Abstract
Consider the closed-loop control system shown in Fig. 8.1. The open-loop transfer function G(s) is assumed to be given as the cascade connection of the controller G c (s) and the plant G p (s), i.e., G(s) = G c (s)G p (s). The system error is defined as e(t) = r(t) − c(t), where \(e(t)=\mathcal {L}^{-1}\{E(s)\}\), \(r(t)=\mathcal {L}^{-1}\{R(s)\}\) and \(c(t)=\mathcal {L}^{-1}\{C(s)\}\).
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Notes
- 1.
The reason for this overshoot is explained in Sect. 8.1.2.
- 2.
This is because the dead zone characteristic presented in Fig. 8.5 is odd. Furthermore, a 0 = 0 for any odd “static” hard nonlinearity.
- 3.
This implies that point (−1, j0) belongs to the polar plot of G(jω)H(jω), i.e., the closed loop system is marginally stable.
- 4.
Note that because of the approximate nature of the method the predicted results are valid only to a certain extent.
- 5.
This is because the saturation characteristic presented in Fig. 8.13 is odd. Furthermore, a 0 = 0 for any odd “static” hard nonlinearity.
- 6.
See Fig. 8.4.
- 7.
This does not mean that another limit cycle with a different amplitude A exists. Recall that N(A) is a linear approximation of a nonlinear function; hence, N(A) changes if A changes and this does not require A to represent an oscillation amplitude in a steady state.
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Hernández-Guzmán, V.M., Silva-Ortigoza, R. (2019). Advanced Topics in Control. In: Automatic Control with Experiments. Advanced Textbooks in Control and Signal Processing. Springer, Cham. https://doi.org/10.1007/978-3-319-75804-6_8
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