# Analyzing Control Loop Behavior in the Frequency Domain

## The frequency domain is a mathematical construct that simplifies the analysis of a control system's performance. It can be used to show how a process operating under the influence of a feedback controller will react to inputs from the controller or a change in the behavior of the process. Frequency domain analysis rests on two fundamental principles.

KEY WORDS

Process and advanced control

Control theory

Process control systems

Process control theory

Tutorial

The frequency domain is a mathematical construct that simplifies the analysis of a control system’s performance. It can be used to show how a process operating under the influence of a feedback controller will react to inputs from the controller or a change in the behavior of the process.

Frequency domain analysis rests on two fundamental principles. The first is linearity, which states that the sum of two signals applied to a linear process will produce an output equal to the sum of the outputs that would have resulted had the two inputs been applied separately. It follows that a linear process will produce a Y % change in the process variable (the process output) following an X % change in the control action (the process input) according to a straight line relationship Y = KX . The steady state gain K will remain constant whether the process is currently running at maximum capacity, minimum capacity, or somewhere in between.

A linear process behaves somewhat differently if the input applied by the controller is not just a simple step change of X %. If the control action continues to oscillate sinusoidally so that the process variable never has a chance to reach a steady state, the apparent gain of the process will generally be less than K . That is, the amplitude of the sine wave coming out of the process will be less than K times the amplitude of the sine wave going in.

An example

The apparent process gain K( ù ) will depend on the frequency ù of the incoming sine wave. Higher frequency sine waves are generally attenuated more severely than lower frequency sine waves so lower gains result at higher frequencies. This phenomenon can be demonstrated with a common child’s toy comprised of a weight hung from a handle by way of a vertical spring. If the child raises and lowers the handle more-or-less sinusoidally, the weight will eventually start oscillating at the same rate as the handle. However, as the child pumps the handle faster and faster, the amplitude of the weight’s oscillations will decrease until the weight no longer moves at all.

The process consists of a handle, a spring, and a weight. A child holding the handle serves as the controller. The process output is the vertical position of the weight as measured by the child’s observations. By moving the handle, the child can input a control action to change the weight’s position.

The weight’s oscillations will also tend to lag behind those of the handle by an ever-increasing margin as the frequency increases. This is another phenomenon shared by all linear processes. The phase of the output lags the phase of the input by an amount that usually increases with frequency. Note, however, that the frequency of the incoming sine wave remains unchanged as it passes through the process. No matter how fast or slow the child pumps the handle, the weight will always oscillate at the same rate, albeit with a different amplitude and phase.

On the other hand, those amplitude and phase changes will be the same every time that a sine wave of the same frequency passes through the process. Thus, amplitude and phase changes can be plotted vs. frequency to produce two fixed curves that are characteristic of the process. These curves comprise the process’s Bode plot , a graphical analysis tool developed by Hendrick Bode (rhymes with ‘roadie’) at Bell Labs in the 1940s. A Bode plot can be derived empirically by exercising the process with a sinusoidal control action at various frequencies or by analyzing the physical characteristics of the process.

Fourier’s Theorem

Of course most real-life processes are not controlled by driving them with strictly sinusoidal inputs. However, the second fundamental principle of frequency domain analysis- Fourier’s Theorem- states that any signal (including a nonoscillatory control action) can be expressed as a 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, then, Fourier Transforms and Bode plots can be used together to predict how a linear process would react to a proposed sequence of control actions. Here’s how:

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

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

Step 3-Use an Inverse Fourier Transform to recombine the modified sine waves back into a single 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 action actually been input to the process.

A Bode plot shows how a sine wave of frequency v radians/sec passing through a linear process will change its amplitude (or magnitude) by a factor of K (v) decibels and lose phase by F (v) degrees. K (v) and v are generally graphed on logarithmic scales. Bode plots vary in shape for different processes, but K (v) always approaches the steady state gain K as v tends to zero. For very high frequencies, K(v) tends towards zero. At some point in between known as the cutoff frequency, the magnitude plot drops to 70.7% of the steady state gain. The frequencies below this point lie within the pass band of the process. The larger the pass band or bandwidth, the more sine waves that can pass through the process with less than 70.7% attenuation. Bandwidth is a common measure of a communication process’s capacity for transmitting data in the form of simultaneous sinusoidal signals.

Note that at no point in this procedure were any individual sine waves actually generated by the controller nor even plotted on paper. All such frequency domain analysis techniques are strictly conceptual. It is merely 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 the 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 much easier in the frequency domain. In the above example, it was merely a matter of multiplication and subtraction to compute the frequency spectrum of the process variable given the Fourier Transform of the proposed control action and the Bode plot of the process.

More Bode plots

Unfortunately, there’s a price to be paid for such computational convenience. It’s not always easy to determine just what mathematical problem needs to be solved in the frequency domain in order to solve the original problem in the time domain.

Stability analysis is a good example. A control engineer experienced with frequency domain analysis can look at a process’s Bode plot and divine from its shape just how aggressive a controller can afford to be without driving the closed-loop system unstable. And once a controller has been designed and applied to the process, the combined Bode plot for the controller and the process operating in series shows whether the closed-loop system would become either more or less stable (and by how much) if the behavior of the process were to suddenly change. These issues are both related to the gain margin and phase margin of the controller-and-process combination, but just what these concepts mean and how they can be read from a Bode plot requires a fairly extensive background in control theory.

On the other hand, resonance analysis is fairly straightforward. Consider the sample Bode plot shown earlier, for example. The magnitude plot shows a distinct peak at the natural frequency . A process with this Bode plot would amplify rather than attenuate an incoming sine wave oscillating at this particular frequency. Any process capable of storing energy will demonstrate this phenomenon known as resonance . If the resonant peak is high enough, the process can actually destroy itself when driven by a sine wave of just the right frequency. This is why soldiers break step when crossing bridges. If the frequency of their marching cadence happens to match the natural frequency of the bridge, it could be forced to oscillate to the point of collapsing.

This sample Bode plot could also represent the spring toy. It has a natural frequency that depends on the mass of the weight, the spring constant, and the friction in the spring. Pumping the handle at this particular frequency will cause the weight to bounce wildly (which is exactly the point of such a toy).

Processing operatiing in series

Frequency domain analysis also affords a simple solution to the problem of how multiple processes will behave when connected in series. Consider, for example, a combined process where the handle of a second spring toy has been hung from the weight of the first. The output of process A (the position of weight A ) is now the input to process B (the position of handle B ).

The output of process A is the input to process B.

Thus, a sinusoidal input of frequency ù applied to process A would generate a sinusoidal input to process B of the same frequency. Process A would attenuate that sine wave by a factor of K A ( ù ) and process B would further attenuate it by a factor of K B ( ù ), for a total attenuation of K A ( ù ) times K B ( ù ). The total phase lag introduced by the two process would be additive; i.e. Ö A ( ù ) plus Ö B ( ù ). Therefore, the Bode plot of the overall system would simply be the product of the two process’s magnitude plots and the sum of the two process’s phase plots. Furthermore, since magnitude plots are generally graphed on a logarithmic scale, the overall magnitude plot can be deduced graphically simply by adding the two individual magnitude plots together point by point.

This can be a particularly powerful analysis tool, especially when attempting to determine the overall behavior of a process operating in series with a controller. Summing the Bode plot of the controller with the Bode plot of the process yields the Bode plot of the combined system. The shape of that combined Bode plot in turn reveals much about the behavior of the closed-loop system.

Unfortunately, the whole technique breaks down if either the process or the controller is not truly linear. Controllers can generally be designed to behave linearly (and almost always are), but considerable care must be taken to ascertain the linearity of the process before frequency domain analysis can even be attempted.

For additional reading, see www.controleng.com/tutorials .

Comments? E-mail Vance VanDoren at controleng@msn.com .

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