Forced Oscillations Conditions in Relay FeedbackControl Systems

Jun FuSchool of Mechanical and

Electrical EngineeringWuhan Textile University

Wuhan, Hubei 430073Email: fujun2008@sina.cn

Wei YuanyuanSchool of Computer Engineering

Wuhan Textile UniversityWuhan, Hubei 430073

Email: weiyy@hotmail.com

Ai Poh LohDept. of Electrical andComputer Engineering

National University of SingaporeEmail: lelohap@nus.edu.sg

Abstract—In the relay feedback system shown in Figure 1,when the magnitude of the forcing signal is larger enough, thephenomenon of forced oscillation occurs, that is, the system syn-chronizes itself automatically with the frequency of the externalsignal. However, the minimal magnitude of the forcing signal forforced oscillation occur can not be obtained easily. An aim of ourresearch work was to determine the minimal magnitude of theexternal forcing signal in order to enforce forced oscillation inthe loop. In this paper, we examined the necessary and sufficientconditions for forced oscillation and led to a solution. Simulationshowed that the solution was the desired minimal magnitude ofthe external forcing signal for forced oscillation in a special limitrange. Although the solution was not a general solution, it didgive the accurate result in some degree and most real systemswere within the special limit range then our result can be used.

I. INTRODUCTION

Auto-tuning of proportional+integral+derivative (PID) con-trollers for single loops systems has been found for more thana decade since it was first proposed by Astrom in [1]. To date,the extension of this technique has not been as successfulfor multi-loop systems as it has been for single loops. Anextensive survey on auto-tuning methods for multivariablesystems can be found in [2].

One auto-tuning method is using relay feedback, it iseasier and more effective, but one of its pre-requisites is thatthe system must be able to sustain steady state oscillationswhen placed under relay feedback control. In single loops,this prerequisite can be easily satisfied, where in the steadystate oscillations exhibit a simple periodic pattern involvingonly one fundamental frequency. In contrast, for multi-loopinteractive systems, such an oscillation pattern is not easilyobtainable when all loops are placed under relay feedbackcontrol. In general, the oscillations exhibit complex switchingsin each fundamental period in some of the loops. This type ofbehavior has been loosely classified as Mode 3 behavior in [3].On the contrary, the case when all loops sustain oscillationswith one and the same fundamental frequency is referred toas Mode 1 behavior.

Much of the research work on auto-tuning of multi-loopcontrollers for multivariable systems have relied on Mode 1behavior to tune their PID controllers. Their basic assumptionis that Mode 1 can be obtained by manipulating the relay

amplitudes in each loop, [4], [5], [6]. This is usually done ona trial and error basis to determine the relay amplitudes. OnceMode 1 behavior is established, the design of multi-loop PIDcontrollers can proceed using the amplitudes and frequency ofthe self-oscillations.

In our approach in [7], a procedure for determining the relayamplitudes to obtain Mode 1 behavior in a 2-input and 2-output system was proposed. This approach was based on aforced oscillation concept [8] where one loop is viewed as thedriving loop and the other as the driven loop. For the sakeof clarify, the phenomenon of forced oscillation is definedas when an external periodic signal imposes its frequencyon a self-oscillation system, damping out the self oscillationsoccurring in it. When this happens, the system synchronizesitself automatically with the frequency of the external signal,undergoing forced oscillations whose frequency is now equalto that of the external signal.

In the context of a 2-input 2-output system under relayfeedback control, the interacting signal from the driving loopcan be viewed as the external signal trying to impose itselfon the driven loop. If the interacting signal has a sufficientlylarge magnitude such that the forced oscillation conditionsare satisfied, then the driven loop undergoes forced oscillationand the overall closed loop system will oscillate with Mode 1behavior. Furthermore, if the magnitude of the forcing signalfor which forced oscillation can be ensured is known, thenrelay amplitudes can be chosen at the outset to ensure Mode1 behavior, thus eliminating the need for any trial and error.

It should be noted that much of the work in forced oscilla-tions can be found in [8] and [9]. The phenomenon of forcedoscillation was described detailed and the necessary conditionsneeded for forced oscillation were given.

However, as the conditions presented are only necessaryand hence cannot provide an accurate minimum, more workshould be done to obtain the accurate minimal magnitude ofthe external forcing signal for forced oscillation.

In this paper, detailed analysis for the necessary conditionsare given based on Tsypkin and Atherton’s works. A analyticand a graphical method, which lead to a consistent result, wereused to explain the constrain led by the necessary conditions.Moreover, an extra condition is proposed so that the necessary

978-1-4244- 7618-3 /10/$26.00 ©2010 IEEE

u(t) c(t) y(t)

-

+x(t)g(s)d

Forcing signal

Fig. 1. Forced oscillation in SISO system.

u1

u2

r11

r21

r12

r22

y1

y2

-

-

++

++

x1

x2

G12

G22

G21

G11d1

d2

Fig. 2. 2-by-2 relay feedback control system.

and sufficient conditions for forced oscillation are obtained.Based on these, the minimal magnitude of the external forcingsignal to ensure forced oscillation can be obtained.

Based on this result, the relay amplitude in the driven loopcan be set accordingly. Then the relay magnitudes in 2-by-2relay feedback control system should be preset and Mode 1oscillation is ensured. Once Mode 1 behavior is established,multi-loop PID controllers can be designed by Z-N method.

The organization of this paper goes as follows. In section A,the analysis of necessary and sufficient conditions for forcedoscillation are given. In section B, the analysis extends to2-by-2 system to ensure mode 1 oscillation happen, then PIcontrollers are designed.

A. Forced Oscillation in Single loop system

In the relay feedback system shown in Figure 1, the phe-nomenon of forced oscillation occurs when external periodicsignal imposes its frequency on a self-oscillation system,damping out the self oscillations occurring in it. When thishappens, the system synchronizes itself automatically with thefrequency of the external signal. In this case, the magnitudeof forcing signal should be larger enough to force SISOsystem oscillate in its frequency. The minimal magnitude ofthe forcing signal for forced oscillation happening is denotedby Rmin. Sufficient and necessary conditions will be examinedto obtain the Rmin.

The SISO relay feedback control system is shown in Figure1. It is assumed that G(s) can be modelled by first order plusdead time transfer function.

G(s) =Ke−Ls

1 + Ts

The external forcing signal, f(t) is assumed to be of the

formf(t) = R sin(ωf t+ θ)

where ωf is the frequency of the forcing signal and θ is thephase shift between u(t) and f(t) where u(t) is the outputfrom the relay.

The output of the linear part G(s), c(t), is a periodicfunction, whose fundamental frequency is ωf . Tf is thecorresponding fundamental period.

c(t) =4

π

infty∑kodd

1

kIm[G(jkωf )e

jkωf t] (1)

From [8], the necessary conditions for forced oscillation are:

4d

π

∞∑kodd

1

kIm[G(jkwf )] +R sin θ = 0 (2)

4d

π

∞∑kodd

Re[G(jkwf )] +R cos θ < 0 (3)

From equations (2) and (3), we get:

R sin θ = −A, A =4d

π

∞∑kodd

1

kIm[G(jkwf )] (4)

R cos θ < −B, B =4d

π

∞∑kodd

Re[G(jkwf )] (5)

According to equation (1), C(t) is a periodic function,whose frequency is ωf , from this point onwards, we assumet = 0 to be the time corresponding to a relay switch from -1to +1. It follows that A = C(0), and B = C(0)

ωf.

After some analysis, we conclude that :

(1) θ locates in the 2nd or 3rd quadrant such that cos θ < 0.(2) If B > 0, Rmin ≥

√A2 +B2.

(3) If B < 0, Rmin = |A|.The necessary conditions (2) and (3) are derived by observ-

ing the geometric requirements associated with the steady stateswitching under forced oscillation conditions. The geometricrequirements are as follows :

x(t−p ) = 0 (6)

dx(t−p )

dt(−1)p+1 < 0 ∀ tp =

πp

ωfp = 1, 2, . . . (7)

While these conditions ensure that switching occurs at everyhalf period, it does not ensure that no further switching takesplace in between each half period.

In order to prevent the additional switchings in betweeneach half period, additional conditions need to be imposed asfollows :

4d

π

∞∑kodd

1

kIm[G(jkwf )e

jkwf t] +R sin(wf t+ θ) < 0, (8)

t ∈ [0,π

wf].

In order for forced oscillation to occur in the loop withfrequency ωf in the configuration in Figure 1, the magnitudeof f(t) must be greater than some minimum Rmin. Rmin

should therefore be the least R which satisfies conditions (2),(3) and (8). Rmin is expected to be related to the linear systemG(s) in some way.

In the following part, the extra condition will be analyzedto see if the conclusion gotten from 1st and 2nd condition isstill correct or only valid within a range.

Recalling that the 3rd condition for forced oscillation isgiven by

C(t) +R sin(ωf t+ θ) < 0, t ∈ (0,π

ωf) (9)

Under forced oscillation conditions, C(t) is the response ofthe linear system to a square wave signal of frequency ωf .The numerical expression of C(t) can be derived from thefollowing analysis.

The differential equation for the process output, C(t), inthis time segment is

T C + C = Ku(t− L) (10)

Case 1: L <Tf

2 = πωf

In this case, B < 0, Rmin = |A|.Let R = |A|. Equation (9) can be re-written as

C(t)−A cos(ωf t) < 0, t ∈ (0,Tf

2.

Denote the left part of the above inequality as F (t), aftersome mathematical analysis, F is proved to be a function ofα and L/T . Denote F as F (α,L/T, ωft), where

α =Tf

Ts, Ts = 2T ln(2e

LT − 1).

Ts is the self-oscillation period of the plant.[10].For a special plant, L/T is known, F is a function of α and

t, then the 3rd equation becomes maxt∈(0,

Tf2

)[F (α, t)] < 0.

The maximum value of F (α, t) also relates with α.As A can be positive or negative, while α is larger or smaller

than 1, two cases will be analyzed separately.Case 1.a: A < 0, in this case, Tf > TS, α > 1.After some mathematical analysis, we can conclude that

Rmin = |A| is sufficient and necessary in range α =Tf

Ts∈

(1, α0), where α0 is the largest α who satisfies all 3 equations.α0 will change with the different L/T , plot the figure in

figure 3.(The upper curve.)Case 1.b: A > 0, in this case, Tf < TS , α < 1.Similarly to case a, α0 can be known. Changed the L/T ,

plot corresponding α0 in Figure3.(The below curve.)From the above analysis, we know that Rmin = |A| is

sufficient and necessary for all α which locate in the areaenclosed by the upper and lower curves in Figure3.

For α > α0, Rmin > |A|, exam a special point, t = t0,which makes c(t0) = 0.

t0 =Tf

2− T ln

eTf2T + 1

eTs2T + 1

0 2 4 6 8 100

2

4

6

8

L/T

α0

αh

αl

A<0

A>0

Fig. 3. Forced oscillations with different frequencies of oscillations.

0 1 2 3 4 50

1

2

3

4

5

6 L/T = 0.1

|A|RexpR

min

0 1 2 3 4 50

5

10

15 L/T = 0.4

α

0 1 2 3 4 50

5

10

15

20 L/T = 0.7

α

0 1 2 3 4 50

5

10

15

20

L/T = 1

α

α

Fig. 4. Compare Rmin ,Rexp,|A|.

Consider equation (8).

y(t0) < 0⇒ Rexp =|A|

sin

(πTf2T

ln eTf2T +1

eTs2T +1

)From the above analysis, we find that for a special system,

Rexp is a function of α. Plot the real Rmin, Rexp and |A|where G(s) = 12.6

1+60se60∗L/T . (changed with α in figure 4 .

From Figure 4, we find that although only one point isexamined to obtain Rexp, Rexp is close to the real Rmin forlower α. Furthermore, as in real plants, L/T always smallerthan 0.1, and the forcing signal period will not 5 times largerthan the self-oscillation period, so |A| is sufficient to ensureforced oscillation happen when L <

Tf

2 .Case 2: Tf

2 < L < Tf

Similarly, we consider a special point, t1 = L− Tf

2 .y(t1) < 0⇒

Rexp =

√√√√[A cos(ωfL)+K+(A−K)e

Tf2T−

LT

sin(ωfL)

]2

+A2

From the above analysis, we find that Rexp is a function ofα. Plot the real Rminα, Rexpα and

√A2 +B2α in figure 5

where G(s) = 12.61+60se

60∗L/T .

As Ts

2T = ln(2eLT − 1) > L

T >Tf

2T , case 2 only happenswhen Ts > Tf . This case is more complexity. Simulations

0.2 0.3 0.4 0.5 0.60.5

1

1.5

0.3 0.4 0.5 0.6 0.72

3

4

5

6

7

0.2 0.4 0.6 0.84

5

6

7

8

9

10

0.4 0.5 0.6 0.7 0.8 0.98

10

12

14

16

18

20

22R

expA2B2R

min

L/T=0.2 L/T=1

L/T=2 L/T=5

α α

α α

Fig. 5. Compare Rmin,Rexp,|A|.

result shows Rexp is still not sufficient for forced oscillation.Further works need to be done for this case.

B. Forced oscillation in 2-by-2 system

For 2-by-2 system, (see Figure 2), we just consider the plantwhich has the moderate interaction and the self-oscillationperiods of loop 1 and loop 2, Ts1, Ts2, are not much further toeach other. (Otherwise the plant will exhibit mode 1 oscillationor mode 2 oscillation under unity relay feedback control, Z-Nmethod can be used directly.)

In order to simplify the analysis, we suppose loop 2 isalways slower than loop 1, that means Ts2 > Ts1. As theplant we considered has moderate interaction and Ts2, Ts1 donot far from each other, the ratio αTs2

Ts1will be less than 5.

For a 2-by-2 system, there are 2 possible kinds of mode1 oscillations with the fundamental frequency of the relayoutputs similar to the self-oscillation frequency of loop 1 andloop2 respectively. To get a stable and easy-measured mode1 oscillation, the slower loop, in this paper we suppose it isalways be loop 2, will be chosen as the forcing loop, whilethe faster loop, loop 1, is chosen as the forced loop.[12]Recalling the analysis for case 1 in section 1, if L/T < 0.1,α = Ts2

Ts1< 5, the magnitude of the forcing signal of loop 1

should be larger than absolute value of A1 of loop 1 where

A1 =4

π

∞∑kodd

1

kIm[G11(jkωf )], ωf =

2π

Ts2

.More attention should be paid on the forcing signal of forced

loop (loop 1), r12. Signal r12 is more like a triangular wavesthan a sinusoid wave, in this paper, fundamental waves ofthe fourier expression of r12 is used as the forcing signal, itsmagnitude, m12 = 4K12

π√

1+T 2ω2

f

, should be larger than d1|A1|,where d1 is the relay magnitude of loop 1.

Use the conclusion in [10], for a SISO relay feedbackcontrol system, (see figure 1)

Ts = 2T ln 2eL/T a = K(1− e−L/T )

where Ts is the self-oscillation frequency, a is the maximumvalue of output y(t).

|A1| and m12 can be known from experiment.Input the square wave whose period is Ts2 to part G11 and

G12 separately, the output is c11(t) and c12(t) respectively,maximum value are a11 a12. A1 is equal to the value ofc11(t) when square wave changes from -1 to 1. m12 =

4K12

π√

1+[ πTf/2T

]2where Tf/2T can be computed from equation

a12 = K12e

Tf2T −1

eTf2T +1

. If K12 is known, Tf

2T = ln K12+a12

K12−a12. Then

m12 is known.Fix the relay magnitude of loop 2, d2 = 1, choose the relay

magnitude of loop 1, d1 = m12

|A1|, do relay feedback control

simulatively, mode 1 oscillation should exhibit as expected.Then Z-N method can be used to design PI controllers.

C. Simulation results

G(s) =

[12.6e−2s

1+60s4e−6s

1+48s5e−4s

1+38s12e−4s

1+35s

]

II. CONCLUSION

For forced oscillation happened in SISO, although we knowthe necessary and sufficient conditions, it is hard to get ananalyzed results due to the complexion of the 3rd conditionand the long range of time. In this paper, for the 3rd inequality,only one special point is examined. Although the conclusionis only necessary but not sufficient for all cases, it is moreaccurate than only consider 1st and 2nd conditions and insome range which include most real system, it is sufficientand necessary.

For applying the conclusion on 2-by-2 system, a practicalmethod for PI controllers designing is given and simulationsresults are acceptable while only few parameters are known.

REFERENCES

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[2] Smail, Menani and Heikki, Koivo A comparative study of recent relay-autotuning methods for multivariable systems. International Journal ofSystems Science, Vol 32, No.4, 2001 pp443-466.

[3] Loh, A. P. and Vasnani, U. V. Necessary Conditions for Limit Cyclesin Multi-loop Relay Systems. IEE Processings D: Control Theory andApplications, Vol 144, 1994, pp 163-168.

[4] Nugent, S. T. and Kavanagh, R. J. Self and Forced Oscillations inMultivariable Relay Control Systems. Automatica, 5, 1969, pp 519-527.

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[7] Loh, A. P. and Fu, J and Tan, W. W. Controller Design for TITO Systemswith Mode 3 Oscillations, Proceedings of Control 2000, September 2000,Cambridge, England.

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