tv-domain design of single- anddouble-valued relay control systems

C.P. Lewis, B.Sc, Ph.D., C.Eng., F.I.E.E.

Indexing terms: Relays, Control systems, Control theory

Abstract: The design of relay control systems is presented having ideal, dead zone and rectangular hysteresistransfer characteristics. The choice of the appropriate describing function model is considered and examples aregiven of limit cycle prediction and digital compensation in the w-domain. The method described is particularlyuseful for the analysis of slowly sampled systems.

List of symbols

N(X) = sinusoidal input describing function (SIDF)for a static nonlinearity

N(X, a) = SIDF of a double-valued nonlinearityN(X, 9) = SIDF of nonlinearity including the effect of a

sample and holdN(X, k) = approximation to NWk(X, (f>); the phase cor-

rected SIDFNS(X, 9) = sampled describing function (SDF) (the SIDF

of a nonlinearity and sampler)NW(X, 9) = w-transform form of the z-transform describ-

ing function (ZDF) or Ts N(X, 9), referred toas the WDF

NWk(X, 9) = WDF for a k, k mode of oscillationNJ^X, 4>) = WDF for a double-valued nonlinearityN2 = z-transform DFD(z) = z-domain form of discrete compensatorD'(vv) = vv-domain form of D(z)D = relay output levelG(s) = s-domain transfer functionG'(vv) = w-domain form of G(s)T = period of a sinusoidal signal7̂ = sampling periodTlc = limit cycle periodX = peak value of the sinusoidal input signald = half width of dead zonek = number of samples per half cycles = Laplace transform complex variableu = real component of wv = imaginary component of ww = w-transform complex variable, w = u +jvx(t) = input signal to nonlinearityy(t) = output signal from nonlinearityy*(t) = impulse sampled form of y(t)z = z-transform complex variable, z = e?Ts

a = phase shift produced by a double-valued non-linearity

b = half width of hysteresis zone9 = sampling anglea = real component of s(o = imaginary component of s(Of = angular frequency of fundamental component

Per-unit values of limit cycle magnitude X/D have beenused throughout.

In simulated results, the scales for x(t) and y(t) are identicalunless otherwise stated.

Paper 4074D (C8), first received 27th September 1984 and in revised form 5thMarch 1985

The author is Reader in Control Engineering at the Department of Systems &Control, Faculty of Engineering, Coventry (Lanchester) Polytechnic, Priory Street,Coventry CV1 5FB, United Kingdom

IEE PROCEEDINGS, Vol. 132, Pt. D, No. 5, SEPTEMBER 1985

1 Introduction

Although much effort has been spent in developingmethods of analysis for nonlinear control systems, thedescribing function (DF) approach used in conjunctionwith simulation is still the most widely used of industrialdesign techniques [1]. In this paper the design of sampled-data relay control systems is considered, a typical systembeing depicted in Fig. 1, which represents a pulse-torqued

Fig. 1 Typical nonlinear sampled-data system

transducer [2]. The method of analysis used here is tocombine the useful properties of a DF model of a nonlin-ear element with a w-domain description of the linear partof that system. It is well known that a bilinear transform-ation from the z to the w-domain is particularly useful forthe design of digital compensators for linear sampled-datasystems [3, 4].

A design method based on this approach has beendescribed by Chen et al. [5] in which the sinusoidal inputdescribing function (SIDF) was used to model the nonlin-earity as if the system was continuous; the w-domaindescription of the sampler and remainder of the systemwas determined in the usual way. It has since been shownby Lewis [6] that this approach has limitations, particular-ly when applied to systems which are sampled slowly. Thereason for this is that the SIDF does not provide a satis-factory model of the effect of the sampling action underslow sampling conditions; this can lead to large errors,particularly in relay control systems where multiple limitcycle conditions cannot be detected. Consequently, a newapproach has been made which makes use of a modifiedform of the sampled describing function (the SIDF of therelay and sampler) or the z-transform DF [7] (ZDF). Thelatter DF does not appear to have been widely used sinceits introduction by Kuo; however, in some cases, it may beused to predict limit cycle conditions when transformed tothe w-domain [6, 8, 9]. These describing functions aredependent not only on the magnitude X, of the sinusoidalsignal to the relay but also on the sampling angle 9,defined in Fig. 2. Here a 2, 2 mode is shown, that is onewhich has a half period of two sampling periods. Ingeneral, the limit cycle condition often met in this type ofsystem has a period 2kTs, where Ts is the sampling period,k E N [10]. It may be shown that if there is only onesampler in the system and there is sufficient filtering then

TSNS(X, 9) = NZ(X,9)

219

where NS(X, 6) is the sampled DF and NZ(X, 0) the ZDF,and this carries through to the w-domain. Of practical

x(t)

y*(t)

Fig. 2 2,2 limit cycle mode waveforms for the ideal relay system

interest is the relay with dead zone, in this case the aboveapproach may be used to predict limit cycles, however thelimit cycles periods may not be an even multiple of thesampling period. Thus Tlc = mTs, k e N, m ̂ 2. In Refer-ences 6 and 8, only single-valued nonlinearities have beenconsidered, in which case the order of the relay samplerand hold combination is immaterial. However, the methodis applicable to double-valued nonlinearities and anexample is considered in Section 3. An important advan-tage of the use of the WDF is in the implementation ofdiscrete compensation, particularly if the system issampled slowly. Further, by the use of a modified contin-uous SIDF model of the relay, which also accounts for thesampling angle, the w-domain design of relay sampled-data systems may be simplified.

2 w-domain form of the z-transform describingfunction (WDF)

The ZDF for a series combination of relay and sampler isdefined [7] as

NZ(X, 9) =y*(t)x*(t)

where x*(t) implies that samples of the input signal areconsidered and cof is the frequency of the fundamentalcomponent. To obtain the w-domain form, a bilineartransformation is used, the most popular being

Z =1 — w

Z, W 6 C

where the relation between the s- and w-domain fre-quencies is

2co = — tan v

here v is referred to as the 'fictitious' frequency

In practice this is a simple procedure as it is shown in theAppendix that for a k, k mode (k samples per half cycle)

z = eJW*>= NWk(X, 9)

where NWk(X, 6) is defined to be the w-domain describingfunction (WDF) of the k, k mode.

It should be noted that the z-transform describing func-tion can only be used in those cases where k ̂ 2, becauseonly information about the samples of the signals withinthe system are processed [10].

If the nonlinearity N in Fig. 1 is an ideal relay havingoutput levels ±D, the various waveforms are as shown inFig. 2. If it is further assumed that the system is limit-cycling in a k, k mode, it is clear that the instant at whichthe sampling occurs, relative to the limit cycle, is deter-mined by the phase shift produced by the linear com-ponents of the system. Thus, the sampling angle 6 e[0, n/k']. In determining the ZDF, it is usual to take a sam-pling instant as the time origin; as the sampled outputsignal of the relay y*(t) is a series of impulses the ZDF isreadily determined. For a 2, 2 mode this has the form

NJX, 0) =D(z + 1)

cof = n/2Ts (1)X(cos 9 + z sin 0)

which on transformation to the w-domain becomes

NW2(X, 6) =X{{1 - w) cos 6 + (1 + w) sin 6}

(2)W=JVf

when eqn. 2 is evaluated at the 'fictitious' fundamental fre-quency vf — tan (n/4) = 1, then

In Fig. 3a and b, graphs of (X/D)\NWk(X, 0)\ and

1.5 "

CD

x" 1.3^ 1 . 2 7z<!<=• 1.2

1.1

1.0

ISDFi

0a

60 -

20

20

= jtan<it/2*)

60 Lb

Fig. 3 Magnitude (a) and argument (b) of the ideal relay w-transformdescribing function against the limit cycle mode k

220 IEE PROCEEDINGS, Vol. 132, Pt. D, No. 5, SEPTEMBER 1985

arg NWk{X, 9) are plotted as functions of k. The usual SIDFvalues for an ideal relay are added as a comparison.

3 Double-valued nonlinearity

This approach may also be used to determine the WDF ofa device having a rectangular hysteresis transfer character-istic followed by a sampler. It may be shown that theoutput waveform of the sampler y*(t) is the delayed formof that produced by the ideal relay. Taking the first sam-pling pulse, following the zero-crossing of x(i) in a positivesense, as the time origin, then the phasor representation ofthis signal X(jco) lags by an extra phase shift a = sin"1

(S/X) where S is the half width of the hysteresis. Hence the2, 2 mode is given by

NW2(X, <t>) =

0 = 0 + s i n " 1 (5/X)

Similarly for the 3, 3 mode,

1.306 ut-l*\-A.\

9 € [0, n/2]

NW3(X, 0) = X

= 9 + sin~i (d/X) 0e[O, n/3]

4 Relay with dead zone

This type of transfer characteristic presents no fundamen-tally new problems, an increase in labour is required as theoutput signal y(t) can take more forms, each of whichrequire the determination of a WDF.

5 Limit cycle conditions for the ideal relay

The limit cycle conditions are found by the solution of

1 + NWk(X, 9)G'(w) = 0

w=j tan (n/2k) keN,k^2 9 e [0, n/k] (3)

which may be sought in the manner used for continuoussystems by finding the intersections of —NWk(X, 9) andG'"1 (/*>)• Here G'(vv) is the transfer function of G{s) trans-formed to the w-domain.

As the phase shift due to the sampling action is a func-tion of k rather than X, and as G'(jv) exists only at distinctvalues of v, determined by k = 2, 3, 4, . . . , it is logical totransfer the effect of the sampler from the describing func-tion plot to G'(jv). Thus a graphical solution is sought of

I/IG'U tan {n/2k})e±J««'2k)-e)] = -\NWk(X, 9)\

9 e [0, n/2k] keN

To show the effect of the sampling action, arcs represent-ing the range of values of 9 are superimposed on theinverse polar locus at each value of k. A limit cycle condi-tion of period 2kT may exist if the describing functionlocus cuts the appropriate arc. In the case of an ideal relay,both the SIDF and the WDF model lie along the negativereal axis.

The principal advantage of employing the w-transformis when discrete compensation is required, as in Fig. 4, in

Fig. 4 Digital compensator added to block diagram

1EE PROCEEDINGS, Vol. 132, Pt. D, No. 5, SEPTEMBER 1985

which case the characteristic equation becomes

1 + NWk(X, 9)D'(w)G'(w) = 0 w = j tan (n/2k) (4)

where D'(vv) represents the discrete compensator D(z) trans-formed to the w-domain.

6 A simplified w-domain model

It can be seen from Fig. 3 that the SIDF, N(X), of the relaymay be used as an approximation to | NWk(X, 9) | forvalues of k ^ 3, without undue error, bearing in mind theapproximate nature of the DF approach. However, it isclear that the phase shift produced by the sampling actioncannot be ignored. Hence, it is proposed that, under theabove restriction, a simple approximation to NWk(X, 9) isN(X)ejM2k-e\ 0 e [0, n/k], keN,k^3, written as N(X, k)for simplicity. In the case of a double-valued nonlinearityNWk(X, 9) is approximated by N(X, a)ej{n/2k-e), 9 e [0, n/k],k e N and k ^ 3. For the ideal relay the limit cycle condi-tion is found by the solution of

1 + N(X, k)G'(w) = 0,w=j tan (n/2k)

Because the phase shift due to the sampling action is afunction of k, the limit cycle conditions are found moreeasily if — N(X) and

G'-i{jv)e±Knl2k-e\ 9 e [0, n/k], v = tan (n/2k)

are plotted as in Fig. 5.

ImG'1jv)

x ( i )3/ V

-2 \ -1

60°/

(jv)

'90°

Fig. 5 Plots of 1/G'(jv) together with arcs for phase shift produced bysampling action and calibrated in terms of 9a Plot of G"l{jv), v = tan (n/2k) in the complex planeb Describing function DF, locus as X —* ooc 0 arc for the 2, 2 modePoints of interest on the negative real axis are:

(i) Prediction of a 2, 2 mode due to the intersection of the 9 arc with the DF,X = 0.42 p.u, 0 = 34°

(ii) Limit cycle predicted by the SIDF(iii) Prediction of a 3, 3 mode, X = 0.84, 0 = 45°. A 4, 4 mode does not exist as

the 6 arc does not reach the negative real axis.

7 Applications

To illustrate the use of the various DF models, four simpleexamples are considered.

7.1 Ideal relayLet N in Fig. 1 be an ideal relay and the linear part of thesystem, including a zero-order hold, be described by thetransfer function

221

G(s) =s2(s

T =

The corresponding w-domain form of G(s) is

G'(w) =0.5(1 - w)(l + 0.165w)

w(l + 2.17w)

In Fig. 5, a plot is shown of 1/G'(jv) to which is added thearcs representing the phase shift produced by the samplingaction and calibrated in terms of 6. The intersection of anarc with the describing function predicts a limit cycle ofperiod 2kT seconds. The calibration of the describing func-tion yields the magnitude X of the limit cycle in the usualmanner.

A comparison of the predicted and simulated values ofthe limit cycles is shown in Table 1. Fig. 6 shows the wave-forms for a 2, 2 mode.

Tutsim

\

70.8

i0.8AD

2D

5s

Fig. 6 Ideal relay (a), hold {b) and system output waveform (c) for the2, 2 mode obtained by simulationThe relay and output waveforms are to the same scale, for clarity the hold waveformis to a larger scale (1.33).

Use was made originally of a hybrid computer [11],here the results were obtained by the use of the Tutsim[12] digital simulation language. Care must be taken wheninterpreting the results using such a language, particularlyin systems involving sampling action, as the simulationprocess is itself discrete.

It may be observed that some of the simulated systemoutput waveforms are biased such that the relay outputhas an asymmetric waveform. This asymmetry is notcarried through to the 'hold' waveform due to the sam-pling action (see Fig. 8).

Table 1: Comparison of predicted and simulated limit cyclevalues

Limit cycle Predicted limit cycle magnitude Simulated valuesperiod, s and sampling angle

Corrected SIDF WDF Xp.u. 6°Ap.u. d° Xp.u. 6°

0.380.79

3445

0.420.84

3245

0.430.84

3243

7.2. Digital compensationAs indicated in Section 5, the effects of digital com-pensation on the limit cycle conditions may be determinedby the use of the WDF. As an example the digital com-pensator having the form

D(z)= l - a z ~ l

is used to suppress the 3, 3 limit cycle found in Section 7.1.In the w-domain the compensator has the form

(1 - a)(l + bw) l + aD'(w) = : b =(1 + w) 1 -a

Fig. 7 shows the effect of D'(vv) on the inverse polar plot ofG'(jv).

(iii)

Fig. 7 Suppression of 3, 3 limit cycle by the use of a digital lead com-pensator D(z) = 1 + az~l

(i) a = 0.28, (ii) a = 0, (iii) a = -2 .5

From the Figure, it is found that the 3, 3 mode is pos-sible for values of a between —2.5 and 0.28, these agreewith the values obtained by Gelb and Vander Velde [10]using a different approach. Simulation was used to verifythe method and typical results are shown in Table 2.Values for the uncompensated case, a = 0, are added for

Table 2: Results of digital compensation on the limit cycleconditions

a

0.30.2

0

-0.3-2.5

Limit cycleperiod, s

4464668

Limit cycle magnitude,

Predicted (WDF)

0.440.430.760.420.841.04.2

p.u.

Simulated

0.470.450.820.420.830.984.3

Fig. 8T = 6 s

Typical limit cycle mode for D(z) = 1 + 0.3~\ X = 0.96,

222 IEE PROCEEDINGS, Vol. 132, Pt. D, No. 5, SEPTEMBER 1985

comparison. In Fig. 8, a typical simulated result is shownfora= -0.3.

7.3 Relay having rectangular hysteresisAs an example, the system of Section 7.1 is considered withthe ideal relay replaced by the device having a rectangularhysteresis transfer characteristic. Owing to the hysteresis,the relay produces a phase shift a and the describing func-tion in the w-domain is complex, in addition to the effectof the sampling action. Plots of —NWk(X, a) andG'-l{jv)ej{nl2k-e\ d e [0, n/k], k e [2, 3], where

NW2(X, a) =

and

are shown in Figs. 9 and 10. The limiting values of S forthe 2, 2 and 3, 3 modes are 0.25 and 0.6, respectively,

ImG~(jv)

ReG"\jv)

G (jv)

Fig. 9 Plots of G"l(jv), v - tan (n/2k) and the 2, 2 mode WDF andSWF for S = 0.2

which agree with the values predicted by Goclowski [13]using a different approach. If the phase-corrected SIDF isused, the limiting values of S are found to be approx-imately 0.2 for the 2, 2 mode and 0.6 for the 3, 3 mode.

Typical predicted values and those obtained by simula-tion are shown in Table 3. It is seen that there is reason-able agreement between the results.

If the continuous model SIDF is used with G'^ijv) topredict the limit cycling condition, then only one limitcycle is predicted, as in Section 7.1.

7.4 Relay with dead zoneIn many practical systems a dead zone is required in thetransfer characteristic to prevent continuous oscillations.

(x.oc) 6=0.6 ! - G " 1 ( J v )

1 2 -w-Nw(X,0f)'U 6=0.2

1.0

ReG'(jv)-1.0 0

Fig. 10 Effect of the rectangular hysteresisS = 0.2 and 0.6 on the 3, 3 and 4, 4 modes; (3) and (4) are the 3, 3 and 4, 4 modelimits

The approach described in Section 7.1 may be used todetermine the limit cycle conditions of such systems. Theonly change in the technique is that it must be recognisedthat the limit cycle periods may be an odd multiple of thesampling period. For the system shown in Fig. 11, Chow

C(s)ZOH r 1

s ( s * 1 )

Fig. 11 System with four possible limit cycle modes

See Chow, Reference 14

[14] lists four possible limit cycle modes, and each moderequires a WDF description for the nonlinearity. (As asingle valued nonlinearity is involved, the use of the WDFrepresentation of the relay and sampler is valid.)

In Table 4 the four modes and their properties are listedtogether with the predicted and simulated results. As acomparison, the results using the phase-corrected form ofthe SIDF are included in the Table. A comparison of theresults from Fig. 12 and those obtained by simulationshows good agreement; for the longer-period modes thecorrected SIDF is useful because it is simple to use. Atypical simulated result is shown in Fig. 13. As in a contin-uous relay control system the dead zone may be used tosuppress limit cycles; in Fig. 12 it is seen that, to suppressall limit cycles, the maximum value of the DF must be lessthan 1.6. Using the SIDF, as an approximation, the dead

Table 4: Predicted and simulated results for the four possible limit cyclemodes of Fig. 11

Table 3: Predicted values forwith rectangular hysteresis

6

0.20.20.20.6

the relay

Period, s Limit cycle magnitude, p.u.

Predicted (WDF)

4 0.426 0.838 1.278 1.28

Simulated

0.450.871.361.33

Mode

2 —

3 -t-

4 • * -

y*{t)

f t ,

n1 1 H i

Period, s

4

4

5

6

|AU**)|

D/x

J2D/X

1.23D/X

1.33D/X

Limit cycle magnitude, p.u.

Chow

0.308

0.43

0.57

0.84

iv-domain CorrectedSIDF

0.3

0.42

0.57

0.84

0.37

0.56

0.81

Simulated

0.43

0.57

0.84

IEE PROCEEDINGS, Vol. 132, Pt. D, No. 5, SEPTEMBER 1985 223

zone is found to be d = 2/nl.6 = 0.4 p.u.; the value foundby Chow and by simulation is 0.53 p.u.

-iyx,e)-3 - 2 -1.6

Fig. 12 Inverse polar plot of G'(jv), v = tan (nTJTlc) and the WDF forthe relay with dead zone indicating limit cycles having periods of 4TS, 5TS

and 6TS

The SIDF has been added to determine the approximate critical value of the deadzone.

Tutsim

A

a

b

10s

T1.UD

1 2D

Fig. 13 Five-second period limit cycle with d = 0.1, X = 057 p.u.a Relay waveform; b —c(t), system output

8 General nonlinear sampled-data systems

Often it is important in nonlinear systems to ensure thatno limit cycles occur, in which case the area between theloci A and B in Fig. 9 could be considered to be a for-bidden zone which —N(X, a) must not enter. The form ofG'~l{jv) may be altered by suitable compensators toensure this condition. The values of fictitious frequency vrequired to draw the 6 arcs being determined by

v = tan T

Conclusions

A simple phase correction to the continuous form of theSIDF for a relay enables prediction of limit cycle modes tobe made in the w-domain; in this domain the effects ofdigital compensation are readily determined. The methodis particularly useful for low-frequency sampling applica-tions, if greater accuracy is required then the w-domain

describing function may be used. The method has beenshown to be applicable to ideal relay, relay with dead zoneand rectangular hysteresis characteristics.

It is believed that the approach could be extended to awide range of nonlinearities because there is a closerelationship between the ZDF and the sampled describingfunction [8], tables of which have been published [10].

10 References

1 ATHERTON, D.P.: 'The development of CAD software for relaysystems'. Proceedings of the 4th Polish-English Seminar on Real TimeProcess Control, Warsaw, May 30-June 3, 1983, pp. 10-19

2 MURTY VEPA, N., and HIGASHIGUCHI MINORU: 'Investiga-tion of pulse-torquing systems', IEEE Trans., 1975, AES-11

3 FRANKLIN, G., and POWELL, J.: 'Digital control of dynamicsystems' (Addison-Wesley, 1980)

4 KUO, B.C.: 'Digital control systems' (Holt-Saunders, 1981)5 CHEN, T.C., HAN, K.W. and THALER, G.J.: 'Stability analysis of

multirate nonlinear sampled-data control systems'. IEE Conf. Publ.794,1981, pp. 229-233

6 LEWIS, C.P.: 'Design of non-linear sampled-data control systems'.Doctoral thesis, Coventry (Lanchester) Polytechnic, UK, 1982

7 KUO, B.C.: 'A z-transform describing function for on-off typesampled-data systems', Proc. Inst. Radio Engrs., 1960, 48, (5), pp.941-942

8 JAMES, D.J.G., and LEWIS, C.P.: 'A w-domain approach to thedesign of sampled relay control systems', Int. J. Control, 1984, 39, (1),pp. 127-134

9 JAMES, D.J.G., and LEWIS, C.P.: 'Limit cycle conditions forsampled relay control systems with transport delay'. Int. Systems Eng.Conf., Wright State University, USA, September 1984

10 GELB, A., and VANDER VELDE, W.: 'Multiple-input describingfunctions' (McGraw-Hill, 1968), Chap. 9

11 LEWIS, C.P.: 'Prediction of limit cycle conditions in non-linearsampled-data systems', Trans. Inst. Meas. & Control, 1979, 1, (4), pp.199-203

12 'Tutsim simulation language handbook'. Twente University of Tech-nology, Netherlands (Micropacs, Lymington, Hampshire)

13 GOCLOWSKI, J.C.: 'Analysis of a sampled-data relay servo withhysteresis'. NEREM Record, 1963, pp. 76-77

14 CHOW, C.K.: 'Contactor servomechanisms employing sampled data',Trans. Amer. Inst. Elect. Engrs., 1954, 73, Pt. II, pp. 51-63

11 Appendix: Equivalence of the describingfunction forms

Transformation from the z to the w-domain is obtained bythe use of

z =1 + w1 - w

(5)

For a k, k mode of oscillation the 'fictitious' frequency inthe w-domain is

w =jv where v = tan (coTJ2)

The limit cycle has the period 2kTs, therefore

w =j tan (n/2k)

Substituting for w in eqn. 5 yields

+jtan(n/2k)z ==

1 -j tan (n/2k)

(6)

If evaluation of the z-transform describing function ismade by the substitution of z = e17*, then z = eJ("//l) as ineqn. 6. Hence,

6)w—j tan (n/2k)

that is, the w-plane describing function is the same as thez-plane describing function, when these are evaluated atthe fundamental frequency in the respective domains.

224 IEE PROCEEDINGS, Vol. 132, Pt. D, No. 5, SEPTEMBER 1985