In a previous problem, your friend, Dr.
Abner Mallity, was working on an altitude control for a biplane.
He devised a model for the aircraft, but is now having second thoughts.
After some experimentation with paper models of the aircraft he came to
the following conclusions.
In the pitch axis there
seem to be two complex poles. He has concluded this by flying numerous
paper aircraft that exhibit pitch oscillations that die out. From
the frequency of the oscillations he has observed, he has concluded that
he can design for an undamped natural frequency of wn
= 1.
From the same observations,
he has concluded that the damping ratio of the aircraft is probably around
z
= 0.4.
After further consideration, Mallity realizes that he chose the model for
the aircraft too quickly and with too little thought. (In other words,
it's not really a good/true description of the aircraft dynamics.)
In retrospect, he realizes that the transfer function he guessed at might
describe the reaction of the pitch of the aircraft, but not the altitude.
Now, Mallity thinks that the pitch controls the rate at which the aircraft
gains altitude. That's probably a closer approximation to the truth.
He now wants to use a description like the one embedded in the closed loop
block diagram below.
Once again, Mallity needs your help. Again, he has a simulation for
this system, and that's shown below. You can experiment with this
simulation - since you can change the gain
to any value you want - to help your understanding
of the system Mallity has proposed. Again, the simulation has been
written to reflect the environment where the president of AAA (Mr. Mel
Lowe) lives, i.e. Worland, Wyoming. That's the Big Horn Mountains
in the backdrop. Worland is just west of the Big Horns (Cloud Peak
goes to 13,125 ft) and is at an elevation of about 4000 ft. The desired
altitude of 7000 ft, used in the simulation, is a reasonable altitude for
this area, except when flying near the Big Horn Mountains.
There is something wrong, either with the system or the simulator.
What could be presented in 8 seconds now seems to take longer. There
is not enough time in the simulation to see the entire response, to the
point where the response settles out. Mallity has gone back to his
simulation-writing friends, and has gotten a simulator with four times
as much time. That's shown below.
Mallity is having serious trouble understanding the behavior in this simulator.
He needs your help understanding two different aspects of the behavior
of this system. He notices the following.
He notices that there
seems to be very little error in the system. Further, the amount
of error does not depend upon the gain in the system.
He also notices that there
is an upper limit to the gain. Above a particular gain value, the
system seems to become unstable.
There also seems to be
a tendency for the system to take longer to settle down as the gain increases.
We have developed some aids for this problem. If you are conversant
with the root locus technique you can click here to see a root locus for
the system.
