PID Controller Example

In this example, we are designing a PID controller whose job is to minimize the effects of a disturbance force on the position, velocity, and acceleration of a block.

Figure 1.

Referring to the figure above, a mass of 2Kg is subjected to a disturbance force, F(t). The PID is required to apply a stabilizing force to damp out the effects of the disturbance and bring the block back to where it was.

The design objective
The objective of the optimization is to determine the parameters of the PID controller so that the sum of the RMS deviations (from initial rest position) of Displacement, Velocity, and Acceleration due to the disturbance is minimized.
The design variables
Three gains of the PID controller, k p , k i , k d MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa aaleaacaWGWbaabeaakiaacYcacaWGRbWaaSbaaSqaaiaadMgaaeqa aOGaaiilaiaadUgadaWgaaWcbaGaamizaaqabaaaaa@3D8B@ are defined as Design Variables (Dv).
DV Value bL, bU
kp 0.0 (0.0, 1.0)
ki 0.5 (0.0, 1.0)
kd 0.0 (0.0, 1.0)
The constraint
1 – kp2 – kd2 – ki2 < 0