  13 August 2011
Author: Giorgos Lazaridis
PID Theory

Tuning a PID controller

The process of setting the P, I and D parameters to obtain an ideal response of the system is called tuning. Over the years, several tuning methods have been described, but we will discuss only the Ziegler-Nichols method and of course the manual tuning. The tuning time is relative to the system type and speed of change. If a system has a rapid response, then the tuning time is very short. On the other hand, a slow response system has a very long tuning time. A fast-response system is for example a robot positioning system. An example of a slow response system could be a large water tank heater. It is obvious that a parameter change in the robot positioning system will have an immediate response at the robot's behavior, while a parameter change in the water tank heater system may need several minutes for the systems to react.

Manual PID Tuning

If someone wants to tune a PID system with this method, then he first must be sure that he understands what each parameter does and how each one affects the system, otherwise this can be a headache. These are the steps for this method:

• The I and D terms are set to zero
• The P term is increased until the system oscillates
• The P term is increased slowly to increase the system response, but it must not become unstable
• When the P term is set to obtain a desired fast response, the I term is increased to stop the oscillations. This will reduce the steady state error but may increase the overshoot. Faster system response may require some amount of overshoot.
• When I is set to desired amount of overshoot and minimal state error, the D term is set until the system achieves an acceptable quick loop to its set-point. Increasing the D term will result in decreasing the overshoot and yields higher gain with stability, but it may cause the system to be very sensitive to noise.

• Tuning a PID system with the ZieglerNichols method

The Ziegler-Nichols method was introduced by John G. Ziegler and Nathaniel B. Nichols in the 1940s. To tune a system with the Ziegler-Nichols method, the engineer must first define the critical P gain, which we will name Kc. This is how its done:

• The I and D terms are set to zero
• The P term is increased until the system oscillates in a stable rate. This will be the Kc
• Once the Kc is defined, then the engineer must measure the oscillation period. It is the same like measuring the period of an AC signal. This will be the Pc. Finally, the user can calculate the values for P,I and D parameters using the following table:

 Control P Ti Td P 0.5Kc - - PI 0.45Kc Pc/1.2 - PID 0.60Kc 0.5Pc Pc/8

If you're wondering what the Ti and Td terms are, these are common parameters in a PID controller. A typical PID controller has three parameters, K, Ti and Td, but there are many different kinds of PID controllers. For example, a PID controller with set-point weighting and derivative filter has six parameters K, Ti, Td, Tf , b and c. Parameters b and c are called set-point weights. They have no influence on the response to disturbances but they have a significant influence on the response to set-point changes, so you do not need to worry about them when tuning a PID system.

Example

Here is an example of a system that we want to tune with the Ziegler-Nichols method. First, we need to find the critical P gain. So, we make the I and D parameters zero and we let the controller operate as a simple proportional controller (only P). Then, we start to increase the parameter P and write down the system's response. Here are 4 typical characteristics of a system's response in relation to the P change:  P is too low and must be increased P is still low  P is too high and made the system unstable This is the value for P that we are looking for. The system has a stable oscillation.

Suppose that the setting for P is 20. So, Kc is 20. Now we can take a closer look at the last chart: The horizontal axis shows the time in seconds. As you can see, the oscillation has a period of about 15 seconds (45 to 60 seconds). So, the second parameter that we need, the Pc, is 15''. To get the PID parameters for our controller, we need to use these values with the Ziegler-Nichols table:

P = 0.60 x Kc => P = 0.6 x 20 => P = 12

Ti = 0.5 x Pc => Ti = 0.5 x 15 => Ti = 7.5

Td = Pc / 8 => Td = 15 / 8 => Td = 1.87

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