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

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

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