Beyond p-only control via root locus by Frank Owen, PhD, PE polyXengineering, Inc. San Luis Obispo, California

PI, PD, PID control P-only control does not modify the root locus of the existing system It simply moves the closed-loop poles along the existing root locus Design of P-only control simply means setting KP to put the poles in a desirable location See other videos in this series to understand P-only controller design via root locus To understand PI and PD control via root locus, we need to look at the structure of those controllers Once PI and PD controller design is understood, PID is nothing new PID can be looked upon at PI with a D component or PD with an I component That is, you design the PI and add the D component to it …or you design the PD and add the I component to it

PI controller structure I control is added to a P-only controller to reduce or eliminate stead-state error Its structure is: 𝐺 𝑃𝐼 = 𝐾 𝑃 + 𝐾 𝐼 𝑠 = 𝐾 𝑃 ∙𝑠+ 𝐾 𝐼 𝑠 = 𝐾 𝑃 𝑠+ 𝐾 𝐼 𝐾 𝑃 𝑠 Thus we are adding… …a zero at − 𝐾 𝐼 𝐾 𝑃 …and a gain …a pole at the origin This changes the root locus of the system…

1st-order system with PI Without the controller: With the controller:

PD controller structure D control is added to a P-only controller to speed up the response Its structure is: 𝐺 𝑃𝐷 = 𝐾 𝑃 + 𝐾 𝐷 ∙𝑠= 𝐾 𝐷 𝑠+ 𝐾 𝑃 𝐾 𝐷 Thus we are adding… …a zero at − 𝐾 𝑃 𝐾 𝐷 …and a gain This also changes the root locus of the system…

1st-order system with PD Without the controller: With the controller: With the zero placed as shown, the CL pole makes the system slower as it migrates along the root locus.

PiD controller structure The entire PID structure is even more complicated Its structure is: 𝐺 𝑃𝐼𝐷 = 𝐾 𝑃 + 𝐾 𝐼 𝑠 + 𝐾 𝐷 ∙𝑠= 𝐾 𝐷 ∙ 𝑠 2 + 𝐾 𝑃 ∙𝑠+ 𝐾 𝐼 𝑠 Thus we are adding… …and two zeros …a pole at the origin The zeros may be added wherever you like, real or complex, by choosing KP , KI , and KD .