Multi-Pulse Oscillations in Relay Systems

MULTI-PULSE OSCILLATIONS IN RELAY SYSTEMS U. M. Rao and D. P. Atherton Dept. of Electrical Engineering, University of New Brunswick, Fredericton, N.B., Canada Abstract. The paper describe s an extension of Tsypkin's method to evaluate multi-pulse oscillations in relay systems. The theory is presented both for a fixed plant transfer function and also for the case where a subsidiary feedback loop, with a time constant and time delay dependent on the relay position, are in parallel with the plant. The nonlinear algebraic equations which give the solution for a particular pulse mode are written in terms of 'A' loci and solved by digital computation. The theory is used to determine multi-pulse oscillations in a satellite attitude control system. The results are c ompared with those from an analog simulation. The approach is also used to investigate multi-pulse forced oscillation mod e s in a simple relay control system. Keywords. Limit cycles; relay control; nonlinear control systems; satellites, artificial; attitude control. INTRODUCTION

The basic mode of oscillation in relay systems has been studied approximately using describing function methods (Kochenburger, 1950; Gelb , 1968; Atherton, 1975a) and exactly using either time domain methods (Hamel, 1949; Bohn, 1961; Chung, 1966) or frequency response methods (Bergen, 1962; Atherton, 1966; Judd, 1974) based on the initial work of Tsypkin (1958). Multi-pulse oscillations in the satellite attitud e control system discussed above have been studied using the describing function method (Miller, 1976; Rao, 1976). Despite the high harmonic content of the relay output waveform, these analyses have yielded quite good results due to the resonance of the plant transfer function.

Many pape rs can be found in the literature on the determination of free and forced oscillations in relay systems. Most of these investigations are concerned with the basic or one pulse mode of oscillation where, for a relay with dead zone, its output is as shown in Fig. 1. In many practical situations, however, multi-pulse oscillations of the form shown in Fig. 2 exist. y(tl

Fig. 1.

Single-pulse output of a relay

Fig. 2.

Multi-pulse output of a relay

In this paper we extend the Tsypkin approach to cover the case of multi-pulse oscillations in a system with a linear plant which may, in addition, contain relay-mode dependent time constants. For a single feedback loop containing a relay with dead zone and exhibiting the basic one pulse oscillation two nonlinear algebraic equations are obtained which may be solved graphically on a polar frequency response plot. The frequency response loci used are summed loci which depend on the response of the plant transfer function to each frequency and its odd harmonic multiples. To analyse various feedback configurations more general summed frequency response loci, known as 'A' loci, have been defined and their summations given for various classes of transfer functions (Atherton, 1966; 1975b) including those with complex poles. It is shown that, similar to the situation for the basic oscillation mode in systems with more than one relay, prediction of multi-pulse operation in a single relay system requires the solution of several nonlinear algebraic equations. Graphical methods are thus not usually applicable and solutions must be obtained computationally. In addition to considering in some detail the satellite problem mentioned above,

The specific situation which initiated this study was the need to determine the possibility of multi-pulse oscillations occurring in a satellite attitude control system where, in addition to the existence of a plant transfer function with a lightly damped resonance, an additional single time constant feedback path existed around the relay. This latter path was also relay-mode dependent in that the network time constant had one value with the relay in the on position and another value when it was in the off position.

1747

U. M. Rao and D. P. Atherton

1748

the method presented is also used to study multi-pulse forced oscillations in relay systems.

which may be written as (2h/n){Im[A (9.-9.,w)] g ~

THEORY

g

In this section we derive the nonlinear algebraic equations which must have a solution if a multi-pulse oscillation exists in the simple feedback loop of Fig. 3 with a fixed plant transfer function. The conditions are then derived for single-pulse and multi-pulse oscillations in a system with a relay-mode dependent transfer function.

_o.l .::u.h!· ()

xl!)

(8)

-Im[A (9.-9.+69.,w)]}

y(t)

G(S)

Cl!)

)

~

~

where 9i wti and the 'A' loci are defined (Atherton, 1966, 1975a) by Re[A (9,w)] g

I

v

k~l(2)k

'"

Im[A (9,w)] ~ I g

sin k9 +

~

(9)

cos k9

(l/k) {vkcosk9-uksink9 } (lO)

k~l(2)

and A (9,w) g

~

(11)

Re[A (9,w)]+jIm[A (9,w) ] g g

A similar expression can be derived for ci(tj+6tj) so that the switching conditions for c(t) in equations (6) and (7) yield n

Fig. 3.

.

Relay control system

IIm[A (9.-9.+69.,w) ]-Im[A (9.-9.,w)] g ~ J ~ g ~ J ~ n(o+6)/2h

(l2)

~~l

Conditions For A Multi-Pulse Oscillation With Fixed Plant Transfer Function

n

IIm[A (9.-9.+69.-69.,w)]-Im[A (9.-9.-69.,w)] g ~ ) ~ ) g ~ ) J ~ n(6-6)/2h (13)

i~l

Figure 2 shows the relay output waveform y(t) which is assumed to have odd symmetry with n pulses per half period. Then n

y(t) ~

I y. (t)

with tl

i~l ~

~

0;

(1)

for j

~

1, ... n.

Similarly evaluating ~i(t) at the switching instants in terms of the 'A' loci the conditions given in equations (6) and (7) become n

and Yi (t) is the symmetrical single-pulse

IRe[A (9.-9.+69.,w)]-Re[A (9.-9.,w)] g ~ ) ~ g ~ )

i~l

waveform shown dotted in Fig. 2. The output c(t) from the linear plant, G(s), which is assumed to satisfy lims+ooG(s)+O is thus n

c(t) ~

i~l ~

I

~

and

(4h/kn) gk sin (kw6 t./2)

Also ~. (t) ~ ~

~

cos {kw[t-t -(6t/2)]+

G(jkw)

gke

I

Hk

~

u

+ jV

k

(3)

(4)

k

(-4hw/n)gksin(kw6t./2)

k~l(2)

~

sin{kw[t-t.-(6t./2)]+
x

~

~

k

}

(5)

For the oscillation waveform y(t) to exist c(t) must satisfy the switching conditions ~

-c(t.)

6 + 6,

-~(t.)

)

-c(t.+6t.) ) ) for j

~

n i~l

g

for j

~

~

~

g

J

> 0

(6)

6 - 6,

-~(t.+6t.)

)

J

< 0

1, .•• n.

~

J

(n/4w) k~ sG(s)

J

(15)

1, ... n.

Thus for a system with an n-pulse oscillation equations (12) and (13) give 2n nonlinear algebraic equations which can be solved for the 2n unknowns tj and 6tj' for j ~ 1, .•• n with tl ~ 0, and w. The slope condition at the switching instants can be checked by substituting the solutions in equations (14) and (15). Oscillations In A system With A Mode Dependent Transfer Function Single pulse oscillation. Let G(s) of Fig. 3 have a time constant and time delay which depend upon the relay position, that is

)

~

)

<

k~l(2) x

and

LRe[A (8.-8.+68.-69.,w)]-Re[A (8.-8.-68.,w)]

(2)

I c.(t)

with c. (t)

(l4)

>-(n/4w) lim sG(s) s+oo

exp(-sT ) O l+sT for Iyl

(7) G(s)

Using (3) at the jth switching instant we obtain

1

0

(16)

exp (-ST f) l+sT

for y

~

0

f

c.(t.) ~ I (2h/kn){u sin[kw(t.-t.+6t.)] k ~ J k~l(2) ~ J ~

The output c(t) of this G(s) for an input of the type given in Fig. 1 is shown in Fig. 4 and is given by

-uksin[kw(t.-t.)] - vkcos[kw(t.-t.+6t.)] ~ J ~ ) ~

c(t) ~ 1 -

+vkCOS[kw(ti-t )] j

(l-a)exp{-(t-T )/T } o 0

(17a)

1749

Multi-Pulse Oscillations in Relay Systems

c(t)

of switching and are given by

b exp{ (-t+lit+t f) /Tf } (17b) for lit + T

f

< t

< rr/w + T

} o + +T )/T] exp[-(71/WT )-(lit-T 1 f o f

where

1 - exp(-lit/T ) 1 1 o ] -+ T T [1 + exp(-71/wT - lit/'r) 0 f

C on

0

1 - exp{-[lit-To+Tf]/T b

a = -b exp{[-71/w + (lit-To+Tf)]/T and liT = (l/To) -

f

}

(l/Tf)

for switching on, and (19) (20)

Expressing c(t) in a Fourier series, we obtain c (t) =

L {A cos kwt + Bk sin kwt} k=1(2) k

(28)

x exp[ (-71/W + lit)/T ] f

(18)

(21)

1

1

1 - exp (-6t/T ) O - UAt/T)]

Coff = -T o - T [1 + exp ( -rr/ wT

(29)

f

for switching off. Further Con and Coff are equal to zero if the discontinuities in c(t) do not occur at the relay switching instants; a condition governed by the values of TO and Tf' b

+ t

+

Fig. 4. kwT 2 ~ 2 {(1-a)f2(To,To,0)-(1-b)f2(To,Tf'lit)} l+k w T o (23)

kwT sin kw(6t+T) - cos kw(6t+T) , (24)

fl (T,T,6t) and f

2

(T,T,6t)

kwT cos kw(6t+T) + sin kw(6t+ T) (25)

Further e(t) can also be expressed as a Fourier series, namely

L

e(t) =

kw{ -~ sin kwt + Bk cos kwt}

k=1(2) It should be noted that since lim sG(s) ~ 0, e(t) has discontinuities at th~+;elay switchon and switch-off instants which must be taken into consideration when applying the conditions for oscillation. Equations (6) and (7), with n = 1, are again the conditions for oscillation. For the given transfer function they can be expressed, in a. form similar to equations (12)-(15), as

{~ (O,NT ,w) - ~ (W6t,wT,W)} must have g

0

Multi-pulse oscillation. Existence of multipulse oscillations, if any, in relay systems, with mode-dependent transfer functions, can be established using a combination of the theories presented in the previous two sections. Again assuming a relay output to have the waveform shown in Fig. 2, the output of the mode dependent plant can be written as n

c(t) =

L

The conditions for oscillation, which are now easily derived, are n

L{~ [w(t.-t.),WT ,w]-~ [w (t.-t.+6t.),WT,W]} i=l g 1 Jog 1 J 1

must have Real Part < (rr/4w)C

(31) onj Imag. Part = -{ (o+6)rr/2h}

for j = 1,2 •.• n

g

{~ (O,wTf,w) - ~ (-w6t,WT,W)} must have g g Real Part < (rr/4w)C off Imag. Part = {(o-6)rr/2h}

L

{A .cos kwt + B .sin kwt} i=l k=1(2) k1 k1 (30)

where Aki and Bki are obtained by substituting 6 t ~ 6ti' T ~ ti + T, a ~ ai and b ~ bi in equations (22)-(25). ai and bi correspond to the values of a and b for pulse widths of 6ti' Similarly, a series expression can be derived for c (t) .

Real Part < (rr/4w)C

on Imag. Part = -{ (o+6)rr/2h}

Output of mode dependent plant with the input of Fig. 1

n

(26)

and

L{A

i=l g

[w(t . -t . +6t.-6t.),wT ,w] f 1 J 1 )

- ~ [w(t.-t.-6t.),WT,w)} g 1 J J (27)

Expressions for ~a(e,WT,W) are derived in the Appendix and are different from Ag(e,W) in the sense that they only satisfy the periodicity conditions of the 'A' locus when To = Tf and TO = Tf· The coefficients Con and Coff are due to the discontinuities in e(t) at the time

must have Real Part «rr/4w)C

offj Imag. Part = {(o-6)rr/2h}

(32)

for j = 1,2, .•• n. Conj and Coffj correspond to values of Con and Coff given by equations (28) and (29) with 6t = litj and the earliep discussion regarding the effect of delays on the discontinuities

U. M. Rao and D. P. Atherton

1750

still applies. Thus, equations (31) and (32) give 2n equations, and can be solved for the 2n unknowns, namely ~ti' ti for i = 1,2 •.• n and W with tl = o. The 2n inequality conditions, resulting from the real parts of the above expressions, may be used to check the satisfaction of the slope conditions at the switching instants. Multi-Pulse Oscillation In A General System Here we consider a system whose plant contains both fixed and mode dependent transfer functions, with the latter of the form considered in the previous section. Since the 'A' loci, Ag (8, w) and Ag (8,WT,W) satisfy the additivity property, conditions for oscillation can easily be written for any plant whose transfer function is a sum of fixed and mode dependent transfer functions.

of the oscillations using the describing function it was found that for certain values of the system parameters a multi-pulse but not a single pulse oscillation could exist. It was therefore of some interest to investigate if such a situation could exist in a system with a fixed rather than a mode dependent pseudo-rate controller time constant, since as far as the authors are aware, no limit cyc les in a single variable autonomous relay system other than those of the one pulse mode have been observed. With the system parameters as given in Table 1 for a range of sensor time delays, T f = To = 2.084 sec, wl = 1.9 rad/sec and S = 0.006, the results liste d in Tabl e 2 were found for the simulated system. TABLE 1

Typical Parameter Values for Hermes

APPLICATION OF THE THEORY In this section two applications of the theory presented above for mUlti-pulse oscillations are given. The first problem considered is the satellite attitude control loop in which this type of oscillation was first observed and originally investigated using the describing function method (Miller, 1976; Rao, 1976). The second example is concerned with these oscillations in a forced nonlinear system. Satellite Attitude Control System The system under consideration is shown in Fig. 5 and i t in corporates a pseudo-rate controller with a mode dependent time constant, a pair of thrusters, flexible spacecraft dynamics and an attitude sensor. The approximate spacecraft transfer function includes the rigid vehicle dynamics and a single highly under-damped fundamental flexible mode. Typical values for parameters of the Hermes satellite are given in Table 1. Since the natural frequency of the rigid body dynamics is so low relative to that of the flexible mode, investigation of the possibility of the high frequency oscillations caused by the flexible appendages may be done neglecting the rigid body dynamics (Miller, 1975). In simulations of this system undertak en in conjunction with the approximate evaluation

Value

Parameter deadband

(6+~)

hysteresis

(21\)

0.1632 deg 0.0223 deg

lag network gain (K fD)

2.964 deg

on-time constant (T ) o off-time constant (Tf)

2.084 sec 39.6 sec

thrust level (F)

1.0764 N

moment arm (L)

0.631 m

space c raft inertia (I) (flexib le + rigid)

1115 N-m sec

flexibl e mode gain (k ) l fl ex ibl e mode damping (s)

7 _7 0.003 - 0.008

flexible mode natural frequency (w ) l conversion factor (Q)

1.9 -

loop time delay (T)

0.0 - 1.25 sec

The exact analysis using a digital computer program based on the above theory gave the r es ults listed in Table 3. The frequencies of oscillation, ti and ~ti's given in this table were in close agreement with those obtained by analog simulation of the system.

I

,--- --------l

!f-:r\_j..--,::r'\....... 0 .. ,--.\

-

f 1

e

Mol

1-

, I

xit)

I

6

-------1 k~ L- _ _ _ _ _

Fig. 5.

2.3 rad/sec

180/n deg/rad

1-------- 1

Gl ls/ =. (T i T ) s+l (j' f

2

Pseudo-rate attitude control of a flexible spacecraft

1751

Multi-Pulse Oscillations in Relay Systems

TABLE 2

Dependence of Oscillation Mode On Sensor Time Delay

Mode

Remarks

1 Pulse

Does not exist for any time delay

2 Pulse

Exists for 0 .52 < Td < 0.66

3 Pulse

Exists for 0.475 < Td < 0.575

4 Pulse

Exists for 0 .475 < Td < 0.555

It is interesting to note from this table that the system cannot support an n = 3 mode although modes which are of lower and higher order, n = 1,2 and n = 4,5 ... , exist. Simulation waveforms are shown for the cases of n = 2 and 4 in Fig. 6. Forced Oscillations

Next, the system with mode dependent pseudorate controller time constant was analysed. C was taken to be 0 . 003 and the sensor time d e lay 0 . 51 secs . The theoretical results, which were veri fied by analog simulation, are given in Table 4. TABLE 3

The conditions for an mth order subharmonic with n pulses in the relay output, when the input r(t) = R sin wft and G(s) is of the

Multi-Pulse Osci llation s In Syste m o f Fig. 5 With Fixed Pseudo-Rate Controller

Mode of Oscillation

Time Delay (sec)

2 Pulse

0.6

3 Pulse

0. 5

4 Pulse

0.5

1.9272

Freq. of Osci llation (rad/sec)

1.9069

2 Pulse

1.9155

3 Pulse

tit 3

t4

6t

sec

msec

sec

msec

t2

6t

ms ec

sec

msec

1.9119

17.89

0.195

16.52

1. 9202

18.75

0.1 2 7

17.86

0 .34 6

16.20

1 9 . 82

0 .1 00

19.15

0 .217

18 . 21

Freq. of Oscillation (rad/sec)

1 Pulse

t3

tit 1

6t

2

0.395

4

1 6 .60

Multi-Pulse Os cillations In System of Fig. 5

TABLE 4 Mode of Osci llation

When a system of the form shown in Fig. 3 is forced with a sinusoidal signal it may support subharmonic oscillations, and the relay output will have a multi-pulse waveform (Atherton, 1975a). The theory developed in the earlie r sections can be extended to solve for this complex subharmonic oscillation.

l

msec

t2

6t

sec

msec

0.125

17.91

2

t3

!.It 3

t4

6t

sec

msec

sec

ms ec

4

t5

!.Its

sec

msec

0.365

18.18

16.68 18.90

does not exist

4 Pulse

1. 9295

21. 20

0 . 085

20 .4 2

0 .178

19.54

0.289

18.35

5 Pulse

1.9358

22 . 00

0.079

21. 33

0.162

20.51

0 . 255

19.48

_. _.~_:_!_~.~_!_.~_~,},. ..-,. _~_·~~;.__~:_-_·~_·.-,-__r1___J. ,,~ :

1

t-

,«, :.tJ'_"___._L_.•

! \: .....

!

.... j.

. ' -.-"

- - '- ~-- -

. -;--.- ;- .. : . -.

I

'· 1

-1 '---'--'_L....-'-'..

:J[t .'J itff~

0.16

e(t) 0

-0.16

Fig. 6.

: - : -~ . -

yet) O~IUU'U~--~-. --,-o,~,-,~lnr,r---.-..~ _ .-,_. ,_-_-L -_J'iJ_'tJ'~~~.

Relay output and input waveforms for n

=

2 and n

4 modes

1752

U. M. Rao and D. P. Atherton

form given in equation (16), can be written as

observed in simulations but, to the authors knowledge, have not previously been predicted theoretically. The theory is developed both for a fixed plant transfer function and also for a case where the transfer function has a relay-mode dependent time constant and time delay. The nonlinear algebraic equations which must have a solution if the system is to sustain an oscillation, are d eri ved in t e rms of 'A' loci, a general form of summed frequency response loci. The analysis is applicable to a large class of systems where the plan t can be represent ed as a summation 'of different types of transfer function s. The method of analysis is also applicable for multi-pulse oscillations in f orced systems. The computer program for solving the nonlinear algebraic equations containing the 'A' loci is written in Fortran and structured so that it can be used for various f o rms of plant transfer functions.

n

r{i [w(t.-t.} ,WT ,w]-i [w(t . -t.+6t.} ,WT,W]} i=l g ~) 0 g ~) ~ + (n/2 h}R'[mwt . +
(33)

)

must have

Real Part < (n /4w) C . on) Imag. Part = -{(o+6 }n/2h)

and n

r{i [w(t.-t.+6t.-6t.} , WTf, w]-i [w (t.-t.- 6t.} , i=l g ~) ~) g ~) ) wT , w]+(n/2h}R ' [mw(t.+6 t.}+

must have

for j

)

(34)

Real Part < (n/4w ) C offj Imag. Part = {( o- 6 )n/2h)

1,2 ... n

where w = wf /m is the subharmonic frequency and R' [mw(t.+6t.)+

)

A satellite attitud e control loop, which includes a relay-mode dependent transfer function, was inve s tigat ed using the theory. In an analog simulati on of the system the p e ri o dic modes obtained agreed closely with the theory. The method was a ls o used to evaluate multi-pulse forced os c illations in a r e lay system and again the r e sults were in good agreement with the analog si mulation.

(1/w)f[(t.+6 t.)-] )

)

+ jr[(t.+6t.}-] )

)

(35)

R{m cos [mw(t.+6t.}+
J

+ j sin[mw(t.+6t.)+
J

yet)

As an example the system of Fig . 3 , wi th G(s} = k/(s+1)3 0 = 0.25, 6 = 0 and h = 1.0 was cons ide r ed . In a simulation tuning the frequency of the forcing functi on r(t} = R s in wft to near multiples of 13 rad/sec., the 180 0 phaseshift frequency for G(j w), resulted for certain input amplitudes in different subharmonic multi-pulse oscillations. The results obtained fr om the analog simulation are compared with the theoretical ones in Tabl e 5. Th e theoretical results are obtained from equations (33)-(35) with ig replaced by the fixed configuration Ag loci for the given plant. Figure 7 shows a simulation waveform for n = 3 .

1

c -1

.

x(t)1

o. 5

..'

· ···· ··r · ·· : · · · · - ··- ~-

-r--- -! ... ~- -- -i-- .; ,j ' --r -- .:- - .; .. --~--~ -+ -:. i 1. .. 1.. I __ •.

I

'I

T TT"F'Tl---'T 'TTTr--r'~T "

::S~FrI=1--~T~~~=l~EFj~:F;::'-Ft~+~F:I~:T.'.· . -~- -

~-

----: -

r

o t-}~~ -\-~-+ -0.5

CONCLUSIONS The paper describes an extension of Tsypkin ' s method to evaluate multi-puls e oscillations in r e lay systems. These oscil lations, termed ' complex oscillations', have earlier been TABLE 5

K

R

4.0

0.5

4.0

3.9

0.5

0.39

wf rad/sec 5

8.33

11.66

Fig. 7.

Multi-pulse output and system forcing function for 5th subharmonic oscillation

Theoretical and Simulation Results For Forc ed Oscillations

m

3 (n= 2 ) 5 (n=3) 7 (n=4)

68

1

8

2

68

2

8

3

68

3

8

4

68

4

0 .998

1.941

1.045

1. 918

0 .959

0.548

1.139

0.698

0.556

1.119

0.712

2.473

0.468

0.355

0.809

0.505

1.695

0.477

2.681

0.289

Theory

0.362

0.805

0.517

1.692

0.483

2.678

0.295

Simuln.

0.933

Theory Simuln. 2 .4 66

0.496

Theory Simuln.

Multi-Pulse Oscillations in Relay Systems

ACKNOWLEDGEMENTS The authors wish to acknowledge the co-operation of Mr. R.A. Miller and Dr. F.R. Vigneron of the Communications Research Centre, Ottawa, in drawing their attention to the satellite attitude control problem and providing them with technical data and some financial support for the project. In addition, the authors wish to acknowledge the partial support of this work by the National Research Council of Canada under Grant No. A1646 and the University of New Brunswick. REFERENCES Atherton, D.P. (1966). Conditions for periodicity in control systems containing several relays. Proc. IFAC, Paper 28E, London. Atherton, D.P. (1975a). Nonlinear Control Engineering. Van Nostrand Reinhold Co., London Atherton, D.P., and S.K. Choudhury. (1975b). An exact solution for oscillations in relay systems with complex poles. Int. J. of Control, 21, 1015. Bergen, A.R. (1962):- A note on Tsypkin's locus. I.R.E. Trans. Automat. Control, AC-7, 78. Bohn,~. (1961). Stability margins and steady state oscillations in on-off feedback systems. I.R.E. Trans. Circuit Theory, CT-8, 127. Chung, J.K-C., and D.P. Atherton. (1966). The determination of periodic modes in relay systems using the state space approach. Int. J. Control, 4, 105. Gelb, A., and W.E. Vander Velde. -(1968). Multiple-Input Describing Functions and Nonlinear System Design. McGraw-Hill, New York. Hamel, B. (1949). Contribution a l'etude mathematique des systemes de reglage par tout-ou-rien. C.E.M.V. Service Technique Aeronautique, l2. Judd, F.F., and P.M. Chirlian. (1974). Graphical analysis and design of limit cycles in autonomous relay control systems. Int. J. Control, 20, 321. Kochenburger, R.J. (1950). A frequency response method for analysing and synthesizing contactor servomechanisms. Trans. A.I.E.E., 69, 270. Miller, R.A., and F.R. Vigneron. (1975). The effect of structural flexibility on the stability of pitch for the communications technology satellite. Canadian Conference on Automatic Control, University of British Columbia, Vancouver, B.C., Canada. Miller, R.A., and F.R. Vigneron. (1976). Attitude stability of flexible spacecraft which use dual time constant feedback lag network pseudo-rate control. AlAA/CASI 6th Communications Satellite Systems Conference, Montreal, 266. Rao, U.M., and D.P. Atherton. (1976). Analysis and design of dual time constant pseudo-rate control loops. CRC Report University of New Brunswick, Fredericton N.B., Canada.

1753

Tsypkin, J.A. (1958). Theorie der Relais Systeme der Automatischen Regelung. R. Oldenbourg-Verlag, Munich. APPENDIX In this section we derive expressions for Ag (8,WT,W) for the single and multi-pulse oscillations in a relay system with mode dependent transfer functions, with the former case being derived as a special case of the latter. The Fourier series expressions for c(t) and c(t) have been given in the main text. Now at the switch-on and switch-off instants, we have n

L

c(t.)

L {Ak.COS

i~l k~l(2)

J

and

n

J

I

L

c(t.+6t.) J

kwt. + Bkisin kwt.} J

1

J

(36)

{Ak.COS kw(t.+6t.) 1 J J

k~l(2)

i~l

+ B .sin kw(t.+6t.)} J

kl

for j

J

1,2 •.. n.

~

Substituting for Aki and Bki' from equations (22)-(25), and expressing the above equations in a form similar to those of equation (8), that is, as n

I

(2h/n)

-c (t.) J

Im{A [(8,-8.+M.),wT,w] 9

i~l

1

1

-A [8.-8,) ,WT ,w]} 9

1

0

(37)

n

I

(2h/n)

Im{A (~. ,WT,W)-A ( •. ,WT ,w)} g

i~l

and

9

1

1

0

n

(2h/n)

-c(t.+6t.) J J

I i~l

Im{A [(8 . -8,+68.-68,), 9 1 1 J

WTf,w] - A [(8,-8.-68.),WT,W]) 9 1 J J n

(2h/n)

L Im{A9 (.~'WTf,w)-A g (~~,WT,W))

i~l

1

1

(38) where .i'.i'~i'~i are given by

. ·i

e.1

e,;

~

e1.

8. + M.

1

~i

e.1

e,

, ;~~ 1

8.

e, -

J

M

1

J

J

1

+ M,

1

J

M,

J

J

(39) one can show that Im A ( •. ,WT

9

1

-

Im A

9

0

,W)~{-S -10(.' ,A )+f,2S1 1(.' ,A ) , 10 0 0 , 10 0

o

,

10

(~,WT,W) ~ 1

+ a,[A 1

j}

A Co 1 (. . ,A

0

0

0

0

{-S 1

-,

S11(.' ,

O(~'f,A ) 1 0

10

,A )-CO 1(.' ,A)} 0 , 10 0

- Af {A f S 1 ,1 (.io,A f )-C ,1(.io,A f )}] O + (l-b.)[A

0

(40)

(41)

S11(~'f,A )-C 1(~'f,A )}] , 1 0 O,10

100

+ b i [ Af O f S 1 ,1

(~if,Af)-CO ,1 (~if' Af)):}

U. M. Rao and D. P . Atherto n

1754

where ~iO = ~i + WTo and ~if = ~i + WT f lm

is obtaine d by letting ~;-+~~ • • (40) n in equatio

,w) Ag (~~,WT f ~

and To -+ T

f

A

g

(~~ , WT, W) is obtaine d by letting

~ i -+ ~

~

-+

i,

TO

-+

T f and b i +>- a i in

Re

Re

Ag ( ~.

~

{co, l(~ '

, WT 0 , W )

A (~., WT, W ) g~

~o

=

{co l(~'f, A ,

~

A locus

, A0 )- A0 SI ,

0

l( ~'

~o

(43)

l( ~ 'f, A 0 ) } )+Sl 1 ~' f, A 0 {A C l(1 + b . [ A0 . , . 2 , 0 1

A f{A fC2,1( ~if' A f)+Sl,1(~if' A f) } ~ Ag

~

Ag ( ~ ~, WT , W )

(45)

j,k

- S j,k+l

2

A S.]+ 2 , k+l (46)

. k( 8 , A) C. k(- 8 , A) = C J, ], 2 = A C - C C j+ 2 , k+l j,k+l j,k and the s p e cific s e ri e s u sed abo v e h a ve t he c l ose d forms S_l, O = rr/4 Sl,l CO ,l

~

n are obtaine d b y _us ing the s ubs titut i o n s give A lm for r e arli e g The e xpressi ons f o r the real and imagina r y l oc i use the s umme d s ine and p arts of the cos i ne s e ri es . (Athert on, 1975a) d e fined by

Ag

S

(4 2 ) ) A l(~'f, )",,\sl 0 ~ 0,

- Af {A f C 2 , 1 ( ~ io' Af) +Sl,l (~io' Af) } ]

Again Re

j m cos m8 2 m=1(2) (l+m A2 )k

L

(44)

o k( 8 , A) s . k(-8, A) = -S J, ],

, A0 :}

' , A )+Sl , l( ~ 1' 0 , A0 ) } - a 1. lA 0 {A 0 C2 , l( ~ ~o 0

(~~ , WTf, W) a nd Re

L

Thes e s e ries satisfy the symmet ry and r e cursive r e lations hips,

equatio n (41) Simila rly for the r e al parts of the on e obtain s

j m sin m8 2 2 k m=l (2) (l+m A )

whe re 0 < 8 < rr and A = wT.

~

i'~i

],

C . k (8 , A) J,

and lm

S. k (8,A)

C2 ,1

2 {rr c o s h[( rr - 28 ) /2A ] }/4 A cos h[rr /2A ] {rr s inh[(rr - 28 ) /2A ] }/4 A cos h[ rr/2A ] (4 7 ) 3 {-rr sinh[( rr - 28 ) /2A ] }/4 A c osh[ rr/2A ]

Al th o ugh the abo ve e quatio n s l ook formid abl e f o r u se the y are easi l y impl e me nt e d o n the di g ital comput er. Fo r th e s ingl e pulse mode n i s s e t e qual t o one.