Parameter
Symbol Typical Values
Inertia Friction
J 1 Nms^2/rad
D 20 Nms/rad
Amplifier*
Ka 10-1000
Resistance
Ra Km L 19
Motor Constant
5 Nm/A
Inductance
1 mH
3.5.1 Theoretical Derivation
1. For the open-loop control, i.e. no feedback as shown in Figure 12, derive an expression for the θ(t) if:
(a) in(t) is a unit step and
(b) the amplifier gain Ka = 10, 100, and 1000.
Din(S) K 0.(s)
Figure 12: Block diagram of the open-loop position control. The open-loop indicates no feedback signal is fed into the system input.
2. For the closed-loop control of which the amplifier gain Kg = 100, calculate the:
(a) time to this peak response is tp,theory = and n1 -2 4 c settling time is ts,thcory= Swn
Hint: Given that the characteristic equation of a second-order system is s^2 + 2ζwn + wn^2 = 0, where ζ is the damping ratio and wn is the natural/undamped frequency of the system. If underdamping, the characteristic equation of the system has complex s-poles. Therefore, the equation can be arranged in the form of (s + (wn + jwn1-))(s + (wn - jwn1-)) = 0, and the corresponding s-poles are -ζwn + jwn√(1-ζ^2) and -ζwn - jwn√(1-ζ^2). Thus, via comparison of 1. the arranged equation to the equation obtained in practice, or 2. the s-poles to those obtained in practice, ζ and wn can be identified.