Ziegler-Nichols method and closed loop characteristic equations.

[CalebP]

Hi Guys,

Attached is a problem from an old exam for a Process Control and Instrumentation unit.

I have tried everything I know (which isn't much, it's not the main assessable portion of the unit).

Other questions similar involve giving us either the characteristic closed loop equation or the controller function and deriving it from there.

We take the characteristic equation and sub in s = wj (j being sqrt(-1)) and can solve for both the I am and Re parts being zero, obtaining the critical value of K and the corresponding w, then using the table becomes quite elementary.

Working backwards to discover G(s) gets tricky due the s being on top (or a 1/s below)

This is the only example I can find where I'm given the overall output function Y(s), so working backwards to find the characteristic equation... I just don't know how, or if it's done differently.

Much of google gives tips of how to effectively use Z-N in real or simulated situations, but this is for an exam. Most of the unit is comprised of sensors, analog and digital conditioning and PLC diagrams, so these control loops/PID are kind of tacked on the end. The last time I touched Laplace was ~5 years ago.

I'm sure there's a really simple solution.

Please help. Thanks.

 

[rude man]

Given the system F(s) plus PI controller gains K_p and K_i, what is the modified transfer function y/u?

From that, determine under what conditions poles are confined to the left-hand plane (hint: use quadratic formula to find the poles as a function of F, K_p and K_i.)

Determine the Z-N values of K_p and K_i from your table.

Use the final-value theorem for the Z-N y/u with a ramp input to determine steady-state output error.

 

[CalebP]

Gah, I really have no idea what I'm doing. I've never seen this stuff before and it's two weeks worth of notes in one unit.

What exactly have I been given in this question? Do I need any knowledge of Laplace to solve this? Is the input u(s) the controller function (involving tauI and tauD, in integral form or Laplace form), or is u(s) the Kp*error(t)...?

I think the first step to solving this is getting it in the form of a closed-loop characteristic equation - something I have no idea how to do. How do I involve the K parameter, is it just Kp for proportional control?

What I've tried is called the CLCE 1 + y(s).K and substituted s = iw. This yield K = ±1 and w = ±1 with the ultimates being Ku=1 and Pu=2pi

Is the final value theorem question trying to get me to so the SS error is zero because PI control is implemented?

 

[rude man]

Gah, I really have no idea what I'm doing. I've never seen this stuff before and it's two weeks worth of notes in one unit.

What exactly have I been given in this question? Do I need any knowledge of Laplace to solve this? Is the input u(s) the controller function (involving tauI and tauD, in integral form or Laplace form), or is u(s) the Kp*error(t)...?

You definitely need a background in Laplace. I can't imagine how you'd be given a problem like this one without that .

u(t) is the input to your compensated system. The system (more commonly called plant) gets modified by the k_p and k_i gains to yield desired response y(t) to an input u(t). The purpose of Z-N gains is purportedly to effect an ideal response characteristic in some way or other.

I think the first step to solving this is getting it in the form of a closed-loop characteristic equation - something I have no idea how to do. How do I involve the K parameter, is it just Kp for proportional control?

What I've tried is called the CLCE 1 + y(s).K and substituted s = iw. This yield K = ±1 and w = ±1 with the ultimates being Ku=1 and Pu=2pi

You should look at Wikipedia's description under "PID Controllers". It's very good.

Is the final value theorem question trying to get me to so the SS error is zero because PI control is implemented?

The question was put to you badly. If you apply a constant input u(t) = c then the output will in the steady-state equal the input, thanks to the presence of the integrator (k_i). So no error in this case.

But if you apply a ramp u(t) = ct, the output will lag the input, and the difference between input u and output y at any given time t is the error, as a percentage of the max. input.

I can't help you after this because any next hints would require a knowledge of manipulating Laplace-transformed variables and I am not permitted to give you an out-and-out solution.

 

[CalebP]

Thanks,

I've done a small amount of Laplace (about 5 years ago :P) but never applied it, nor was that class a prereq.

The old exam was a 2008 edition so maybe I don't even need to know this for the unit. If you say it's needed then I'm guessing I don't have to know this stuff.

Thanks for your trouble.