Why Nonlinear Analysis?

An Oscillating System

Another Nonlinearity - A Saturating Amplifier
    Saturating Amplifier Example

Summary

Problems

        Control system analysis is based on the concept that the system is linear.  That's not always true, but the reason that designers do that is that there are very few analytical techniques that can be used to predict the behavior of nonlinear systems.  Prediction is the key here.  When control systems are designed it is imperative that users and customers know how the system will behave in all situations.  Linear analysis techniques are good in that way because you can guarantee stability, for example.  You would know that the system is stable and that the stability of the system was in no way dependent upon the conditions in the system, either the input(s) or the initial state of the system.

        Nonlinear systems present a problem in that regard.  Here are some facts about nonlinear systems.

• Nonlinear systems can exhibit instability when certain inputs are applied but may be well behaved (stable) for all other inputs.
• Nonlinear systems often have limit cycles, in which sustained oscillations occur but only at a particular amplitude and frequency.
• Linear systems might oscillate at a particular frequency, but oscillation at a particular amplitude is a phenomenon peculiar to nonlinear systems.
• There are very few analytical techniques that can be used to predict behavior of nonlinear systems.  For comparison, many universities teach courses in Linear Systems, and in those courses general techniques for analysis of linear systems are taught.  General techniques for analysis of nonlinear systems are hard or impossible to find and we are often left to use very specific techniques for special situations and we are not able to make general statements.
• To make predictions of nonlinear system behavior designers often use simulations.  However, doing a simulation of a nonlinear system only tells the designer about that situation.  There could be other situations in which the system misbehaves, and the designer will only find that out if s/he does a simulation for that specific situation.

        We will begin by examining a very simple system.  The system is nonlinear and has the following characteristics.

• The controller is a "bang-bang" controller.  It is either full ON in one direction or full ON in the other direction.  This is the kind of control action that you get from a very high gain saturating amplifier, for example.  In control parlance, the controller is an "ideal relay".  That's shown in the block diagram below.
• The output of the controller drives a linear system.
• The output of the linear system is fed back to a comparator.
• The system looks much like a linear control system except for the nonlinear controller.

Under certain conditions this system will oscillate - i.e. exhibit a limit cycle.  We are going to determine conditions that will tell us when the system oscillates.  We will build on what we know about the system, and we will make a few assumptions about what might happen in the system.

        Let us consider a situation with no input to the system.  (We say that we have an autonomous system.)  Here is the whole list of things we will assume.

• The system is autonomous, i.e. it has no input.
• The system oscillates.
• At any point in the system there will be a periodic signal.
• G(s) is a system that is a low-pass filter.
• In actuality it may be a motor, or some other device that consumes large chunks of power, but the important point is the form of the transfer function.  When we say that the system is a low-pass filter we mean that it has a DC gain, and that the frequency response drops off as you go up in frequency.

        Let's start by asssuming that the system is oscillating.  Remember, we are working on the system shown below.

Assuming no input and a sinusoidal output, the input to the controller (the ideal relay) is a sinusoid.  That's the signal shown below.

Now, if the input signal to the ideal relay is a sinusoid, the output surely is not a sinusoid.  The output of the ideal relay is +A when the input is positive, and -A when the input is negative.  A is some constant.  The output is, in fact, a square wave.  Here is what it will look like.  In the example square wave, the output saturates at +10.5v and -10.5v, which is what you would get with a power operational amplifier with +12 and -12v supplies.

Now, you have to consider what happens to the square wave as it goes through the transfer function, G(s).  Remember that we are assuming that G(s) is the transfer function of a low-pass filter.  The periodic signal out of the ideal relay can be described by a Fourier Series.  If we have a square wave that "flips" between +A and -A, the Fourier series is given by:

• When the system oscillates, the output of the ideal relay - also the input to G(s) - is periodic.
• Periodic signals can be represented by a Fourier Series representation.
• G(s) is a low-pass filter.
• G(s) will selectively filter out the higher frequencies in the Fourier Series, and it will slectively pass the lower frequencies.

Now, using this block diagram (Simulink representation) it is possible to calculate the response of the system and get a graph of that response.  Here is the calculated response as shown on the scope in the diagram.

Notice that the response of the system is oscillatory.  And, that's with no input.  There are several interesting aspects of the oscillations.

• The output seems to be pretty much sinusoidal.  Upon closer examination we might find deviations from a sinusoidal shape, but overall, it looks pretty good.
• The output is at a definite frequency.
• The output attains a definite amplitude.  Starting from zero it even looks like the amplitude asymptotically approaches a final value, and that would be a clear indication that the system will only oscillate at a single, pre-determine amplitude.

        First, let's examine the transfer function, G(s).  For this system, G(s) is given below.

This system has two time constants, an electrical time constant (faster) and a mechanical time constant (slower).

        Now, let's combine two of the things we have noticed here.

• First, the output of the ideal relay is a square wave, and it it periodic.
• Being periodic the square wave has a Fourier Series representation.
• As the Fourier Series square wave is input to G(s), G(s) acts as a low-pass filter.
• That implies that signals will be attenuated as they pass through G(s).
• The higher frequency signals will be attenuated more.
• If G(s) has a good drop-off, the output of G(s) may well be pretty much the fundamental of the Fourier Series.  Higher frequency terms will be attenuated, possible pretty severely.
• If G(s) filters out everything but the fundamental (and we will assume that for now) then, we can focus on the fundamental.
• Since the system oscillates with no input, the output of the system will be the input of the system - after we account for the sign change in the comparator.
• That means that the output magnitude has to equal the input magnitude.
• If also means that the output phase has to be shifted by -180^o.

        So, let's look at the transfer function again.  First, we will expand the denominator.  We do that because we will be better able to compute the point where phase shifts by exactly -180^o.  (Click here if you want a the details on these computations in a shorter web page.)

        Since we are interested in frequency response we let s = jw.  Then the frequency response function, G(jw), becomes:

We want to compute the frequency at which the phase shifts by exactly -180^o. Notice that we always get exactly -90^o from the integrator term, so we need to compute the frequency for which we get  -90^o from the quadratic term.  We'll get exactly -90^o from the quadratic term when the real terms disappear.  To do that we must have:

or:

This should give us the frequency of oscillation of the closed loop system.  Let's check that for the example system.  In the example system, we had:

• t_e= 0.1
• t_m= 1.0
• Computed oscillation frequency: w_o=  3.16 rad/sec, or f = 0.5Hz.
• Measured oscillation frequency:  5 cycles from t = 7 sec. to t = 18 sec. (estimated from the graph above).
• f_meas = 5/11 = 0.45Hz.
• Given the accuracy of our estimates, this looks to be pretty close.

        The amplitude of the oscillations can be computed fairly easily in this situation.  We know the Fourier Series of the relay output.  That is repeated below.

The fundamental is 4A/p.  Let us work on only the fundamental frequency component for a while.  That component is at a frequency of:

We can compute how the fundamental component changes as it passes through G(s).  Remember that the frequency response is given by:

And, at the oscillation frequency, w_o, the magnitude of the frequency response can be computed as:

Then, note that at the oscillation frequency, w_o, we have:

And, for the time constants for the example system, this works out to be 0.091.  That means that the output of G(s) will have a fundamental component with a magnitude of:

For the system we simulated, we chose A = 10.5, which is about what you would get from a saturating operational amplifier with +12v and -12v supplies.  That yields an amplitude of 1.215v.  Comparing that to the response, the amplitude seems to be on the money.

So, it seems like we have a good method for predicting the magnitude and frequency of oscillations in this kind of system.  There are just a few unresolved questions.

• We didn't account for the higher frequency components in the signal.
• We may not be able to proceed as easily if we don't have an ideal relay.  We need to be able to handle other kinds of nonlinearities.

So, comparing the first few components of the output signal we have (after computing another point):
 

So, the output isn't a pure sine wave, but the harmonic content is not that large.  Our assumption that the output was sinusoidal was a fairly good assumption.

        So, our conclusion has to be that this is a reasonable way to predict oscillations in a system with an ideal relay.  Here are the main points in the argument.

• The controller is a "bang-bang"  - i.e. an "ideal relay" - controller.  It is either full ON in one direction or full ON in the other direction.
• The output of the controller drives a linear system.
• The output of the linear system is fed back to a comparator.

The process you use to analyze the system is as follows:

• Assume that the relay output is a symmetrical square wave.
• Determine where G(jw) has a phase shift of -180^o.  Call that frequency w_o.
• If the phase shift never reaches -180^o, then the system probably does not oscillate.
• Calculate how large the fundamental signal is coming out of the transfer function G(s) at w_o.
• Check that harmonics are not large because there is an assumption that the output is a sine wave, and anything else might modify the relay input in a way that invalidates our assumption of a symmetrical square wave out of the relay.

        If the nonlinearity is not a ideal relay, then the analysis above must be modified.  Another common situation would be a saturating amplifier.  In that case, the block diagram would look like the one below.

Now, if we want to determine whether the system oscillates we need to account for the saturating amplifier.  However, the output of the saturating amplifier is more complex than a square wave, and the Fourier Series for the output will now have a fundamental component that has a magnitude dependent upon the magnitude of the input sinusoid.  The problem now is to incorporate that fact into our analysis.

        Let's assume that there is a sinusoid at the input of the nonlinearity - the saturating amplifier.  Here is a plot of the output of the saturating amplifier against the input amplitude, A.  In the plot the saturation levels are -10.5v and +10.5v.

Shown below is an example of a sine wave that saturates at +10.5 and at -10.5v.  You can set the amplitude of the sine wave by typing in a value in the text box provided.  Be prepared for a little "noise" when you type.  The sine wave can react instantaneously.  Click the button to start after you type in your first value, but you can change amplitude "on the fly".

We want to be able to compute the fundamental component of a general signal like the one above, where the size of the sine part is variable, and the saturation takes place at a fixed level, but at at different angle within the sine wave as the size/amplitude of the sine part varies.

        Let's consider what we are going to do.

• We will assume that the signal resembles the one above.  In particular:
• We assume that they system is operating so that the saturation region is reached.
• The symmetry of the signal above will ensure that it is composed of only sines, with no cosine components.
• The symmetry of the signal above also ensures that only odd harmonics are present.
• We will represent the signal by the following:
• SatSine(t) = Asin(2pft) for 0 < t < t_s.
• SatSine(t) = A_sat for t_s < t < T/4.
• We will compute the sine component of the fundamental.
• We will assume what is a standard/typical representation of the Fourier Series in which the typical term is a_ncos(2pnft) + b_nsin(2pnft).
• Givne that, the fundamental component is give by the expression below.

Actually, we can simplify the integral a little bit because we only need to integrate over the first quarter cycle if we take advantage of the symmetry of the signal, SatSine(t).  That would give us the following.

Now, we can evaluate these two integrals.  First note:

Then, use that identity in the first integral and we have:

Then, doing the integration, we find:

Now, note the following:

• There are two sine and cosine terms that are zero.
• We get sin(0) from the first term at the bottom limit.
• We get cos(p/4) from the second term at the top limit (T/4).
• ts is given by:
• t_s = sin^-1(A_sat/A)/2pf

Then, we can use the trigonometric identity:

to simplify the middle term.  Doing that we get:

Now, we can take advantage of what we know about some of these quantities.  For example, we know:

Then, finally, we have:

or:

Now, sum up what we know.

• If the input to a saturating element is a sinusoidal signal, then we have:
• The output is a sine wave as long as the saturating element does not saturate.  For our calculations above, we have assumed a gain of 1.0 in this situation.
• When the input is large enough to saturate the saturating element, the output has a Fourier series and the first term has the magnitude given above as b₁ above.
• In doing the analysis above, we assumed that the saturating element had a gain of 1.0 until it saturated, and then saturated at a value of A_sat.

Now, let's note the following.

• First, we assume that the input is zero, and that we want to know whether or not the system will oscillate.
• Secondly, we note that, if the system oscillates, there will be a sort of "steady state" in the system.  In other words, the system may oscillate, but the oscillations will not grow or decay.
• If the oscillations do not grow or decay, then as we go around the loop (say we start from the input to the saturating element) the signals may get larger or smaller, but by the time we get around the entire loop we have to have the same signal we started with.
• We are going to make the same assumptions we made for the system we examined earlier - the ideal relay system.  Those assumptions are:
• The output is periodic, and thus can be represented with a Fourier Series.
• G(s) is a low pass system and filters out any harmonics, so that the output of G(s) is only the fundamental in the Fourier Series.

The term on the right - the "equivalent gain" is the describing function.  It is a function of the input amplitude, A, and we will represent it as DF(A).  It's not really a gain because it is not constant.  It depends on the input amplitude, A.  Here is a plot of the describing function plotted against A, the input amplitude to the nonlinearity.

Note the following about this function.

• For small amplitude inputs the gain is one.
• A gain of 1.0 is what you expect since the system is operating linearly and the gain is one in the linear region.
• For amplitudes larger than the saturation level the "gain" decreases.
• A decrease in gain should be expected because the amplitude of the output (at the fundamental frequency) continues to increase, but reaches a limit.  (In the limit, as the input amplitude gets larger and larger, the output will look like the output of an ideal relay.  However, while the output amplitude does not increase, the input increases and the ratio of output - at the fundamental - to the input goes down - just like the graph shows.

Let's go back to the block diagram of the system.  That's shown again below.

Now, treat the nonlinearity as a "variable gain" using the Describing Function.  That means that we will focus on the equivalent block diagram below.

Then, taking into account the gain of the describing function, the gain of G(jw) and the change in sign in the comparator/subtractor in the loop, we have to have the following for sustained oscillations.

or

            Let's see how this can be applied.  Let's look at a particular system.  It is similar to what we have been considering, but it has a gain of 50 in front of the saturation.  That will simulate an amplifier that saturates at -10.5 and +10.5v.  Here is the system.

First, we need to consider how we can plot G(jw).  The plot of choice is the Nyquist plot - not a Bode' plot, not a Nichols Chart, whatever.  The Nyquist plot for the transfer function we are using as an example is shown below.  This plot does not contain the gain of 50 in the amplifier.

The transfer function, G(s), that is plotted here is:

• t_e= 0.1
• t_m= 1.0

• Plot the frequency response, G(jw).  In this example, we will include the gain of 50 in the plot.
• When we plot the frequency response, the plot is on a complex plane, and we plot the imaginary part agains the real plot with frequency as a parameter.
• Plot -1/DF(A) superimposed on the plot with the frequency response.
• Determine if there is a point where the plot of the frequency response and the negative inverse describing function plot intersect.
• If there is an intersection, then there will be oscillations in the system.  (At least the method predicts that there will be oscillations in the system.
• The frequency at which the frequency response intersect determines the frequency of the oscillations.
• The amplitude at which the negative inverse describing function intersects determines the amplitude of the oscillations.
• For the above method to work, G(s) should be a low-pass filter.

The frequency response intersects the negative inverse describing function at around -4.8.  At that point we can do the following computations.

• The system should oscillate.
• The frequency of oscillation - as determined from the frequency response - is 0.5 Hz.  Here we note that we computed the frequency at which G(jw) has a phase shift of -180^o above, and found that frequency to be 0.5 Hz.
• The amplitude is found by determining when the describing function has a value of 1/4.8.  That's about about .208.  To determine that, we need to examine the plot of the describing function.

Estimating the point at which DF(A) = 0.21, that looks to be around A = 60.  That gives us the input to the saturation element.  That's the output of an amplifier with a gain of 50, so the input to the amplifier is 1.2, and that also has to be the magnitude of the output.  We aren't concerned about the sign here, only the magnitude of the oscillations.
• Summarizing our predictions, we have:
• The system oscillates.
• The frequency of oscillation is 0.5 Hz.
• The amplitude of the oscillations at the system output is 1.2.

Compare this simulation with the block diagram of the system just for reference.

Then, examine the computed response of the system.  That is shown below.

We can compare the computed response with our predictions.  Here is the comparison

• The system oscillates.  That's what we predicted.
• The predicted frequency of oscillation is 0.5 Hz.
• Measuring the frequency, we count 10 periods in 20 seconds, for a frequency of 0.5 Hz.  That's what we predicted.
• Notice that we examine the portion where we seem to be getting sustained oscillations.
• The predicted amplitude of the oscillations at the system output is 1.2.
• The oscillations seem to have an amplitude of 1.25 or a little more compared to the 1.2 that was predicted.  That's close but not right on.  We'll accept that.

        We can summarize what has been presented in this lesson.

• Nonlinear systems can exhibit sustained oscillations at a particular amplitude and frequency.
• A Describing Function is a kind of nonlinear gain that determines the ratio of the fundamental of a periodic output of a nonlinearity when the nonlinearity is excited by a sinusoidal input.
• Using the describing function and the frequency response of the linear portion of the system - which must have a low-pass transfer function it is possible to determine the following:
• Whether oscillations occur, and if they do occur to find:
• Amplitude of the oscillations
• Frequency of the oscillations
• To determine the above, plot the frequency response of the linear portion of the system using a Nyquist plot, and on the same plot superpose a plot of the negative inverse of the describing function.  The intersection will permit determination of amplitude and frequency of the oscillations.

        There are several points that should be noted.

• The linear portion of the system is assumed to be a low-pass.  That filters out any harmonics in the periodic signal, and makes the assumption of a sinusoidal input to the nonlinearity realistic.  That's what allows you to use a describing function.
• The nonlinearities we have examined here are "memoryless".  In other words, the output depends only upon the present value of the input.  Nonlinearities that have hysterisis may not depend only upon the present outut, and in those cases, the describing function might be complex to account for phase shift in the fundamental.

• Problem Nonlinear1P01