PID Control


Joined Jan 28, 2005
Originally posted by Iodem_Asakura@Jul 19 2005, 11:03 PM
Have someone information about PID Control Theory?
[post=9183]Quoted post[/post]​
There are volumes of information on the Internet that describe the technique referred to as PID.

Here's just one example of some introductory material available on the Internet:

PID Description

PID is a powerful approach to controlling feedback systems. If you are getting into Control Systems Design you will do well to study this technique.



Joined Nov 17, 2003
The fundamental principle behind using PID control is to obtain the best elements of derivative and integral control.

By studying the characteristic equation of simple proportional control we can see that from a steady state point of view, the feedback gain needs to be large in order to make the response fast and steady state error small, this is fairly intuitive, considering the closed-loop feedback of such a system is given by:

GC / 1+GC

If the gain is high: GC / 1+ GC → 1 i.e. zero steady state error

Sadly, making the feedback gain large has an undesirable effect on the transient behaviour, causing a low damping ratio, large peak overshoot and oscillatory action. This is because the damping ratio, ς, and natural frequency, ωn, are linked via the equation:

ς =a/(ωn)

Where a is a constant

This is graphically proved via a root-locus diagram, where increasing the gain causes the angle, ø, between the locus and the imaginary axis to decrease. The damping ratio is derived by: ς = Sin ø . This can also be proved mathematically by analysing the characteristic equation.

So if we introduce derivative feedback to the proprtional system (PD control) we can obtain an extra parameter which can be used to change the damping ratio independant of the natural frequency. Again this can be proved by mathematically analysing the characteristic equation

If we now introduce integral feedback to the proportional system (PI control) we can bring the steady state error to zero. It is worth noting that the steady state error under proportional control can only be zero when the gain is infinite (an impossibility). The integral control allows the steady state error to be zero when the the gain is finite, this is regulated by the expression; Kp/Ti (where Kp is the controller gain). The problem with PI control is that as the gain increases the damping ratio decreases.

So when we consider each form of control in turn we can see that by bring the three main elements, proportionality, differentiation and integration, together we can obtain a control system where the specific parameters are easily adjusted to obtain a system of the required specification.

For those that like the Maths, PID control is defined by:

u(t) = Kp[e(t) + 1/Ti(∫ e(τ)dτ + Td e'(τ))]


Kp is the controller gain
Ti is the integral parameter
Td is the derivative parameter
e(t) is the input error
u(t) is the system output

Roughly speaking, a small value of Ti brings the steady state error to zero, but reduces the damping ratio. Td can be used to increase the damping ratio independantly of Ti, and increasing Kp increases the natural frequency and thus the response of the system.

I can explain the Maths more if it would help you understand the concepts, but what I have detailed above is fundamentally the details of PID control.