Astrom Opt PID IEE Kosarac Olja

8/17/2019 Astrom Opt PID IEE Kosarac Olja

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PID CONTROL

Design of PID controllers based on constrainedoptimisation

H.Panagopoulos, K.J.Åström and T.Hägglund

Abstract: A new design method for PID controllers is presented, based on optimisation of load disturbance rejection with constraints on robustness to model uncertainties. The design alsodelivers parameters to deal with measurement noise and set point response. Thus, the formulationof the design problem captures four essential aspects of industrial control problems, leading to aconstrained optimisation problem which can be solved iteratively.

1 Introduction

The PID controller is today’s most commonly used controlalgorithm [1]. At the moment, there exist many differ-ent methods to find suitable controller parameters. Themethods differ in complexity, flexibility, and in the amount of process knowledge used. Depending on the application,there is a need to have several types of tuning method.There are simple, easy to use methods which require littleinformation, e.g. the method described in [2], as well asmore sophisticated methods which require more informa-tion and more computations.

There are several reasons to look for better methods todesign PID controllers. One is the significant impact it may

have because of the widespread use of the controllers.Another is the benefit emerging auto-tuners and tuningdevices can derive from improved design methods.

This paper describes a new design method for PIDcontrollers, where the primary design goal is to obtaingood load disturbance responses. This is done by minimis-ing the integrated control error IE . Robustness is guaran-teed by requiring that the maximum sensitivity be less thana specified value M s . Measurement noise is dealt with byusing filtering. Good set point response is obtained byusing a structure with two degrees of freedom.

The specifications are expressed in terms of a number of parameters for which good default values can be found. Inthe simplest case good default values can be given to all

parameters. The user simply supplies a model of the process and the design parameter, which is the maximumsensitivity, M s . Consequently, the method provides all the

parameters of the PID controller: controller gain k , integraltime T i , derivative time T d and set point weight b. Inaddition, the filters of the measured signal and the set

point are delivered.For related work, see for example [3–6].

2 Problem formulation

The design problem is illustrated in Fig. 1. A process withtransfer function G ( s) is controlled with a PID controller with two degrees of freedom. The transfer function G c( s)describes the feedback from process output y to controlsignal u, and G ff ( s) describes the feed forward from set

point y sp to u. Three external signals act on the controlloop, namely set point y sp , load disturbance l and measure-ment noise n.

The design objective is to determine the controller parameters in G c( s) and G ff ( s) so that the system behaveswell with respect to changes in the three signals y sp , l and n, as well as in the process model G ( s). Hence, the

specification will express requirements on load disturbance response robustness with respect to model uncertainties measurement noise response set point response

2.1 Process and controller structures

The design problem is formulated so as to apply to a widevariety of systems. Consequently, the process is assumed to

be linear, time invariant, and specified by a transfer func-tion G ( s), which is analytic with finite poles, and possiblyan essential singularity at infinity. The description coversfinite dimensional systems with time delays and infinite

dimensional systems described by linear partial differentialequations.

Initially the controller is described by

uðt Þ ¼ k ðby spðt Þ yðt ÞÞ þ k i

ð t 0

ð y spðtÞ yðtÞÞd t

þ k d dyðt Þ

dt

where k , k i , k d and b are controller parameters.It proves beneficial to replace the signals y and y sp with

their filtered values y f and y sp f . The filtered signals are

generated by

Y f ð sÞ ¼ F yð sÞY ð sÞ

Y f spð sÞ ¼ F spð sÞY spð sÞ

# IEE, 2002

IEE Proceedings online no. 20020102

DOI: 10.1049/ip-cta:20020102

Paper received 23rd November 2001

H. Panagopoulos is with the M-real Corporation, Technology Center Örnsköldsvik, Örnsköldsvik, Sweden

K.J. Åström and T. Hägglund are with the Department of AutomaticControl, Lund Institute of Technology, Box 118, Lund S-221 00, Sweden

32 IEE Proc.-Control Theory Appl., Vol. 149, No. 1, January 2002


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where the filters are low pass filters of first or second order.The controller can thus be characterised by four parametersk , k i , k d , b, and two filters F y and F sp .

2.2 Load disturbance attenuation

The primary design goal is to achieve good rejection of load disturbances. There are no detailed assumptions madeabout the load disturbances except that they are low-frequency. The most common performance criteria is tominimise IAE , the integrated absolute control error, at load disturbances. We have used the integrated error IE instead.The IE criteria is a good approximation of IAE , and the twocriteria are identical for non-oscillatory loops. By combin-

ing the IE minimisation with a robustness criterion, well-damped control loops are obtained.

The reason for using IE is that its value is directly related to the controller parameters, and that it enables simple and ef ficient design algorithms. If a unit step load disturbanceis applied at the process input, the IE value becomes [7]:

IE ¼

ð 10

eðt Þdt ¼ 1

k i

The integral gain k i is thus inversely proportional to theintegrated error when a unit step load disturbance isapplied at the process input. By maximising the integralgain k i the effect of the load disturbance l on the output y isminimised.

2.3 Robustness

Sensitivity to modelling errors can be expressed as thelargest value of the sensitivity function, i.e.

M s ¼ maxo

1

1 þ GG cðioÞ

The quantity M s is simply the inverse of the shortest

distance from the Nyquist curve of the loop transfer function to the critical point 1. Typical values of M s arein the range 1 to 2.

The sensitivity can also be expressed as the largest valueof the complementary sensitivity function, i.e.

M p ¼ maxo

GG cðioÞ

1 þ GG cðioÞ

ð1Þ

Typical values of M p are in the range 1.0 to 1.5.Another possibility is to use the H

1-norm

g ¼ maxo

1 þ jGG cðioÞj1 þ GG cðioÞ

ð2Þ

which is discussed in detail in [8].

2.4 Measurement noise

In traditional PID controllers the filters are often onlyapplied on the derivative term. A common choice of derivative term is

Dð sÞ ¼ kT d s

1 þ sT d N

Y ð sÞ ¼ k d s

1 þ s k d kN

Y ð sÞ

where N is a number in the range 2 – 10. This will reduce

the high-frequency gain to k (1 þ N ). Since the filter is onlyapplied on the derivative term and not on the proportionalterm, the high-frequency gain can never be made smaller than k .

In this study all terms in the controller are filtered. For first-order filters with filter time constant T f , the high-frequency gain becomes k d =T f . For second-order filters thehigh-frequency gain goes to zero as the frequency goes toinfinity. A nice feature of the design is to provide asystematic way to determine T f .

2.5 Set point response

The transfer function relating set point to process output is

given by

G spð sÞ ¼GG ff

1 þ GG c F y F sp

¼ k i þ bks

k i þ ks þ k d s2

GG c1 þ GG c F y

F sp

When the controller has been designed to give good attenuation of disturbances, the parameter b and the filter

F sp can be chosen to give an appropriate set point response.This is done by determining the maximum of G sp:

M sp ¼ maxo

jG spðioÞj

2.6 Tuning parameter

The tradeoff between performance and robustness variesamong different control problems. Therefore, it is desirableto have a design parameter to change the properties of theclosed-loop system.

For the proposed design method the robustnessconstraint is a good measure of the performance of thesystem. It has been shown in [9] that the variable M s fulfillsall the requirements of a good design parameter.

3 Difficulties of PID design

The solution of the PID design can be formulated as a parameter optimisation problem: maximise integral gain k isubject to the constraints that, (a) the closed loop system isstable, and (b) the Nyquist curve of the loop transfer function is outside a circle with centre at s ¼ C and radius R. The constraint (b) can be formulated as

C þ k i

oðk i o

2k d Þ

G ðioÞ

2

R2 ð3Þ

The constraint that the maximum sensitivity is smaller than M s corresponds to C ¼ 1 and R ¼ 1= M s . It is also possibleto introduce constraints on the complementary sensitivity

[9]. The solution to the optimisation problem gives good PIcontrollers as shown in [9]. For the PID controller the

problem is the lack of constraints, which prohibits theoptimisation of three parameters. This is understood by

Gff y sp u

G

–Gc

l

n

y

Fig. 1 Conceptual block diagram describing design problem

IEE Proc.-Control Theory Appl., Vol. 149, No. 1, January 2002 33


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investigating the robustness constraint in (3). Theconstraint defines a set of controller parameters k , k i and k d such that the maximum sensitivities are less than

prescribed values. An example of such a set is shown inFig. 2. The Figure shows that the largest value of k i occursat a ridge, which means that the solution is very sensitive to

parameter variations. This phenomenon that occurs for most transfer functions G ( s) illustrates one dif ficulty of using derivative action.

3.1 Geometric interpretation

Assuming that the process has positive low-frequency gain,and introducing

G ðioÞ ¼ r ðoÞeijðoÞ ¼ aðoÞ þ ibðoÞ

we find that the constraint in (3) can be written as

r 2

R2 k þ

aC

r 2

2þ

r 2

o2 R2 k i o

2k d þ obC

r 2

2 1 ð4Þ

For fixed k d and o this represents the exterior of ellipseswith centre in k ¼ aC =r 2 and k i ¼o

2k d 7obC =r 2. The

ellipse has its axes parallel to the coordinate axes. Thehorizontal half axis has length R=r and the vertical half axishas length o R=r . This is illustrated in Fig. 3. When o

ranges from 0 to 1 the ellipses have an envelope whichdefines one boundary of the sensitivity constraint. Sincethe constraint is quadratic in k i the envelope has two

branches; only one branch corresponds to stable closed-loop systems. To have a stable closed-loop system it must also be required that k i is positive.

Fig. 4 shows the envelope for different values of k d . The

Figure illustrates what happens if the integral gain k i ismaximised for a fixed derivative gain k d , where k d ¼ 0corresponds to PI control. It also shows that the integralgain can be increased substantially by introducing deriva-tive action. Notice that the maximum of k i on the envelopecoincides with the maximum of k i on the locus of thelowest vertex of the ellipses for small values of k d . Thislocus is shown in dashed lines in the Figure. Also noticethat the envelopes have corners for larger values of k d . TheFigure shows that the maximum of k i will typically occur at a corner of the envelope. When this happens the optimumis very sensitive to parameter variations. There are alsoother drawbacks as will be illustrated by an example.

3.2 Example

Consider the system with the transfer function

G ð sÞ ¼ 1

ð s þ 1Þ4

ð5Þ

00

0.5

0.2

0.4

0.6

0.8

1.0

k i

k 1.0

1.5 0 0.5

1.0 1.5

2.0 2.5

3.0 3.5

k d

Fig. 2 Geometric illustration of robustness constraint (3). Volume represents parameters for PID controllers such that maximum sensitivity is less

than 1.4. Curve is computed for process G(s) ¼ 1=(s þ 1)4

k i

k

R/r

R/r

Fig. 3 Graphical illustration of sensitivity constraint (4)

k d = 01.0

0

0.6

0.8

0.4

0.2

k d = 1 k d = 2

k d = 3 k d = 3.1 k d = 3.31.0

0

0.6

0.8

0.4

0.2

0 0.5 1.51.0–0.5 0 0.5 1.51.0–0.5 0 0.5 1.51.0–0.5

Fig. 4 Cuts of robustness region in Fig. 2 for constant derivative gain k d . Curves computed for PID control of process G(s) ¼ 1=(s þ 1)4. Dashed line shows

locus of lowest vertex of ellipses



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Solving the constrained optimisation problem we find that the controller gains are k ¼ 0.925, k i ¼ 0.9 and k d ¼ 2.86.Fig. 5 shows the Nyquist diagram of the loop transfer function. Notice that the Nyquist curve has a cusp. Thiswill always occur when the maximum occurs at a ridge,

because when there is a maximum at a ridge the Nyquist curve has a tangency to the sensitivity circle at twofrequencies. This happens when the controller has a

phase lead that increases rapidly, a typical behaviour of acontroller with complex zeros. It is possible to get a veryrapid increase in phase if the relative damping of the zerosis small. The closed-loop system has two pairs of complex

poles with relative damping z ¼ 0.24 and z ¼ 0.42, whichmeans that the time responses can be expected to beoscillatory.

For comparison we have also designed a controller byanother method which gives the parameters k ¼ 1.14,k i ¼ 0.51 and k d ¼ 1.14. The Nyquist curve for this control-ler is shown dashed in Fig. 5. This controller has two pairsof complex poles with relative damping z ¼ 0.59 and z ¼ 0.65, which is more reasonable. The Bode plot for the loop transfer functions obtained with the controller isshown in Fig. 6. The Figure shows that the controller that

maximises k i has a large phase lead above the gain cross-over frequency.

Fig. 7 shows the responses of the closed-loop system tosteps in the setpoint and the load disturbance. The Figureshows that the controller which maximises k i givesresponses that are quite oscillatory, indicating that thedesign is not too good. The integrated absolute error for a unit step load disturbance, IAE , gives a quantitativeassessment of the performance. For the controller that maximises integral gain we find that IAE ¼ 3.05 whichcan be compared with IAE ¼ 2.43 for the controller shown

by a dashed curve in Fig. 7.The example shows how it is possible to increase

integral gain substantially by the introduction of derivativeaction. To obtain large integral gains it is, however,necessary to introduce a lot of phase lead. By evaluatingtime and frequency responses of the obtained closed loopsystems we have found that the large phase lead required tomaximise integral gain has severe drawbacks. In the next Section we will introduce additional constraints to avoid these drawbacks.

1.0

0.5

0

–0.5 y

–1.0

–1.5

–2.0–2.0 –1.5 –1.0 –0.5 0 0.5

x

k k k = 0.925, = 0.9, =2.86 i d k k k = 1.14, = 0.511, =1.14 i d

_1

Fig. 5 Nyquist curve of loop transfer function for PID control of processG(s) ¼ 1=(s þ 1)4

101

|

(

) |

L

i ω 100

10–1

10–1

ω

100

10–1

ω

100

–100

a r g

(

)

L

i ω

–120

–160

–180

–140

–200

k k k = 0.925, = 0.9, =2. 86 i d

k k k = 1.14, = 0.511, =1. 14 i d

Fig. 6 Bode plot of loop transfer function for PID control of processG(s) ¼ 1=(s þ 1)4

1.5

k k k = 0.925, = 0.9, =2.86 i d k k k = 1.14, = 0.511, =1.14 i d

1.0

0.5

0

1.5

1.0

0.5

0

0 10 20 30 40 50

1.5

1.0

0.5

0

0 10 20 30 40 50

2.0

y

u u

0.4

0.2

0

–0.2

y

step in setpoint step in load disturbance

Fig. 7 Time responses for PID control of process G(s) ¼ 1=(s þ 1)4



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4 PID design

Having shown that a direct generalisation of the designmethod for PI controllers in [9] is not suitable for PIDcontrollers, we will now make some modifications. One

possibility is to introduce additional constraints which prevents excessive phase leads of the controller. After several attempts, it has been found that good controllersare obtained by maximising integral gain if the followingconstraints are added to the robustness constraint.

_vv €ww €vv _ww

ð_vv2 þ _ww2Þ3=2

<>>:

ð9Þ

where o0 is the frequency where the sensitivity function

has its maximum. Reasonable values of N are in theinterval 2 – 10. In Table 1 the influence of the filter on theloop transfer function at the critical frequency o0 is shown

by calculating the arg F y(io0). Note that the special choicesof T f in (9) for the first- and second-order filters will giveapproximately the same amount of influence of the looptransfer function at o0 for a certain value of N .

The insertion of a filter will modify the loop transfer function, but with suitable values of N these changes arekept small. Nevertheless, it is possible to take thesechanges into account by repeating the design with processG replaced by F yG . The results of such a repeated designare shown in Table 1, where the process is given in (5).

The trade-off between filtering capacity and loss of

control performance is demonstrated in the Table. Alarge value of M n is obtained when the IAE value issmall. The advantage of the proposed design method isits systematic way to determine measurement noise fi lters,such that the noise level is reduced.

6 Set point response

To obtain a complete design it remains to find a suitablevalue of the set point weight b. One possibility is toconsider the transfer function from set point to output,and to make sure that the maximum magnitude of this

transfer function is not larger than 1. This gives a set point response without overshoot for most systems. For a

Table 1: Properties of PID controllers obtained for system F y G with G (s ) ¼¼1/(s 1 1)4 and M s ¼¼1.4 for different filter of

order n ¼¼1, 2

n 1 1 1 2 2 2 7

N 2 5 10 2 5 10 1

arg F y (i o0) 50. 12.57 6.35 59. 12.69 6.36 0.

K 0.7895 0.9152 1.0009 0.6496 0.8161 0.9671 1.140

T i 2.8334 2.3322 2.2867 2.8285 2.4033 2.2963 2.227

T d 1.2979 1.0406 1.0130 1.3083 1.0716 1.0180 0.999

o0 0.5091 0.6695 0.7227 0.4703 0.6226 0.7011 0.790IAE 4.2425 3.0711 2.7821 4.9524 3.4774 2.8825 2.4209

M n 1.1392 3.7584 7.9840 1.0726 2.0550 6.1308 1



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controller with setpoint weighting, the response to setpoint changes is given by the transfer function

G spð sÞ ¼G ð sÞG ff ð sÞ

1 þ G ð sÞG cð sÞ F yð sÞ

¼ k i þ bks

k i þ ks þ k d s2

G ð sÞG cð sÞ

1 þ G ð sÞG cð sÞ F yð sÞ

ð10Þ

A straightforward search procedure can be used to deter-

mine if there is a value of b that makes the maximum of jG sp(io)j equal to 1. Such a procedure will give a suitablevalue of b if one exists.

Only nonnegative values of b are allowed, since negativevalues may result in inverse step responses in the controlsignal, which is an undesirable way to reduce overshoots. If jG sp(io)j is larger than 1 for b ¼ 0, a low-pass filtering of the set point may be used to reduce the magnitude of jG sp(io)j further.

The set point filter F sp( s) can be determined in thefollowing way. Let m s be the maximum of the transfer function in (10), and let o sp be the frequency where themaximum occurs. A first-order filter

F sp ¼ 1

1 þ sT sp

has the magnitude 1=m s at the frequency o sp if the timeconstant is

T sp ¼ 1

o sp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 s 1

p

Feeding the set point through a low-pass filter designed in this way will reduce the magnitude at the frequencyo sp to 1.

7 Numerical solution of design problem

Finally, a brief discussion of the numerical solution of theoptimisation problem for the design of PID controllers in(6) is given. The optimisation problem is non-convex and the solution is obtained numerically with the OptimisationToolbox in Matlab 5, which requires substantial calcula-tions. However, with today’s personal computers there areno major limitations.

As for most numerical optimisation routines it is impor-tant to have good initial conditions and a suitable searchinterval. A natural choice of initial conditions will be thecontroller parameters k and k i from the PI design in [9],that is,

½ k 0 k 0i k 0d ¼ ½ k k k k i 0

and a suitable search interval is given by

o start ¼ oo0=2

o stop ¼ ð oo180 þ oo270Þ=2

ō 0 is the frequency at which the sensitivity function fromthe PI design has its maximum. ō 180 and ō 270 are thefrequencies where the argument of the loop transfer func-tion from the PI design is 180 and 270, respectively.

The following procedure is used to solve the design problem:

(i) Obtain a model of the process in terms of a transfer function.(ii) Choose the design parameter expressed by the maxi-mum of the sensitivity function M s .

(iii) Determine the number of constraints, as it differsdepending on the considered system.(iv) Make a PI design to obtain the initial values [k ¯ k ¯i 0]and the frequencies [ō 0 ō 180 ō 270].(v) Solve the design problem with the Optimisation Tool-

box in Matlab 5.(vi) Verify that the resulting controller parameters fulfilthe constraints. If not, adjust the initial values or settings inthe accuracy of the numerical routine.

8 Examples

The design method has been tested on a number of examples which illustrate its properties. The followingtransfer functions have been considered:

G 1ð sÞ ¼ 1

sð s þ 1Þ3 ; G 2ð sÞ ¼

e5 s

ð s þ 1Þ3

G 3ð sÞ ¼ 1

ð s þ 1Þð1 þ 0:2 sÞð1 þ 0:04 sÞð1 þ 0:008 sÞ

G nð sÞ ¼ 1ð s þ 1Þn ; n ¼ 4 to 7; G 8ð sÞ ¼ 1 2 s

ð s þ 1Þ3

The first seven models capture typical dynamics encoun-tered in the process industry. Model G 1 is an integrating

process and G 2 models a process with long dead time.Model G 8 has a zero in the right half-plane, which isuncommon in process control, but it has been included todemonstrate the wide applicability of the design procedure.

Figs. 8 and 9 show the responses to changes in set point and load. The details of the design calculations and simulations are summarised in Table 2. Note that the PIDcontroller obtained is compared to the corresponding PIcontroller to show the amelioration of the PID design.

Although models G 1 – G 8 represent processes with largevariations in process dynamics, Figs. 8 and 9 show that theresulting closed-loop responses for a load disturbance

become similar for each value of M s . This is important because it means that the proposed design procedure givesclosed-loop systems with desired and predictable proper-ties.

There is also a clear similarity between the responsesobtained with the different values of the tuning parameter

M s , which indicates its suitability as a tuning parameter.Responses obtained with M s ¼ 1.4 show little or no over-shoot, as is normally desirable in process control. Faster responses are obtained with M s ¼ 2.0. The settling time at

load disturbances, t s , is significantly shorter with the larger value of M s . On the other hand, these responses areoscillatory with larger overshoots. This can be seen fromthe comparison between IE and the integrated absoluteerror IAE in Table 2.

The controller gain k varies significantly with the design parameter M s: it is larger for designs when M s ¼ 2.0 thanfor those when M s ¼ 1.4. However, integral time T i is fairlyconstant for the stable processes, i.e. all processes except G 1 . The derivative time T d is usually larger for designswith M s ¼ 1.4 than for those with M s ¼ 2.0. In all cases thePID design generates a controller with complex zeros for

M s ¼ 2.0. Thus, the controller will not be realisable inserial form.

For M s ¼ 2.0 the M p values are large. Consequently, theovershoots would be significant if the set point weight isb ¼ 1. However, acceptable set point responses areobtained by suitable choices of either the set point



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1.0

0.5

0

2.0

y

1.5 1.0

0.5

0

y

1.5

–0.10

0.10

u

1.0

0.5

0

u

1.5

0

–0.05

0

0.05

50 100 150 250200 0 20 40 60 80 100 120 140 160

1.0

0.5

0

y

1.5

1.0

0.5

0

y

1.5

42

–2

u

10

1.0

0.5

0

u

1.5

0

68

0 1 2 3 4 5 6 7 8 9 20 40100 30 50 60

1.0

0.5

0

y

1.5

1.0

0.5

0

y

1.5

1.0

0.5

0

u

1.5

1.0

0.5

0

u

1.5

0 10 20 30 40 50 60 70 80 900 10 20 30 40 50 60 70 80

1.0

0.5

0

y

1.5

1.0

0.5

0

y

1.5

1.0

0.5

0

u

1.5

1.0

0.5

0

u

1.5

0 10 20 30 40 50 60 70 80 90 100 110 0 10 20 30 40 50 60 70 80 90 100

a b

c d

e f

g h

Fig. 8 Comparison between PID (solid line) and PI controller (dashed line) for M s ¼ 1.4, showing step response followed by load disturbance of closed loop system

a system G 1 e system G 5b system G 2 f system G 6c system G 3 g system G 7

d system G 4 h system G 8



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1.0

0.5

0

y

1.5

1.0

0.5

0

y

1.5

–0.2

0.6

u

1.0

0.5

0

u

1.5

0

0.2

0

0.4

50 100 150 0 20 40 60 80 100 120

1.0

0.5

0

y

1.5

1.0

0.5

0

y

1.5

10

–10

u

30

1.0

0.5

0

u

1.5

0

20

0 1 2 3 4 5 6 7 8 20 40100 30 50 70

1.0

0.5

0

y

1.5

1.0

0.5

0

y

1.5

1.0

0.5

0

u

1.5

1.0

0.5

0

u

1.5

0 10 20 30 40 50 60 70 80 900 10 20 30 40 50 60 70

1.0

0.5

0

y

1.5

1.0

0.5

0

y

2.0

1.0

0.5

0

u

1.5

1.0

0.5

0

u

1.5

0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 70

a b

c d

e f

g h

60

–0.5

1.5

60

Fig. 9 Comparison between PID (solid line) and PI controller (dashed line) for M s ¼ 2.0, showing step response followed by load disturbance of closed loop system

a system G 1 e system G 5b system G 2 f system G 6c system G 3 g system G 7

d system G 4 h system G 8



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weight b or the filter F sp . According to Table 2, it is not always enough to set b ¼ 0 to obtain a small overshoot;filtering may also be needed.

9 Conclusions

PID controllers were designed to capture demands on load disturbance rejection, set point response, measurement noise and model uncertainty. Good load disturbanceresponses were obtained, minimising the integrated control

error IE . Robustness is guaranteed by requiring a maxi-mum sensitivity of less than a specified value M s . Measure-ment noise is dealt with by filtering. Good set point response is obtained by using a structure with two degreesof freedom.

The design procedure has been applied to a variety of systems, stable and integrating, with long dead times and with right half-plane zeros. The method can also be used todevelop simple tuning rules which are significant improve-ments of the Ziegler – Nichols rules [10].

10 References

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2 ZIEGLER, J.G., and NICHOLS, N.B.: ‘Optimum settings for automaticcontrollers’, Trans. ASME , 1942, 64 , pp. 759 – 768

3 SCHEI, T.S.: ‘Automatic tuning of PID controllers based on transfer function estimation’, Automatica, 1994, 30, (12), pp. 1983 – 1989

4 VAN OVERSCHEE, P., and MOOR, B.D.: ‘Optimal PID control of chemical batch reactor ’. Proceedings of 1999 European control confer-ence, Karlsruhe, Germany, 1999

5 LANGER, J., and LANDAU, I.D.: ‘Combined pole placement =sensitivity function shaping method using convex optimization criteria’,

Automatica, 1999, 35 , (6), pp. 1111 – 11206 KRISTIANSSON, B., and LENNARTSSON, B.: ‘Optimal PID control-

lers including roll off and Schmidt predictor structure’. Proceedings of 14th world congress of IFAC, Beijing, P.R. China, 1999, Vol. F, pp. 297 – 302

7 ÅSTRÖM, K.J., and HÄGGLUND, T.: ‘PID controllers: theory, designand tuning’ (Instrument Society of America, North Carolina, 1995)

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1 loop shaping’, Int. J. Robust Nonlinear Control , 2000, 10,

pp. 1249 – 12619 ÅSTRÖM, K.J., PANAGOPOULOS, H., and HÄGGLUND, T.: ‘Design

of PI controllers based on non-convex optimization’, Automatica, 1998,34, (5), pp. 585 – 601

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Table 2: Properties of obtained controllers for system G 1–G 8 with different values of design parameter M s

Process M s k T i T d b T sp IE IE =IAE o0 t s M p

G 1(s ) 1.4 0.324 6.59 2.35 0.00 0.17 20.4 0.67 0.77 15.3 1.53

2.0 0.680 4.50 2.27 0.00 0.08 6.61 0.69 0.91 37.8 1.50

G 2(s ) 1.4 0.325 3.55 1.69 0.69 0.00 10.9 0.99 0.26 48.2 1.00

2.0 0.555 3.21 1.74 0.00 1.61 5.80 0.64 0.29 39.2 1.32

G 3(s ) 1.4 15.96 0.212 0.15 0.00 0.26 0.013 0.57 19.1 2.99 1.52

2.0 43.13 0.189 0.13 0.81 0.00 0.0044 0.75 25.6 3.35 1.65

G 4(s ) 1.4 1.14 2.23 1.00 0.00 0.27 1.95 0.81 0.79 17.4 1.09

2.0 2.27 1.91 0.98 0.00 0.53 0.84 0.61 0.95 15.3 1.62

G 5(s ) 1.4 0.784 2.68 1.24 0.00 0.51 3.41 0.84 0.54 23.9 1.04

2.0 1.47 2.33 1.25 0.00 0.72 1.59 0.56 0.63 21.0 1.55

G 6(s ) 1.4 0.635 3.12 1.47 0.00 0.44 4.92 0.88 0.42 27.5 1.01

2.0 1.15 2.74 1.49 0.00 0.97 2.38 0.56 0.48 23.3 1.50

G 7(s ) 1.4 0.552 3.57 1.69 0.00 0.25 6.44 0.90 0.34 32.7 1.00

2.0 0.982 3.14 1.73 0.00 1.24 3.20 0.57 0.39 28.2 1.47

G 8(s ) 1.4 0.312 2.25 0.80 0.60 0.00 7.22 0.86 0.51 30.5 1.00

2.0 0.542 2.07 0.79 0.00 1.03 3.82 0.62 0.58 22.4 1.31


Astrom Opt PID IEE Kosarac Olja

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