Root locus analysis
This is a technique used as a stability criterion in the field of classical control theory developed by Walter R. Evans which can determine stability of the system.
The root locus plots the poles of the closed loop transfer function in the complex s-plane as a function of a gain parameter (see pole–zero plot).
[1][2][3][4][5][6][7][8][9] In addition to determining the stability of the system, the root locus can be used to design the damping ratio (ζ) and natural frequency (ωn) of a feedback system.
Lines of constant damping ratio can be drawn radially from the origin and lines of constant natural frequency can be drawn as arccosine whose center points coincide with the origin.
By selecting a point along the root locus that coincides with a desired damping ratio and natural frequency, a gain K can be calculated and implemented in the controller.
This is often not the case, so it is good practice to simulate the final design to check if the project goals are satisfied.
The root locus of a feedback system is the graphical representation in the complex s-plane of the possible locations of its closed-loop poles for varying values of a certain system parameter.
The points that are part of the root locus satisfy the angle condition.
The value of the parameter for a certain point of the root locus can be obtained using the magnitude condition.
Typically, a root locus diagram will indicate the transfer function's pole locations for varying values of the parameter
and the use of simple monomials means the evaluation of the rational polynomial can be done with vector techniques that add or subtract angles and multiply or divide magnitudes.
The vector formulation arises from the fact that each monomial term
The polynomial can be evaluated by considering the magnitudes and angles of each of these vectors.
So to test whether a point in the s-plane is on the root locus, only the angles to all the open loop poles and zeros need be considered.
It turns out that the calculation of the magnitude is not needed to determine if a point in the s-plane is part of the root locus because
The root locus only gives the location of closed loop poles as the gain
a horizontal running through that zero) minus the angles from the open-loop poles to the point
Note that these interpretations should not be mistaken for the angle differences between the point
The plot of the root locus then gives an idea of the stability and dynamics of this feedback system for different values of
Given the general closed-loop denominator rational polynomial the characteristic equation can be simplified to The solutions of
to this equation are the root loci of the closed-loop transfer function.
Given we will have the characteristic equation The following MATLAB code will plot the root locus of the closed-loop transfer function as
varies using the described manual method as well as the rlocus built-in function:
The following Python code can also be used to calculate and plot the root locus of the closed-loop transfer function using the Python Control Systems Library[14] and Matplotlib[15].The root locus method can also be used for the analysis of sampled data systems by computing the root locus in the z-plane, the discrete counterpart of the s-plane.
The equation z = esT maps continuous s-plane poles (not zeros) into the z-domain, where T is the sampling period.
A diagonal line of constant damping in the s-plane maps around a spiral from (1,0) in the z plane as it curves in toward the origin.
The Nyquist aliasing criteria is expressed graphically in the z-plane by the x-axis, where ωnT = π.
Many other interesting and relevant mapping properties can be described, not least that z-plane controllers, having the property that they may be directly implemented from the z-plane transfer function (zero/pole ratio of polynomials), can be imagined graphically on a z-plane plot of the open loop transfer function, and immediately analyzed utilizing root locus.
The idea of a root locus can be applied to many systems where a single parameter K is varied.
For example, it is useful to sweep any system parameter for which the exact value is uncertain in order to determine its behavior.
