Observation on Determining the Sampling Period in Digital Control

Copyright © IFAC Large Scale Systems. Rio Patras , Greece. 1998

OBSERVATION ON DETERMINING THE SAMPLING PERIOD IN DIGITAL CONTROL

Petraq Marango Polytechnic University Department of Electronics Automatic Control Division Tirana, Albania e-mail: [email protected]

Abstract: The sampling period is one of the parameters that characterises digital control of the industrial processes. It directly effects to the establishment of the signal with no deformity and to the very dynamics of the closed-loop. According to the theoretical and experimental observations, the paper presents and discuss over the band width of sampling in digital control, the problem of enlarging and decreasing of sampling period, etc. Especially it observes the influence of the value of sampling period to stability and performance of closed-loop of digital control. Copyright © 1998 IFAC. Keywords: digital control, sampling periods, stability, performance.

The starting points in determining the value of the sampling period are the recommended values of the upper and lower limit band. The principle according to which the frequency of sampling should be larger in magnitude than twice the maximal frequency of the power spectrum (Karapici, 1988), will serve as a base for the establishment of the original undisturbed signal. This will also be appropriate in the case of control of industrial processes. An example would be a fIrst order object, used frequently as a mathematical model for industrial processes in the thermal fIeld. In this case the sampling frequency (Agalliu, 1986) will be: I' Js ~

2 I'

_ 0.3184 J3dB - - - -

time constant of the fIrst order model. The limitation of the frequency band with the decrease in amplitude 3dB, ensures a signal reproduction with minimal undisturbed. The estimation of step-response for the closed-loop system, up to 95% of the established value for a time of (3-4)To, will have 'n' samples: n=

(3-4)To +1=2 1t . To

(2)

The information that can achieved by this determination of the sampling period is a minimal one. It is an insufficiency in this case for the estimation of the dynamic of the digital control. This situation can be improved by the enlargement of the samples number of the converter AID during the time of the step-response.

(1)

To

where fs is sampling frequency, f3dB is frequency with the decrease in amplitude of 3dB and To the

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This simple analysis yields the conclusion that the sampling period T requires as a condition the fulfilment of:

(5)

If the resonance frequency /, of the closed loop is

(3)

known, then it is recommended (Schoeme, 1969) that for the sampling period:

or it is all the same

n· TO

T=-n

n>-l

1 1 --"?T"?--

(4)

8/,.

(6)

16/,.

Again there are recommendation with a wide range (Terao, 1975) for defining of the sampling period:

For an object of order higher than one, the constant will equal To the highest constant of the time for which there is a decrease of 3dB.

(7)

The inequality T --( To ·n is not to be examined in deeper detail, since a persistent decrease of value of the sampling period causes several other problems in an industrial process. Selecting a sampling period of a few seconds in an oven, whose model involves a time constant of 30-40 minutes, means that you will increase considerably the number of the samples to be worked on. This becomes the source of routines, noises and delays that are practically undesirable. Furthermore, in a similar industrial process, we need to be carefully aware not only of the electronic features, but also of the mechanic, pneumatic, hydraulic, etc., which hold a primary position in industrial production lines to be controlled. The very inertia of these constructions does not allow solving by decreasing the sampling period, in order to increase the number of samples.

The value of the sampling period effects directly the stability of the closed-loop of control system. From a mathematical point of view this is clarified considering that such value is part of the values of the coefficients of the characteristic equation of the closed-loop system, and as a result it effects the distribution of poles in the complex plane. Referring again to the case of the first order process, including it in a closed-loop with unit negative feedback and a PI controller: l+d Z-1

G(Z) = K

In determining the value of the lower limit of the sampling period, the following factors are to be noted. While carrying out contemporary the measurement and control of a finite number of variable, the sampling period hasn't to be smaller that the total time of the commutation of the different inputs of the computer linked to these variables. As such we can mention the control of the state of cells, the samples of values from the process, their manipulation, the feedback to command the process, etc. We also have to take into consideration the data of the AID converter, such as the minimal time defined by it and the value allowed to us by the very syntax of the computer for carrying out the input samples. It is clear that these values are to be searched in the accompanying catalogue of the used converter.

where:

G(s) = k p ( 1+ d1 =- k~

(8)

1 I

l-Z-

and K

~) = k p + k(r;

The process is represented by model with the two parameters: ko (gain of the process) and 1'0 (time constant), and also the sampling period T, generating so the following analog part of the mathematical model by using a zero-older hold (Marango, 1997):

(9)

For selection of the sampling period in literature, different reasoning paths with wide ranges of definition are to be found. In many industrial processes, recommendation refer to the duration of the delay time (Lee, 1965) or the time constant, presented by the process in approximation of the its step-response according to the first order model and delay time Td :

where

k = ko (1- d) and d = ex p( -

~)

These results show that the term Kk of the transfer function of the closed-loop depends on the parameters of the process ko, To ' on the adjustment coefficient of the PI controller and also on the sampling period.

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By applying the PI controller with compensation of the pole of the process model ( Bloch and Brenac, 1982) in a corrected open-loop we will achieve for the closed loop: GfZ)-

\:

Kk

- Z - (1 - Kk)

2

Sampling period (sec) 4 5 6 7

3

! 1

I

I

kp

i

(11) !

!

2 APERIODIC

3

Having a same pole as the value, the stability of the closed-loop will be ensured for the condition:

Izl = 11- Kkl-< 1

4 5

or

6

(12) 0-< Kk -< 2

7 UNSTABLE

8

According to (12) the coefficient K of the controller, for the stability limit will be solved:

9 1

(13)

While the sampling period approaches infmitely:

--

Fig.I. The influence of sampling period to the stability of closed-loop system. (14)

The coefficient of the controller had been modified in the limits k p = 1+10, while the sampling period according to -

And while it decreases infinitely:

T

= 0.0144 + 050 .

To Based on the study, is observed that for having a stable closed-loop digital system with an acceptable performance, we can acknowledge:

(15)

1 T5.-To 5

The above calculations show that the value of the sampling period influences greatly the stability of the closed-loop of digital control. So, decreasing it infinitely will allow the system to remain continually stable, while increasing it infmitely reveals a upper limit value beyond which the system turns unstable. A series of experiments have been carried out, for a thermal process model with To = 14 sec. ko = I and change of the parameter K . These experiments verify the existence of the upper limit of stability, in our case in the value of the adjustment of the controller 2 K =k;;' also the decrease of oscillations distancing

(16)

In the industrial processes controlled by the computer, the "adjustment" of the digital controller or its "periodic watch" yields no problems. It consists only in "writing" and "rewriting" the lines of a program in a specific language. After having determined the values of the coefficients of the controller, as a result of the used method of synthesis is possible to achieve small modifications of the sampling period. This will cause small changes in the in the character of the step-response, without changing the selected adjustment of the digital controller. This becomes considerably self evident in the study. For instance, in the case of a sampling period represented by T = 0242 To there is a controller coefficient k p =4 .

from this zone. It becomes evident that, for a certain value of the adjustment coefficient of the controller, the movement in the realised experiments reveals that the increase of the sampling period is accompanied with the increase of oscillations in the system, until the limit of the zone is reached. These conclusions are shown in the Fig.l .

If is required a decrease of the time-duration of the step-response of the process, and successively accept an overshoot of approximately 5%, it is necessary only to switch to another sampling period

T=0265To·

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regulateurs numeriques sur microprocesseur, Project de Speciale Automatique, LAG - INP de Grenoble. Cadzow, J.A and Martens, H.R. (1970).DiscreteTime Systems and Computer Control Systems, Prentice-Hall, Englewood Cliffs, New Jersey. Karapici, G. (1988) . .Automatika e Sistemeve Elektroenegjitike. SHBLU, Tirane. Larminant, de, P., Thomas, Y. (1977). Automatique des systemes lineares, Flammarion Sciences. Lee, W. T. (1965). Plant Regulation by on-line Digital Computers, SIT - Symposium on Digital Control. Ljung, L. (1981). Reglertori - Moderna analys och syntesmetoder, Lund. Lowe, E.l., Hidelen, A.E. (1971). Computer Control in Process Industri, Pete Pelegrinus,

The above features and observations, based on the possibilities created by a modest laboratory, are theoretically and experimentally verified for the chosen model. Referring to these considerations, and also to the literature experience (Ljung, 1981; Lowe and Hidelen, 1971; Cadzow and Martens, 1970) it is possible to recommend the following limits for choosing the sampling period of digital control, in accordance with the nature of the industrial process, table 1. Table 1. Some recommendation values for the limit choosing of sampling period. Process Speed Pressure Level Temperature

Min. Sampling Period 1.0 sec. 1.0 sec. 8.0 sec.

Max. Sampling Period 0.5 sec. 2.0 sec. 2.0 sec. 10.0 sec

Ltd. Marango, P. (1997). Relatively Simple Coordinates for PI - Digital Adjustments, Life Cycle Approaches to Production Systems, ICMSNOE Proceedings, p.508-513. Schoeme, A. (1969). Proressrchensysteme der Verfahrensindustrie. Carl Hauser Verlag, Muenchen, page 128. Terao, M. (1975) Quantization and Sampling Selection for Efficient DOe. Instrumentation Technology, Vo1.l4, Nr.8, p.49.

REFERENCES Agalliu, J. (1986). Matjet elektronike. SHBLU, Tirane Bloch, S., Brenac, A. (1983). Mise en oeuvre edivers

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