#### YukiWong

Joined Dec 5, 2011
27
Hi all,
I have an first order plus time delay system with K=58.46, T(time constant)=3.08, Td(Time delay)=0.06 and i intend to design a PI Controller to this system by any method, do you guys know any method that can help me work out the parameter clearly of Kc and Ti or other parameter that is needed.

Thank you very much Last edited:

#### Papabravo

Joined Feb 24, 2006
20,400
There is not enough information to do much with this problem, because you have no way of evaluating your choices. You need some kind of metric or "cost function" which you can evaluate for the answer that you seek. If properly chosen such a function will yield a unique "optimal" result.

We usually know that a result is optimal because any small change makes the "cost function" worse.

• YukiWong

#### t_n_k

Joined Mar 6, 2009
5,455
Another comment on the numbers given.

A transport delay of 0.06 sec will probably have little consequence for the response of a first order system with a time constant of 3.08 sec. The 3.08 sec time constant will tend to dominate the overall behavior.

• YukiWong

#### YukiWong

Joined Dec 5, 2011
27
There is not enough information to do much with this problem, because you have no way of evaluating your choices. You need some kind of metric or "cost function" which you can evaluate for the answer that you seek. If properly chosen such a function will yield a unique "optimal" result.

We usually know that a result is optimal because any small change makes the "cost function" worse.
Hi Papabravo,

Thanks for your reply. My system is from a DC Motor. How to find or calculate the "cost function"?

#### steveb

Joined Jul 3, 2008
2,436
The traditional approach of looking at gain margin and phase margin in the open loop response should work. Instead of a cost function (or maybe this would be the cost function) use the traditional rules about the acceptable margins.

I think the delay time will play into the optimization process even though it is small relative to the time constant. The reason is that a first order system with no delay is trivial to stabilize since even infinite proportional gain will be stable and gives perfect response. At higher frequency the time delay represents a significant phase shift, and so this becomes the key limitation for setting the high frequency gain and zero location in the PI filter.

EDIT: Personally, I use 12 dB for gain margin and 45 degrees for phase margin. These numbers typically leave sufficient margin for tolerances and uncertainties and result in time domain responses that are acceptable without too much overshoot (or ringing), and reasonable response times or usually obtained.

Last edited:
• YukiWong

#### Papabravo

Joined Feb 24, 2006
20,400
This is an example of a heuristic "cost function" where you pick one or more measurable values and you write a set of contraints as inequalities. For example

1. Gain Margin > 12 dB
2. Phase Margin < 45 degrees
Using these criteria you can evaluate your choices for the parameters and find the set of feasible solutions. In this case the solution is not liable to be unique. It also does not tell how much better you might be able to get.

You should research the Nichols chart for more insight

http://en.wikipedia.org/wiki/Nichols_plot

#### steveb

Joined Jul 3, 2008
2,436
This is an example of a heuristic "cost function" where you pick one or more measurable values and you write a set of contraints as inequalities. For example

1. Gain Margin > 12 dB
2. Phase Margin < 45 degrees
Using these criteria you can evaluate your choices for the parameters and find the set of feasible solutions. In this case the solution is not liable to be unique. It also does not tell how much better you might be able to get.
There is a typo there, Phase margin > 45 degrees is needed

I usually use the constraint that the solution should give gain and phase margin about equal to those values, rather than greater than. However, I place these constraints under the worst case tolerance conditions after doing a sensitivity analysis. The numbers tend to work out near each other for many systems, and you can just use the worst case as the constraint. The benefit of equating is that the system ends up being nearly as fast as possible, without unwanted problems.

Of course there is no magic about 12 dB and 45 degrees, it is just a general rule and another designer might make a different choice based on experience with particular systems.

If the above constraints are used with the > sign, then additional constraints should be imposed. Rise time, allowed overshoot and other constraints will help, otherwise there is a huge range of allowed values and no basis to make a choice. If the system doesn't need to push any speed limits, then more margin is a good idea, so knowing these constraints is helpful in that it allows you to maximize gain margin and phase margin, which makes the system more robust.

However, as this particular problem is stated, the implication is to use some type of standard design strategy. The above gain and phase margin approach is one such example. If more constraints were given and it turned out that they can't be met using a PI filter, then a more advanced compensator would need to be considered. But, this problem is pretty much forcing you down one path.