Frequency Domain Analysis Explained

01 April 2006

Control engineers should think about sine waves to analyse, understand, and improve feedback control systems.

Predicting the future behaviour of a process is key to the analysis of feedback control systems. Knowing how the controlled process will react to the controller’s efforts allows the controller to choose the course of action required to drive the process variable towards the setpoint.

Linear processes are particularly predictable since a combination of two control efforts applied simultaneously to the process will produce a process variable equal to the sum of the outputs that would have resulted had the two control efforts been applied separately. A linear process also demonstrates a
constant, steady-state gain K. That is, if B is the value of the process variable when the control effort is zero, then the process variable eventually will settle out at a value of Y=KX+B when the control effort is fixed at a value of X. This relationship yields a straight line when Y is plotted against X (hence the expression ‘linear process’).

Linear processes also respond to non-constant inputs in predictable ways. Most importantly, sinusoidal
inputs always yield sinusoidal outputs. In fact, if the input from the controller happens to be a sine wave with frequency w, then the process variable output by the process also will be a sine wave with the same frequency. Although real-life controllers rarely generate purely sinusoidal control efforts, this phenomenon is the basis for frequency domain analysis of a process’ behaviour.

Linear process example
A simple example of frequency domain analysis is illustrated in figure 1. This linear process consists of a weight hanging from a handle-mounted spring. The position of the weight is controlled by moving the handle up and down.

Anyone who has ever played with such a toy knows that if the handle is moved in a more-or-less sinusoidal
manner, the weight will start oscillating at the same rate, though out of synchronisation with the handle. Only at relatively low frequencies where the spring doesn’t stretch will the handle and weight move in lock step.

At higher and higher frequencies, the weight will begin to oscillate more than the handle yet lag further and further behind it. At the natural frequency of the process, the weight’s oscillations will reach their maximum height. The natural frequency is determined by the mass of the weight and the stiffness of the spring.

Above the natural frequency, the amplitude of the weight’s oscillations will decrease and its phase will grow more negative (that is, oscillations will grow smaller and smaller and lag further behind). At very high frequencies, the weight will move only slightly, in exactly the opposite direction of the handle.

Bode plots
All linear processes demonstrate similar behaviour. They transform a sinusoidal input into a sinusoidal output with the same frequency but a different amplitude and phase. Just how much the amplitude and phase change depends on the gain and phase lag of the process. Gain is the factor by which the process amplifies the sine wave en route from input to output, and the phase lag is the degree by which the sine wave is delayed.

Unlike the steady-state gain K, the gain and phase lag of the process vary depending on the frequency of the
incoming sine wave. The weight-andspring process does not change the amplitude of a low frequency sine
wave much. It is said to have a low frequency gain of one. Near the natural frequency, the gain is greater than one since the amplitude of the output is greater than the amplitude of the input. The high frequency gain of the process is almost zero since the weight barely oscillates at all when the toy is shaken rapidly.

The process’s phase lag is an additive factor. In this example, it starts near zero for low frequency inputs since the weight and the handle oscillate in synch when the handle is moved very slowly. The phase lag drops to -180 degrees at high frequencies where the handle and weight move in opposite directions
(hence the expression ‘180 degrees out of phase’ used to describe any situation involving complete opposites).

Figure 2 shows the complete spectrum of gains and phase lags for the weightand-spring process at all
frequencies between 0.01 and 100 radians per second. This is an example of a Bode plot, a graphical analysis tool developed by Hendrick Bode at Bell Labs in the 1940s. It can be used to determine the amplitude and phase of the output that results when the process is driven by a sinusoidal input with a
particular frequency. To get the output amplitude, simply multiply the input amplitude by the gain shown at that frequency. To get the output phase, add the phase lag to the input phase.

Fourier’s Theorem
Gains and phase lags shown in a process’ Bode plot are its defining characteristics. They tell an experienced control engineer everything he needs to know about the behaviour of the process and how it will respond in the future not only to sinusoidal control efforts but to any control effort.

Such an analysis is made possible by Fourier’s Theorem, which states that any continuous sequence of measurements or signal can be expressed as an infinite sum of sine waves. Mathematician Joseph Fourier proved his famous theorem in 1822 and produced an algorithm known as the Fourier Transform for computing the frequency, amplitude, and phase of each sinusoid in that sum from measurements of the original signal.

Theoretically, Fourier
Transforms and Bode plots can be used together to predict how a linear process would react to a proposed sequence of control efforts. Here’s how:

1 Use the Fourier Transform to mathematically decompose the proposed control effort into its theoretical sine wave components or frequency spectrum.

2 Use the Bode plot to determine how each of those sine waves would have been modified had it been passed through the process by itself. That is, apply the appropriate amplitude and phase changes dictated by each sine wave’s frequency.

3 Use an inverse Fourier Transform to recombine the modified sine waves into one signal.

Since the inverse Fourier Transform is essentially an addition operation, the linearity of the process will
guarantee that the combined effect of the theoretical sine waves computed in step 1 will be the same as if they had remained summed together. Thus, the combined signal computed in step 3 will represent the process variable that would have resulted had the proposed control efforts been input to the process.

Note that at no point in this procedure are any individual sine waves generated by the controller nor plotted on paper. All such frequency domain analysis techniques are conceptual. It is a matter of mathematical convenience to translate signals from the time domain into the frequency domain with the Fourier Transform (or the closely related Laplace Transform), solve the problem at hand using Bode plots and other frequency domain analysis tools, then transform results back into the time domain.

Most control-design problems that can be solved in this manner can also be solved by direct manipulations in the time domain, but the calculations are generally easier in the frequency domain. In the above example, it was a matter of multiplication and subtraction to compute the frequency spectrum of the process variable given the Fourier Transform of the proposed control efforts and the Bode plot of the process.

Adding sine waves
It is not altogether obvious that adding up the right combination of sine waves will yield a signal with a desired shape as Fourier posited. An example may help.

In Figure 3, consider again the spring and weight. This time the person holding the weight is also sitting on a board that is moved up and down (like a child’s ‘teeter-totter’ or ‘see-saw’) and a ship on the open ocean. Suppose that the ship happens to be rising and falling on the waves in a sinusoidal manner at a frequency of w and amplitude of A. Also suppose that the teeter-totter is oscillating at three times that frequency and one-third that amplitude while a child bounces the toy at five times that frequency and one-fifth that amplitude. Figure 3 shows what those three sinusoidal motions would look like if they were observed separately.

Now suppose that the child is sitting on the end of the teeter-totter, which in turn is bolted to the deck of the ship. If the three individual sine waves happen to line up just right, the spring and weight total movement would approximate a square wave as shown in Figure 4.

This isn’t exactly a practical example, but it does demonstrate that the sum of a base sine wave plus one-third of its third harmonic plus onefifth of its fifth harmonic approximates a square wave with a frequency of w and an amplitude just under A. The approximation gets even better when one-seventh of the seventh harmonic is added plus one-ninth of the ninth harmonic. In fact, Fourier’s Theorem shows that if such a
summation were to be continued ad infinitum, the total would match a square wave of amplitude A exactly.
Fourier’s Theorem can also be used to decompose a nonperiodic signal into an infinite sum of sine waves.

—Vance VanDoren

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