Control system with equation C = A*x + B*dx/dt

[zsero]

Homework Statement

This question came up in a computer science / robotics exam and I still don't know the solution for it. I figured out that it's classical mechanics related, so I thought this might be the best place to ask it.

Suppose a control system is described by the equation C = A*x + B*dx/dt, where B is proportional to the mass of the robot. The behaviour of the system can be characterised by the steady state (e.g. the asymptotic velocity of the robot) and the half-life time of the decrease of the distance to the steady state. Explain how the steady state and the half-life change if the mass of the robot is doubled.

Homework Equations

C = A*x + B*dx/dt

The Attempt at a Solution

I've figured out that the steady state doesn't change, as it happens when dx/dt is 0, thus B is not affecting the solution.

And this is how far I understand it. Can you explain to me, what kind of movement is this, what is the real-life meaning of the steady state and half-life for this movement and that how to calculate the change in the half-life?

 

[tiny-tim]

hi zsero!

write it Bdx/dt = C - Ax, then solve it by "separating the variables"

 

[zsero]

hi zsero!

write it Bdx/dt = C - Ax, then solve it by "separating the variables"

OK, I arrive at the following equation:

-B/A*ln(x) = c*t + k1

My problem here is that I don't understand the meaning of the equations and that what is asked by half-life and steady state.

How can I get the half-time from this equation?

 

[tiny-tim]

-B/A*ln(x) = c*t + k1

ok, now multiply by -1/B and then e-to-the on both sides …

x = x_oe^-ACt/B (where x_o = e^-kA/B)

does that look familiar? ​

 

[zsero]

Thanks for the help! Actually, I'm not a Physics student, so I don't really know, but I'd guess it's the exponential delay. So if B doubles then the constant becomes half, thus the half-life becomes double. Is this correct?

 

[tiny-tim]

good guess!

but you really should make yourself familiar with half-life (and exponential decay generally) …

look it up in wikipedia or the pf library ​

 

[zsero]

Thanks for the help!