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This paper addresses the root locus (locus of positive gain) and the complementary root locus (locus of negative gain) of biproper transfer functions (transfer functions with the same number of poles and zeros). It is shown that the root locus and complementary root locus of a biproper transfer function can be directly obtained from the plot of a suitable strictly proper transfer function (transfer function with more poles than zeros). There exists a lack of sources on the complementary root locus plots. The proposed procedure avoids the problems pointed out by Eydgahi and Ghavamzadeh, is a new method to plot complementary root locus of biproper transfer functions, and offers a better comprehension on this subject. It also extends to biproper open-loop transfer functions, previous results about the exact plot of the complementary root locus using only the well-known root locus rules.

The analysis of the location of the closed-loop poles in the

In the design of control systems, for changing time constants, and when large loop gain is required in inner loops, the complementary root locus is useful tool [

There exists a lack of sources on the complementary root locus and many textbooks about control systems present the root-locus method and its application in control design, but do not present information about complementary root locus [

In [

For instance, in [

In this paper an alternative procedure to plot the complementary root locus of transfer functions that are biproper is presented. The main idea was to show that the root locus and complementary root locus of a biproper transfer function can be directly and exactly obtained from the plot of the root locus and complementary root locus of a suitable strictly proper transfer function (transfer function with more poles than zeros). This procedure removes the problem cited above, because the plots are done only with strictly proper open-loop transfer functions. The proposed method extends the results presented in [

Given the feedback system described in Figure

A closed-loop controlled system.

Now, define

In this section a solution of the problem stated in Section

From (

Therefore, the characteristic equation of this feedback system is

Note that if

In the case where

The next theorem offers an alternative method to solve the problem stated in Section

The root-locus and complementary root-locus plots (

From (

The proof of Theorem

An alternative proof of Theorem

Figure

Relation between the gains

Figure

The characteristic equations (

In order to exemplify the results of the above theorem, the next examples are presented.

Consider the open-loop transfer function, that is, biproper, in Figure

It is easy to plot the root locus (

The root locus plot of Example

Now, the complementary root locus (

The complementary root locus plot from (

Another method to plot this complementary root locus, using only the root-locus construction rules, was proposed in [

Hence, defining

Now, we can easily plot the root locus of the above equation, in the complex plane

The root-locus plot in the plane

The complete root-locus plot (

The complete root-locus plot of (

Now, from Figure

The root-locus plot of (

The complementary root-locus plot of (

The root locus and complementary root locus plots presented in this paper, that illustrate the proposed method, were obtained by using the software MATLAB. The main contribution of this paper is an alternative analysis of the complementary root locus of biproper transfer functions, without using the theorems presented in [

Consider the open-loop transfer function

Eydgahi and Ghavamzadeh [

Now, one can use the results of Theorem

Therefore, from (

The root-locus plot from (

The complementary root-locus plot from (

The complete root-locus plot from (

Hence, from Figure

The root-locus plot from (

The complementary root locus of (

The complementary root-locus plot from (

A new method for plotting complementary root locus of biproper open-loop transfer functions has been presented in this article. The main idea was to show that this plot can be directly obtained from the root locus and complementary root locus of a suitable strictly proper transfer function. With this method it is not necessary to perform the additional analysis proposed in [

The proposed method allows an exact plot of complementary root locus, of biproper open-loop transfer functions, using only well-known root locus rules. From the method presented in [

The authors gratefully acknowledge the partial financial support by FAPESP, CAPES, and CNPq of Brazil.