Abstract
This chapter deals with the frequency response of continuous time linear time invariant dynamical systems. Eight interactive tools are included, to understand the frequency response of linear systems of different orders. An interactive tool to explain nonminimum phase systems is included, as well as another one devoted to fitting models in the frequency domain.
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Notes
- 1.
As pointed out in Chap. 2, a signal, in a broad sense, is any physical quantity that evolves over time.
- 2.
Infinite series converging punctually to a piecewise periodic and continuous function. Fourier series is the basic mathematical tool of Fourier analysis used to analyze periodic functions through the decomposition of said function into an infinite sum of much simpler sinusoidal functions (as a combination of sines and cosines with integer frequencies).
- 3.
- 4.
Recall that \(e^{j\phi }=\cos \phi + j \sin \phi \).
- 5.
This is simplest definition for low-pass filters, but there are other definitions depending on the filtering characteristics (band-pass, high-pass) of the system.
- 6.
Band-stop filters passes most frequencies unaltered, but attenuates those in a specific range to very low levels. It is the opposite of a band-pass filter.
- 7.
Attribute of a control system that shows low sensitivity to effects that were not considered in the analysis and design phase of the system, e.g. disturbances, unmodeled dynamics, noise, etc.
- 8.
The modulus operator is kept for \(|\omega |\), but it could be removed as only positive frequencies have physical meaning. Negative values of the frequency will be used to build Nyquist diagrams in the framework of the Nyquist stability criterion in Chap. 6. Also, notice that if the transfer function contains the factor \((j\omega )^{-n}\), the log magnitude becomes \(-20n\log {(|\omega |)}\).
- 9.
\(20\log {(|k|)}\) if \(k\ne 1\).
- 10.
The profile of the phase plot and that of the asymptotes could change if the transfer function has negative static gain (adding \(180^{\circ }\)) and/or RHP poles. This can be analyzed in Sect. 4.8 devoted to NMP systems.
- 11.
Depending on the sign of the static gain and if the poles are in the LHP or RHP.
- 12.
If the transfer function contains the factor \((j\omega )^{n}\) the log magnitude becomes \(20n\log {(|\omega |)}\).
- 13.
Note that in the case of real zeros, their associated time constant is called \(\beta _\ell \) in this text, instead of also using \(\tau _\ell \) as for the poles.
- 14.
\(\kappa \) is associated with the complete transfer function, but here its sign is considered in the form in which poles and zeros are represented.
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Guzmán, J.L., Costa-Castelló, R., Berenguel, M., Dormido, S. (2023). Frequency Response. In: Automatic Control with Interactive Tools. Springer, Cham. https://doi.org/10.1007/978-3-031-09920-5_4
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