Alternative Methods vs Iterative Design — Control Theory

[ltkach2015]

QUESTION (2/2)

1) What are some alternatives to iterative design in control theory?


2) I have a certain plant transfer function PTF(s) that is higher order than two and non-unity numerator.
I want certain characteristics such as a certain damping ratio (zeta).
So I want approximate it as second order (specially second order underdamped approximation)
I introduce controller whose transfer function is K(s) in the feed-foward direction, and a unity gain in (-) feedback
PTF(s) has open loop zeros which will become closed loop zeros (and a part of the denominator of the closed loop transfer function). So, I want to cancel the closed loop zeros with the higher order closed loop poles.
I impose the "Angle Design Criteria" in-conjuction with the second order approximation constraint, therefore I determine that the system would need a phase lead angle (e.g. -75.89 degrees)
This implies that the difference of controller's pole angles and zero angles is negative.
There must exist either only: one zero and no poles, or some combination that yields a negative angle.

So I introduce lag compensator which will increase the order of my system and introduce another closed loop zero
case1) K(s) = (s+z1); no poles, so I using angle criteria I can determine the zero location
or
case2) K(s) = (s+z2)/(s+p2); I set the zero arbitrarily, and determine pole location from angle criteria

So my OLTF = K(s)*PTF(s); (OLTF denotes open loop transfer function)

both cases may not satisfy the second order approximation constraints.

How can I analytically solve for pole and zero location that satisfy angle criteria and second order approximation constraints? (given the compensator chosen) Thank you.

 

[nilesh_pat]

One alternative to iterative design in control theory is the use of analytical design techniques. Analytical techniques involve the use of mathematical equations and algorithms to design controllers based on desired performance characteristics. These techniques can be used to solve for pole and zero locations that satisfy angle criteria and second order approximation constraints. For example, the root locus technique can be used to graphically identify the stable operating point for a given system and then determine the optimal controller parameters, while the frequency response method can be used to analyze the response of a system to a sinusoidal input and then design the controller accordingly. Other analytical methods include the state space approach and the loop shaping approach.