# need the mathematical models of various PID structures/forms

#### numericman

Joined Dec 2, 2011
21
hello.I'm a new comer to process control.i want to compare different algorithms used for the digital implementation of Pid controllers.my personal study tells me that i have to discretize the various pid models using different numerical methods.i would then write a computer program for the various discrete pid algorithms.but i can't find the mathematical model for any of the PID forms(parallel,series,noise suppressing etc).all i can see is either their continuous time function that expresses their output as a sum of the individual P,I,D terms or their transfer functions relationship.surely this aren't their models? i suppose their model should be in the form of a differential equation.please an explicit explanation coupled with model equations would be greatly appreciated.many thanks.

#### steveb

Joined Jul 3, 2008
2,436
hello.I'm a new comer to process control.i want to compare different algorithms used for the digital implementation of Pid controllers.my personal study tells me that i have to discretize the various pid models using different numerical methods.i would then write a computer program for the various discrete pid algorithms.but i can't find the mathematical model for any of the PID forms(parallel,series,noise suppressing etc).all i can see is either their continuous time function that expresses their output as a sum of the individual P,I,D terms or their transfer functions relationship.surely this aren't their models? i suppose their model should be in the form of a differential equation.please an explicit explanation coupled with model equations would be greatly appreciated.many thanks.
Note that it is not a differential equation you want. In discrete time the equivalent equation to a differential equation in continuous time is called a difference equation.

There are established methods to convert between continuous time and discrete time systems. The most common is the bilinear transform (also called the Tustin method). I placed a link to this below, but many control theory books talk about it too.

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

The basic idea of the Tustin method is similar to the trapezoidal rule for numerical integration. Assuming one has a linear system in the continuous time domain, you can take the Laplace transform. Then replace all s-variables with (2*(z-1))/(T*(z+1)), where the z-variable is the Z-transform frequency variable, and T is the sample time. Then you can transform back to the discrete time domain.

For the case of the most common PID form, you don't need to go through this formal process. The basic PID is the sum of a gain, derivative and integral, so you can just make a simple model for each piece and then add them up. So the measured variable is subtracted from the reference signal, which creates negative feedback. This difference signal is called an error signal. The error signal is input into each of the three pieces of the PID, then the output of each piece is added together.

1. The gain part is straightforward. Just multiply by the gain whether in continuous time or discrete time.

2. The integral part can be done directly with the Tustin Method considering that an integral is 1/s in the Laplace domain. This just amounts to the trapezoidal rule for integration. Then you can multiply by the integral gain before you add it in to the final control signal

3. The derivative is also simple since a derivative is s in the Laplace domain. Converting using the Tustin rule is more accurate than a simple Euler derivative (x(n)-x(n-1))/T. Again, you can then multiply by the derivative gain before you add it in to the final control signal

4. You may want to put a simple low pass filter on the output of the derivative before you add it back in to the final control signal. This is called bandlimited differentiation, and it helps to control problems due to noise, since noise is amplified by a derivative.

If you are using a different variation on the PID theme, then you may be able to break it into pieces, or it might be easier to make one transfer function and use the formal Tustin method directly on the whole system. I'd have to see the particular variation to know which is easier. But, once you have the process down, it's relatively easy, and remember that Matlab has tools that can do these conversions automatically for you.

#### numericman

Joined Dec 2, 2011
21
many thanks steveb for taking time to reply me.you're too kind.but here is the thing.i submitted my literature review to my professor.in it i gave the various block diagrams of the different pid structures/modifications;ideal,parallel,series,IMC,etc.i included their various time and laplace domain mathemtical representation as well as their transfer functions.my professor said i had failed to include their model equations.i had though that their various time and laplace domain mathemtical representation or transfer functions were the model equations?this informed my earlier question about a differential equation.please can you explain what he means with a clear example.so sorry i wasn't explicit the first time.

#### steveb

Joined Jul 3, 2008
2,436
many thanks steveb for taking time to reply me.you're too kind.but here is the thing.i submitted my literature review to my professor.in it i gave the various block diagrams of the different pid structures/modifications;ideal,parallel,series,IMC,etc.i included their various time and laplace domain mathemtical representation as well as their transfer functions.my professor said i had failed to include their model equations.i had though that their various time and laplace domain mathemtical representation or transfer functions were the model equations?this informed my earlier question about a differential equation.please can you explain what he means with a clear example.so sorry i wasn't explicit the first time.
Can you attach pictures or pdfs of what you submitted to your professor? It is difficult to figure out what your professor may have objected to without seeing the details.

#### numericman

Joined Dec 2, 2011
21
good day Steve.thanks for your reply.i've done alot more reading and i think i can obtain the differential model equations.in any case if it doesn't work i'll let you know.
my research topic is actually 'numerical experimentation with pid controllers.I'm suppose to do 2 of the following,
compare algorithms used for the digital implementation of pid controllers;

compare available software used for the auto tuning of pid controllers.

my supervisor tells me to bring along TEN CASE STUDY on the topic.please where can i get them? i've search the net for weeks to no avail.emphasis for now is on obtaining ten case study on 'algorithms used for the digital implementation of pid controllers'. Many thanks