I am taking a Controls class and am stuck on a homework problem. I don't want the solution, but rather perhaps a hint....
I have a controls equation:
y" + 4.2y' + 9y = C
This equation is the result of a equation modified by control function B[g1 y(t), g2 y'(t)].
The original equation is y" - k1*y = k2 * B[]. I am give zeta = 0.7 and omega = 3--from which y" + 4.2y'+9y = C was obtained.
Now the problem states:
determine if y(t) goes to 0, for all C values and all initial conditions. If not, determine what steady-state value it does go to. and then try to modify function B[] to overcome the problem.... Note: C is not known, or measurable.(it represents wind disturbance, but is assumed constant by the problem statement).
Here's what I have so far:
Y(s) = C/s[s^2+4.2s+9] = C/s(s+ (-2.1+j2.1))(s + (-2.1-j2.1));
applying final value theorem:
sY(s) as s->0, gives
sC/s(s+r1)(s+r2),
then lim s->0 gives C/(r1*r2), where r1,r2 are the roots of the characteristic equation (-2.1 +/- j2.1).
so the steady state value is:
C/8.8
If I've screwed anything up to this point, please let me know.
Now, this is the part that I don't understand:
how to modify B[y,y'] from B = (-9-k1)/k2*y(t) -4.2/k2 * y'(t) to overcome the steady state error?
I see that if I can differentiate the entire equation that C would drop out, but I don't think that's really an option. this would give:
y''' + 4.2y" + 9y' = 0.
I know I can't just add -C to B[] because C is unknown.
Any hints?
Thanks,
Bo