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ENGR5001 / Week 5
Old spec 5.2 examples, PD and PV controlTutorialOld spec ENGR2022:112 min

ENGR202 (old spec) Control 5.2 PD, PV and PID Control, PD and PV examples (slides 4-9)

The plant is X=ωn2s2+2ζωns+ωn2UX = \dfrac{\omega_n^2}{s^2 + 2\zeta\omega_n s + \omega_n^2}U with unity negative feedback and error e=d−xe = d - x.

Formulas you may need
  • PD: u=kpe+kddedtu = k_pe + k_d\dfrac{de}{dt}; PV: u=kpe−kvdxdtu = k_pe - k_v\dfrac{dx}{dt} (learn this)
  • Compare the closed-loop characteristic equation with s2+2ζclωcls+ωcl2s^2 + 2\zeta_{cl}\omega_{cl}s + \omega_{cl}^2 (learn this)
  1. (a)
    PD control u=kpe+kd de/dtu = k_pe + k_d\,de/dt. Find the closed-loop Transfer Function, characteristic equation and steady state output for a unit step.
  2. (b)
    PV control u=kpe−kv dx/dtu = k_pe - k_v\,dx/dt (velocity feedback of the output in an inner loop). Find the closed-loop Transfer Function and steady state output, and say how it differs from PD.
  3. (c)
    For a plant with ζ=0.5\zeta = 0.5 and ωn=1\omega_n = 1 rad/s, choose PV gains so that the closed loop has natural frequency 2 rad/s and damping ratio 0.7. What is the steady state output for a unit step?