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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: 353-1-4024875

Fax: 353-1-4024992

e-mail: aodwyer@dit.ie

Abstract: The ability of proportional integral (PI) and proportional integral derivative (PID) controllers to

compensate many practical industrial processes has led to their wide acceptance in industrial applications. The

requirement to choose either two or three controller parameters is perhaps most easily done using tuning rules. A

summary of tuning rules for the PI and PID control of single input, single output (SISO) processes with time

delay are provided in this report. Inevitably, this report is a work in progress and will be added to and extended

regularly.

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 their

wide 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 controllers

perform well for processes with benign dynamics and modest performance requirements [2, 3]. It has been stated

that 98% of control loops in the pulp and paper industries are controlled by SISO PI controllers [4] and that, in

process control applications, more than 95% of the controllers are of PID type [3]. The PI or PID controller

implementation has been recommended for the control of processes of low to medium order, with small time

delays, when parameter setting must be done using tuning rules and when controller synthesis is performed

either once or more often [5]. However, Ender [6] states that, in his testing of thousands of control loops in

hundreds of plants, it has been found that more than 30% of installed controllers are operating in manual mode

and 65% of loops operating in automatic mode produce less variance in manual than in automatic (i.e. the

automatic controllers are poorly tuned); this is rather sobering, considering the wealth of information available in

the literature for determining controller parameters automatically. It is true that this information is scattered

throughout papers and books; the purpose of this paper is to bring together in summary form the tuning rules for

PI and PID controllers that have been developed to compensate SISO processes with time delay. Tuning rules for

the variations that have been proposed in the ‘ideal’ PI and PID controller structure are included. Considerable

variations in the ideal PID controller structure, in particular, are encountered; these variations are explored in

more detail in Section 2.

2. PID controller structures

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

as follows:

U s G s E s

( ) ( ) ( )= (1)

with G s K

T s

c c

( ) ( )= + +1

(2)

and with K

= proportional gain, T

= integral time constant and T

= derivative time constant. If T

= ∞

and T

= 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 measured

value constant, and less than the desired value). The introduction of integral action facilitates the achievement of

equality between the measured value and the desired value, as a constant error produces an increasing controller

output. 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 PID

controller 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 been

defined 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 will

influence the value of the model parameters, which will in turn affect the controller values determined from the

tuning rules. The modelling strategy used in association with each tuning rule, as described in the original

papers, is indicated in the tables. Of course, it is possible to use the tuning rules proposed by the authors with a

different 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 are

provided in Appendix 2.

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

K e

( ) =

− τ

Method 1: Parameters obtained using the tangent and point method (Ziegler and Nichols [8], Hazebroek and

Van den Waerden [9]); Appendix 2.

Method 2: K

, τ

assumed known; T

estimated from the open loop step response (Wolfe [12]); Appendix

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: K

, T

, τ

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 frequency

determined using a relay in series with the closed loop system in a master feedback loop. The

model gain is obtained by the ratio of the integrals (over one period) of the process output to the

controller 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 proportional

control (Hwang [2]); Appendix 2.

Method 10: Parameters obtained from two points estimated on process frequency response using a relay and

a relay in series with a delay (Tan et al. [39]); Appendix 2.

Method 11: T

and τ

are determined from the ultimate gain and period estimated using a relay in series

with the process in closed loop; K

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: T

and τ

estimated from K

, T

determined using Ziegler-Nichols ultimate cycle method;

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

Method 15: T

and τ

estimated from K

, T

determined using a relay autotuning method; K

estimated

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

Method 16: )j(G

135

ω ,

135

ω and K

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

process in closed loop; estimates for T

and τ

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: K

estimated from the open loop step response. T

90%

and τ

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: T

and τ

estimated from relay autotuning method (Lee and Sung [163]); K

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 6000

controller (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:

τ and

T obtained from an open loop step test as follows: )tt(4.1T

%33%67m

−= ,

m%67m

T1.1t −=τ .

K assumed known (Chen and Yang [23a]).

Method 26:

τ and

T obtained from an open loop step test as follows: )tt(245.1T

%33%70m

−= ,

%70%33m

t498.0t498.1 −=τ .

K 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 s

K e

( ) =

− τ

)

Method 1: τ

assumed known; K

determined from the slope at start of the open loop step response

(Ziegler and Nichols [8]); Appendix 2.

Method 2: K

, τ

assumed known.

Method 3: Parameters estimated from the ultimate gain and frequency values determined from an experiment

using 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 PI

control (Rotach [77]); Appendix 2.

Method 5: Parameters are estimated from the servo closed loop transient response under proportional

control (Srividya and Chidambaram [80]); Appendix 2.

Method 6:

K and

T 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 the

process in closed loop; the crossover frequency estimate is obtained by also using an amplitude

dependent gain (Pecharroman and Pagola [165]); Appendix 2.

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

( )

G s

K e

s sT

( ) =

− τ

)

* Method 1: Method of moments (Astrom and Hagglund [3]).

Method 2: K

, T

, τ

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:

K and

T 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 s

( ) =

K e

T s T s

m m m

−

+ +

2 1

( )( )

K e

T s T s

m m

−

+ +

1 1

1 2

)

Method 1: K

, τ

or K

,ξ

, τ

assumed known.

Method 2: Parameters estimated using a two-stage identification procedure involving (a) placing a relay in

series with the process in closed loop and (b) placing a proportional controller in series with the

process in closed loop (Sung et al. [139]); Appendix 2.

* Method 3: Parameters obtained in the frequency domain from the ultimate gain, phase and frequency

determined using a relay in series with the closed loop system in a master feedback loop. The

model gain is obtained by the ratio of the integrals (over one period) of the process output to the

controller output. The other parameters are obtained from the frequency domain data (Hwang

[160]).

Method 4: T

and τ

estimated from K

, T

determined using a relay autotuning method; K

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

90− and

180− ,

respectively (Wang et al. [143]); Appendix 2.

* Method 9: Parameter estimates back-calculated from discrete time identification method (Wang and

Clements [147]).

Method 10:

K ,

T and

τ are determined from the open loop time domain Ziegler-Nichols response

(Shinskey [16], page 151);

T assumed known.

Method 11: Parameters estimated from two points determined on process frequency response using a relay

and 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 = .

τ and

T obtained from an open loop step test as follows:

)tt(794.0T

%33%701m

−= ,

%70%33m

t937.0t937.1 −=τ .

K assumed known (Miluse et al.

[27b]).

Method 15:

K and

T 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

) model (G s

( ) =

τ−

)

Method 1: K

, T

, τ

assumed known.

G. Second order system (repeated pole) plus integral plus time delay (SOSIPD) model (G s

( ) =

( )

sT1s

τ−

)

Method 1:

K and

T are estimated from estimates of the ultimate and crossover frequencies (Pecharroman and

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

Method 2: K

, T

, τ

assumed known.

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

( ) =

( )( )( )

K e

sT sT sT

m m m

−

+ + +

1 1 1

1 2 3

Method 1: K

, τ

known.

I. Unstable first order lag plus time delay model (G s

( ) =

K e

−

)

Method 1: K

, τ

known.

Method 2: The model parameters are obtained by least squares fitting from the open loop frequency

response of the unstable process; this is done by determining the closed loop magnitude and

phase values of the (stable) closed loop system and using the Nichols chart to determine the

open loop response (Huang and Lin [154], Deshpande [164]).

J. Unstable second order system plus time delay model (G s

( ) =

( )( )

K e

sT sT

m m

−

− +

1 1

1 2

)

Method 1: K

, τ

known.

Method 2: The model parameters are obtained by least squares fitting from the open loop frequency

response of the unstable process; this is done by determining the closed loop magnitude and

phase values of the (stable) closed loop system and using the Nichols chart to determine the

open loop response (Huang and Lin [154], Deshpande [164]).

K. Second order system plus time delay model with a positive zero (G s

( ) =

(

)

( )( )

K sT e

sT sT

m m

1 1

1 2

−

+ +

− τ

)

Method 1: K

, τ

known.

L. Second order system plus time delay model with a negative zero (G s

( ) =

(

)

( )( )

K sT e

sT sT

m m

1 1

1 2

+ +

− τ

)

Method 1: K

, τ

known.

M. Fifth order system plus delay model (

( )

G s

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

a s a s a s a s a s

m s

( )

+ + + + +

−

1 2

4 5

1 2

)

Method 1: K

, τ

known.

N. General model with a repeated pole (G s

K e

( )

− τ

)

* Method 1: Strejc’s method

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

P. Delay model ( )e)s(G

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, the

tuning 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 cover

techniques 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 design

process.

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

cycling methods).

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