Consider the shown negative feedback system. Let H(s=1, Ws=0, Vs)=0 and G(s) represent a spring-mass-damper system which can be modeled by the following transfer function:
Where, ζ = 0.01 is the damping ratio and ωn = 5 is the natural frequency. The desired output is a step input, R(s) = 1/s.
Controller E(s) De(s)
Plant
F(s)
RO
G(s)
OY
Sensor
X(s)
B(s)
H(s)
Figure 1
Let Dc(s) = Kp + Ka/s + iKi. Find the stability ranges of the PID control.