Your boss wants you to design an autopilot control system for the pitch angle, t
ID: 2083108 • Letter: Y
Question
Your boss wants you to design an autopilot control system for the pitch angle, theta, of a new commercial airplane. One of your team members comes up with the following dynamic equation for the pitch angle theta: 5 theta(t) + 40theta(t) + 75theta(t) = theta_d(t) where theta_d (t) is the desired input pitch angle. a) Find the transfer function P(s) = theta(s)/theta_d(s) for the pitch angle dynamics. (b) Is the open-loop system stable? Please explain your answer. (c) Apply a gain K in the forward path such that the open loop transfer function is KP(s) as shown in Figure la. Use the Final Value Theorem to determine the steady-state value of theta(t) as t rightarrow infinity when an input of theta_d = 5u(t) is applied. (u(t) is a unit step input) (d) Based on your answer in (c), what is the steady-state error in the open-loop system? (e) Find theta(t) using the inverse Laplace transform when an input of theta_d = 5u(t) is applied to the system. Based on part (c), does your answer make sense? Please explain why it does or does not. (f) Assume you can measure the pitch angle, theta, with a gyro in the feedback path as shown in Figure 1b. Use the final Value Theorem to determine the steady-state value of theta(t) as t rightarrow infinity when an input of theta_d = 5t(t) is applied to the closed-loop system. (u(t) is a unit step input) (g) Based on your answer in (f), what is the steady-state error in the closed-loop system? (h) Based on your answers in (d) and (g), which system has better performance with regard to steady-state error for a gain K? Please explain your answer.Explanation / Answer
a) We are given following differential equation of the system :
5d2/dt2 + 40 d/dt + 75(t) = d(t)
Taking laplace transform of both sides we get,
5S2(s) + 40S(s) +75(s) =d(s)
(5S2 + 40S +75)(s) =d(s)
Therefore transfer function P(s)
P(s) = (s) /d(s) = 1 / (5S2 + 40S +75) .............1
This is the required transfer function of the system
b)Given system is stable.
Poles of the system lie at -5 & -3
It is clear that both poles of this system lie in the left half of the s-plane hence system is stable.
c)Now gain of K is applied to transfer function and input d(t) = 5u(t) is applied.
therefore d(s) = 5 / S2
(s) = KP(s)U(s)
(s) = K[1 / (5S2 + 40S +75)][5/S2]
now by final value theorem,
= lim S (s)
t->infinity S->0
= lim KS[1 / (5S2 + 40S +75)][5/S2]
S->0
= lim 5K /( 5S3 + 40S2 +75S)
S->0
=
therefore steady state value is infinity i.e. output is unstable and goes on increasing and doesn't reach steady state.
d) steady state error = d - = 5u(t) - = -
From result in c it is clear that steady state error is infinity
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