# Frequency-Response and Frequency-Domain Models

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**DOI:**https://doi.org/10.1007/978-1-4471-5102-9_236-2

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## Abstract

A major advantage of using frequency response is the ease with which experimental information can be used for design purposes. Raw measurements of the output amplitude and phase of a plant undergoing a sinusoidal input excitation are sufficient to design a suitable feedback control. No intermediate processing of the data (such as finding poles and zeros or determining system matrices) is required to arrive at the system model. The wide availability of computational and system identification software tools has rendered this advantage less important now than it was years ago; however, for relatively simple systems, frequency response is often still the most cost-effective design method. The method is most effective for systems that are stable in open loop but can also be applied to unstable systems. Yet another advantage is that it is the easiest method to use for designing dynamic compensation as seen in the **Cross-Reference** article: [1] “Classical Frequency-Domain Design Methods”.

## Keywords

Frequency response Magnitude Phase Bode plot Stability Bandwidth Resonant peak Phase margin (PM) Gain margin (GM) Disturbance rejection bandwidth## Introduction

**frequency response,**or response to a sinusoid. First we express the sinusoid as a sum of two exponential expressions (Euler’s relation):

*u*and output

*y*, where we define the transfer function

*s*=

*jω*in

*G*(

*s*), then the response to

*u*(

*t*) =

*e*

^{jωt}is

*y*(

*t*) =

*G*(

*jω*)

*e*

^{jωt}; similarly, the response to

*u*(

*t*) =

*e*

^{−jωt}is

*G*(−

*jω*)

*e*

^{−jωt}. By superposition, the response to the sum of these two exponentials, which make up the cosine signal, is the sum of the responses:

*G*(

*jω*) is a complex number that can be represented in polar form or in magnitude-and-phase form as

*G*(

*jω*) =

*M*(

*ω*)

*e*

^{jφ(ω)}or simply

*G*=

*Me*

^{jφ}. With this substitution, Eq. (3) becomes for a specific input frequency

*ω*=

*ω*

_{o}

*G*(

*s*) has a sinusoidal input with magnitude

*A*, the output will be sinusoidal at the

*same*frequency with magnitude

*AM*and will be shifted in phase by the angle

*φ*.

*M*is usually referred to as the

**amplitude ratio**or

**magnitude**, and

*φ*is referred to as the

**phase**, and they are both functions of the input frequency,

*ω*. The frequency response can be measured experimentally quite easily in the laboratory by driving the system with a known sinusoidal input, letting the transient response die, and measuring the steady-state amplitude and phase of the system’s output as shown in Fig. 1. The input frequency is set to sufficiently many values so that curves such as the one in Fig. 2 are obtained. Bode suggested that we plot \(\log |M|\)

*vs*\( \log \omega \) and

*φ*(

*ω*)

*vs*\(\log \omega \) to best show the essential features of

*G*(

*jω*). Hence such plots are referred to as Bode plots. Bode plotting techniques are discussed in Franklin et al. (2019).

We are interested in analyzing the frequency response not only because it will help us understand how a system responds to a sinusoidal input but also because evaluating *G*(*s*) with *s* taking on values along the *jω* axis will prove to be very useful in determining the stability of a closed-loop system. Evaluating *G*(*jω*) provides information that allows us to determine closed-loop stability from the open-loop *G*(*s*).

*ζ*.

**bandwidth**, defined to be the maximum frequency at which the output of a system will track an input sinusoid in a satisfactory manner. By convention, for the system shown in Fig. 4 with a sinusoidal input

*R*, the bandwidth is the frequency of

*R*at which the output

*Y*is attenuated to a factor of 0.707 times the input (If the output is a voltage across a 1-

*Ω*resistor, the power is

*v*

^{2}and when |

*v*| = 0.707, the power is reduced by a factor of 2. By convention, this is called the half-power point.). Figure 5 depicts the idea graphically for the frequency response of the

*closed-loop*transfer function

**resonant peak**

*M*

_{r}.

Bandwidth is a measure of speed of response and is therefore similar to time-domain measures such as rise time and peak time or the *s*-plane measure of dominant root(s) natural frequency. In fact, if the *KG*(*s*) in Fig. 4 is such that the closed-loop response is given by Fig. 3a, we can see that the bandwidth will equal the natural frequency of the closed-loop root (i.e., *ω*_{BW} = *ω*_{n} for a closed-loop damping ratio of *ζ* = 0.7). For other damping ratios, the bandwidth is approximately equal to the natural frequency of the closed-loop roots, with an error typically less than a factor of 2.

*ζ*and

*ω*

_{n}. If we consider the curve for

*ζ*= 0.5 to be an average, the rise time (

*t*

_{r}) from

*y*= 0.1 to

*y*= 0.9 is approximately

*ω*

_{n}

*t*

_{r}= 1.8. Thus we can say that

*t*

_{r}and

*ω*

_{n}. Most systems being analyzed for control systems design are more complicated than the pure second-order system, so designers use Eq. (8) with the knowledge that it is a rough approximation only. For a second-order system, the bandwidth is inversely proportional to the rise time,

*t*

_{r}. Hence we are able to link the time and frequency domain quantities in this way.

The definition of the bandwidth stated here is meaningful for systems that have a low-pass filter behavior, as is the case for any physical control system. In other applications the bandwidth may be defined differently. Also, if the ideal model of the system does not have a high-frequency roll-off (e.g., if it has an equal number of poles and zeros), the bandwidth is infinite; however, this does not occur in nature as physical systems don’t respond well at infinite frequencies.

For most open-loop systems with high gain at low frequencies, \(\mathcal {S} (s) \) for a disturbance input has very low values at low frequencies, indicating an ability to attenuate low-frequency disturbances. The response to the input or disturbance grows as the frequency of the input or disturbance approaches *ω*_{DRB}. The **disturbance rejection bandwidth** *ω*_{DRB} characterizes the speed of the recovery from a disturbance, describing the frequency where the response to a disturbance has been reduced to 0.707 of the input disturbance, defined by Fig. 5. As such, the time to recover from a disturbance is inversely proportional to *ω*_{DRB}. Considering that \(\mathcal {T}(s)+\mathcal {S}(s)=1\), disturbances are attenuated so that \(\mathcal {S}(s)\rightarrow 0\) when the output tracks the input \(\mathcal {T}(s)\rightarrow 1\), as illustrated in Fig. 5. Therefore, an increase in bandwidth also increases the disturbance rejection bandwidth and typically *ω*_{DRB} < *ω*_{BW}. The maximum value of \(\mathcal {S}(s)\) is referred to as the disturbance rejection peak, *M*_{d}, which represents the maximum possible overshoot during a recovery from a disturbance.

For analysis of either \(\mathcal {T}(s)\) or \(\mathcal {S}(s)\), it is typical to plot their response versus the frequency of the input as shown in Fig. 5. Frequency response for control system design can either be evaluated using the computer, or it can be quickly sketched for simple systems using the efficient methods described in Franklin et al. (2019). The methods described next are also useful to expedite the design process as well as to perform sanity checks on the computer output.

## Neutral Stability: Gain and Phase Margins

In the early days of electronic communications, most instruments were judged in terms of their frequency response. It is therefore natural that when the feedback amplifier was introduced, techniques to determine stability in the presence of feedback were based on this response.

Suppose the closed-loop transfer function of a system is known. We can determine the stability of a system by simply inspecting the denominator in factored form (because the factors give the system roots directly) to observe whether the real parts are positive or negative. However, the closed-loop transfer function is usually not known; in fact, the whole purpose behind understanding the root-locus technique (see Franklin et al., 2019) is to be able to find the factors of the denominator in the closed-loop transfer function, given only the open-loop transfer function. Another way to determine closed-loop stability is to evaluate the frequency response of the *open-loop* transfer function *KG*(*jω*) and then perform a test on that response. Note that this method also does not require factoring the denominator of the closed-loop transfer function. In this section we will explain the principles of this method.

*K*is larger than 2. The neutrally stable points lie on the imaginary axis – that is, where

*K*= 2 and

*s*=

*j*1.0. Furthermore, all points on the root locus have the property that

*s*=

*jω*, so

*K*defined such that a closed-loop root falls on the imaginary axis) will satisfy the conditions of Eq. (9). Figure 7 shows the frequency response for the system whose root locus is plotted in Fig. 6 for various values of

*K*. The magnitude response corresponding to

*K*= 2 passes through 1 at the same frequency (

*ω*= 1 rad/sec) at which the phase passes through -180

^{∘}, as predicted by Eq. (9).

*K*less than the value at the neutrally stable point will result in a stable system. At the frequency

*ω*where the phase \(\angle G(j\omega )=-180^{\circ }\) (

*ω*= 1 rad/sec), the magnitude |

*KG*(

*jω*)| < 1.0 for stable values of

*K*and > 1 for unstable values of

*K*. Therefore, we have the following trial stability condition, based on the character of the open-loop frequency response:

*KG*(

*jω*)| crosses the magnitude (=1) once, the most common situation. However, there are systems for which an increasing gain can lead from instability to stability; in this case, the stability condition is

### Phase Margin

*ω*

_{1}, \(|G(j\omega _{1})|=\frac {1}{ K}\). How much more

*phase*could the system tolerate (as a time delay, perhaps) before reaching the stability boundary? The answer to this question follows from Eq. (9), i.e., the phase margin (PM) is defined as

### Gain Margin

*ω*

_{2}, \(\angle G(j\omega _{2})=-180^{\circ }\). How much more

*gain*could the system tolerate (as an amplifier, perhaps) before reaching the stability boundary? The answer to this question follows from Eq. (10), i.e., the gain margin (GM) is defined as

*KG*(

*jω*)| crosses magnitude (=1) more than once or where an increasing gain leads to instability. A rigorous way to resolve these situations is to use the Nyquist stability criterion as discussed in Franklin et al. (2019).

## Closed-Loop Frequency Response

*KG*(

*jω*)| shows the typical behavior

*ω*

_{c}is the crossover frequency where |

*G*(

*jω*)| = 1. The closed-loop frequency-response magnitude is approximated by

In the vicinity of crossover, where |*KG*(*jω*)| \(=1, |\mathcal {T} (j\omega )|\) depends heavily on the PM. A PM of 90^{∘} means that \( \angle G(j\omega _{c})=-90^{\circ }\), and therefore \(|\mathcal {T} (j\omega _{c})|=0.707\). On the other hand, PM = 45^{∘} yields \(| \mathcal {T}(j\omega _{c})|=1.31\).

*ω*

_{c}, though usually it is less than 2

*ω*

_{c}; thus

Another specification related to the closed-loop frequency response is the resonant-peak magnitude *M*_{r}, defined in Fig. 5. For linear systems, *M*_{r} is generally related to the damping of the system. In practice, *M*_{r} is rarely used; most designers prefer to use the PM to specify the damping of a system, because the imperfections that make systems nonlinear or cause delays usually erode the phase more significantly than the magnitude.

It is also important in the design to achieve certain error characteristics, and these are often evaluated as a function of the input or disturbance frequency. In some cases, the primary function of the control system is to regulate the output to a certain constant value in the presence of disturbances. For these situations, the key item of interest for the design would be the closed-loop frequency response of the error with respect to disturbance inputs and associated disturbance rejection bandwidth *ω*_{DRB} as well as the disturbance rejection peak *M*_{d} defined in Fig. 5.

## Summary and Future Directions

The frequency response methods are the most popular because they easily account for model uncertainty and can be measured in the laboratory. A wide range of information about the system can be displayed in a Bode plot. The dynamic compensation can be carried out directly from the Bode plot. Extension of the ideas to multivariable systems has been done via singular value plots. Extension to nonlinear systems is still the subject of current research.

## Cross-References

## Bibliography

- Franklin GF, Powell JD, Emami-Naeini A (2019) Feedback control of dynamic systems, 8th edn. Pearson Education, New YorkzbMATHGoogle Scholar