PI/PID Controller Design from the Laplace Domain
We can also formulate discrete PID controllers directly from the Laplace domain. Here, the ideal PID algorithm is written as:
Now we can apply either the backward difference or bilinear transformation methods to get an equivalent discrete PID controller.
Say we apply the backward difference method. Then,
For deciding about PID parameters. There is a good table from the book (A First Course in Fuzzy and Neural Control by Hung T. Nguyen Nadipuram R. Prasad, Carol L. Waler, Elbert A. Walker).
In the following picture you can find it.
In general, it's useful to think about every controller as a mass-spring-damper system. In this system we have different parameters which will affect the response of out system.
In order to control this system it's necessary to change the values of mass and spring coefficient and damping factor. How to do this?
We use a kind of PD controller. P is proportional gain; this gain will be added to Kspring ,so as a resulting system in closed loop we have (kspring+kp); this new k is able to change our natural frequency and the response of our system. How?
Let's first defind Kspring. Kspring is a coefficient which indicates the strength of the spring when it's expanded x from it's rest position. In the following link http://en.wikipedia.org/wiki/Hooke's_law you can get more info and good video also is provided in this regard.
As a result when Ktotal=kspring+kp is LARGER then the spring is more stiff and strong. So it's hard to make it oscillating so the natural frequency is larger. So our system is more stable when is affected by NOISE.
What is the limit of Ktotal?
It's important to have large kp but if we increase kp we need more actuation capabilities and the system natural frequency must be high enough from our closed loop natural frequency.
We have other parameter; it's kd our damping factor. This parameter is like friction. Friction is able to damp our system. If we increase Kd it means we make more friction in our system. As you can see in the Table2.1 above. If we increase Kd overshoot will be damped and also settling time will be decrease. These are important parameters in PID controller.
As a result we can write a mass spring system as
In short: Kp LARGE --- Sensitivity LOW -- Stiffness LARGE -- Natural freq. LARGE.