Design a PID Controller for System G(s) w/ 0% Overshoot & <1s Settling Time

[215]

Homework Statement

Design a controller for the system G(S) such, that the step response to close loop transfer function has an overshoot of 0% and a settling time of less than 1 sec.

Homework Equations

G(s) = \frac{1}{10s^2 + 5s + 10}
Controller has to be a PID controller.

The Attempt at a Solution

First as the assignment mentions, the overshoot has to be 0% which means that we interested in a critical damped system ζ = 1
since the settling time has to be lower than 1 sec, i can deduce that ω_n has to be less than 4. which means that, my poles shall be on the real axis and less than -4.

I chose i want my poles to be at -5 and -7, and using solve i am able to see that a PD controller consiting of d = 115 and p = 340 would do the job.
but the step reponse is showing my about 12.5 percent overshoot, but the settling time matches. the close loop transfer function i end up with
G(s) = \frac{115s + 340}{10s^2 + 120s + 350}
the poles are places where they should be.. so really don't see where i am going wrong with this ? please help my.

 

[UltrafastPED]

You need some I in order to creep up on the boundary.

Try for a solution with just PI; your initial goal is to satisfy the 0% overshoot requirement.

Then start adding in some D to reduce the total time.

 

[donpacino]

I chose i want my poles to be at -5 and -7, and using solve i am able to see that a PD controller consiting of d = 115 and p = 340 would do the job.
but the step reponse is showing my about 12.5 percent overshoot, but the settling time matches.

Having poles at -5 and -7 would result in an overdamped system. To have a critically damped system you want matching poles ie. 2 poles at -5.

The fact that your system is under-damped (overshoot), indicates that your poles have imaginary components to them and something is wrong with your solve function.

 

[215]

Yes.. but even for an overdamped system it doesn't resemble the charateristica of an overdamped system...
In[959]:= Solve[10 x^2 + 120 x + 350 == 0, x]

Out[959]= {{x -> -7}, {x -> -5}}

There is no complex component.

 

[donpacino]

If a system has overshoot its poles have an imaginary component to them.

For designing the controller I would follow ultrafastPED's advice. Design a PI controller first then add your d competition.

I would recommend using the Zeigler Nichols PID tuning method as a good starting point

 

[215]

I just don't understand how come i canont determine my PID values, from where i want my poles placed..
I've solved ch. eq. for all possible controllers and PD seems to be the only one capable of given me poles at those two locations.

 

[donpacino]

ok. If you look at a second order system with one zero, if the zero is equal to or less than the smallest pole, it can begin to dominate the equation. That is where your overshoot is coming from. It is also due to the fact that you are adding a hgih D component. I would recommend using a PI or PID controller.

It is very hard to get a critically damped PD controlled system.

 

[215]

how would zero be able to affect a system... as far I've understood is it only the poles which affects the system.

 

[donpacino]

In the case of your system there is a zero at -3. Due to the fact that the zero is ~ half that of the smallest pole, it can influence the system response. As the smallest zero approaches the smallest pole, it will have less and less of an effect on the system.

The idea that only poles effect the system is only true assuming a pole is dominant.

 

[215]

but how come do they affect the system.. mean if a zero is dominant, would it then act as an dominant pole??.. But still the zero does not contain a imaginary part, which still make no sense why it should oscillate, and create overshoot.

 

[donpacino]

the concept that system will have an oscillatory component if the poles have an imaginary component only holds true for poles.

http://seit.unsw.adfa.edu.au/staff/sites/valu/TEACHING/CT1/effects.pdf
look at page 14.

Note: You really won't learn about this type of stuff until you get to grad school.

 

[215]

I can see that it changes, but how can i prevent it? i mean shouldn't they have some conditions aswell??

 

[donpacino]

The zero is created by the D portion of the PD controller. I recommend using a PI or PID controller.
You are almost always going to get an over-damped step response with a PD controller.

 

[215]

but it's not possible to make one using a PI which have these values.

 

[rude man]

Have you tried the Ziegler-Nichols method for PID controller gains determination? You should be able to come up with a pid controller giving you zero overshoot for your plant transfer function.

I don't know if the 1 sec. settling time is reached thereby.

 

[215]

since this is just trial model it would be possible to do it. but since the real model cannot become marginal stable, is the method unuseable.