+ + −ω ω τ ε ε

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

872 2114 1 0466 0 0647

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .87

87

2 2

1

0 0609 1 197 0 323=

+

− + −ω ω τ ω τ

6 [ ]

( )K KK K Kc H

H m H m

m H m

( ). . .

882 20 794 1 0 541 0126

1= −

− +

+

ω τ ω τ,

( )( )T

K K K

Ki

c H m

H m H m H m

( )( )

. . .88

88

2 2

1

0 0078 1 8 38 197=

+

+ −ω ω τ ω τ

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

892 20 738 1 0415 0 0575

1= −

− +

+

ω τ ω τ,

( )( )T

K K K

Ki

c H m

H m H m H m

( )( )

. . .89

89

2 2

1

0 0124 1 4 05 0 63=

+

+ −ω ω τ ω τ

( ) [ ]( )

K KK K Kc H

m H m

H m

( ). . . .

902115 0 564 1 0 959 0 773

1= −

− +

+

ω τ ε ε,

( )[ ]( )( )T

K K K

Ki

c H m

H m H m

( )( )

. . ln . .90

90

2

1

0 0335 1 0947 1 19 1 07=

+

+ + −ω ω τ ε ε

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

912 2107 1 0 466 0 0667

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .91

91

2 2

1

0 0328 1 2 21 0 338=

+

− + −ω ω τ ω τ

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

922 20 789 1 0 527 011

1= −

− +

+

ω τ ω τ,

( )( )T

K K K

Ki

c H m

H m H m H m

( )( )

. . .92

92

2 2

1

0009 1 9 7 2 4=

+

+ −ω ω τ ω τ

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

932 20 76 1 0 426 0 0551

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .93

93

2 2

1

0 0153 1 4 37 0 743=

+

+ −ω ω τ ω τ


Kc( )94 7 Ti

( )940471. KK

u

m uωξ

τ τ≤ + −

0649 058 0005

1 1

2

. . .m

m

m

mT T,

ξτ τ

> + +

0613 0613 0117

1 1

2

. . .m

m

m

mT T

Servo – nearlyminimum IAE, ISE,ITAE – Hwang [60]

(continued)

Kc( 95) Ti

(95)0471. KK

u

m uωξ

τ τ≤ + −

0649 058 0005

1 1

2

. . .m

m

m

mT T,

ξτ τ

> + +

0613 0613 0117

1 1

2

. . .m

m

m

mT T

0 6173. Ku 0 38. Tu 015. Tu T Tm m m2 2 + τ = 0.25

0 6766. Ku 0 33. Tu 019T. u T Tm m m2 2 + τ = 0.5

Minimum IAEregulator – ultimatecycle - Shinskey [16]

– page 151.Model: Method 10.

0 7874. K u 0 26. Tu 0 21. Tu T Tm m m2 2 + τ = 0.75

Direct synthesisπ

τT

A Km

m m m

1 ,2 1Tm , 0 5 1. Tm

Gain and phasemargin – Hang et al.

[35]Model: Method 4

A m mo= =15 30. , φ

Am mo= =2 0 45. , φ

A m mo= =30 60. , φ

A m mo= =4 0 67 5. , .φ

A m mo= =5 0 72. , φ

Sample A m m,φprovided. Model has a

repeated pole ( Tm1)

ωp m

m m

T

A K1 1

24 12

1

ωω τ

πpp m

mT− + Tm 2

Gain and phasemargin – Ho et al.

[140]

Model: Method 1Am m

o= =2 0 45. , φ A m mo= =30 60. , φ A m m

o= =4 0 70. , φ

T Tm m1 2> . Sample

A m m,φ provided( )

( )ω

φ π

τp

m m m m

m m

A A A

A=

+ −

−

05 1

12

.

Kc( )96 Ti

( )96 Td( )96Gain and phase

margin – Ho et al.[141]

Model: Method 1 Am mo= =2 0 45. , φ A m m

o= =30 60. , φ A m mo= =4 0 70. , φ

2ξτ

mm

mT≥ ,

τm

mT≤ 1

Sample A m m,φprovided *

Kc( )97 Ti

( )97 Td( )97Gain and phase

margin – Ho et al.[142]

Model: Method 1 Am mo= =2 0 45. , φ A m m

o= =30 60. , φ A m mo= =4 0 60. , φ

ω τ ξn m m< 2 ; Sample

A m m,φ provided

7 ( ) [ ]

( )K K

K K Kc Hm H m

H m

( ). . . .

942122 0 55 1 0 978 0 659

1= −

− +

+

ω τ ε ε,

( )[ ]( )( )T

K K K

Ki

c H m

H m H m

( )( )

. . ln . .94

94

2

1

0 0421 1 0969 1 2 02 111=

+

+ + −ω ω τ ε ε

[ ]( )

K KK K Kc H

H m H m

m H m

( ). . .

952 2111 1 0467 0 0657

1= −

− +

+

ω τ ω τ,

( )( )TK K K

Ki

c H m

H m H m H m

( )( )

. . .95

95

2 2

1

0 0477 1 2 07 0 333=

+

− + −ω ω τ ω τ

KA K Tc

m mm

m

m

( )96

1

2 2= + −

ππξ π τ

, ( )T Ti m m m( )96

12

2= + −π

πξ π τ , ( )

T TT Td

m

m m m m

( )96 12

1 12 2=

+ −π

πξ π τ

KT

A Tcp m

m

m

p mp m

( )97 12

1

22= + −

ωπ

πξω

π ω τ , T TTi m

m

m pp m

( )9712

1

22= + −

π

πξω

π ω τ ,

T T

T Td

m

m

pm m p m

( )97 1

1 12 2

=

+ −

π

πξω

π ω τ

*( ) ( )

φ

π π πτ ξ π

m

m m

mm m m

m

TA A A

A<

+ +

− − −2

1

28 1 2 1

4


ξτ

m m

m m

TK

12 1ξ m mT

05 1. Tm

mξ

ξ m > 0 7071. or

0 0507071

2 1015

12

..

.<−

<τ

ξ

m

m mT

0 7071

2 11 1

12

.,

τ

ξξm

m m

mT −

> ≥

or 0 05 0151

. .< <ξ τm m

mT,

ξ τξm m

mmT 1

1 1> <,

Wang et al. [143]

Model: Method 8

Minimum of

2 0735812

1TK

eT

Km

m

Tm m

m m m

m m

−τ

ξ ξτ

,. 2 1ξ m mT

05 1. Tm

mξξ m ≤ 0 7071. and

0 15 11

. ≤ ≤ξ τm m

mT

Kc( )98 8 Ti

( )98 0 25 98. ( )Ti

ξ m ≤ 1 or

ξ m > 1 with

05. π φ− >m

31

1

2τξ ξm

mm mT

+ −

Gain and phasemargin – Leva et al.

[34]

Model: Method 5

Kc( )99 Ti

( )99 Td( )99

ξ m > 1 with

31

1

2τξ ξm

mm mT

− −

< 0 5. π φ− ≤m

31

1

2τξ ξm

mm mT

+ −

,

φ m = 70 0 at least

8 ( )

( )K

TK

T T

T T Tc

cn i

m

cn m m cn m

i d cn i cn

( )982

12 2 2 2

12

2 2 2 2

1 4

1=

+ +

− +

ω ω ξ ω

ω ω,

( )( )

ωτ

φξ τ π φ

π φ τcn

mm

m m m m

m m m

T

T= − +

−

− −

−14 07

2 05

0 51 1

212 2

. tan.

.

TT

Ticn

cn m mm cn m

cn m

( ) tan . . tan98 1 12

12

205 05

21

= + − −−

−

ωω τ φ π

ξ ωω

, K TK T

Tz

ccn m

m d

cnm

m m

cn

( )( )

99 12

99

2

12

2

2 2

1 2 1 1=

+ − −

+ω

ω ξ ξ

ω,

z

T

cn

m cn mcn m

m m

=

− + ++ −

−

ω

φ π ω τω

ξ ξtan . tan05

1

1 1

2

, TT z

zi

m m m

m m

( )991

2

2

1

1=

+ + −

+ −

ξ ξ

ξ ξ, T

T

T zd

m

m m m

( )99 1

12 1

=+ + −

ξ ξ


1047 1.ξ

τm m

m m

TK 2 1ξm mT 05 1.

Tm

mξ Am = 3 , φm = 600

2 094 1.ξ

τm m

m m

TK 2 1ξm mT 05 1.

Tm

mξ A m = 15. , φm = 300

1571 1.ξ

τm m

m m

TK 2 1ξm mT 05 1.

Tm

mξ A m = 2 , φm = 450

0 785 1.ξ

τm m

m m

TK 2 1ξm mT 05 1.

Tm

mξ A m = 4 , φm = 6750.

Gain and phasemargin – Wang and

Shao [144]

Model: Method 8

Authors quotetuning rule for

Am = 3 , φm = 600 ;

other tuning rulesobtained usingauthors method 0 628 1.

ξτ

m m

m m

TK 2 1ξm mT 05 1.

Tm

mξ A m = 5 , φm = 720

Pemberton [145]Model: Method 1

23

1 2( )T TK

m m

m m

+τ

T Tm m1 2+ T TT T

m m

m m

1 2

1 2+

Pemberton [24]

Model: Method 1

( )T TKm m

m m

1 2+τ

T Tm m1 2+ T TT T

m1 m

m1 m

2

2+

01 101

. .≤ ≤τm

mT;

0 2 102

. .≤ ≤τm

mT

Pemberton [145]

Model: Method 1

23

1 2( )T TK

m m

m m

+τ

T Tm m1 2+ T Tm m1 2

4+ 01 10

1

. .≤ ≤τm

mT

0 2 102

. .≤ ≤τm

mT

Suyama [100]Model: Method 1

T TK

m m

m m

1 2

2+

τ T Tm m1 2+T T

T Tm1 m

m1 m

2

2+

Smith et al. [146]Model: Method not

stated( )

T TK

m m

m m

1 2++λ τ T Tm m1 2+

T TT T

m1 m

m1 m

2

2+

Chiu et al. [29]

Model: Method 6( )

λλτ

T TK

m m

m m

1 2

1++ T Tm m1 2+

T TT T

m1 m

m1 m

2

2+λ variable; suggestedvalues are 0.2, 0.4, 0.6

and 1.0.

( )λ

λτT

Km

m m

1

1 + Tm1 Tm2

T Tm m1 2> . λ = pole of

specified closed loopoverdamped response.

Wang and Clemens[147]

Model: Method 9 ( )2

11λξ

λτm m

m m

TK + 2 1ξ m mT

Tm

m

1

2ξUnderdamped

response

Gorez and Klan[147a]

Model: Ideal( )m1mmm

1mm

T2KT2

τ+ξξ

2 1ξ m mTTm

m

1

2ξNon-dominant time

delay

mm

2m1m

K)TT(368.0

τ+ T Tm m1 2+ T T

T Tm1 m

m1 m

2

2+Closed loop overshoot

= 0%

mm

2m1m

K)TT(514.0

τ+ T Tm m1 2+ T T

T Tm1 m

m1 m

2


= 5%

mm

2m1m

K)TT(581.0

τ+ T Tm m1 2+ T T

T Tm1 m

m1 m

2


= 10%

Miluse et al. [27a]


Overdampedprocess;

2m1m TT >

mm

2m1m

K)TT(641.0

τ+ T Tm m1 2+ T T

T Tm1 m

m1 m

2


= 15%


mm

2m1m

K)TT(696.0

τ+ T Tm m1 2+ T T

T Tm1 m

m1 m

2


= 20%

mm

2m1m

K)TT(748.0

τ+ T Tm m1 2+ T T

T Tm1 m

m1 m

2


= 25%

mm

2m1m

K)TT(801.0

τ+ T Tm m1 2+ T T

T Tm1 m

m1 m

2


= 30%

mm

2m1m

K)TT(853.0

τ+ T Tm m1 2+ T T

T Tm1 m

m1 m

2


= 35%

mm

2m1m

K)TT(906.0

τ+ T Tm m1 2+ T T

T Tm1 m

m1 m

2


= 40%

mm

2m1m

K)TT(957.0

τ+ T Tm m1 2+ T T

T Tm1 m

m1 m

2


= 45%

Miluse et al. [27a](continued)

mm

2m1m

K)TT(008.1

τ+ T Tm m1 2+ T T

T Tm1 m

m1 m

2


= 50%

mm

1mm

KT736.0

τξ 1mmT2ξ

m

1m

2Tξ

Closed loop overshoot= 0%

mm

1mm

KT028.1

τξ 1mmT2ξ

m

1m

2Tξ


mm

1mm

KT162.1

τξ 1mmT2ξ

m

1m

2Tξ


mm

1mm

KT282.1

τξ 1mmT2ξ

m

1m

2Tξ


mm

1mm

KT392.1

τξ 1mmT2ξ

m

1m

2Tξ


mm

1mm

KT496.1

τξ 1mmT2ξ

m

1m

2Tξ


mm

1mm

KT602.1

τξ 1mmT2ξ

m

1m

2Tξ


mm

1mm

KT706.1

τξ 1mmT2ξ

m

1m

2Tξ


mm

1mm

KT812.1

τξ 1mmT2ξ

m

1m

2Tξ


mm

1mm

KT914.1

τξ 1mmT2ξ

m

1m

2Tξ


Miluse et al. [27a]


Underdampedprocess;

15.0 m ≤ξ<

mm

1mm

KT016.2

τξ 1mmT2ξ

m

1m

2Tξ


Miluse et al. [27b]Model: Method 14

mm

1m

KT736.0

τ1mT2 1mT5.0 2m1m TT =

Seki et al. [147b]Model: Method not

specifiedc2

0

2c

0c cos)j(G θω

ωω

c

c2

c tan2tan

ω

θ++θiT5.0

‘Re-tuning’ rule. 0ω =

freq. when controlledsystem goes unstable

(crossover freq.);

cω = new crossover

freq. ; =θc phase lag


44 2 2

1350

+ β β

ωG jp ( )

4 1

135 0

+ ββ ω

44

1

135 0+ β ω1 2≤ ≤β ;

τm mT< 0 25 1.

3

G jp u( )ω

3 2.ω u

08.ω u

ω τu m ≤ 016.

Autotuning –Landau and Voda

[148]


19.

( )G jp uω

4ω u

1ω u

016 0 2. .< ≤ω τu m

Robust

( )T T

Km m m

m m

1 2 052 1

+ ++

. ττ λ

T Tm m m1 2 05+ + . τ ( )T T T T

T Tm m m m m

m m m

1 2 1 2

1 2

0 5

05

+ +

+ +

.

.

τ

τ

λ varies graphicallywith τm m mT T( )1 2+0 1 101 2. ( )≤ + ≤τ m m mT T

τm m mT T( )1 2+ λ τm m mT T( )1 2+ λ τm m mT T( )1 2+ λ

0.1 0.50 1.0 0.29 10.0 0.140.2 0.47 2.0 0.22

Brambilla et al. [48] –values deduced from

graph

Model: Method 1

0.5 0.39 5.0 0.16

mm

mm

KT00.1

τξ

mm2 τξm

m

2ξτ 14.3A m = ,

0m 4.61=φ ,

00.1Ms =

mm

mm

KT22.1

τξ

mm2 τξm

m

2ξτ 58.2A m = ,

0m 0.55=φ ,

10.1Ms =

mm

mm

KT34.1

τξ

mm2 τξm

m

2ξτ 34.2A m = ,

0m 6.51=φ ,

20.1Ms =

mm

mm

KT40.1

τξ

mm2 τξm

m

2ξτ 24.2A m = ,

0m 0.50=φ ,

26.1M s =

mm

mm

KT44.1

τξ

mm2 τξm

m

2ξτ 18.2A m = ,

0m 7.48=φ ,

30.1Ms =

mm

mm

KT52.1

τξ

mm2 τξm

m

2ξτ 07.2A m = ,

0m 5.46=φ ,

40.1Ms =

Chen et al. [53]

Model: Method 1

mm

mm

KT60.1

τξ

mm2 τξm

m

2ξτ 96.1A m = ,

0m 1.44=φ ,

50.1Ms =

Lee et al. [55]

Model: Method 1( )

TK

i

m m2λ τ+ ( )2 2

2 21

2 2

ξ λ τλ τm m

m

m

T − −+

T TTTi m mm

i

− + −2 112

ξ

( )τλ τm

i mT

3

6 2 +

Desired closed loop

response = ( )

e

s

m s−

+

τ

λ 1 2


( )T

Ki

m m2λ τ+ ( )T Tm mm

m1 2

2 222 2

+ −−+

λ τλ τ

T T TT T

Ti m mm m

i− − +1 2

1 2

( )−+

τλ τm

i mT

3

6 2

Desired closed loop

response = ( )

e

s

m s−

+

τ

λ 1 2

( )T

Ki

m mλ τ+ ( )2

21

2

ξ τλ τm m

m

m

T ++

( )T T

T

Ti m m

mm

m

i−

−+

26

1

12

3

ξ

τ

λ τ

Desired closed loop

response = ( )

es

ms−

+

τ

λ 1

Lee et al. [55](continued)

( )T

Ki

m mλ τ+ ( )T Tm m

m

m1 2

2

2+ +

+τ

λ τ

( )T T T

T T

Ti m m

m mm

m

i

− +

−+

( )1 2

1 2

3

6

τ

λ τDesired closed loop

response = ( )

es

ms−

+

τ

λ 1


K esT sT

ms

m m

m−

+ +

τ

1 11 2

or K eT s T s

ms

m m m

m−

+ +

τ

ξ12 2

12 1 - filtered controller

G s KT s

T sT sc c

id

f

( ) = + +

+1 1 1

1. 1 tuning rule.


Robust

Hang et al. [35]

Model: Method 4( )2

21T

Km m

m m

++

τλ τ

Tm1 m+ 05. τ TT

m1 m

m1 m

ττ2 +

Model has a repeatedpole ( )Tm1 .

λ τ> 025. m .

( )Tfm

m

=+

λτλ τ2


K esT sT

ms

m m

m−

+ +

τ

1 11 2

or K eT s T s

ms

m m m

m−

+ +

τ

ξ12 2

12 1 - filtered controller

G s KT s

T s b sa sc c

id( ) = + +

++

1 1 11

1

1

. 1 tuning rule.


RobustJahanmiri and Fallahi

[149]

Model: Method 6

2 1ξτ λm m

m m

TK ( )+

λ τ ξ= +0 25 01 1. .m m mT

2 1ξ m mT Tm

m

1

2ξ

b m1 0 5= . τ

a m

m1 2

=+

λττ λ( )


K esT sT

ms

m m

m−

+ +

τ

1 11 2

or K eT s T s

ms

m m m

m−

+ +

τ

ξ12 2

12 1 - classical controller

G s KT s

T sT sN

c ci

d

d( ) = +

+

+

11 1

1. 7 tuning rules.



1 Kc( )100 τ τ

m

T

em

m151

1 5. .−

−

1 0 9 12

+ −

−

. eTm

mτ

0 56 112 1

..

τ τm

T

em

m−

+

−

0 6 2. Tm

Tm

m

2 3τ

≤Minimum IAE -

Shinskey [59] – page159

Model: Open loopmethod not specified

25 1. TK

m

m mτ τm mT+ 0 2 2. τm mT+ 0 2 2.Tm

m

2 3τ

>

0800 1. TK

m

m mτ( )15 2. Tm m+ τ ( )060 2. Tm m+ τ T

Tm

m m

2

2

025+

=τ

.

0770 1. TK

m

m mτ( )12 2. Tm m+ τ ( )0 70 2. Tm m+ τ T

Tm

m m

2

2

0 5+

=τ

.

** Minimum IAE -Shinskey [59]

0833 1. TK

m

m mτ( )075 2. Tm m+ τ ( )060 2. Tm m+ τ T

Tm

m m

2

2

0 75+

=τ

.

** Minimum IAE -Shinskey [17]

059. K u 0 36T. u 0 26. Tu τ m

mT 1

0 2= . , TT

m

m

2

1

0 2= .

** Minimum IAEShinskey [17]

085 1. TK

m

m mτ087 1. TK

m

m mτ100 1. TK

m

m mτ

125 1. TK

m

m mτ

198. τm

2 30. τm

2 50. τm

2 75. τm

086. τm

165. τm

200. τm

2 75. τm

τ m

mT 1

0 2= . ,TT

m

m

2

1

01= .

TT

m

m

2

1

0 2= .

TT

m

m

2

1

0 5= .

TT

m

m

2

1

10= .

1 K

eKT

T Tc T

m m

m

m

m

m

m

m

m

( )

.2

. .

100

1

1

2 2

2

100

48 57 1 1 0 34 0 21

=

+ −

+ −

−τ τ

τ τ


αKm

βτm χτm

Minimum ISE –McAvoy and

Johnson [83] –deduced from graph

Model: Method 1

ξ m τm

mT 1

α β χ ξ m τm

mT 1

α β χ ξ m τm

mT 1

α β χ

1 0.5 0.7 0.97 0.75 4 0.5 3.0 1.16 0.85 7 0.5 5.4 1.19 0.851 4.0 7.6 3.33 2.03 4 4.0 22.7 1.89 1.28 7 4.0 40.0 1.64 1.14N = 201 10.0 34.3 5.00 2.7

1 0.5 0.9 1.10 0.64 4 0.5 3.2 1.33 0.78 7 0.5 5.9 1.39 1.041 4.0 8.0 4.00 1.83 4 4.0 23.9 2.17 1.17 7 4.0 42.9 1.89 1.37N = 101 10.0 33.5 6.25 2.43

Direct synthesis

Astrom et al. [30] –Method 13

3

13

1 2

KT Tm

m

m m

++

τ T Tm m1 2+ T TT T

m1 m

m1 m

2

2+N = 8 - Honeywell

UDC6000 controller

ατ

TK

m

m m

1Tm1 Tm2

ξ mρ α ξ m

ρ α ξ mρ α

6 ≥ 0 02. 0.51 3 ≥ 004. 0.50 2 ≥ 006. 0.451.75 0.27 0.46 1.75 0.13 0.42 1.75 0.07 0.391.75 0.07 0.36 1.5 0.33 0.44 1.5 0.17 0.381.5 0.11 0.33 1.5 0.08 0.28 1.0 0.50 0.40

Smith et al. [26] –deduced from graph

[ ( )ρ τ= +m m mT T1 2 ]

Model: Method notstated

1.0 0.25 0.46 1.0 0.17 0.48 1.0 0.13 0.49

N = 10


K esT sT

ms

m m

m−

+ +

τ

1 11 2

or K eT s T s

ms

m m m

m−

+ +

τ

ξ12 2

12 1 - alternative classical

controller G s KT s

NT sT sc c

i

d

d

( ) = +

++

1 1 1

1. 1 tuning rule.


******

08 10 7

20 3. . .T Tm m

mτ 05 1 2. T Tm m+ τm m mT T1 23 N=10Hougen [85]

Model: Method 1 084 10 8

20 2. . .T Tm m

mτ 0 53 1 31 2. .T Tm m+ ( )008 1 20 28

..τm m mT T N=30


K esT sT

ms

m m

m−

+ +

τ

1 11 2

or K eT s T s

ms

m m m

m−

+ +

τ

ξ12 2

12 1 - alternative non-

interacting controller 1: U s KT s

E s K T sY sci

c d( ) ( ) ( )= +

−1 1

. 3 tuning rules.



1 Kc( )101 τ τ

m

T

em

m0 5 1 4 11

15. . [ ].+ −

−

1 0 48 12

+ −

−

. eTm

mτ

0 42 11 1

..2

τ τm

T

em

m−

+−

0 6 2. Tm

Tm

m

2 3τ

≤Minimum IAE -

Shinskey [59] – page159

Model: open loopmethod not specified 333 1. T

Km

m mττm mT+ 0 2 2. τm mT+ 0 2 2. Tm

m

2 3τ

>

118 1. TK

m

m mτ 2 20. τm 0 72. τm

τ m

mT 1

0 2= . ,TT

m

m

2

1

01= .

125 1. TK

m

m mτ 2 20. τm 110. τm

TT

m

m

2

1

0 2= .

167 1. TK

m

m mτ 2 40. τm 165. τm

TT

m

m

2

1

0 5= .

Minimum IAE -Shinskey [17] – page119. Model: method

not specified.

25 1. TK

m

m mτ 2 15. τm 2 15. τm

TT

m

m

2

1

10= .

Minimum IAE -Shinskey [17] – page121. Model: method

not specified.085. K u 035. Tu 017. Tu

τ m

mT 1

0 2= . ,TT

m

m

2

1

0 2= .

1 K

eKT

T Tc

m m

m

m

m

m

m

m

m

( )

.

. .

101

1 5T

1

2 2

2

100

38 40 1 1 0 34 0 21

=

+ −

+ −

−τ τ

τ τ


K esT sT

ms

m m

m−

+ +

τ

1 11 2

or K eT s T s

ms

m m m

m−

+ +

τ

ξ12 2

12 1 - series controller

( )G s KT s

T sc ci

d( ) = +

+1 1 1 . 1 tuning rule.



Least mean squareerror - Haalman [23]Model: Method 1

23

1TKm

m mτTm1 Tm2

T Tm m1 2> . Maximum

sensitivity = 1.9, Gainmargin = 2.36, Phase

margin = 500


K esT sT

ms

m m

m−

+ +

τ

1 11 2

or K eT s T s

ms

m m m

m−

+ +

τ

ξ12 2

12 1 - non-interacting

controller

+−

+= )s(Y

NsT1

sT)s(E

sT1

1K)s(Ud

d

ic . 2 tuning rules.


Servo – Min. IAE -Huang et al. [18]Model: Method 1

Kc( )102 2 Ti

( )102 Td( )102

0 12

1

< ≤TT

m

m

;

0 1 11

. ≤ ≤τ m

mT; N =10

2 KK T

TT

TT Tc

m

m

m

m

m

m m

m

m

m

( ) . . . . .102

1

2

1

2

12

1

0.08651 7 0636 66 6512 137 8937 122 7832 261928= + − − +

τ τ τ

+

+

−

+

+

1 336578 30098 10 9347 141511 29 40681

2 64052

1

1 03092

1

2 3452

1

105702

K TTT

TT

TT

T

m

m

m

m

m

m

m

m

m

m

m

. . . . .. . . .

ττ

+

−

+

− −1 34 3156 701035 152 63922

1 1

0 94502

1 1

0 9282

1

2

1

0 8866

KTT T

TT T T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

. . .. . .

τ τ τ

+ −

− + + +

−1

47 9791 57 9370 10 4002 6 7646 7 34531

2

1

0 8148 0 4062

1

2

1

2

12

K TTT

e e eTm

m

m

m

m

T

T

T

T

T m

m

m

m

m

m

m m

m. . . . .. .

τ ττ τ

T TT

TT T

T

T

TT Ti m

m

m

m

m

m

m

m m

m

m

m

m

m

( ) . . . . . . .1021

1

2

1 1

2

2

12

2

1

2

1

3

0 9923 0 2819 02679 14510 06712 0 6424 2 504= + − −

− +

+

τ τ τ τ

+

+

+

−

+

TTT T T

TT

TT T T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

2

1 1

2

1

2

1

22

1

3

1

4

1

2

1

4

2 5324 2 3641 2 0500 18759 08519. . . . .τ τ τ τ

+ −

−

−

−

T TT T T

TT T

TT

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

2

1 1

3

1

2

2

1

2

1

2

1

3

2

1

4

13496 34972 2 4216 31142. . . .τ τ τ

+

+

+

+

+

TT

TT T

TT T T

TT

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

1

5

2

1 1

4

2

1

2

1

3

1

2

2

1

3

2

1

5

05862 0 0797 0 985 12892 12108. . . . .τ τ τ τ

T TT

TT T

T

T

TT Td m

m

m

m

m

m

m

m m

m

m

m

m

m

( ) . . . . . . .1021

1

2

1 1

2

2

12

2

1

2

1

3

0 0075 03449 0 3924 00793 27495 0 6485 0 8089= + + −

+ +

+

τ τ τ τ

+ −

+

−

−

T TT T T

TT

TT Tm

m

m

m

m

m

m

m

m

m

m

m

m1

2

1 1

2

1

2

1

2

2

1

3

1

4

9 7483 3 4679 58194 10884. . . .τ τ τ

+

−

−

+

T TT T T

TT T

TT

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

2

1 1

3

1

2

2

1

2

1

2

1

3

2

1

4

12 0049 14056 3 7055 100045. . . .τ τ τ

+

−

−

+

TT

TT T

TT T T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

1

5

2

1 1

4

2

1

2

1

3

1

2

2

1

3

0 3520 6 3603 32980 7 0404. . . .τ τ τ τ

+

−

+

+

+

TT

TT

TT T

TT T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

1

2

1

4

2

1

5

1

6

2

1 1

5

2

1

6

14294 69064 0 0471 11839 17087. . . . .τ τ τ


Servo-Min. IAE -Huang et al. [18]

Kc( )103 3 (N=10) Ti

( )103 Td( )103

0 4 1. ≤ ≤ξm ; 0 05 11

. ≤ ≤τm

mT

+

−

−

+

TT

TT T

TT T

TT T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

1

4

2

1

2

1

3

2

1

3

1

2

2

1

4

1

2

1

5

1 7444 12817 21281 15121. . . .τ τ τ τ

3 KK T T Tc

m

m

mm m

m

m

m

m

(.9009

. . . . .103)

1 1 1

11 81727 32 9042 319179 38 3405 0 2079= − − + +

+

−τ ξ ξ τ τ

+

+

− + −

1 29 3215 359456 214045 51159 2193811

01571

1

1 22340 1311 1 9814 1 737

K T Tm

m

m

m

mm m m. . . .

. .. . .τ τ ξ ξ ξ

+ −

+

−

−1 17 7448 268655 52 9156

1

0 1303

1

1 2008

1

11207

K T T Tmm

m

mm

m

m

m

mm. . .

. ..ξ τ ξ τ τ ξ

+ − − + − +

1 22 4297 33331 85175 15312 089061

0 3626 11 1

K Te e e T

m

m

mm

T T m m

m

m

m m

m m

m. . . . ..τ ξ ξτ

τξ

τ ξ

T TT T T Ti m

m

mm

m

mm

m

mm

m

mm

( . . . . . . .103)1

1 1

2

1 1

2211731 6 3082 06937 85271 24 7291 67123 7 9559= + − +

− −

+

τξ

τξ

τξ

τξ

+ −

−

+

+

TT T T Tm

m

m

m

mm

m

mm

m

m1

1

3

1

4

1

32

1

32 3937 271372 166 9272 363954. . . .τ τ

ξτ

ξτ

+ −

− − +

TT Tm m

m

mm

m

mm m1

2

1

23

1

3 494 8879 22 6065 16084 29 9159. . . .ξ τ ξ τ ξ ξ

+

−

−

+

TT T T Tm

m

mm

m

mm

m

m

m

mm1

1

5

1

42

1

3

1

2349 6314 84 3776 938912 1101706. . . .τ ξ τ ξ τ τ ξ

+ −

− −

−

+

TT T Tm

m

mm m

m

mm

m

mm1

1

4 5

1

6

1

56251896 19 7569 12 4348 117589 55268. . . . .τ ξ ξ τ ξ τ ξ

+

−

−

+

TT T T Tm

m

mm

m

mm

m

mm

m

mm1

1

42

1

33

1

24

1

568 3097 17 8663 225926 95061. . . .τ ξ τ ξ τ ξ τ ξ

T TT T T Td m

m

mm

m

mm

m

mm

m

m

( ) . . . . . . .1041

1 1

2

1

2

1

3

00904 08637 01301 4 9601 14 3899 07170 125311= + − +

+ + −

τξ

τξ

τξ

τ

+ −

−

− −

TT T Tm m

m

mm

m

mm

m

m1

1

22

1

3

1

4

42 5012 214907 69555 12 3016. . . .ξ τ ξ τ ξ τ

+

+

+ +

TT T Tm m

m

mm

m

m

m

mm m1

1

32

1

2

1

3 4102 9447 7 5855 19 1257 17 0952. . . .ξτ

ξτ τ

ξ ξ

+

−

−

+

TT T T Tm

m

mm

m

mm

m

m

m

mm1

1

5

1

42

1

3

1

23108688 17 2130 1100342 50 6455. . . .τ ξ τ ξ τ τ ξ

+ −

− −

−

+

TT T Tm

m

mm m

m

mm

m

mm1

1

4 5

1

6

1

5616 7073 16 2013 0 0979 109260 54409. . . . .τ ξ ξ τ ξ τ ξ


Regulator – Min. IAE- Huang et al. [18]Model: Method 1

Kc( )104 4 Ti

( )104 Td( )104

0 12

1

< ≤TT

m

m

;

0 1 11

. ≤ ≤τ m

mT; N =10

+

+

−

+

TT T T Tm

m

mm

m

mm

m

mm

m

mm1

1

42

1

33

1

24

1

529 4445 216061 241917 62798. . . .τ ξ τ ξ τ ξ τ ξ

4 KK T

TT

TT Tc

m

m

m

m

m

m m

m

m

m

( ) . . . . .104

1

2

1

2

12

1

0.80581 01098 86290 766760 33397 11863= − + − +

−τ τ τ

+

+

−

−

1 231098 203519 52 0778 1210331

0 6642

1

2 1482

2

1

0 8405

2

1

2 1123

K T TTT

TTm

m

m

m

m

m

m

m

m

. . . .. . . .

τ τ

+

+

−

− −1 9 4709 13 6581 19 4944

1

2

1

0 5306

2

1 1

1 0781

2

1 1

0 4500

K TTT

TT T

TT Tm

m

m

m

m

m

m

m

m

m

m

m

m

. . .. . .

τ τ τ

+ −

− + + −

128 2766 191463 8 8420 7 4298 114753

1

2

1

11427

21

2

1

2

12

K TTT

e e eT

m

m

m

m

m

T

T

T

T

T m

m

m

m

m

m

m m

m. . . . ..

ττ

τ τ

T TT

TT T

T

T

TT Ti m

m

m

m

m

m

m

m m

m

m

m

m

m

( ) . . . . . . .1041

1

2

1 1

2

2

12

2

1

2

1

3

00145 2 0555 0 7435 4 4805 12069 02584 7 7916= − + + −

+ +

+

τ τ τ τ

+ −

+

−

−

+

T TT T T

TT

TT T

TT Tm

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

2

1 1

2

1

2

1

2

2

1

3

1

4

2

1 1

3

6 0330 3 9585 30626 7 0263 7 0004. . . . .τ τ τ τ

+ −

−

+

+

T TT T T

TT

TT Tm

m

m

m

m

m

m

m

m

m

m

m

m1

2

1

2

1

2

1

2

1

3

2

1

4

1

5

2 7755 15769 31663 2 4311. . . .τ τ τ

+ −

−

−

+

−

T TT

TT T

TT T T

TT T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

2

1

5

2

1 1

4

2

1

2

1

3

1

2

2

1

3

1

2

1

5

0 9439 2 4506 0 2227 19228 0 5494. . . . .τ τ τ τ

T TT

TT T

T

T

TT Td m

m

m

m

m

m

m

m m

m

m

m

m

m

( ) . . . . . . .1041

1

2

1 1

2

2

12

2

1

2

1

3

0 0206 09385 0 7759 23820 2 9230 3 2336 72774= − + + −

+ −

+

τ τ τ τ

+ −

+

+

−

T TT T T

TT

TT Tm

m

m

m

m

m

m

m

m

m

m

m

m1

2

1 1

2

1

2

1

2

2

1

3

1

4

9 9017 2 7095 61539 111018. . . .τ τ τ

+

+

−

−

T TT T T

TT T

TT

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

2

1 1

3

1

2

2

1

2

1

2

1

3

2

1

4

10 6303 57105 7 9490 6 6597. . . .τ τ τ

+

−

−

+

TT

TT T

TT T T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

1

5

2

1 1

4

2

1

2

1

3

1

2

2

1

3

80849 4 4897 7 6469 21155. . . .τ τ τ τ

+

+

−

+

−

TT

TT

TT T

TT T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

1

2

1

4

2

1

5

1

6

2

1 1

5

2

1

6

50694 4 1225 2 274 0519 11295. . . . .τ τ τ

+

+

−

−

TT

TT T

TT T

TT T

TTm

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m1

1

4

2

1

2

1

3

2

1

3

1

2

2

1

4

1

2

1

5

2 2875 0 9524 16307 09321. . . .τ τ τ τ


Regulator - Min. IAE -Huang et al. [18]Model: Method 1

Kc( )105 5 Ti

( )105 Td( )105 [N=10] 0 4 1. ≤ ≤ξm ; 0 05 1

1. ≤ ≤

τm

mT

5 KK T T T Tc

m

m

mm m

m

m

m

mm

m

m

( . . . . . .105)

1 1 1

3 0.0861 357307 1419 14023 6 8618 0 9773 555898= − − + +

−

+

τ ξ ξ τ τ ξ τ

+ −

+ + +

−

− −1 33093 538651 114911 08778 29 8822

1

2

1

2 3

1

1 6624

1

0 6951

K T T T Tmm

m

m

m

mm m

m

m

m

m

. . . . .. .

ξ τ τ ξ ξ τ τ

+

− − − +

− −1 53535 16 9807 254293 01671 0 0034

1

0 476211197 14622 58981

2 1208

K T Tm

m

mm m m m

m

m

. . . . ..

. ..

τ ξ ξ ξ ξ τ

+ − − − − + +

1 250355 54 9617 01398 82721 63542 104791

30836

1

1 2103 11 1

K T Te e e T

m

m

mm

m

mm

T T m m

m

m

m m

m m

m. . . . . .. .τ ξ τ ξ ξτ

τξ

τ ξ

T TT T T Ti m

m

mm

m

mm

m

mm

m

m

( . . . . . . .105)1

1 1

2

1

2

1

3

02563 118737 16547 161913 9 7061 35927 195201= + − −

− + +

τξ

τξ

τξ

τ

+ −

+

− −

TT T Tm m

m

mm

m

mm

m

m1

1

22

1

3

1

4

14 5581 2 939 0 4592 34 6273. . . .ξ τ ξ τ ξ τ

+

+

+ −

TT T Tm m

m

mm

m

m

m

mm m1

1

32

1

2

1

3 450 5163 8 9259 8 6966 6 9436. . . .ξ τ ξ τ τ ξ ξ

+

−

−

+

TT T T Tm

m

mm

m

mm

m

m

m

mm1

1

5

1

42

1

3

1

2327 2386 20 0697 42 2833 85019. . . .τ ξ τ ξ τ τ ξ

+ −

+ −

+

−

TT T Tm

m

mm m

m

mm

m

mm1

1

4 5

1

6

1

5612 2957 8 0694 7 7887 2 3012 2 7691. . . . .τ ξ ξ τ ξ τ ξ

+

+

−

+

TT T T Tm

m

mm

m

mm

m

mm

m

mm1

1

42

1

33

1

24

1

588984 102494 54906 4 6594. . . .τ ξ τ ξ τ ξ τ ξ

T TT T T Td m

m

mm

m

mm

m

mm

m

m

( . . . . . . .105)1

1 1

2

1

2

1

3

0 021 33385 0185 05164 0 9643 08815 0584= − + + −

− − +

τξ

τξ

τξ

τ

+ −

+

+ −

TT T Tm m

m

mm

m

mm

m

m1

1

22

1

3

1

4

12513 13468 2 3181 52368. . . .ξτ

ξτ

ξτ

+

+

− −

TT T Tm m

m

mm

m

m

m

mm m1

1

32

1

2

1

3 4153014 119607 2 0411 31988. . . .ξτ

ξτ τ

ξ ξ

+

−

−

−

TT T T Tm

m

mm

m

mm

m

m

m

mm1

1

5

1

42

1

3

1

2334675 0 8219 15 0718 18859. . . .

τξ

τξ

τ τξ

+

+ −

+

−

TT T Tm

m

mm m

m

mm

m

mm1

1

4 5

1

6

1

560 4841 2 2821 0 9315 0529 06772. . . . .

τξ ξ

τξ

τξ

+ −

+

−

+

TT T T Tm

m

mm

m

mm

m

mm

m

mm1

1

42

1

33

1

24

1

514212 71176 2 3636 0 5497. . . .τ

ξτ

ξτ

ξτ

ξ


K esT sT

ms

m m

m−

+ +

τ

1 11 2

or K eT s T s

ms

m m m

m−

+ +

τ

ξ12 2

12 1 - ideal controller with

set-point weighting ( ) [ ] [ ]U s K F R s Y sKT s


ii c d d( ) ( ) ( ) ( ) ( ) ( ) ( )= − + − + − . 1 tuning rule.


Ultimate cycle06. Ku

FK KK Kp

m u

m u

=−+

131617

.05. Tu

Fi = 10125. Tu

F Fd p= 2

Repeated pole

01 3. < <τm

mT;

10% overshoot

Oubrahim andLeonard [138]

Model: Method notspecified 06. Ku

FK Kp

m u

=+

3829 35

05. Tu

Fi = 10125. Tu

F Fd p= 2

Repeated pole

01 3. < <τm

mT;

20% overshoot


K esT sT

ms

m m

m−

+ +

τ

1 11 2

or K eT s T s

ms

m m m

m−

+ +

τ

ξ12 2

12 1 - non-interacting

controller [ ] ( )U s K bTs

R s Y s c T s Y sci

d( ) ( ) ( ) ( )= +

− − +1

. 1 tuning rule.


Direct synthesis

Kc( )105a 6 154 105. ( )Ti

a Td( )105a

b = 0.198c K Kc

am= −( )105 1 ;

zero overshoot

Kc( )105a 127 105. ( )Ti

a Td( )105a

b = 0.289c K Kc

am= −( )105 1 ;

minimum IAE

Hansen [150]

Model: Method 1

Kc( )105a 175 105. ( )Ti

a Td( )105a

b = 0.143c K Kc

am= −( )105 1 ;

conservative tuning

6 ( )[ ]

( )[ ]K

T T T T

K T T T Tc

a m m m m m m

m m m m m m m m

( ).

. .

105 1 2 1 22 3

1 2 1 22 3 2

2 0 5

9 0 5 0167=

+ + +

+ + +

τ τ

τ τ τ,

( )[ ]( )[ ]

TT T T T

T T T Ti

a m m m m m m m

m m m m m m

( ). .

.105 1 2 1 2

2 3

1 2 1 22

3 05 0167

05=

+ + +

+ + +

τ τ τ

τ τ,

( )[ ]( )[ ]T

T T T T

K T T T T

T TKd

a m m m m m m

m m m m m m m m

m m m

m

( ).

. .105 1 2 1 2

2 2

1 2 1 22 3

1 22 05

3 0 5 0167=

+ + +

+ + +−

+ −τ τ

τ τ τ

τ


K esT sT

ms

m m

m−

+ +

τ

1 11 2

or K eT s T s

ms

m m m

m−

+ +

τ

ξ12 2

12 1 - ideal controller with

filtered derivative G s KTs

T ssTN

c ci

d

d( ) = + +

+

11

1. 2 tuning rules.


Direct synthesisωp m1

m m

T

A K

1

24 1

2

1ω

ω τ

πpp m

mT− +

Tm2Hang et al. [151]

Model: Method 1A m m

o= =2 0 45. , φ

A m mo= =30 60. ,φ

A m mo= =4 0 67 5. , .φ A m m

o= =5 0 72. , φ

Sample A m m,φprovided. N = 20.

T Tm m1 2>( )

( )ω

φ π

τp

m m m m

m m

A A A

A=

+ −

−

0 5 1

12

.

RobustHang et al. [151]

Model: Method 1( )T

Km

m m

1

λ τ+ Tm1 Tm2 N = 20. T Tm m1 2>

Table 80: PID tuning rules – SOSPD model K e

T s T sm

s

m m m

m−

+ +

τ

ξ12 2

12 1- Two degree of freedom controller:

)s(Rs

NT1

sTK)s(E

sNT1

sTsT

11K)s(U

d

dc

d

d

ic

+

β+α−





)b105(cK 7 )b105(

iT )b105(dT

0.1Tm

m ≤τ

; 0.1m =ξ



et al. [10]



)c105(cK )c105(

iT )c105(dT

0.1Tm

m ≤τ

; 5.0m =ξ



et al. [10]Minimum ITAE -Pecharroman and

Pagola [134a]

Model: Method 15

uK7236.0 uT5247.0 uT1650.05840.0=α , 1=β ,

N = 10,0

c 65.139−=φ1K m = ; 1Tm = ;

1m =ξ

7

−τ

+=2

m

mc

)b105(c

]02295.0T

[

6978.0389.1

K1

K ,

τ+

τ−τ+=3

m

m

2

m

m

m

mm

)b105(i T

231.1T

434.3T

104.402453.0TT

τ−

τ+

τ−

τ+=

4

m

m

3

m

m

2

m

m

m

mm

)b105(d T

7962.0T

359.2T

741.2T

852.103459.0TT ,

3

m

m2

m

m

m

m

T07345.0

T1371.0

T1285.06726.0

τ+

τ−

τ−=α ,

2

m

m

m

m

T02724.0

T2679.08665.0

τ+

τ−=β

−τ

+=2

m

mc

)c105(c

]01147.0T

[

5013.03363.0

K1

K ,

τ+

τ−τ+−=3

m

m

2

m

m

m

mm

)c105(i T

054.2T

522.5T

858.402337.0TT

τ+

τ−

τ+=

3

m

m

2

m

m

m

mm

)c105(d T

2826.0T

161.1T

023.203392.0TT ,

3

m

m2

m

m

m

m

T1699.0

T5680.0

T05413.06678.0

τ+

τ−

τ−=α ,

2

m

m

m

m

T1212.0

T1205.08646.0

τ−

τ−=β


uK803.0 uT509.0 uT167.0585.0=α , 1=β ,

N = 10, 0c 146−=φ

uK727.0 uT524.0 uT165.0584.0=α , 1=β ,

N = 10, 0c 140−=φ

uK672.0 uT532.0 uT161.0577.0=α , 1=β ,

N = 10, 0c 134−=φ

uK669.0 uT486.0 uT170.0550.0=α , 1=β ,

N = 10, 0c 125−=φ

uK600.0 uT498.0 uT157.0543.0=α , 1=β ,

N = 10, 0c 115−=φ

uK578.0 uT481.0 uT154.0528.0=α , 1=β ,

N = 10, 0c 105−=φ

uK557.0 uT467.0 uT149.0504.0=α , 1=β ,

N = 10, 0c 93−=φ

uK544.0 uT466.0 uT141.0495.0=α , 1=β ,

N = 10, 0c 84−=φ

uK537.0 uT444.0 uT144.0484.0=α , 1=β ,

N = 10, 0c 73−=φ

uK527.0 uT450.0 uT131.0477.0=α , 1=β ,

N = 10, 0c 63−=φ

uK521.0 uT440.0 uT129.0454.0=α , 1=β ,

N = 10, 0c 52−=φ

uK515.0 uT429.0 uT126.0445.0=α , 1=β ,

N = 10, 0c 41−=φ

uK509.0 uT399.0 uT132.0433.0=α , 1=β ,

N = 10, 0c 30−=φ

uK496.0 uT374.0 uT123.0385.0=α , 1=β ,

N = 10, 0c 19−=φ

uK480.0 uT315.0 uT112.0286.0=α , 1=β ,

N = 10, 0c 10−=φ


Pagola [134b]

=φc phase


1K m = ; 1Tm = ;

101.0 m <τ<

Model: Method 15

uK430.0 uT242.0 uT084.0158.0=α , 1=β ,

N = 10, 0c 6−=φ

Table 81: PID tuning rules - PDI 2 model 2

sm

m seK

)s(Gmτ−

= - controller


11(K)s(U dcc

ic −−++= . 1 tuning rule.



Model: Method 1 2mmK75.3 τ m4.5 τ m5.2 τ b = 0.167

Table 82: PID tuning rules – SOSIPD model (repeated pole) 2

1m

sm

)sT1(seK m

+

τ−


)s(Rs

NT1

sTK)s(E

sNT1

sTsT

11K)s(U

d

dc

d

d

ic

+

β+α−






Model: Method 2

)d105(cK 8 )d105(

iT )d105(dT

0.1Tm

m ≤τ

;



et al. [10]

uK672.1 uT366.0 uT136.0601.0=α , 1=β ,

N = 10, 0c 164−=φ

uK236.1 uT427.0 uT149.0607.0=α , 1=β ,

N = 10, 0c 160−=φ

uK994.0 uT486.0 uT155.0610.0=α , 1=β ,

N = 10, 0c 155−=φ

uK842.0 uT538.0 uT154.0616.0=α , 1=β ,

N = 10, 0c 150−=φ

uK752.0 uT567.0 uT157.0605.0=α , 1=β ,

N = 10, 0c 145−=φ

uK679.0 uT610.0 uT149.0610.0=α , 1=β ,

N = 10, 0c 140−=φ

uK635.0 uT637.0 uT142.0612.0=α , 1=β ,

N = 10, 0c 135−=φ

uK590.0 uT669.0 uT133.0610.0=α , 1=β ,

N = 10, 0c 130−=φ


Pagola [134a]

=φc phase


1K m = ; 1Tm = ;

101.0 m <τ<

Model: Method 1

uK551.0 uT690.0 uT114.0616.0=α , 1=β ,

N = 10, 0c 125−=φ

8

+τ

+=002325.0

T

5667.01778.0

K1

K

m

mc

)d105(c ,

τ−

τ+

τ−τ+=4

m

m

3

m

m

2

m

m

m

mm

)d105(i T

835.4T

70.13T

98.14T

16.112011.0TT

τ+=

m

mm

)d105(d T

3620.0262.1TT , 6666.0=α , 2

m

m

m

m

T03845.0

T09750.08206.0

τ+

τ−=β


uK520.0 uT776.0 uT087.0609.0=α , 1=β ,

N = 10, 0c 120−=φ

Minimum ITAE -Pecharroman andPagola [134a] –

continueduK509.0 uT810.0 uT068.0

611.0=α , 1=β ,

N = 10, 0c 118−=φ

Table 83: PID tuning rules - SOSPD model with a negative zero ( )

( )( )K sT e

sT sTm m

s

m m

m1

1 13

1 2

−

+ +

− τ

- controller with filtered

derivative G s KT s

T sT sN

c ci

d

d( ) = + +

+

11

1. 1 tuning rule.


Robust

( )T T T

Km m m

m m

1 2 3+ −+λ τ

T T Tm m m1 2 3+ − ( )T T T T T T

T T Tm m m m m m

m m m

1 2 1 2 3 3

1 2 3

− + −

+ −

T T Tm m m1 2 3> >

N=10; [ ]λ τ= Tm m1 ,Chien [50]

Model: Method 1

( )2 2 3ξ

λ τ

T T

Km m

m m

−

+2 1 3ξT Tm m− ( )T T T T

T Tm m m m

m m

12

1 3 3

1 3

2

2

− −

−

ξ

ξN=10; [ ]λ τ= Tm m1 ,


( )( )K sT e

sT sTm m

s

m m

m1

1 13

1 2

−

+ +

− τ

- classical controller

G s KT s

T sT sN

c ci

d

d( ) = +

+

+

11 1

1. 1 tuning rule.


Robust

( )T

Km

m m

2

λ τ+Tm2 Tm1 T Tm m1 2>

N=10; [ ]λ τ= Tm m1 ,Chien [50]

Model: Method 1( )T

Km

m m

1

λ τ+Tm1 Tm2 T Tm m1 2>

N=10; [ ]λ τ= Tm m1 ,


( )( )K sT e

sT sTm m

s

m m

m1

1 13

1 2

−

+ +

− τ

- series controller with


T sT sN

c ci

d

d( ) = +

+

+

11

11

. 1 tuning rule.


Robust

( )T

Km

m m

2

λ τ+Tm2 T Tm m1 3− T T Tm m m1 2 3> >

N=10; [ ]λ τ= Tm m1 ,Chien [50]

Model: Method 1( )T

Km

m m

1

λ τ+Tm1 T Tm m2 3− T T Tm m m1 2 3> >

N=10; [ ]λ τ= Tm m1 ,

Table 86: PID tuning rules - SOSPD model with a positive zero ( )

( )( )K sT e

sT sTm m

s

m m

m1

1 13

1 2

+

+ +

− τ

- ideal controller

G s KT s

T sc ci

d( ) = + +

1 1

. 1 tuning rule.



Minimum IAE, ISEand ITAE – Wang et

al. [97]

Model: Method 1

( )ρ

p q p

p0 1 1

02

−,

p T T Tm m m m0 1 2 3= + + − τq T Tm m m1 1 2 05= + + . τ

qpp1

1

0

− ,

p T Tm m m2 1 205= . τ1

p q pp q p

pp

0 2 2

0 1 1

1

0

−−

− ,

p T Tm m1 1 2= +

( )0 5 0 51 2 3. .τ τm m m m mT T T+ −

q T Tm m2 1 2=

( )+ +0 5 1 2. τ m m mT T

01 11

. ≤ ≤τm

mT

01 12

1

. ≤ ≤TT

m

m

01 13

1

. ≤ ≤TT

m

m

1

ρ τ τ τ= − + − + − +

35550 3 6167 21781 55203 14704 04685 14746

1

2

1

3

1 1 1

2

1

3

1

. . . . . . .m

m

m

m

m

m

m

m

m

m

m

m

m

mTTT

TT T T

TT

TT

+ − − −

TT T

TT

TT

m

m

m

m

m

m

m

m

2

1 1

2

1

3

1

04685 0 4918 03318. . .τ + − +

TT T

TT

TT

m

m

m

m

m

m

m

m

3

1 1

2

1

3

1

14746 0 3318 2 5356. . .τ, minimum IAE

ρ τ τ τ= − + − + − +

39395 3 2164 16185 58240 10933 02383 13508

1

2

1

3

1 1 1

2

1

3

1

. . . . . . .m

m

m

m

m

m

m

m

m

m

m

m

m

mTTT

TT T T

TT

TT

+ − − −

TT T

TT

TT

m

m

m

m

m

m

m

m

2

1 1

2

1

3

1

02383 06679 0 0564. . .τ + − +

TT T

TT

TT

m

m

m

m

m

m

m

m

3

1 1

2

1

3

1

13508 0 0564 2 5648. . .τ, minimum ISE

ρ τ τ τ= − + − + − +

32950 34779 25336 55929 14407 0 5712 15340

1

2

1

3

1 1 1

2

1

3

1

. . . . . . .m

m

m

m

m

m

m

m

m

m

m

m

m

mTTT

TT T T

TT

TT

+ − − −

TT T

TT

TT

m

m

m

m

m

m

m

m

2

1 1

2

1

3

1

05712 03268 05790. . .τ + − +

TT T

TT

TT

m

m

m

m

m

m

m

m

3

1 1

2

1

3

1

15340 05790 2 7129. . .τ, minimum ITAE


( )( )K sT e

sT sTm m

s

m m

m1

1 13

1 2

+

+ +

− τ

- ideal controller with filtered

derivative G s KT s

T sT sN

c ci

d

d( ) = + +

+

11

1. 1 tuning rule.


Robust

( )

T TT

T

K T

m mm m

m m

m m m

1 23

3

3

+ ++ +

+ +

τ

λ τ

λ τT T

TTm m

m m

m m1 2

3

3

+ ++ +

τλ τ

TT

m m

m m

3

3

τλ τ+ +

++ +

+ +

T T

T TT

T

m m

m mm m

m m

1 2

1 23

3

τ

λ τ

T T Tm m m1 2 3> >

N=10; [ ]λ τ= Tm m1 ,Chien [50]

Model: Method 1

( )

2 13

3

3

ξτ

λ τ

λ τ

TTT

K T

mm m

m m

m m m

++ +

+ +

2 13

3

ξτ

λ τT

TTmm m

m m

++ +

TT

m m

m m

3

3

τλ τ+ +

++

+ +

T

TTT

m

mm m

m m

32

13

3

2ξτ

λ τ

T T Tm m m1 2 3> >

N=10; [ ]λ τ= Tm m1 ,


( )( )K sT e

sT sTm m

s

m m

m1

1 13

1 2

+

+ +

− τ


G s KT s

T sT sN

c ci

d

d( ) = +

+

+

11 1

1. 1 tuning rule.


Robust

( )T

K Tm

m m m

2

3λ τ+ +Tm2 Tm1 T T Tm m m1 2 3> >

N=10; [ ]λ τ= Tm m1 ,Chien [50]

Model: Method 1( )

TK T

m

m m m

1

3λ τ+ +Tm1 Tm2 T T Tm m m1 2 3> >

N=10; [ ]λ τ= Tm m1 ,


( )( )K sT e

sT sTm m

s

m m

m1

1 13

1 2

+

+ +

− τ

- series controller with


T sT sN

c ci

d

d( ) = +

+

+

11

11

. 1 tuning rule.


Robust

( )T

K Tm

m m m

2

3λ τ+ + Tm2T

TTmm m

m m1

3

3

++ +

τλ τ

N=10; [ ]λ τ= T, m , T =

dominant timeconstant

Chien [50]

Model: Method 1

( )T

K Tm

m m m

1

3λ τ+ + Tm1T

TTmm m

m m2

3

3

++ +

τλ τ

N=10; [ ]λ τ= T, m , T =

dominant timeconstant

Table 90: PID tuning rules - TOLPD model ( )( )( )

K esT sT sT

ms

m m m

m−

+ + +

τ

1 1 11 2 3

- ideal controller

G s KT s

T sc ci

d( ) = + +

1 1

. 1 tuning rule.


Minimum performance indexStandard formoptimisation -

binomial - Polonyi[153]

Model: Method 1

1 46

2 2

3

1− +

T TT

Tm

m

m

m

m

mτ τ4 6 2Tm m mτ τ+ τm ( )T Tm m m1 210> + τ

Standard formoptimisation –

minimum ITAE -Polonyi [153]

Model: Method 1

1 213 4

2 2

3

1− +

.

.T T

TTm

m

m

m

m

mτ τ2 7 3 4 2. . Tm m mτ τ+ τm ( )T Tm m m1 210> + τ

Table 91: PID tuning rules - TOLPD model - 3

m

sm

m )sT1(eK

)s(Gm

+=

τ−

– Two degree of freedom controller:

)s(Rs

NT1

sTK)s(E

sNT1

sTsT

11K)s(U

d

dc

d

d

ic

+

β+α−

+++= . 1 tuning rule.






)e105(cK 2 )e105(

iT )e105(dT

0.1Tm

m ≤τ

;



et al. [10]

2

−τ

+=003273.0

T

275.14020.0

K1

K

m

mc

)e105(c ,

τ−

τ+

τ−τ+=4

m

m

3

m

m

2

m

m

m

mm

)e105(i T

146.4T

77.11T

86.12T

647.73572.0TT

τ−τ+=2

m

m

m

mm

)e105(d T

04000.0T

2910.08335.0TT , 2

m

m

m

m

T04773.0

T2509.06661.0

τ+

τ−=α ,

2

m

m

m

m

T03621.0

T2303.08131.0

τ+

τ−=β

Table 92: PID tuning rules - unstable FOLPD model K e

sTm

s

m

m−

−

τ

1- ideal controller G s K

T sT sc c

id( ) = + +

1 1

.

3 tuning rules.


Direct synthesis

De Paor andO’Malley [86]

Model: Method 1

( )TK

T Tm

mm m m mcos 1−

τ τ +

( )TK

TT T T

m

m

m m

m mm m m m

11

−−

ττ τ τsin

( )

1

10 75T

TTm

m m

m mm

−

ττ

φtan .

,

( )φτ

ττ τm

m m

m mm m m m

TT

T T=−

− −−tan 1 11

11

12T

TT Tm

m m

m m i

ττ−

τm

mT< 1

Chidambaram [88]

Model: Method 1

1 13 0 3K

T

m

m

m

. .+

τT

Tmm

m

25 27−

τ046. τm

τm

mT< 0 6.

11650 245

..

KT

m

m

mτ

0176 0 36

0 179 0 324 0 1612

. .

. . .

+

− +

τ

τ τ

m

mm

m

m

m

m

TT

T T

0176 0 36. .+

τm

mmT

T τm

mT< 06.

11650 245

..

KT

m

m

mτ

0176 0 36 25. .+

τm

mmT

T 0176 0 36. .+

τm

mmT

T 0 6 08. .≤ ≤τm

mT

Valentine andChidambaram [154] -

dominant poleplacement

Model: Method 1

11650 245

..

KT

m

m

mτ

0176 0 36

012 01

. .

. .

T

T

m m

m

m

+

−

ττ 0176 0 36. .+

τm

mmT

T 08 1. < ≤τm

mT


sTm

s

m

m−

−

τ

1- non-interacting controller

U s KT s

E s K T ssTN

Y sci

c d

d( ) ( ) ( )= +

−

+1 1

1. 2 tuning rules.


Servo tuning Minimum performance indexHuang and Lin [155]

- minimum IAE

Model: Method 2

− − + +

−1 0 433 0 2056 0 3135

1

K T Tm

m

m

m

m

. . ..0251

τ τT

T Tmm

m

m

m

− + +

00018 08193 7 7853

6 6423

. . ..

τ τT

Tem

m

m

Tm

m− + +

0 0312 1 6333 00399

7 6983

. . ..

ττ 0 01 0 8. .≤ ≤

τ m

mT; N=10

Regulator tuning Minimum performance indexHuang and Lin [155]

- minimum IAE

Model: Method 2

− + +

−1

0 2675 01226 0 87811

K T Tm

m

m

m

m

. . ..004

τ τT

T Tmm

m

m

m

00005 2 4631 957952 9123

. . ..

+ +

τ τT

Tmm

m

0 0011 0 4759. .+

τ0 01 0 8. .≤ ≤

τ m

mT; N=10


sTm

s

m

m−

−

τ

1- classical controller

G s KT s

T sT sN

c ci

d

d( ) = +

+

+

11 1

1. 1 tuning rule.


Regulator tuning Minimum performance index0 9091. T

Km

m mτ170. τm 0 60. τm τm mT = 01.

TK

m

m mτ1 90. τm 0 60. τm τm mT = 0 2.

08929. TK

m

m mτ2 00. τm 0 80. τm τm mT = 0 5.

08621. TK

m

m mτ2 25. τm 0 90. τm τm mT = 0 67.

Shinskey [16] -minimum IAE – page

381.

Method: Model 1

08333. TK

m

m mτ2 40. τm 100. τm τm mT = 0 8.

Table 95: PID tuning rules - unstable SOSPD model ( )( )

G sK e

sT sTmm

s

m m

m

( ) =− +

− τ

1 11 2

- ideal controller

G s KT s

T sc ci

d( ) = + +

1 1

. 2 tuning rules.




Kc( )106 3 Ti

( )106 Td( )106 Tuning rules

developed from K Tu u,

Robust

TT

Tmm

m11

2

2

2λλ

λ

+

+

λλ

TT

mm

122+

+

λλ

λλ

TT

TT

mm

mm

12

12

2

2

+

+

+

τm mT = 02. λ = [ . , . ]0 5 19T Tm mKm uncertainty = 50%

τm mT = 0 4. λ = [ . , . ]1 3 19T Tm m

τm mT = 02. λ = [ . , . ]0 4 4 3T Tm m

τm mT = 0 4. λ = [ . , . ]11 4 3T Tm m

Rotstein and Lewin[89]

Model: Method 1

Km uncertainty = 30%

τm mT = 0 6. λ = [ . , . ]2 2 4 3T Tm m

λ determinedgraphically – sample

values provided

3 ( )

( )( )

KK

T T

T T T TT T T

cm

m m

m m m m m

m m m m m

( ) .106 1 22

1 2 1 2

1 2 1

0.65

2

1111 1

1

=

++

− −

τ

τ τ

,

( )( )( )T

T T T TT T Ti m

m m m m

m m m m m

( )106 1 2 1 2

1 2 1

0.65

2 1= ++

− −

τ

τ τ,

( )( )( )T

T T T TT T Td m

m m m m

m m m m m

( ) .106 1 2 1 2

1 2 1

0.65

05 1= ++

− −

τ

τ τ


G sK e

sT sTmm

s

m m

m

( ) =− +

− τ

1 11 2


G s KT s

T sTN

sc c

i

d

d( ) = +

+

+

11 1

1. 2 tuning rules.



( )[ ]( )

bTaT

T

K aT T

mm

m m

m m m m

12

2

22

1 2

14

4

++

− +

τ

τ

( )( )

4

41 2

1 2

T T

aT Tm m m

m m m

τ

τ

+

− + Tm2

( )0 2≤ ≤τm

dT N;

01 033. .TTN

Tmd

m≤ ≤

Minimum ITAE -Poulin and

Pomerleau [82], [92] –deduced from graph

Model: Method 1( )τm d mT N T+ 1 a b ( )τm d mT N T+ 1 a b

0.05 0.9479 2.3546 0.30 1.6163 2.66120.10 1.0799 2.4111 0.35 1.7650 2.73680.15 1.2013 2.4646 0.40 1.9139 2.81610.20 1.3485 2.5318 0.45 2.0658 2.9004


0.25 1.4905 2.5992 0.50 2.2080 2.9826

0.05 1.1075 2.4230 0.30 1.6943 2.70070.10 1.2013 2.4646 0.35 1.8161 2.76370.15 1.3132 2.5154 0.40 1.9658 2.84450.20 1.4384 2.5742 0.45 2.1022 2.9210

Input step loaddisturbance

0.25 1.5698 2.6381 0.50 2.2379 3.0003


G sK e

sT sTmm

s

m m

m

( ) =− +

− τ

1 11 2

- series controller

( )G s KT s

T sc ci

d( ) = +

+1 1 1 . 1 tuning rule.


Direct synthesisHo and Xu [90]

Model: Method 1

ωp m

m m

T

A K1 157

12

1

. ω ω τp p mmT

− − Tm2

( )( )

ωφ

τp

m m m m

m m

A A A

A=

+ −

−

157 1

12

.


G sK e

sT sTmm

s

m m

m

( ) =− +

− τ

1 11 2

- non-interacting controller

U s KT s

E s K T ssTN

Y sci

c d

d( ) ( ) ( )= +

−

+1 1

1. 2 tuning rules.


Servo tuning Minimum performance indexHuang and Lin [155]

- minimum IAE

Model: Method 2

Kc( )107 4 Ti

( )107 Td( )107

T Tm m2 1≤ ;

0 05 0 41

. .≤ ≤τm

mT

4 KK T T

TT

TT

TTc

m

m

m

m

m

m

m

m

m

m m

m

( ).344

. . . . . .107

1 1

1

2

1

2

1

0.995

2

12

1 10 741 13363 0099 727 914 708 481 9 915= − − +

+ −

+

−τ τ τ

−

− + − −

1 84 273 90 959 9 034 2 386 16 30411031

2

1

0 9972 1

2

1

2

12

KT T

TT

e e em

m

m

m

m

m

m

TTT

TT

m

m

m

m

m m

m. . . . .. .

τ τ

τ τ

T TT T

TT

TT

TTi m

m

m

m

m

m

m

m

m

m m

m

( ).

. . . . . .1071

1 1

212

2

1

2

1

0.985

2

12

149 685 141418 88 717 17 29 20518 12 82= − − −

− +

−

τ τ τ

+

+ + − +

TT

TT

Te e em

m

m

m

m

m

m

TTT

TT

m

m

m

m

m m

m1

1

0 2862

1

198823 611 0 000805 141702 2 032 10 0061

2

1

2

12

. . . . .. .

ττ

τ τ

T TT T

TT

TT

TTd m

m

m

m

m

m

m

m

m

m m

m

( ) . . . . . .1071

1 1

2

2

1

2

1

2

2

12

0 4144 15805 142 327 0 7287 01123 18 317= − + −

+ +

−

τ τ τ

+

−

+

+

+

TT

TT

T

T

T

T Tmm

m

m

m

m m

m

m m

m

m

m1

1

32

1

3 22

13

22

13

1

4

48695 10542 204 009 47 26 396349. . . . .τ τ τ τ

+ −

+

−

+

−

TT

T

T

T

T

T

TT Tm

m m

m

m m

m

m m

m

m

m

m

m1

32

14

23

14

22

2

14

2

1

4

1

5

138 038 52155 646 848 19 302 4731 72. . . . .τ τ τ τ

+

−

−

+

−

TT

T

T

T

T

T

T

T

TTm

m m

m

m m

m

m m

m

m m

m

m

m1

42

15

24

15

32

2

15

22

3

15

2

1

5

425789 289 746 841807 1313 72 37688. . . . .τ τ τ τ

+

−

+

+

TT

TT

TT

TTm

m

m

m m

m

m m

m

m m

m1

1

6 52

16

25

16

42

2

16

6264 79 161469 204 689 25706. . . .τ τ τ τ

+ −

+

−

T TT

TT

TTm

m m

m

m m

m

m

m1

22

4

16

2

12

3

2

1

6

791857 648 217 5 71. . .τ τ


Huang and Lin [155]- minimum IAE -

continuedModel: Method 2

Kc( )108 5 Ti

( )108 Td( )108

T T Tm m m1 2 110< ≤ ;

0 05 0251

. .≤ ≤τm

mT

5 KK T T

TT

TT

TTc

m

m

m

m

m

m

m

m

m

m m

m

( ) . . . . . .108

1 1

0.3055

2

1

2

1

0.5174

2

22

1 1302 85914 34 82 10 442 22 547 14 698= − − + +

+ −

−

−τ τ τ

−

−

+ − +

152 408 5147 53 378 0 000001 0 2861

1 0077

2

1

0 9879

2 1

2

1

2

12

KT T

TT

e e em

m

m

m

m

m

m

T

T

T

T

Tm

m

m

m

m m

m. . . . .. .

τ τ

τ τ

T TT

TT T

TT

TTi m

m

m

m

m

m

m

m

m

m m

m

( ) . . . . . .1081

1

2

1 1

2

2

1

2

2

12

72 806 268 746 4 9221 246819 0 6724 151351= − −

+

+

+

τ τ τ

+ −

−

−

−

+

TT

TT

TT

TT Tm

m

m

m

m

m m

m

m m

m

m

m1

1

3

2

1

3 22

13

22

13

1

4

6914 46 00092 795465 14 27 558017. . . . .τ τ τ τ

+

+

+

−

T TT

TT

TT

TTm

m m

m

m m

m

m m

m

m

m1

32

14

23

14

222

14

2

1

4

1417 65 0 4057 55536 0001119. . . .τ τ τ

+ − + − +

T e e eT

TTm

T

T

T

T

T m

m

m

m

m

m

m

m

m m

m1

1

19 056

2

1

7 3464

44 903 0 000034 15694 6787781

2

1

2

12

. . .. .τ τ

τ

T TT T

TT

TT

TTd m

m

m

m

m

m

m

m

m

m m

m

( ).

. . . . . .1081

1 1

11798

2

1

2

1

0.1064

2

12

175515 86 2 348 727 0 008207 55619 0 0418= − +

− −

+

τ τ τ

+

+

− + −

TT

TT

Te e em

m

m

m

m

m

m

T

T

T

T

Tm

m

m

m

m m

m1

1

0 0355

2

1

0 0827

278959 0 005048 187 01 0000001 001491

2

1

2

12

. . . . .. .

ττ

τ τ


Regulator tuning Minimum performance indexHuang and Lin [155]

– minimum IAE

Model: Method 2

Kc( )109 6 Ti

( )109 Td( )109

T Tm m2 1≤ ;

0 05 0 41

. .≤ ≤τm

mT

6 KK T T

TT

TT

TTc

m

m

m

m

m

m

m

m

m

m m

m

( ). .

. . . . . .109

1 1

1164

2

1

2

1

2 54

2

12

1 174167 31364 0 4642 103069 83916 66 962= − − − +

− −

−

−τ τ τ

−

− + + +

159 496 7079 23121 126 924 26 9441

1 065

2

1

1014

2 1

2

1

2

12

KT T

TT

e e em

m

m

m

m

m

m

T

T

T

T

Tm

m

m

m

m m

m. . . . .. .

τ τ

τ τ

T TT T

TT

TT

TT Ti m

m

m

m

m

m

m

m

m

m m

m

m

m

( ) . . . . . . .1091

1 1

2

2

1

2

1

2

2

12

1

3

0 008 2 0718 6 431 0 4556 0 7503 2 4484 18686= + +

+ +

+ −

τ τ τ τ

+ −

−

+

+

+

T TT

TT

TT T

TTm

m

m

m m

m

m m

m

m

m

m m

m1

2

1

3

22

13

22

13

1

4

23

14

2 9978 21135 12 822 39 001 22848. . . . .τ τ τ τ

+ −

−

+

T TT

TT

TTm

m m

m

m m

m

m

m1

23

14

22 2

14

2

1

4

4 754 0 527 164. . .τ τ

T TT T

TT

TT

TT Td m

m

m

m

m

m

m

m

m

m m

m

m

m

( ) . . . . . . .1091

1 1

2

2

1

2

1

2

2

12

1

3

0 0301 11766 4 4623 05284 14281 4 6 11176= − + −

+ −

+ +

τ τ τ τ

+

−

−

−

−

T TT

TT

TT T

TTm

m

m

m m

m

m m

m

m

m

m m

m1

2

1

3 22

13

22

13

1

4

23

14

10886 5 0229 05039 98564 7528. . . . .τ τ τ τ

+ −

+

−

T TT

TT

TTm

m m

m

m m

m

m

m1

23

14

22

2

14

2

1

4

2 3542 9 3804 01457. . .τ τ


Huang and Lin [155]- minimum IAE –

continuedModel: Method 2

Kc( )110 7 Ti

( )110 Td( )110

T T Tm m m1 2 110< ≤ ;

0 05 0251

. .≤ ≤τm

mT

7 KK T T

TT

TT

TTc

m

m

m

m

m

m

m

m

m

m m

m

( ).

. . . . . .110

1 1

21984

2

1

2

1

0.791

2

12

1 1750 08 1637 76 1533 91 7 917 6187 6 451= − + +

− +

−

τ τ τ

−

+ − − +

10002452 13729 1739 77 0 000296 2 3111

32927

2

1

10757

2 1

2

1

2

12

KT T

TT

e e em

m

m

m

m

m

m

T

T

T

T

Tm

m

m

m

m m

m. . . . .. .

τ τ

τ τ

T TT T

TT

TT

TT Ti m

m

m

m

m

m

m

m

m

m m

m

m

m

( ) . . . . . . .1101

1 1

2

2

1

2

1

2

2

12

1

3

51678 57 043 1337 29 01742 01524 7 7266 6011 57= − +

+ −

+ −

τ τ τ τ

+

−

+

+

+

T TT

TT

TT T

TTm

m

m

m m

m

m m

m

m

m

m m

m1

2

1

3

22

13

22

13

1

4

23

14

0 0213 65 283 0 0645 913552 274 851. . . . .τ τ τ τ

+

−

−

− +

T TT

TT

TT

e emm m

m

m m

m

m

m

TTT

m

m

m

m1

23

14

22 2

14

2

1

4

0 003926 2 0997 0001077 49 007 0 0000261

2

1. . . . .τ ττ

+

T em

TTm m

m1 0 2977

2

1.τ

T TT T

TT

TT

TT Td m

m

m

m

m

m

m

m

m

m m

m

m

m

( ) . . . . . . .1101

1 1

2

2

1

2

1

2

2

12

1

3

0 0605 4 6998 29 478 0 0117 0 0129 0 6874 140135= − + −

+ −

+ +

τ τ τ τ

+

−

−

−

+

T TT

TT

TT T

TTm

m

m

m m

m

m m

m

m

m

m m

m1

2

1

3 22

13

22

13

1

4

23

14

0 002455 14712 01289 238 864 06884. . . . .τ τ τ τ

+

−

−

T TT

TT

TTm

m m

m

m m

m

m

m1

23

14

22

2

14

2

1

4

0 007725 01222 0 000135. . .τ τ

Table 99: PID tuning rules – general model with a repeated pole G sK e

sTmm

s

mn

m

( )( )

=+

− τ

1- ideal controller

G s KT s

T sc ci

d( ) = + +

1 1

. 1 tuning rule.


Direct synthesisSkoczowski andTarasiejski [156]

Model: Method 1

( )≤

+

+

−

ω ω

ωg m

m

g m

n

g d

T

K

T

T

1

1

2 2 1

2 2

1

Tm nn

Tm−+

12

, n ≥ 2

1 ωπ

τπ

φ

πτ

πφ

g

mm

mm

mm

mm

T n nn

Tn

b

T n nn

Tn

=

− − − ++

++

±

− + ++

+−

2

2 2

2 22 4 4 2

2 4 34 2 2 2

with

b T n nn

Tn

n n nn

Tn

mm

mm m

m

mm= − − +

++

+

+ + −

− + ++

+−

2 22

22 22 4 4 2

4 2 12

4 34 2 2 2

πτ

πφ

πφ

πτ

πφ( )

Table 100: PID tuning rules – general stable non-oscillating model with a time delay - ideal controller

G s KT s

T sc ci

d( ) = + +

1 1

. 1 tuning rule.


Direct synthesis)a110(

cK 2 )a110(iT )a110(

dT)a110(

cK )a110(iT

τ−

τ

ar

m

ar

m)a110(i T

1T

TGorez and Klan

[147a]Model: not specified

)a110(cK )a110(

iT )a110(iT25.0

2 m

)a110(i

)a110(i)a110(

cT

TK

τ+= ,

2

T2

T211

TTar

m

2

ar

m

ar)a110(

i

τ−

τ++

= ,

( )

τ

+−τ

−−++=)a110(

i

m)a110(c

ar

2m

aa)a110(

iar

crcr

)a110(i)a110(

i

ar)a110(d

T3

21K1

T2TT

TT

TTT

TT

arT = average residence time of the process (which equals mmT τ+ for a FOLPD process, for example); aaT =

additional apparent time constant; m)a110(i

m)a110(ccr

T21K1T τ

τ+−=

Table 101: PID tuning rules – fifth order model with delay

( )G s



m ss m

( )( )

=+ + + + +

+ + + + +

−1

11 2

23

34

4 5

1 22

33

44

55

τ

– ideal controller G s KT s

T sc ci

d( ) = + +

1 1

.

1 tuning rule.


Direct synthesisMagnitude optimum- Vrancic et al. [159]Model: Method 1

3

Kc( )112 Ti

( )112 Td( )112

3

( ) ( ) ( )( ) ( ) ( )[ ]

Ka a b a b a a a b a b a a b a b a b

K a b a a a b a b b b a b a b a b Tc

m m m

m m m m m d

112 13

12

1 1 2 1 2 2 1 3 3 12

1 1 2 2 1 12 3

12

1 1 2 1 12

3 1 2 3 1 12

1 12 3

1 12

2 05 0167

2 0333=

− + − + + − + − − + + − +

− + + − − + + − + − + − − +

τ τ τ

τ τ τ τ

. .

.

( ) ( ) ( )( ) ( )[ ]

Ta a b a b a a a b a b a a b a b a b

a a b a b a b a b Ti

m m m

m m m d

112 13

12

1 1 2 1 2 2 1 3 3 12

1 1 2 2 1 12 3

12

1 1 2 2 1 12

1 1

2 05 0167

05=

− + − + + − + − − + + − +

− − + + − + − − +

τ τ τ

τ τ τ

. .

.

Td( )112 … see attached sheet with δ = 0.

Table 102: PID tuning rules – fifth order model with delay

( )G s



m ss m

( )( )

=+ + + + +

+ + + + +

−1

11 2

23

34

4 5

1 22

33

44

55

τ

– controller with filtered derivative

G s KTs

T sT sN

c ci

d

d( ) = + +

+

11

1. 1 tuning rule.


Direct synthesisMagnitude optimum- Vrancic et al. [73]Model: Method 1

Kc(113) 4 Ti

(113) Td(113) 8 20≤ ≤N

4

( ) ( ) ( )

( ) ( ) ( ) ( )K

a a b a b a a a b a b a a b a b a b

K a b a a a b a b b b a b a b a b TTN

a bc

m m m

m m m m m dd

m

113 13

12

1 1 2 1 2 2 1 3 3 12

1 1 2 2 1 12 3

12

1 1 2 1 12

3 1 2 3 1 12

1 12 3

1 12

2

1 1

2 0 5 0 167

2 0 333

=− + − + + − + − − + + − +

− + + − − + + − + − + − − + − − +

τ τ τ

τ τ τ τ τ

. .

.

( ) ( ) ( )

( ) ( )T

a a b a b a a a b a b a a b a b a b

a a b a b a b a b TTN

im m m

m m m dd

113 13

12

1 1 2 1 2 2 1 3 3 12

1 1 2 2 1 12 3

12

1 1 2 2 1 12

1 1

2

2 0 5 0 167

0 5

=− + − + + − + − − + + − +

− − + + − + − − + −

τ τ τ

τ τ τ

. .

.

T see attached sheetd(113) =

4. Conclusions

The report has presented a comprehensive summary of the tuning rules for PI and PID controllers that

have been developed to compensate SISO processes with time delay. Further work will concentrate on evaluating

the applicability of these tuning rules to the compensation of processes with time delays, as the value of the time

delay varies compared with the other dynamic variables.

Table 103: Tuning rules published by year and mediumYear Journal articles Conference

papers/correspondence

Books/ Ph.D.thesis

Total

1942 1 11950 1 11951 1 11952 1 11953 3 31954 1 11961 1 11964 1 11965 1 11967 1 1 21968 1 11969 2 21972 2 21973 1 1 21975 2 21979 1 11980 2 21982 1 11984 2 1 31985 1 1 21986 1 11987 1 11988 3 4 2 91989 4 3 71990 4 1 51991 5 51992 1 2 31993 5 4 1 101994 5 2 2 91995 14 2 1 171996 8 6 2 161997 11 4 1 161998 6 7 131999 12 1 1 142000 1 16 17

1941-1950 2 0 0 2

1951-1960 6 0 0 61961-1970 5 0 3 81971-1980 7 0 2 91981-1990 15 7 7 291991-2000 68 44 8 120TOTAL 103 51 20 174

List of journals in which tuning rules were published and number of tuning rules published

Advances in Modelling and Analysis C, ASME Press 1AIChE Journal 4Automatica 12British Chemical Engineering 1Chemical Engineering Communications 5Chemical Engineering Progress 1Chemical Engineering Science 1Control 1Control and Computers 1Control Engineering 7Control Engineering Practice 4European Journal of Control 1Hydrocarbon Processing 2Hungarian Journal of Industrial Chemistry 1IEE Proceedings, Part D 9

(including IEE Proceedings - Control Theory and Applications, Proceedings of the IEE, Part 2)IEEE Control Systems Magazine 1IEEE Transactions on Control Systems Technology 4IEEE Transactions on Industrial Electronics and Control Instrumentation 1IEEE Transactions on Industrial Electronics 1IEEE Transactions on Industry Applications 1Industrial and Engineering Chemistry Process Design and Development 2Industrial Engineering Chemistry Research 12International Journal of Adaptive Control and Signal Processing 1International Journal of Control 3International Journal of Electrical Engineering Education 1International Journal of Systems Science 1Instrumentation 1Instrumentation Technology 2Instruments and Control Systems 2ISA Transactions 2Journal of Chemical Engineering of Japan 2Journal of the Chinese Institute of Chemical Engineers 3Pulp and Paper Canada 1Process Control and Quality 1Thermal Engineering (Russia) 2Transactions of the ASME 7

(including Transactions of the ASME. Journal of Dynamic Systems, Measurement and Control)Transactions of the Institute of Chemical Engineers 1

Classification of journals in which tuning rules were published and number of tuning rulespublished

Chemical Engineering Journals 32(AIChE Journal, British Chemical Engineering, Chemical Engineering Communications, Chemical EngineeringProgress, Chemical Engineering Science, Transactions of the Institute of Chemical Engineers, Hungarian Journalof Industrial Chemistry, Industrial and Engineering Chemistry Process Design and Development, Industrial

Engineering Chemistry Research, Journal of Chemical Engineering of Japan, Journal of the Chinese Institute ofChemical Engineers)Control Engineering Journals 53(Automatica, Control, Control and Computers, Control Engineering, Control Engineering Practice, EuropeanJournal of Control, IEE Proceedings, Part D, IEE Proceedings - Control Theory and Applications, Proceedings ofthe IEE, Part 2, IEEE Control Systems Magazine, IEEE Transactions on Control Systems Technology,International Journal of Adaptive Control and Signal Processing, International Journal of Control, InternationalJournal of Systems Science, Instrumentation, Instrumentation Technology, Instruments and Control Systems,ISA Transactions, Process Control and Quality)Mechanical Engineering Journals 8(Advances in Modelling and Analysis C, ASME Press, Transactions of the ASME, Transactions of the ASME.Journal of Dynamic Systems, Measurement and Control)Electrical/Electronic Engineering Journals 4(EE Transactions on Industrial Electronics and Control Instrumentation, IEEE Transactions on IndustrialElectronics, IEEE Transactions on Industry Applications, International Journal of Electrical EngineeringEducation)Trade Journals 5(Hydrocarbon Processing, Pulp and Paper Canada, Thermal Engineering (Russia))

5. References

1. Koivo, H.N. and Tanttu, J.T., Tuning of PID Controllers: Survey of SISO and MIMO techniques.

Proceedings of the IFAC Intelligent Tuning and Adaptive Control Symposium, Singapore, 1991, 75-80.

2. Hwang, S.-H., Adaptive dominant pole design of PID controllers based on a single closed-loop test.

Chemical Engineering Communications, 1993, 124, 131-152.

3. Astrom, K.J. and Hagglund, T., PID Controllers: Theory, Design and Tuning. Instrument Society of America,

Research Triangle Park, North Carolina, 2nd Edition, 1995.

4. Bialkowski, W.L., Control of the pulp and paper making process. The Control Handbook. Editor: W.S.

Levine, CRC/IEEE Press, Boca Raton, Florida, 1996, 1219-1242.

5. Isermann, R., Digital Control Systems Volume 1. Fundamentals, Deterministic Control. 2nd Revised Edition,

Springer-Verlag, 1989.

6. Ender, D.B., Process control performance: not as good as you think. Control Engineering, 1993, September,

180-190.

7. Astrom, K.J. and Wittenmark, B., Computer controlled systems: theory and design. Prentice-Hall

International Inc., 1984.

8. Ziegler, J.G. and Nichols, N.B., Optimum settings for automatic controllers. Transactions of the ASME, 1942,

November, 759-768.

9. Hazebroek, P. and Van der Waerden, B.L., The optimum adjustment of regulators. Transactions of the

ASME, 1950, April, 317-322.

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Appendix 1: List of symbols used (more than once) in the paper.

a a1 2, = PID filter parameters

A m = gain margin

b = setpoint weighting factor

b1 = PID filter parameter

c = derivative term weighting factor

E(s) = Desired variable, R(s), minus controlled variable, Y(s)

FOLPD model = First Order Lag Plus time Delay model

FOLIPD model = First Order Lag plus Integral Plus time Delay model

G sc ( ) = PID controller transfer function

G jp( )ω = process transfer function at frequency ω

G jp( )ω = magnitude of G jp( )ω , ( )∠G jp ω = phase of G jp( )ω

IAE = integral of absolute error

IMC = internal model controller

IPD model = Integral Plus time Delay model

ISE = integral of squared error

ISTES = integral of squared time multiplied by error, all to be squared

ISTSE = integral of squared time multiplied by squared error

ITAE = integral of time multiplied by absolute error

ki = integral gain of the parallel PID controller

kp = proportional gain of the parallel PID controller

kd = derivative gain of the parallel PID controller

Kc = Proportional gain of the controller

Km = Gain of the process model

Ku = Ultimate gain

K25% = proportional gain required to achieve a quarter decay ratio

m = multiplication parameter in the lead controller

Ms = closed loop sensitivity

N = determination of the amount of filtering on the derivative term

PP = perturbance peak = peak of system output when unit step disturbance is added

R(s) = Desired variable

RT= recovery time = time for perturbed system output (when a unit step disturbance is added) to come to its final

value

SOSPD model = Second Order System Plus time Delay model

Td = Derivative time of the controller

TCL = desired closed loop system time constant

Tf = Time constant of the filter in series with the PID controller

Ti = Integral time of the controller

Tm = Time constant of the FOLPD process model

T T Tm m m1 2 3, , = Time constants of the higher order process models

Tu = Ultimate time constant

T25% = period of the quarter decay ratio waveform

TOLPD model = Third Order Lag Plus time Delay model

TS = settling time

U(s) = manipulated variable

Y(s) = controlled variable

λ = Parameter that determines robustness of compensated system.

ξ m = damping factor of an underdamped process model, ξ = damping factor of the compensated system

κ = 1 K Km u

φ = phase lag, φm = phase margin, φω = phase lag at an angular frequency of ω

=φc phase corresponding to the crossover frequency

τm = time delay of the process model, ( )τ τ τ= +m m mT

ω = angular frequency, ωu = ultimate frequency, ωφ = angular frequency at a phase lag of φ

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