Root Locus Practical Sketching Rules for Fractional-Order Systems

Table 1

Practical rules for RL sketching of FO systems as defined in (11).

	Fractional-order case, (11)
Rule  1	RL is symmetrical about the real axis.
Rule  2	If 𝑙<∑^𝑛𝑝=1𝛼𝑝<𝑙+2, 𝑙= 1, 3, 5, …,; then the number of branches is 𝑙+1.
Rule  3	Mark all open-loop poles and zeros on the s-plane.
Rule  4	If necessary, use the angle condition (1) to determine the open-loop poles that have RL branches departing from them.
Rule  5	Compute the asymptotes centroid, 𝜎, and angle, 𝜑, according to the following expressions: 𝜎=−∑^𝑛𝑝=1𝑎𝑝𝛼𝑝+∑^𝑚𝑞=1𝑏𝑞𝛽𝑞∑^𝑛𝑝=1𝛼𝑝−∑^𝑚𝑞=1𝛽𝑞,; 𝜑=(2ℎ+1)⋅180^∘∑^𝑛𝑝=1𝛼𝑝−∑^𝑚𝑞=1𝛽𝑞,ℎ=0,±1,±2,….
Rule  6	Points in the real axis belong to the RL if, for all poles and zeros seen to the right, 𝛿=∑^𝑛𝑝=1𝛼𝑝−∑^𝑚𝑞=1𝛽𝑞 is an integer odd number.
Rule  7	Find the intersection of the RL with the imaginary axis making 𝑠=𝑗𝜔 in the characteristic equation and solving it in order to determine 𝐾 and 𝜔.
Rule  8	Compute the breakaway and break-in points using the characteristic equation and determining 𝑑𝐾/𝑑𝑠=0.
Rule  9	Determine the departure and arrival angles using the angle condition (1).