Well, I’ve been playing a time-wasting computer game… an old DOS game called Ascendancy…. It required that I re-install Windows on my Mac, which is the only reason I haven’t been wasting even more time! Grrr!
Still, I looked at a little math during the week.
I may have found a way that I can approach elementary classical control theory…. For example, if you specify that the closed loop response should be first-order, and that the plant is second-order, you can solve for the control law– and discover that it is PID.
Okay… but we can also turn that around. If the closed loop response should be first-order, and the control law is PID, then we “discover” that the plant is second-order.
Of somewhat more interest is the case where the control law is something other than PID, such as lead or lag, or PID with a low pass filter.
I don’t know about you, but I like seeing that PID control is the exact solution for a second-order plant and a first-order closed loop response. This may not be all that useful in the real world, but conceptually it helps me construct a framework for various control laws.
The model in my head is: PID is an exact solution to the second-order approximation of any given plant; the better that approximation, the better solution PID control gives us. That last statement is, of course, pure conjecture on my part.
I do, however, find myself wondering if I can derive any of the common parameter-setting rules for PID from the parameters of the corresponding second-order plant. And, again, of somewhat more interest is: what about rules for the parameters of other control laws? I’ll let you know.
I also found an interesting function in Mathematica… DualLinearProgramming. It is used in the “linear programming tutorial”… and nowhere else. In particular, “find selected function” will not find it.
I was surprised to see that not only does it solve the dual of a primal linear program, but in fact it simultaneously solves both the dual and the primal. I know that it can be used for some sensitivity analysis, but I’m not sure yet just how much. In particular, I want to know if it will tell me “shadow prices”. Finding the optimal solution is a fine start, but half the point of linear prgramming — it seems to me — is the sensitivity analysis.
You may infer correctly that I picked up linear programming during the week… I don’t think I touched fiber bundles at all – that’s what happens to them, I get excited by something more applied.
And that’s the week that was.
I will be trying to write the next quaternions post today.