What do you want? I told you the procedure I use. Do the math yourself or admit you don't really know if your method is better than cascaded closed loop control because you don't know how how to implement cascaded control properly.
The key is what I call alpha or 30 from your equation. For a critically damped response, the inner loop closed loop poles must be at -30/2 or more negative. The outer loop poles must be at 2/3 of the inner loop poles or greater in magnitude.
This shoots holes in George's ascertion that the inner loop must be 3 times greater than the outer loop. George must not do motion control. OK, we know he doesn't. George's ascertion that the inner loop must be 3 times greater than the outer loop may apply to the few types of systems that he is familiear with but not to all systems. The math must be done on a case by case basis. JCH's system is much different from a temperature control system.
The point is do the math or stop misleading by spreading false myths. Question everything. I don't believe anything until I prove if for myself. Question me too but I have posted links to pdf files for the last five years that show the techniques I use. It is time that someone else puts some effort into finding the truth because I think this news group is just a social club where people post opinions and not hard facts or math.
I have JCH's problem worked out in great detail. JCH, if the inner loop is set to the slowest response where the inner loop poles are at
-30/2 and the outer loop poles are set at 2/3 of that or -10 the response is still much faster that what you have shown.
Here is my cascaded loop response using your equation with
Now do the math your self and see if you get the same results. It should be obvious if you truly know what your are doing.
This news group should consider this a thumb in its eye for not backing up anything they said with real facts or calculations. Okay, I haven't either but atleast I have done the calculations. I am just showing the results. This can't be a one way street. Someone worthy is going to have to show me that they have put some effort into this before I spill more info.
Peter Nachtwey