# How does frequency-response and lead-lag compensators fit in with PID or State Space control

I am new to control theory, as my background is in statistics. In watching a wonderful set of lectures by John Rossiter, and am feeling pretty comfortable with my understanding of the basic concepts such as feedback, PID controllers, lead-lag controllers, and state-space control.

My question is in regards to frequency response and subsequently lead-lag compensators. I am not really clear on how this idea/design fits in with other control concepts such as PID or state-space control. Now this could just be because I am looking at a textbook or some videos, so there is just an artificial demand to ensure some topics are covered--at the sacrifice of the flow of the material. I might be falling into that hole.

I understand that the idea of frequency response is that the input is a sin wave, and the output is a shifted sin wave, that can either have a change in amplitude or change in phase. Now in the previous video chapters, the focus was on transfer functions, PID control, and such. So in those previous videos, the input function was always a unit step or step function. Correspondingly, the discussions were about minimizing the error between the target and system by introducing feedback. Hence I am a little lost about why the sudden shift to looking at different frequencies and corresponding change in phase and amplitude?

Now I understand the math behind these frequency-response ideas. The math itself is pretty simple. But I feel like I understand the math without understanding the intuition or intent in using frequency-response methods or designing a lead-lag compensator. I can make up some intuition for myself, but this is not my area of expertise and hence I am not sure I am on the right track.

My own sense is that this might be a way to introduce more complicated input signals. Technically I could decompose any input signal as a sequence of step functions or even dirac delta functions. I suppose the authors of the textbooks want to limit the amount of functional analysis requirements in their books :). The other thing I can think of is that these lead-lag compensators are common in some discipline like electrical engineering where input sin waves might be more common? But I don't know whether any of these suggestions are accurate or even in the ballpark.

I would really appreciate it if someone could explain how frequency-response and these lead-lag compensators fit in to the world of controllers. Are they only used in specific contexts or perhaps I am missing the connection to the wider field of control. Thanks.

• Lead-Lag and PID are both a type of system that plays the role of a "compensator" in a feedback-control-loop. State space is an analysis technique, among many. You can use state space to understand either PID or Lead Lag, and this is often done for education. But the real benefit of state space is that it is good to get at systems with higher dimensionality (in particular aerospace and robotics). Whereas PID and Lead/Lag tend to apply to single-input/single-output systems (often in process control), and thus simpler means of analysis are used in actual practice Jan 14 at 21:34
• As far as the particular diferences of PID vs Lead-Lag. Lead-Lag gives you exactly one pole and one zero. But it has finite gain at DC. Whereas PID gives you infinite gain at DC, i.e., via tha integrator part of it. PID also gives you two zeros. Analysis methods like Bode, Root Locus can make the effects of these things somewhat intuitive, but you really have to also work thru the algebra and do lots of examples to develop that intuition. Jan 14 at 21:40

PID and Lag-lead compensator have something in common.

1. Lag is comaparable to Integral term as it can reject steady state error but its frequency response are difference. You can realize lag as RC low-pass filter
2. Lead is comparable to derivative term. By introduce derivative or lead into th system, we make it oscillate faster and increse loop gain.You can realize lag as RC high-passfilter

You can either use PID or Lag-lead compensator or mixed but in design process we need to find suitable parameter to make control system stable and meet the system requirement.

When we talking about frequency response because we use Nyquist stabillity as design criteria but we can use root locus as well.

Note about input response we usually used impulse response, step response and ramp response.

Impulse response contain every frequency so we can show that system is atable for all frequency.

Step response also contain all frequency but it easier to use since step input is commonly use with real system. it can show over-shoot steady state error settling time ,etc.

Some system doesn't has steady state error with step-input but steady state error can occur with ramp input or disturbance so sometime is useful to show how control system response to ramp input.