The original work is at here. I borrow this existing implementation and test it with some tuning procedure I have read.

There are many propose method to tune a PID controller, and here I write down my finding in following these procedure to tune a PID controller

1. Online trial and error method

Adopt from link

Here is the procedure

• Find K_P: Set τ_I=∞, and τ_D=0. Find K_C such that the system is critically stable. Denote this value at K_{P, max}

• Find τ_I: Set K_P=0.5K_{P, max}, and τ_D=0. Find τ_I such that the system is critically stable. Denote this value at τ_{I, min}

• Find τ_D: Set K_P=0.5K_{P, max}, and τ_I=2τ_{I, min}. Find τ_D such that the system is critically stable. Denote this value at τ_{D, max}

• The tuning result is K_P=0.5K_{P, max}; τ_I=2τ_{I, min}; τ_D=0.5τ_{D, max}

Here are tuning steps

Step	Result
1 (KP=2; τI=∞;τD=0)
2 (KP=1; τI=0.012;τD=0)
3 (KP=1; τI=0.006;τD=0.006)
4 (KP=1; τI=0.006;τD=0.003)

2. Ziegler-Nicholas method

Detail please refer to link

• Setting K_I=0, and K_D=0, and find ultimate gain K_u and oscillation periodT_u

• Setting K_P, τ_I and τ_D according to this table

Ziegler-Nicholas in action

Step	Result
Set KP st. system oscillate. Ku=2/1 and Tu=0.02*2
P controller (KP=1; KI=0; KD=0)
PI controller (KP=0.9; KI=27; KD=0)
PD controller (KP=1.6; KI=0; KD=0.008)
classic PID (KP=1.2; KI=60; KD=0.006)
Pessen Integral Rule (KP=1.4; KI=87.5; KD=0.0084)
some overshoot (KP=0.66; KI=33; KD=0.0088)
no overshoot (KP=0.4; KI=20; KD=0.00533333)

1. I run it in both windows and Linux and find that the tuning value is quite difference. It may because of the different implementation of 3rd parties packages.