Qualitative graphical representation of Nyquist plots

doi:10.1016/j.sysconle.2015.06.005

Systems & Control Letters

Volume 83, September 2015, Pages 53-60

https://doi.org/10.1016/j.sysconle.2015.06.005 Get rights and content

Abstract

In this paper, the procedure for manually drawing the Nyquist plot of a generic transfer function is revised and, in particular, two novel parameters ( $Δ_{τ}$ , $Δ_{p}$ ), which allow to simplify this process, are presented. Thanks to these parameters, the analysis of the frequency response at low and high frequencies is considerably enhanced, with a very little effort. These parameters allow to predict initial and final directions of the polar curve in the vicinity of initial and final points and consequently the sectors of the complex plane where the plot starts/ends. In many cases it is possible to obtain a qualitative Nyquist plot, able to correctly predict the stability properties of the closed-loop system, by simply joining the initial and final tracts found with the proposed procedure. Moreover, the analysis based on these parameters can aid to correctly interpret the plots obtained with computer programs which often, in particular when poles at the origin are present, hide the behavior of the frequency response in the area close to the origin of the complex plane.

Introduction

In classical control theory, Nyquist stability criterion plays a central role for discussing the stability of closed-loop systems. Its effectiveness is strictly related to the correct drawing of the polar plots of the loop transfer function $G (j ω)$ . However, in many applications the exact plot is not essential and a rough sketch reproducing the shape of the polar plot of $G (j ω)$ may be sufficient, provided that this plot correctly reproduces the encirclements of the loop transfer function about the critical point $- 1 + j 0$ and the intersections with the real axis. The rules for drawing Nyquist plots seem well settled and are illustrated in all controls textbooks with no substantial differences, see e.g. [1], [2], [3] among many others. They are based on the analytical computation of the frequency response $G (j ω)$ for $ω = 0$ , $ω = \infty$ and for those frequencies where the diagram intersects the real and imaginary axes. In more complex cases, i.e. high-order systems, polar plots are obtained indirectly, by means of a sort of “translation” of Bode diagrams in the polar plane. Unfortunately, standard approaches are characterized by some drawbacks. They may require complex calculations. Moreover, an analysis based on Bode diagrams (with the phase starting/ending at an integer multiple of $π / 2$ ) may lead to the wrong conclusion that the polar plots always start from or end to a point located in the real or imaginary axis (see for instance the figures reported in textbook [2], and in particular Figs. 8–33). Indeed, as highlighted by [4], when the polar plots start at infinity the locus generally does not approach a coordinate axis but tends to get further and further away from these axes.

The use of CACSD (Computer Aided Control System Design) programs is the option that provides the best results in terms of accuracy. But, also in these cases some problems may arise. In fact, often the obtained polar plots may result unreadable because of the large span in the magnitude over the entire frequency range, that hides the local behavior of the curve, in particular in the region enclosed by the unit circle. This situation is quite common when systems owning poles on the imaginary axis are considered. In order to cope with this problem in [5], [6] a logarithmic scaling of the magnitude of the frequency response is proposed, that allows to magnify the parts of the polar plot close to the origin without losing the diagram overview. This approach gives good results in particular with respect to the problem of detecting the intersection with the (negative) real axis, but it is characterized by two important drawbacks: the proposed plotting technique needs a numerical analysis software (Matlab functions have been developed by the authors) and it requires to arbitrarily set the minimum value of the magnitude that can be represented in the diagram.

In this paper, the method for manually drawing polar plots is revised. The proposed approach has a great value from an educational point of view for a twofold reason. On the one hand, the definition of two novel parameters ( $Δ_{τ}$ , $Δ_{p}$ ) contributes to greatly simplify the drawing procedure and to improve the correctness of the Nyquist plot, in particular in the regions “far from” and “close to” the origin. On the other hand, these two parameters and the related considerations can also be the key to correctly interpret many polar plots obtained with CACSD programs. This allows to correctly analyze the stability of the system in a feedback configuration on the basis of the Nyquist criterion. Moreover, the use of parameters $Δ_{τ}$ and $Δ_{p}$ can be useful to study the dynamic behavior of a linear system in feedback configuration with a nonlinear static element. In this case, these parameters used together with well-known methods such as circle criterion [7], Popov criterion [7] or describing function method [8], may be conclusive to assess the stability of the nonlinear feedback system.

The paper is organized as follows. In Section 2 the proposed parameters for a qualitative evaluation of Nyquist plots are defined and their meaning explained. In Section 3 the method for drawing Nyquist plots by exploiting the proposed parameters is illustrated step by step, and some numerical examples are provided in Section 4. Concluding remarks are given in the last section.

Section snippets

Qualitative graphical analysis of the frequency response in the complex plane

Consider a transfer function $G (s)$ , without time-delays, expressed as follows: $G (s) = \frac{K}{s^{h}} \frac{\prod_{i = 1}^{p} (β_{i, m_{i}} s^{m_{i}} + β_{i, m_{i} - 1} s^{m_{i} - 1} + \dots + β_{i, 1} s + β_{i, 0})}{\prod_{i = 1}^{q} (α_{i, n_{i}} s^{n_{i}} + α_{i, n_{i} - 1} s^{n_{i} - 1} + \dots + α_{i, 1} s + α_{i, 0})},$ where $α_{i, 0} \neq 0$ , $β_{i, 0} \neq 0$ , $\sum_{i = 1}^{p} m_{i} = m$ and $\sum_{i = 1}^{q} n_{i} = n$ . Note that from (1) it is possible to obtain all the standard expressions of rational polynomial transfer functions, i.e.

•
Polynomial form: $G (s) = \frac{b_{m} s^{m} + b_{m - 1} s^{m - 1} + \dots + b_{1} s + b_{0}}{s^{h} (a_{n} s^{n} + a_{n - 1} s^{n - 1} + \dots + a_{1} s + a_{0})}$ .
•
Zero-pole-gain form: $G (s) = ρ \frac{(s - z_{1}) (s - z_{2}) \dots (s - z_{m})}{s^{h} (s - p_{1}) (s - p_{2}) \dots (s - p_{n})}$ .
•
Time-constant form: $G (s$

Qualitative drawing of the Nyquist plot

The qualitative drawing of the Nyquist plot of a generic transfer function $G (s)$ having the structure given in (1) can be done using the following procedure.

1.
Initial point. The initial point of the diagram can be determined by computing magnitude $M_{0} = | G_{0} (j ω) |$ and phase $φ_{0} = arg {G_{0} (j ω)}$ of the approximating function $G_{0} (j ω)$ for $ω \to 0^{+}$ .
2.
Phase shift (lead or lag) for $ω ≃ 0^{+}$ . For $ω ≃ 0^{+}$ the Nyquist plot starts with a phase shift $Δ φ_{0}$ which is concordant with the sign of parameter $Δ_{τ}$ defined in (5). Therefore,

Numerical examples

In order to highlight the significance of the two parameters $Δ_{τ}$ and $Δ_{p}$ , a few examples, that appear in the literature,⁴ have been taken into account. For the sake of simplicity, the considered transfer functions are supposed to be characterized by a positive static gain $μ$ and positive time constants $τ_{i}^{'}$ , $τ_{i}$ , but it is worth noticing that the correctness of the proposed procedure does not depend on these

Conclusion

In this paper, the use of two novel parameters $Δ_{τ}$ and $Δ_{p}$ for improving and simplifying the standard techniques for qualitative drawing of the Nyquist plot is proposed. The two parameters can be computed with simple calculations involving only additions and divisions, and in many cases are sufficient to draw qualitative plots that provide all the information needed for assessing the stability of linear and nonlinear systems with a feedback control. Moreover, the two parameters can be helpful for

References (8)

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Feedback Control of Dynamic Systems
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R.C. Dorf et al.
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(2008)
P. Ferreira
Concerning the Nyquist plots of rational functions of nonzero type
IEEE Trans. Educ.
(1999)

There are more references available in the full text version of this article.

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Qualitative graphical representation of Nyquist plots

Abstract

Introduction

Section snippets

Qualitative graphical analysis of the frequency response in the complex plane

Qualitative drawing of the Nyquist plot

Numerical examples

Conclusion

Feedback Control of Dynamic Systems

Automatic Control Systems

Modern Control Systems

Concerning the Nyquist plots of rational functions of nonzero type

IEEE Trans. Educ.