Tuning methods for fractional-order controllers

摘要

A method for tuning a fractional-order proportional-integral (PI) controller $$ C\left(s\right)=K_p\left(1+K_i\frac{1}{s^{\alpha }}\right)$$includes deriving values for K_p, K_i, and α that satisfy a flat phase condition represented by $$ \angle \frac{dG\left(s\right)}{ds}{{}_{s={\mathrm{jw}}_c}=\angle G\left(s\right)}_{s={\mathrm{jw}}_c}.$$K_p is derived to ensure that a sensitivity circle tangentially touches a Nyquist curve on a flat phase. K_i and α are derived to ensure that a slope of a Nyquist curve is approximately equal to the phase of an open loop system at a given frequency. A method for tuning a PI controller $$ C\left(s\right)=K_p\left(1+K_i\frac{1}{s^{\alpha }}\right)$$where the gain crossover frequency w_c is known includes obtaining measurements of ∠P(jw_c) and |P(jw_c)| of an unknown, stable real plant P(s) using iterative relay feedback tests, and approximating the derivative of the plant phase, ∠P(jw_c), provided by the relationship $$ s_p\left(w_c\right)=w_c\frac{d\angle P\left(\mathrm{jw}\right)}{dw}{❘}_{w_c}\approx \angle P\left({\mathrm{jw}}_c\right)+\frac{2}{\pi }\left[\ln K_g-\ln P\left({\mathrm{jw}}_c\right)\right],$$wherein |K_g|=P(0) is the static gain of the real plant P(s).