A new and simple method to construct root locus of general fractional-order systems

Fundamentals of fractional‐order LTI circuits and systems: number of poles, stability, time and frequency responses

“…In Table III (case a < 0, K > 1), the system oscillates when 0:25π ¼ sec À1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1:25 þ p osc p , then the value of p osc = 0.75. Figure 6(b) shows the number of unstable poles function against p, where the system moves from unstable to stable behavior when p = 0.75, and the frequency response is shown in Figure 6(c), which is consistent with [37].…”

“…which can be written in the form D s ð Þ ¼ 0:8s 22α þ 0:5s 9α þ 1 þ p; α ¼ 0:1: (66) Figure 6(d) shows the number of unstable poles function against p, where the system switches from stable to unstable behavior when p = 1.316, which is consistent with [37].…”

“…In Tables II and III, the number of physical poles functions (N*) when a = À 1is Example (3): [37,38]: Consider the following characteristic equation: D s ð Þ ¼ s 2 À 3s 1:5 À 2s þ 2 þ p ð Þs 0:5 þ 12 À p; p ≥ 0: (61) This example can be written in the form D s ð Þ ¼ s 4α À 3s 3α À 2s 2α þ 2 þ p ð Þs α þ 12 À p; α ¼ 0:5: (62) Figure 6(a) shows the number of unstable poles function (40) against p, where the system is always unstable, which is consistent with [37,38]. In Tables II and III, the number of physical poles functions (N*) when a = À 1is Example (3): [37,38]: Consider the following characteristic equation: D s ð Þ ¼ s 2 À 3s 1:5 À 2s þ 2 þ p ð Þs 0:5 þ 12 À p; p ≥ 0: (61) This example can be written in the form D s ð Þ ¼ s 4α À 3s 3α À 2s 2α þ 2 þ p ð Þs α þ 12 À p; α ¼ 0:5: (62) Figure 6(a) shows the number of unstable poles function (40) against p, where the system is always unstable, which is consistent with [37,38].…”

“…Example (5): [37]: Consider the following characteristic equation: D s ð Þ ¼ 0:8s 2:2 þ 0:5s 0:9 þ 1 þ p; p ≥ 0;…”

Review of fractional PID controller

“…There is a method to construct the root locus of a fractional order control system [71,72] . Based on the root locus, one can perform an analysis of the system.…”