Matt,

I went through your calculations and now think I=20
understand why they differ from the MathCad results.

At 05:55 PM 3/14/2010, you wrote:
>Thanks everybody for all their comments. It should take me a while yet
>to parse all that information.
>The Inyo looks like it might fit inside a pressure cooker? That might
>help to isolate it from barometric pressure variation.
>**********
>I=92ll try to expand on how I derived the transfer function:
>
>1.) Neither the ground nor the mass is stationary from the perspective
>of an inertial frame.

Correct.

>2.) The only forces that can act on the mass are from the spring and
>from the feedback transducer.

And the force resisting the acceleration of the=20
mass as it is forced to follow ground motion. The=20
spring force (variation) is designed to be negligible.

>3.) Both the spring force and the feedback transducer force depend
>only on the distance between the ground and mass (and derivative and
>integral of that distance).
>
>Those statements gave me this equation of motion for the mass:
>X(s) is the mass position from an inertial frame.
>Y(s) is the mass position from the intertial frame.

One is ground position.......?

>         F =3D ma          (Newton=92s Law)
>         F =3D F(s)[Y(s) =AD X(s)]   (From  2,3)
>So:     M * s ^ 2 * X(s) =3D F(s) [Y(s) =AD X(s)]
>
>Then it=92s just algebra to get the transfer function.
>
>Now F(s) =3D K_m  +  K_p  +  K_i / s  +  s * K_d

I think here you are deriving an expression for=20
the gain *around* the loop, the 'loop gain' ,=20
which is an important concept, but it is not the instrument response.

What needs to be considered is that the=20
instrument's output is taken following Q, the=20
'forward' portion of the loop.  To get the loop=20
gain you multiplied Q by G*(1/R_p + C +=20
1/(T*R_i)), the 'feedback' portion of the loop,=20
which for simplicity we can call B. So the loop=20
gain is just Q B.  In the configuration I=20
described, if Q B >> 1, which is assured (at all=20
but the highest frequencies) by design, by making=20
Q large enough, the instrument response will=20
closely approximate 1/B.  Interestingly, that=20
means that it doesn't depend on the spring=20
characteristics or on Q (so long as it is high=20
enough), but only on B -- which is the whole=20
point of using feedback.  The 'feedback.pdf'=20
reference is a simple explanation of how that works.

>Where K_m is the mechanical spring constant, K_p, K_i, and K_d are the
>constants of the PID controller. For example, K_p =3D Q*G/R_p where Q is
>the position sensor sensitivity in V/m, G is N/m, and R_p is the
>proportional feedback resistor. Likewise, K_d=3D G*Q*C, and K_i=3DQ*G/(T
>*R_i) where T is the integrator time constant.

Regards,
Brett



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