# clarification on key concepts of control theory

#### curious_mind

##### Member level 5
I have few questions to get my basics right on control theory. I am assuming that I do not use any tools for visualization. I am looking for answers to improve my analytical skills

1. When do we choose time domain or frequency domain method to design a control system?
2. I see that bode plot is sufficient in most cases to assess stability, then what is the role of routh hurwitz, nyquist and polar plot?
3. Can root loci be used when multiple parameters are variable (not just gain)?
4. Does State space directly models the differential equation without any transformation (i guess to preserve the initial states) and where does the general form of state equations come from? (books do not provide the basis)
5. Difference between compensator (lead,lag) vs PID controller (no clarity is provided anywhere)
6. Can I have only I controller without P and D?
7. For design based on state space methods, what are the ways to check stability. (is bode plot,nyquist,routh hurwitz or root loci applicable)?

#### FvM

##### Super Moderator
Staff member
General objection, all of your questions are answered somewhere in control theory literature. I'll try to answer some elementary points.

1. Performance criteria of a control system are often specified in time domain, e.g. settling time to x percent error, complete analysis can be better done in frequency domain, in so far the methods are complementary.
2. Bode criterion is simple and descriptive but doesn't work for systems, e.g. if the magnitude graph crosses A=1 multiple times. Nyquist will still work then.
5. Different approaches ending up in similar solutions. The compensator concept focusses on cancelling the unwanted process transfer function by a circuit with inverse transfer function. The process can be regulated either with oder without control loop feedback. PID controller uses an universal parameterizable transfer function to regulate a process with arbitrary transfer function in closed loop feedback.
6. Yes, if it fits the control process. A typical process like PT2 can't be stable operated with pure I controller. You can easily see why in Bode diagram.

#### curious_mind

##### Member level 5
Hi FvM

I have one additional question. I was just reading somewhere that for a double integrator plant, lead compensator to use for stabilization. I was wondering as why do they not use a PID controller? (reasoning was not provided in books)

#### FvM

##### Super Moderator
Staff member
Sketch the bode diagrams and try yourself. PID transition frequency is usually choosen in the "horizontal" P branch of the magnitude curve. But with a double integrator plant this would end up in zero phase margin. Stability can be only achieved by shifting the transition frequency to the D branch. In so far you can use a PID or PD controller.

#### curious_mind

##### Member level 5
In simulink using auto tuning method for a double integrator plant, I found response was critically damped. Where as using a lead compensator, I found that overshoots was around 20 %. ( this is adjustable based on pole zero separation and the gain). There was no way of minimizing overshoots in lead compensator based design. PD was obviously noisy