HOMEWORK SET NO. 1
Problem 1:
(a) Describe the project topic that you have selected, present the equations of motion (mathematical model) for the open-loop system, and describe the control objectives as quantitatively as possible.
(b) For this project topic, present analyses and/or simulation results to show the open-loop system behavior. Also discuss why you think this may be a problem where adaptive control could be useful.
Problem 2:
Problem 1.5 from your textbook.
Problem 3:
A servomechanism is described by the continous-time transfer function:
Y(s)/U(s) = G(s) = K/[s(Ts+1)]
Show that the equivalent discrete-time transfer function, when u(t) is generated by a zero order hold and the sampling rate is D, is given by:
H(z) = [Kd(z-b)]/[(z-1)(z-a)]
where
Kd = K(Ta - T + D)
b = [Ta + aD - T]/[Ta - T + D]
a = exp(-D/T)
Problem 4:
Consider a system with the plant model x(t+1) = u(t), the disturbance model h(t+1) = h(t), and the output equation y(t) = x(t) + h(t). Express the system in state variable form, then show that the corresponding difference equation form is:
y(t) = y(t-1) + u(t-1) - u(t-2)
Problem 5:
For the servo system in Problem 3 above, write the discrete-time system equations in difference equation form, and also in the regression form:
y(t+1) = jT(t)q
where
j is the measurement vector and q is the parameter vector.