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Frequency-Response and Frequency-Domain Models

  • Abbas Emami-NaeiniEmail author
  • Christina M. Ivler
  • J. David Powell
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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

A very common way to use the exponential response of linear time-invariant systems (LTIs) is in finding the frequency response, or response to a sinusoid. First we express the sinusoid as a sum of two exponential expressions (Euler’s relation):
$$\displaystyle \begin{aligned} A\cos (\omega t)=\frac{A}{2}(e^{\,j\omega t}+e^{-j\omega t}). \end{aligned} $$
(1)
Suppose we have an LTI system with input u and output y, where we define the transfer function
$$\displaystyle \begin{aligned} G(s)=\frac{Y(s)}{U(s)}. \end{aligned} $$
(2)
If we let s =  in G(s), then the response to u(t) = ejωt is y(t) = G()ejωt; similarly, the response to u(t) = ejωt is G(−)ejωt. By superposition, the response to the sum of these two exponentials, which make up the cosine signal, is the sum of the responses:
$$\displaystyle \begin{aligned} y(t)={\frac{A}{2}}[G(j\omega )e^{\,j\omega t}+G(-j\omega )e^{-j\omega t}]. {} \end{aligned} $$
(3)
The transfer function G() is a complex number that can be represented in polar form or in magnitude-and-phase form as G() = M(ω)e(ω) or simply G = Me. With this substitution, Eq. (3) becomes for a specific input frequency ω = ωo
$$\displaystyle \begin{aligned} y(t) &={\frac{A}{2}}M\left( e^{\,j(\omega t+\varphi )}+e^{-j(\omega t+\varphi )}\right) , \notag \\ &=AM\cos (\omega t+\varphi ), \end{aligned} $$
(4)
$$\displaystyle \begin{aligned} M &=|G(j\omega )|=|G(s)|{}_{s=j\omega _{o}}{}\\ &=\sqrt{\{\text{Re}[G(j\omega _{o})]\}^{2}+\{\text{Im}[G(j\omega _{o})]\}^{2}}, \end{aligned} $$
$$\displaystyle \begin{aligned} \notag \\ \varphi &=\angle G(j\omega )=\tan ^{-1}\left[ \frac{\text{Im} [G(j\omega _{o})]}{\text{Re}[G(j\omega _{o})]}\right] . \end{aligned} $$
(5)
This means that if an LTI system represented by the transfer function 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(). Hence such plots are referred to as Bode plots. Bode plotting techniques are discussed in Franklin et al. (2019).
Fig. 1

Response of \(G(s)= \frac {1}{(s+1)}\) to the input \(u= \sin 10\,t.\) (Source: Franklin, Powell, Emami-Naeini, Feedback Control of Dynamic Systems, 8th Ed., Ⓒ 2019, p-334, Reprinted by permission of Pearson Education, Inc., Upper Saddle River, NJ)

Fig. 2

Frequency response for \(G(s)= \frac {1}{s+1}.\) (Source: Franklin, Powell, Emami-Naeini, Feedback Control of Dynamic Systems, 8th Ed., Ⓒ 2019, p-102, Reprinted by permission of Pearson Education, Inc., Upper Saddle River, NJ)

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 axis will prove to be very useful in determining the stability of a closed-loop system. Evaluating G() provides information that allows us to determine closed-loop stability from the open-loop G(s).

For the standard second-order system
$$\displaystyle \begin{aligned} G(s)=\frac{1}{(s/\omega _{n})^{2}+2\zeta (s/\omega _{n})+1}, {} \end{aligned} $$
(6)
the Bode plot is shown in Fig. 3 for various values of ζ.
Fig. 3

Frequency responses of standard second-order systems (a) magnitude, (b) phase. (Source: Franklin, Powell, Emami-Naeini, Feedback Control of Dynamic Systems, 8th Ed., Ⓒ 2019, p-338, Reprinted by permission of Pearson Education, Inc., Upper Saddle River, NJ)

A natural specification for system performance in terms of frequency response is the 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 v2 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
$$\displaystyle \begin{aligned} \frac{Y(s)}{R(s)}\overset{\varDelta }{=}\mathcal{T}(s)=\frac{KG(s)}{1+KG(s)}. \end{aligned} $$
(7)
The plot is typical of most closed-loop systems in that (1) the output follows the input \(\left ( |\mathcal {T}|\cong 1\right ) \) at the lower excitation frequencies and (2) the output ceases to follow the input \( \left ( |\mathcal {T}|<1\right ) \) at the higher excitation frequencies. The maximum value of the frequency-response magnitude is referred to as the resonant peak Mr.
Fig. 4

Unity feedback control system

Fig. 5

Definitions of bandwidth and disturbance rejection bandwidth, resonant peak, and disturbance peak. (\(\mathcal {T}(s)\) is blue, \(\mathcal {S}(s)\) is red)

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., ωBW = ωn 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.

For a second-order system, the time responses are functions of the pole-location parameters ζ and ωn. If we consider the curve for ζ = 0.5 to be an average, the rise time (tr) from y = 0.1 to y = 0.9 is approximately ωntr = 1.8. Thus we can say that
$$\displaystyle \begin{aligned} t_{r}\cong \frac{1.8}{\omega _{n}}. {} \end{aligned} $$
(8)
Although this relationship could be embellished by including the effect of the damping ratio, it is important to keep in mind how Eq. (8) is typically used. It is accurate only for a second-order system with no zeros; for all other systems, it is a rough approximation to the relationship between tr and ωn. 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, tr. 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.

In many cases, the designer’s primary concern is the error in the system due to disturbances rather than the ability to track an input. For error analysis, we are more interested in the sensitivity function, \(\mathcal {S} (s)=1-\mathcal {T}(s)\), rather than \(\mathcal {T}(s)\). In a feedback control system, as in Fig. 4, the sensitivity can be interpreted as either the error response to a reference input or the system response to an output disturbance:
$$\displaystyle \begin{aligned} \mathcal{S}(s)=\frac{E(s)}{R(s)}=\frac{Y(s)}{Y_{d}(s)}=\frac{1}{1+KG(s)}. \end{aligned}$$

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 ωDRB. The disturbance rejection bandwidth ωDRB 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 ωDRB. 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 ωDRB < ωBW. The maximum value of \(\mathcal {S}(s)\) is referred to as the disturbance rejection peak, Md, 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() 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.

Suppose we have a system defined by Fig. 6a and whose root locus behaves as shown in Fig. 6b; that is, instability results if K is larger than 2. The neutrally stable points lie on the imaginary axis – that is, where K = 2 and s = j1.0. Furthermore, all points on the root locus have the property that
$$\displaystyle \begin{aligned} |KG(s)|=1\quad \text{and}\quad \angle G(s)=-180^{\circ }. \end{aligned}$$
At the point of neutral stability, we see that these root-locus conditions hold for s = , so
$$\displaystyle \begin{aligned} |KG(j\omega )|=1\quad \text{and}\quad \angle G(j\omega )=-180^{\circ }. {} \end{aligned} $$
(9)
Thus a Bode plot of a system that is neutrally stable (i.e., with 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).
Fig. 6

Stability example: (a) system definition; (b) root locus. (Source: Franklin, Powell, Emami-Naeini, Feedback Control of Dynamic Systems, 8th Ed., Ⓒ 2019, p-355, Reprinted by permission of Pearson Education, Inc., Upper Saddle River, NJ)

Fig. 7

Frequency-response magnitude and phase for the system in Fig. 6. (Source: Franklin, Powell, Emami-Naeini, Feedback Control of Dynamic Systems, 8th Ed., Ⓒ 2019, p-356, Reprinted by permission of Pearson Education, Inc., Upper Saddle River, NJ)

Having determined the point of neutral stability, we turn to a key question: Does increasing the gain increase or decrease the system’s stability? We can see from the root locus in Fig. 6b that any value of 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()| < 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:
$$\displaystyle \begin{aligned} |KG(j\omega )|<1\quad \text{at}\quad \angle G(j\omega )=-180^{\circ }. {} \end{aligned} $$
(10)
This stability criterion holds for all systems for which increasing gain leads to instability and |KG()| 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
$$\displaystyle \begin{aligned} |KG(j\omega )|>1\quad {\mathrm{at}}\quad \angle G(j\omega )=-180^{\circ }. {} \end{aligned} $$
(11)
Based on the above ideas, we can now define the robustness metrics gain and phase margins:

Phase Margin

Suppose at ω1, \(|G(j\omega _{1})|=\frac {1}{ K}\). 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
$$\displaystyle \begin{aligned} \text{PM}=\angle G(j\omega _{1})-(-180^{\circ }). \end{aligned} $$
(12)

Gain Margin

Suppose at ω2, \(\angle G(j\omega _{2})=-180^{\circ }\). 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
$$\displaystyle \begin{aligned} \text{GM}=\frac{1}{K|G(j\omega _{2})|}. \end{aligned} $$
(13)
There are also rare cases when |KG()| 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

The closed-loop bandwidth was defined above. The natural frequency is typically within a factor of two of the bandwidth for a second-order system. We can help establish a more exact correspondence by making a few observations. Consider a system in which |KG()| shows the typical behavior
$$\displaystyle \begin{aligned} \begin{array}{rcl} |KG(j\omega )| &\displaystyle \gg &\displaystyle 1\quad \text{for}\quad \omega \ll \omega _{c}, \\ |KG(j\omega )| &\displaystyle \ll &\displaystyle 1\quad \text{for}\quad \omega \gg \omega _{c}, \end{array} \end{aligned} $$
where ωc is the crossover frequency where |G()| = 1. The closed-loop frequency-response magnitude is approximated by
$$\displaystyle \begin{aligned} |\mathcal{T}(j\omega )|{=}\left\vert \frac{KG(j\omega )}{1+KG(j\omega )} \right\vert {\cong} \left\{ \begin{array}{ll} 1, & \omega \ll \omega _{c}, \\ |KG|, & \omega \gg \omega _{c}. \end{array} \right. {} \end{aligned} $$
(14)

In the vicinity of crossover, where |KG()| \(=1, |\mathcal {T} (j\omega )|\) 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 \(| \mathcal {T}(j\omega _{c})|=1.31\).

The exact evaluation of Eq. (14) was used to generate the curves of \(|\mathcal {T}(j\omega )|\) in Fig. 8. It shows that the bandwidth for smaller values of PM is typically somewhat greater than ωc, though usually it is less than 2ωc; thus
$$\displaystyle \begin{aligned} \omega _{c}\leq \omega _{{}_{BW}}\leq 2\omega _{c}. \end{aligned} $$
(15)
Fig. 8

Closed-loop bandwidth with respect to PM. (Source: Franklin, Powell, Emami-Naeini, Feedback Control of Dynamic Systems, 8th Ed., Ⓒ 2019, p-386, Reprinted by permission of Pearson Education, Inc., Upper Saddle River, NJ)

Another specification related to the closed-loop frequency response is the resonant-peak magnitude Mr, defined in Fig. 5. For linear systems, Mr is generally related to the damping of the system. In practice, Mr 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 ωDRB as well as the disturbance rejection peak Md 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

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

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  • Abbas Emami-Naeini
    • 1
    Email author
  • Christina M. Ivler
    • 2
  • J. David Powell
    • 3
  1. 1.Electrical EngineeringStanford UniversityStanfordUSA
  2. 2.Shiley School of EngineeringUniversity of PortlandPortlandUSA
  3. 3.Aero/Astro DeptStanford UniversityStanfordUSA

Section editors and affiliations

  • J. David Powell
    • 1
  1. 1.Aero/Astro DeptStanford UniversityStanfordUSA