Limit Cycle Prediction in Free Structured Nonlinear Systems

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LIMIT CYCLE PREDICTION IN FREE STRUCTURED NONLINEAR SYSTEMS

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P. McNamara and D. P. Atherton

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Abstract: '!he paper outlines scrne interesting methods for the accurate prediction of limit cycles in a large range of single-variable and multivariable nonlinear free structured systems. Free structured systems enable a greater range of systems to be analysed than the IIDre usual situation with one nonlinear block and one linear block. The methods also enable all limit cycle predictions made to be accurately verified and refined until they are effectively exact. A CAD package is briefly discussed which implements the limit cycle prediction methods. Keywords:

Limit cycles.

Nonlinear systems.

1NI'RODUCT1ON

C.A.D.

Stability.

virtually exact limit cycle predictions to be made for a wide range of systems. '!he techniques described have been implemented in a userfriendly CAD package which is normally used on a PRIME 550 or VAX 11/ 780 computer, and is also being IrOdified to run on a single user IoIOrkstation. '!he program is written in FORTRAN 77 and makes extensive use of the NAG mathematical library and GINO graphics.

To determine the stability of a nonlinear system there are two main theoretical a[:proaches available: absolute stability methods and limit cycle prediction methods, although in principle one has to check in the latter case that no other forms of instability exist. Generally absolute stability methods determine conditions on the sector or slope bounds of any nonlinearities prese nt for which the system is asymptotically stable in the large. '!hese conditions are mathematically rigorous, tot unfortunately are usually far too conservative to be of significant practical use and are only a[:plicable to a limited category of system configurations. This conservatism is often particularly apparent for systems where the types of nonlinearities present are known reasonably accurately. Since instability in nonlinear systems frequently produces limit cycle behaviour, often a IIDre practical approach to control system design is to find the conditions necessary for a system to sustain a limit cycle. These conditions then accurately determine the stability limits of the system.

LIMIT CYCLE PREDICTION METHODS Consider the free structured system shown in Fig . 1 wi th b blocks, each wi th m inputs and m outputs. Since it is free structured, each element of each of these blocks may be linear or nonlinear, and thus we use a short form notation [Bl] to represent the matrix of nonlinear and linear elements in block one. If x is a vector of periodic signals, each with th€ same fundamental per iod, at the input to block one, then the vector of output signals of block one, .y!. is given by

.y!. = This paper presents scrne frequency domain methods for the accurate prediction of limit cycles in autonomous free structured nonlinear continuous and discrete systems. By free structured systems we mean a system such as that shown in Fig . 1, where the system is defined Of up to ten SlUare blocks, each with a maximum of five inputs and five ootputs. Each element of each block can be either linear, nonlinear, zero, unity, or a A wide range of sample and zero-order hold. systems can be rearranged into this form including: single-input single-output (S1SO) systems with a number of nonlinearities, S1SO systems with multiple feedback paths, multipleinput multiple-output (MIMJ) systems where the nonlinearities can be in any number of the system blocks, and nonlinear sampled data systems. '!he limit cycle prediction methods to be discussed include: a describing function a[:proach, a harmonic balance method, a method based on the Gershgorin eigenvalue bounding theorem, and an iterative method which enables the accurate verification of all limit cycle predictions. '!he iterative method can also often be used to converge to IIDre accurate, if not exact, solutions. '!he canbination of these methods enables

[Bl]

(I)

~

Therefore in order to find conditions for a limit cycle, x, to exist in the system we must solve the set-of nonlinear differential equations given by

-x

[Bb] ."

[B2] [Bl]

~.

(2)

Several computational techniques are used here to obtain solutions to this equation and these are explained in the following sections. Describing Function Method '!he describing function (DF) method, Atherton (1975, 1981} , is widely used for the analysis of nonlinear systems and it is well known that the validity of the a[:proach depends on the filtering properties of the particular system. '!he describing function a[:proach for a continuous system of the form shown in Fig.l, requires each nonlinearity in the system to be replaced Of its describing function. '!he implicit assumption in the approach is that, ...nen the system limit cycles, the input to each nonlinearity is almost

25

O. 1'. i-.Ic:\amara alld D. 1' ..-\!hcrtoll

~(j

sinusoidal.

'!bis means that we are assuming that yb-l are sinusoidal vectors in Fig.l. In this way we obtain a quasilinearised approximation to equation (2) and we then seek the conditions recessary for sinusoidal limit cycles at the input to each nonlinearity to be sustained. '!be resulting set of nonlinear equations can be solved ut a Newton Raphson method, NAG (1982). In simpler systems with a llDre specific structure the resulting equations can also be solved graphically, as for instance in Mees (1973), Gray and Nakhla (1981), and Atherton (1981). In general, however, for free structured systems a graphical solution, though desirable, is not possible because of the difficulty of separating the amplitude dependent and frequency dependent effects. Though fairly straightforward, the validity and therefore the usefulness of the DF method in general for free structured systems, is limited because of the restrictive assumption that the input to each nonlinearity should be sinusoidal.

.!J.Y!.r ••• ,

Harmonic Balance Method One way to improve the accuracy of the DF method, in any system, is to include the effects of the higher harmonics, neglected ut the OF approach, into the analysis. '!berefore we assume that the input signal to each nonlinearity is made up of a number of integrally related harmonics and then seek the condi tions necessary to sustain the set of harmonics in the system. This approach is conceptually straightforward but there are some very difficult problems to overcome before such a method can be implemented. The first of these problems is that it is very difficul t to express analytically the relationship between a given set of harmonics at the input to a nonlinearity and the resulting set of harmonics at the nonlinearity output. This problem is complex even in single variable systems with only one oonlinearity, but for free structured systems of the type shown in Fig.l, it becomes, analytically at least, virtually impossible. The computational method used here to overcome this problem is to make use of the fast Fourier transform (FFT), and is best illustrated by consider ing the system shown in Fig. 2, where n is a time invariant and frequency independent nonlinearity, and g is a linear element with known transfer function. To apply the harmonic balance method to this system one assumes that x in equation (2) can be represented ut its Fourier series so that x(t)

l:

akcos(kwt + k) and

-c(t)

With this approach nonlinearities are easily deal t wi th in the time domain and linear elements, with known transfer functions, in the frequency domain. This method has also been extended to the much llDre complex case of free structured systems of the sort shown in Fig.l. With the result that given a periodic signal vector x = [xI ••• Xrol T at the input to the first block 'Of the free structured system shown in Fig.3, then the per iodic vectors.1l., ••• , yb-l, and c = [Cl ••• Cml T , can be calculated exactly. In order to be able to predict limit cycles in the system shown in Fig.l we have to find the conditions necessary for the system to be able to sustain an oscillation in one or llDre (or all) of the loops. We then apply the harmonic balance method ut looking at the system of Fig. 3 with the loops opened at the input to block one, in the same way as was done for Fig. 2. To do this we assume the existence of a periodic input signal x,and determine the conditions necessary for c=-x to be satisfied. Assume first that the vectors and c are periodic and that each component xi(tT of X; and ci (t) of 52 can be expressed as the FourIer series

x

xi(t) =

k=o

bkcos (kwt + 8 k )

aikcos(kwt + ik);for i=l, •• , m

(5)

L q,kcos(kwt + 8ik );for i=l, •• , m

(6)

k=o

where the fundamental frequency w is the same for all m loops. If the limit cycle vector x can be accurately approximated as the sum of p harmonics in each loop then the following 2p equations (assuming for the rroment that there are no d.c terms) must hold in each of the m loops. For i aik

l:

L k=o

(3)

k=o e(t)

look-up table type procedures, the corresponding array of samples of the resulting nonlinearity output signal, y(t), is computed. '!be FIT is then applied to y(t) and the resulting set of harmonics of y are multiplied by the complex gain of the linear plant at the corresponding frequencies to form the spectrum of c. '!be spectrum of c can be transformed ut an inverse FIT back into the time domain if the waveform c(t) is required. By using the FIT in this way we are able to c0mpute the exact periodic signals y(t), and c( t) , resulting from a periodic signal x(t). The Newton Raphson algorithm then uses the difference between the harmonics in e and those in x to produce a new estimate of the 2p unknowns of x, and this iterative process continues.

(4)

To predict a limit cycle it is then assumed that x(t) and c(t) can be accurately represented, or approximated, by only p harmonics. '!berefore we must find the set of 2p values w, ai' ••• , 2, ••• , p such that x(t) = -c(t), where <1>1 is set to zero as a reference point, without loss of generality. In the software which is written for p(lO we use a Newton Raphson method, NAG (1982) to solve the 2p nonlinear equations given by equating equations (3) and (4). '!bis method requires some initial guesses for the 2p unknowns. Given these initial values, the loop is opened ut removing the link between e and x in Fig.2 and an array of equally spaced samples of x (t) c:onputed over one period. Using simple

Hk

••• , m

o o

for k for k

1,

P

1, ••• , p.

(7)

In many cases only the odd harmonics will be nonzero (for IT-symmetric oscillations) in ...nich case if a balance is sought for p odd harmonics then the right hand side of equation (7) is changed to "for k = 1, 3, 5, ••• ,2p-l". Elquation (7) means that there are 2pn unknowns to be found and ...nere both odd and even harmonics are being balanced these unknowns are: in loop 1 w, all' ..• ' alp" ~12' ••• , lp; and in each loop, i, of the remalnlng m-l loops, ail, ••• , ail?' il, ••• , il? '!be Fhase of the fundamental In loop 1, II is set to zero as a reference point, and .,) the Fhase of the signals in all of the loops are obtained relative to that of xl in loop l. These equations can be left in the polar form as in equation (7) and solutions found for w, aik and ik. Alternatively, if we write

Limit CH'le Prediction in Free Structured :\onlinear Syste ms aik

cos(kwt+~ik)

= Aik cos kwt + Bik sin kwt, and

union of the discs

bik cos(kwt+8ik) = Wik cos kwt + Zik sin kwt (8) then equation (7) can be expressed in its rectangular form and solutions found for w, Aik and Bik. In using the software, the rectangular form of equation (7) was often found to have the better convergence properties with the Newton Raphson routine €!11ployed, and was implemented in preference to the IXllar form. In addition, it was found that the Newton Raphson algorithm would often iterate fairly quickly to approximate solutions with residues (Le. the right hand side of equation (7) for a given set of solution estimates) around 0.01 but then take rruch longer to arrive at the ideal situation where equation (7) was satisfied to within machine precision. Often this extra accuracy is almost meaningless in practice and so to improve the speed of convergence, the following simple method was used. Instead of requiring the harmonics in -x and c to match exactly , _ allow them to be- slightly different. Therefore, rather than solve the rectangular form of equation (7) ~ instead solve the following. For i = 1, m : Aik - Wik: ( 6 for k :Bik - Zik: ( 6 for k

1, ••• , P 1, .•. , P

(9)

Here 6 is known as the residue acceptance limit, and usually a value in the range 0.00001(6(0.001 provided successful and quick convergence to a sufficiently accurate limit cycle solution. In the CAD implementation of these methods, 6 is actually controlled interactively by the user.

m

: >. - a kk : (j~k : a kj :

A Gershgorin Banding Method Firstly, Gershgorin's theorem may be stated as follows. Given an m x m matrix A with elements akj, all of the eigenvalues >. of A lie in the

(10)

Because A and AT have the same eigenvalues, one can alternatively sum down the columns, so all the eigenvalues also lie in the union of discs m

:>. - a kk : \~k :a jk :·

(11)

If the system of Fig.l is sufficiently 'low pass', then x will be almost sinusoidal, and for many systems-an approximate single block model of the system is a OF equivalent, where the effecti ve OF of each loop, not of each nonlinear i ty , is used. This model defines the relationship bet_en a sinusoid in any loop input (of the <:pen loop version of Fig.l) and the resulting sinusoids at the output of the loops. It should be noted that the assumption that x is sinusoidal does not place any constraints on-the inter-block signals, E, ... , yb-l, or on the internal block signals. This model can be expressed as an m x m matrix, T, where each element tik(ak, w) is an amplitude and frequency dependent gain, and k is the number of the loop. Therefore each element of a particular column of T is found by injecting sinusoids of various amplitudes and frequencies into the corresponding input of the <:pen loop version of Fig.1. If _ inject a sinusoid xk=ak cos (wt+h) into loop k, and calculate the fundamental sinusoids at the outputs, 5:;., then _ have

-q Given that ~ have a harmonic balance method which can accurately predict limit cycles in free structured sytems, then there are still three main problems to be dealt with before _ can realistically use such a method in practice. The first of these is that the Newton Raphson method requires initial estimate s for the unknown limit cycle parameters. In some simple examples it was found that these estimates can be just guesses as the region of convergence of the Newton Raphson method may be quite large. But, in general, this is not the case and a more reliable method to generate initial estimates is required. The second of the three problems is how many harmonics do _ need to solve in order to obtain an accurate limit cycle prediction? We therefore need to know the accuracy of any limit cycle prediction results obtained, whether the DF or harmonic balance methods are used. This is a very important lXlint since even in systems such as Fig. 2, the DF method (equivalent to a single harmonic balance) can fail to predict limit cycles which do exist and in rare cases, it can predict a limit cycle which does not exist. Finally, _ need a method which can give us some idea as to the number and location of limi t cycles in a particular system. Ideally such a method would have a simple graphical interpretation, but here, because of the problem of separating the nonlinear and linear effects, this is not possible. The first two problems are solved by the iteration method discussed after the next section, whilst the last problem is solved for both 5150 and MIM) systems by a method based on Gershgorin' s eigenvalue banding theorem discussed briefly in the next section.

27

bi cos(wt + 8i), and (12)

For a sinusoidal limit cycle to exist at x then T

x

(13)

where the negative feedback has been absorbed into T, as defined in equation (12). Therefore for x to be a sinusoidal limit cycle then T IlUSt have- one or more unit eigenvalues. From Gershgorin's theorem then either the row sum (14)

or the column sum m

:>. - t .. (a . ,w): 11

1

( L

(15)

kii

IlUSt be satisfied for >. =1. For this work only the column sum, equation (15) is used since its evaluation depends on only one amplitude, ai, for a particular column, whereas equation (14) involves all of the amplitudes simultaneously. If a unit eigenvalue is contained within any or all of the discs defined by equation (15), then a sinusoidal limit cycle may occur. To implement this method graphically is difficult . So in practice the search for a unit eigenvalue is carried out numerically for each loop i, i.e. column i of T, by using the Newton Raphson method to find the range of amplitudes ai, and frequencies w for which equation (15) contains a unit eigenvalue. If such ranges exist, they are generally easy to locate in this manner. The frequency and amplitude ranges obtained this way are often conservative but serve as very useful starting points for the harmonic balance method.

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1'. ~1r:'-Jall1ara and D. P ..\thertoll

Iteration Method The use of the iteration nethod to determine the accuracy of a given limit cycle prediction is best illustrated by considering the free structured system of Fig.l wi th the loops cpened in the error path as shown in Fig.3. In the figure ~ = [elk ••• entk]T and 29< = [xlk ••• Xmk] T where k is the number of iterations that have been carried out. Assume that an approximate limit cycle solution vector, denoted as ~ has already been estimated and that the accuracy of that solution is to be checked. The approximate solution vector ~ might be sinusoidal or made up of several harmonics (in each loop) • The first step is to inject ~ into the system of Fig. 3 and then using the FFT based methods described above, calculate the exact periodic signal vectors.:L!J<, ••• , ~ and ~ where k is equal to one. If ~ is an exact limit cycle solution then the fedback signal vector ~ will be equal to ~ . If, however, ~ is not an exact limit cycle solution then there will be some difference between ~ and ~. The ccrnparison between ~ and ~ is made graphically by plotting the signals in each loop together. Thus, for k equal to one, xlk is plotted with elk' x2k is plotted with e2k' and so on. If the m graphs show that eik = xik for i = 1, ••. ,m, then ~ is an exact limit cycle solution. Any difference between ~ and ~ is an indication of the accur acy of ~. This iteration process can be continued with the fedback error, ~, becoming the next input ~. Therefore ~ is the initial approximate limit cycle solution vector and thereafter the iteration process is given by 29<+1

=

~

-[Bb]

[B2] [Bl] 29<

approach is to use the harmonics resulting in ~, when, for example, ~ was taken as a sinusoidal vector, as the initial estimates for a harmonic balance. This enables the estimate for the limit cycle frequency as well as for the amplitudes and phases of the harmonics, to be iterated by the Newton Raphson nethod. So, for instance, one might start with a single sinusoidal limit cycle estimate and then, by using iteration, obtain estimates for the first and third harmonics. The amplitudes and [tlases of these harmonics can then be transferred automatically to the harmonic balance section, and a two harmonic balance solution, starting from these estimates, is then found. Iteration, with ~ now the two harmonic balance solution, might then show, for example, that this two harmonic limit cycle approximation was not an exact solution. In this case, one could use the harmonics in ~ as initial estimates for a first, third and fifth harmonic balance. This process of iteration and harmonic balancing is repeated until effectively exact limit cycle solutions are found. In cases where iteration is not contractive, iteration can still provide useful spectral information which can be used as initial estimates for harmonic balancing. In addition, in some cases, a slight modification of the system structure can turn a noncontractive system into a contractive one. For example, in a SISO system of the type shown in Fig. 2, if the nonlinear i ty is a cubic, then it can be shown that equation (16) is not contractive. However, if the system is pole shifted Holtzman (1970) then contraction can occur. By making use of all of the methods described here, limit cycle solutions can be refined from an initial very rough approximation provided by the Gershgor in method, to almost exact solutions.

(16)

for k iterations Iohere k = 1,2,3, In many examples it is found that equation (16), with the frequency chosen correctly, is contractive and will converge from an initial limit cycle solution estimate 2Sl to an exact limit cycle solution after i iterations of equation (16). In this case .!i is an exact limit cycle solution, and 29<+1 is equal to 29< for all k ) i. It is important to note that iteration, as defined by equation (16), does not alter the frequency (of 29<), Iohich remains constant for all k iterations. However, iteration does successively modify the amplitudes and [tlases of all of the harmonics in 29<. Because of this, it is often p:lssible to start the iteration method with ~ equal to a vector of sinusoids for example, at a known limit cycle frequency and then iterate, using equation (16), to an exact and p:lssibly very distorted limit cycle solution. Convergence can also occur from an ~ vector which is made up of several harmonics. The actual conditions necessary for convergence to occur are difficult to determine analytically, and are not discussed further here. However, details of proofs, using the contraction mapping theorem, for some systems can be found in McNamara and Atherton (1985) and McNamara (1986). In practice, the easiest nethod to determine whether equation (16) is contractive for a particular system and a given 2Sl is to actually try iteration to see if contraction occurs. This is relatively easy to determine from the waveforms obtained. When, as is often the case, equation (16) is contractive then virtually exact limit cycle solutions can sometimes be easily found by altering only the frequency of the limit cycle approximation ~, and repeating the sequence of iterations. In other cases, even when equation (16) is contractive, a more efficient

APPLICATIONS Some examples showing software are given below

applications

of

the

Example 1 This example is the well known Fi tt's example (Atherton (1981)) with n(x) x 3 , and G(S)=10S2/(S4+0.04s 3+2.0206s 2 +0.0404s+O.980302) This plant has two pairs of lightly damped p:lles near to frequencies of 0.9 and 1.1 radians/ second. This system is a well known counterexample to Aizerman' s conjecture. Although the DF method fails to predict any limi t cycles for this system, the methods of this paper show that there are a large number of limit cycles. As already mentioned, the convergent nature of the iteration method can be used to provide good initial estimates for the magnitudes and [tlases of the harmonics for the harmonic balance method. In this case it can be shown by using the contraction mapping theorem that in order for iteration as defined by equation (16) to converge, the system needs to be pole shifted. Therefore the nonlinearity is changed to xLrx, and the linear plant is changed to GI (s)=G(s)/(l+rG(s)). The limit cycles of the pole shifted and original systems are of course the same. If a suitably large value of r

is chosen then iteration of the pole-shifted system converges. This enables the harmonic balance method to be used. A large number of limit cycle solutions were found in this way and some of these are as

Limit (\de Prediction in Free Structured '\onlincar Sntcms follows. 'l11e first of these, shown in Fig.4, was found by harmonic balancing. 'l11e spectrum of this waveform is shown in Table 1. Table 1 Spectrum of first limit cycle with f u ndamental frequency 0 3005 rads/sec n 1 3 5 7 9 11 13

15 17 19

Amplitude 1.129 0.212 0.119 0.1 0.102 0.109 0.089 0.015 0.164 0.069

Phase (deq.) 0 179.3 -1.5 177 .1 -4.6 173.6 -9.3 30.4 176.0 179.4

Table 2 Spectrum of second limit cycle with fundamental frequency 0 384 rads/sec

,

n 1 3 5 7 9 11

Amplitude 0.773 0.108 0.014 0.077 0.166 0.101

Phase (deg) 0 178.2 -33.8 11.8 -172.4 -168.3

The next limi t cycle shown in Fig. 5 is one which according to simulation is unstable. 'l11e spectrum of this waveform is shown in Table 2. It should be noted that in Fig.4 (and Fig.5) there are actually two waveforms plotted together. 'l11e first is the input signal to block one, 29< while the second is the resulting fedback error signal 29<+1. In this case for k=l, there is almost no difference between the two waveforms, so that they appear as one line on the graph. This is an indication of the high accuracy of the results. The complete results for this system show that it has a number of limi t cycles, some of which are stable and some unstable according to simulation, with frequencies between 0.02 rads/sec
slope. Nonlinearities nl2 and ~I are ideal relays with heights of ±I, and n22 is an ideal saturation with lD1it slope and saturation levels of ±l. The linear block has GI I (5)=5/(5 3+25 2+5), and GI2 (5)=0.4/(5 2+25+1), ~I (5)=0.3/(5 2+5) G22 (5)=6/(5 3+35 2+25). 'l11e DF method predicts a limit cycle with w=1.294 rads/sec, with al=3.5, bre accurate solutions can then be obtained using harmonic balancing to give a two sinusoid solution with w=1.278 rads/sec, with in al=3.6, ~I=O, a3=0.05, ~3=175° in loop 1 and al=3.35, ~1=160, a3=0.07, ~3=16lo in loop 2. Iteration shows that these solutions are exact. Example 4 As a final example the limit cycles in a 11ll1ti-

variable sampled data system are studied. 'l11e system is a 2-by-2 MIMO system consisting of a sample and zero order hold in each loop, followed by nonlinear and linear blocks. The nonlinear block has elements nIl, n22 which are ideal saturations with unit slope and limit levels of ±l and elements n12' n21 ideal relays with unit height. The elements of the linear block are Gl l (5)=10/(5+3), GI2 (5)=10/(5+1)3, G21 (s)=10/(s2+.3s+1), G22 (5)=10/5(5+1) • 'l11ere are a wide range of sampling frequencies for which there are a variety of subharmonic limit cycles p:lssible. For example, if both samplers are synchronised and have sampling periods of T I =T 2=0.5 seconds then both second and fourth subharmonic limit cycles are p:lssible. The second subharmonic has p:lsitive heights of h=4.l9 in loop 1 and h=0. 084 in loop 2. 'l11e fourth subharmonic is shown in Fig. 8 with the crosses marking the sampling instants. If a I1llltirate system is considered with the sampler in loop one having a period of T I =0.5/3 seconds and in loop 2 T2=0.5 seconds, then a limit cycle is obtained which is a sixth subharmonic in loop 1, and a second subharmonic in loop 2. That is the limit cycle shown in Fig.9, with frequency 2n rads/sec in both loops. CONCWSIONS This p3.per has outlined oome methods and their software implementation for the prediction of limit cycles in nonlinear free structured systems. The methods include the describing function method, the harmonic balance method and the iteration method. The iteration method is very important in that it enables the accuracy of any limit cycle solution estimate to be determined. In addition because convergence often occurs, it enables almost exact limit cycle predictions to be rrade for a wide variety of complex control systems. Stability rrargins for a system can be assessed by evaluating the anount of a:lditional gain or phase shift required to cause a system to limit cycle. This facility is particularly useful for design since the effects of including ronlinearities in a system designed using a linearised approach can be easily assessed. The methods have been implemented in a single CAD p3.ckage which is easy to use, rrakes extensive use of graphics facilities, and includes on-line help. In a:ldition the methods presented are being extended to the ITDdelling of practical nonlinear systems. For example, given a simple system with one ronlinearity and one linear element, it is possible to compute the spectra of all of the signals in the loop due to a sinusoidal input to the open loop system. By

o.

30

1'. l\1c"lama ra alld D. P. Atil e rt o ll

varying the amplitude and frequency of the injected sinusoid, plots of the rragnitudes and phases of the harmonics at the output of the system can be canputed. These can then be ccmpared with experimentally obtained data. The model of the system can the n be altered until the spectra it produces, are in reasonable agreement with those obtained expe rimentally . This approach rreans that once a set of harmonic experimental data i s obtained, most of the modelling IoUrk can be carried out interactively on the computer.

Atherton, D.P., (1981). 'Stabililty of Nonlinear Systems'. Research Studies Press. Holtzman, J.M., (1970). 'Nonlinear System Theory - A Functional Analysis Approach'. Prentice-Hall. Numerical Algorithms Group. (1982). 'NAG Fortran Library Manual'. Vol. 1, 005NBF. (1973) • 'Describing Functions, Mees, A.I., circle criteria and multiple-loop feedbac k systems'. Proc . lEE. 120. pp.126-l30. Gray, J.O., and Nakhla, N.B., (1981 ) . 'Prediction of limit cycles in multivariable nonlinear s ystems'. Proc. lEE. 128. Part D. No.5. pp.23324l. McNamara, O.P., and Athe rton, D.P. (1985). 'An iterative rrethod f or limit cycle determination'. Contro l 85 . lEE Inte rnational Conference, Cambridge . UK. McNarrara, O.P. (1986). ' Canputer-Aided Design of Nonlinear Contro l Systems using Describing Function Based Methods', DPhil Thesis, University of Susse x, England.

ACKNCWLE[x;EMENl'S The authors wish t o acknowl edge the support of this research Dj the United Kingdom Scie nce and Engineering Rese arch Council. REFERENC ES

Atherton, D.P., (197 5). ' Nonlinear Engineering I . Van Nost r and Rienhold.

knock 1

Control

Bl oc k b

nIx)

G (s)

~ Fig. 2. F~g .

1.

Single l oop nonlineilc system .

Fn,o! struc t u red n
t.:

Ik

Block 2

(!nlk

Fl.g . 3 .

xmk ' - - - - '

Open loop version of Fig. 1.

free structured system of

".,\.I .... '~" ,..II- .I tltct. J,I . . l

h ... , ,ltc. ''1-'- ",..,1 , . - '/ .IlocL 1,1.,.1 f1+1IOc>tno .. ",..IC-Ol

~lI&nbT .. II,..LI_O)

Fig. 4 . First. Liloit c y c le in thE':" Fitt's

Fig. 5 .

excur.ple . 1.
th~

Second lim1 t cycle in thF! Fitt's example.

Fig. 6 .

t'I

lqIl(-Q I

High frequency limit eXample 2 .

,-l..,Z I -

cycl~

OI

...

, .~

Fig. 7.

Low frequency limi t cycle

in

~exaDlple

2.

Fig. 8.

Limit cycle in sampled data system

Fig. 9.

Limi t. cycle in mul ti - r ate sampled dat.a s yst..em

in