The Construction of Asymptotic Bode Plots: A New Direct Method

TL;DR

This work introduces a direct method for constructing asymptotic Bode plots of a transfer function $G(s)$ by using generalized approximating functions $G_k(s)$, built from critical frequencies and a systematic set of properties. The method yields a piecewise-linear magnitude plot with outer and inner segments and a stepwise phase plot derived from recursive phase updates, without requiring detailed factor-by-factor analysis. Two case studies illustrate the approach, showing close agreement with actual Bode plots and demonstrating its efficiency and scalability relative to classical methods. Overall, the direct method offers a more systematic, potentially more precise, and time-saving framework for initial qualitative frequency-domain analysis, especially as the number of transfer-function factors grows.

Abstract

Bode plots represent an essential tool in control and systems engineering. In order to perform an initial qualitative analysis of the considered systems, the construction of asymptotic Bode plots is often sufficient. The standard methods for constructing asymptotic Bode plots are characterized by the same drawbacks: they are not systematic, may be not precise and time-consuming. This is because they require the detailed analysis of the different factors composing the considered transfer function, meaning that more and more intermediate steps are required as the number of factors increases. In this paper, a new method for the construction of asymptotic Bode plots is proposed, which is based on the systematic calculations of the so-called generalized approximating functions and on the use of well defined properties. The proposed method is referred to as a direct method since it allows to directly draw the asymptotic Bode magnitude and phase plots of the complete transfer function without requiring the detailed analysis nor the plots construction of each factor. This latter feature also makes the proposed direct method more systematic, potentially more precise and less time-consuming compared to standard methods, especially when dealing with a large number of factors in the transfer function. The comparison of the proposed direct method with the standard approaches is performed, in order to examine the benefits offered by the direct method.

Figures (6)

• Figure 1: Graphical representation of the asymptotic Bode magnitude plot of function $G(s)$ in \ref{['Gs_Fact']}. The latter has a piecewise behavior composed of a sequence of $r-1$ inner linear segments $G_k(\omega)=|G_k(s)|_{s=j\omega}$ for $\omega \in \mathcal{B}^\omega_{k}$ in \ref{['B_omega_k']} computed as in Property \ref{['bode_magn_prop']}, where $r=\hbox{dim}(\Omega_c)$ in \ref{['omega_c_defini']}, and $2$ external linear segments.
• Figure 2: Graphical representation of the stepwise Bode phase plot of function $G(s)$. The latter has a piecewise behavior composed of a sequence of $r+1$ horizontal segments $\varphi_k(\omega)$ computed as in Property \ref{['critical_gain_defini_prop']}, where $r=\hbox{dim}(\Omega_c)$ in \ref{['omega_c_defini']}, and of $r$ vertical segments obtained by connecting the endpoints of the horizontal segments $\varphi_k(\omega)$.
• Figure 3: Asymptotic Bode plots of function $G(s)$ in \ref{['Gs_example']} obtained using the direct method. (a) Asymptotic magnitude plot and actual magnitude plot. (b) Stepwise phase plot, asymptotic phase plot and actual phase plot. The critical frequencies are $\omega_1\!=\!2$ rad/s, $\omega_2\!=\!30$ rad/s, and $\omega_3\!=\!200$ rad/s, while the corresponding critical frequencies for the construction of the asymptotic phase plot are $\omega^a_1$, $\omega^b_1$, $\omega^a_2$, $\omega^b_2$, $\omega^a_3$, and $\omega^b_3$.
• Figure 4: Asymptotic Bode plots of function $G(s)$ in \ref{['Gs_example_2']} obtained using the direct method. (a) Asymptotic magnitude plot and actual magnitude plot. (b) Stepwise phase plot, asymptotic phase plot and actual phase plot. The critical frequencies are $\omega_1=0.1$ rad/s, $\omega_2=2$ rad/s, $\omega_3=8$ rad/s, and $\omega_4=80$ rad/s, while the corresponding critical frequencies for the construction of the asymptotic phase plot are $\omega^a_1$, $\omega^b_1$, $\omega^a_2$, $\omega^b_2$, $\omega^a_3$, $\omega^b_3$, $\omega^a_4$, and $\omega^b_4$.
• Figure 5: Asymptotic Bode plots of function $G(s)$ in \ref{['example_2']} obtained using the direct method. (a) Asymptotic magnitude plot and actual magnitude plot. (b) Stepwise phase plot, asymptotic phase plot and actual phase plot. The critical frequencies are $\omega_1=1$ rad/s and $\omega_2=5$ rad/s, while the corresponding critical frequencies for the construction of the asymptotic phase plot are $\omega^a_1$, $\omega^b_1$, $\omega^a_2$ and $\omega^b_2$.