Locus Plotting: Significance and How it Works

The root locus of a feedback system is a graphical representation in the complex s-plane of the possible positions of its closed-loop poles for various values of a certain system parameter. 

This depiction shows the probable locations of the poles for the feedback system.

 This representation shows the possible locations of the poles for the system as a whole for different values of the parameter.

The angle condition can be satisfied by all of the points that are a component of the root locus. 

By using the magnitude condition, one may determine the value of the parameter that corresponds to a certain point on the root locus.

The root locus can be used to design the damping ratio (ζ) and natural frequency (ωn) of a feedback system, in addition to identifying whether or not the system is stable. 

It is possible to draw lines with a constant damping ratio radially out from the origin, and lines with a constant natural frequency can be drawn as arccosines whose centre points coincide with the origin.

 Calculating and implementing a gain K in the controller is made possible by selecting a location along the root locus that coincides with a desired damping ratio and natural frequency. 

This allows for the calculation of gain K. 

The majority of control textbooks include more complex methods of controller design utilising the root locus. 

For instance, lag, lead, PI, PD, and PID controllers can be created essentially using this technique.

It is assumed in the formulation of the damping ratio and the natural frequency that the overall feedback system may be accurately modelled by a system of the second order.

That is, it is assumed that the system possesses a dominant pair of poles. 

Since this is not always the case, it is standard procedure to do a simulation of the completed design in order to determine whether or not the project goals have been met. 

Developing a Locus Plot

In order to do an analysis of the negative feedback loop, the root locus plot is utilised. 

This plot illustrates the trajectories of the closed-loop poles as the feedback gain K is varied from 0 to. 

Please write the transfer function for the system and draw, by hand, the root locus with respect to K for each of the functions G(s) that are given below. 

Please provide a concise explanation of the rules that you employed in order to generate your sketch for each system, as well as calculations for any asymptotes.

•  G(s) (8+2)(8+6) s(s+1)(8 + 5) (s + 10) = 8+5 
•  G(s) = (8+30) (s²+28+10)
•   G(s) = 8+1 s(s²+28+10) 
• G(s) = 8+2 8²(8² +68+25)
•  G(s) = 2 (s+1)(8+3)(8+4)

The control system’s point of origin

The depiction of the roots of the characteristic p equation of the closed-loop system as a function of the variation of one system parameter is known as the root locus.

This plot is used to analyse how the gain affects the closed-loop system.

It is a method that determines how the poles move across the S-plane in response to alterations made to one of the control parameters.

In the process of determining the root locus, which is part of the control system, we will analyse the position of the roots, as well as their movement and the information linked with it. 

This information will be utilised to provide feedback on the functioning of the system.

An Illustration of a Root Locus

Determine the equation of the locus of a point such that the sum of its distances from (0, -1) and (0,1) is equal to 3.

Solution

Let there be a point on the specified locus identified by the coordinates P(x, y).

It should be obvious that

A = (0, −1) B = (0, 1)

Given the aforementioned conditions,

The sum of the distances from point P to A and point P to B is equal to 3.

As a result, we may write it as:-

PA + PB = 3

By rearranging the equation that was just given,

PA = 3 − PB

Now, bringing both sides of the equation back to their original form,

PA²=(3-PB)²

PA² =9+ PB² – 6PB

x²+(y+1)² = 9+x²+(y−1)-6PB

(y+1)² -(y-1)²-9= -6PB 4y – 9 = – 6PB

Again, bringing both viewpoints into balance,

16y²−72y+81 = 36PB² 16y²−72y+81

 = 36(x²+(y−1)²)

16y²−72y+81 = 36x²+36y²−72y+36 36x²+20y² = 45

The equation given above can be expressed as follows:

(x²/a²)+(y²/b²)=1

An ellipse can therefore be defined using the equation that was just presented.

As a result, the equation of locus is as follows:

The equation for an ellipse is 36x² + 20y² = 45.

Conclusion

The term “root locus analysis” refers to a graphical method that is used in the fields of control theory and stability theory.

 This is a method that can identify whether or not a system is stable, and it was developed by Walter R.

 Evans is specifically for use as a stability criterion in the field of classical control theory.

 The poles of the closed-loop transfer function are plotted in the complex s-plane as a function of a gain parameter using the root locus. For further information, see the pole-zero plot.

When compared to other methods, the Root Locus Technique in control systems is relatively simple to put into action.

We are able to make an accurate prediction of how well the overall system will do with the assistance of the root locus.

A more effective method of indicating the parameters is provided by the root locus.