See Source File
http://straightlinecontrol.com/pid_algorithms.html

SECOND FORM OF THE PID ALGORITHM

The second form of the algorithm is called "noninteracting, or "parallel" or "ideal" or "ISA" . I understand one manufacturer refers to this as "interacting", which serves to illustrate that terms by themselves may not tell you what the algorithm is. This form is used in most textbooks, I understand. I think it is unfortunate that textbooks do not at least recognize the different forms. Most if not all books written for industry users rather than students recognize at least the first two forms. The basic difference between the first and second forms is in the way derivative is handled. If the derivative term is set to zero, then the two algorithms are identical. Since derivative is not used very often (and shouldn't be used very often) perhaps it is not important to focus on the difference. But it is important to anyone using derivative, and people who use derivative should know what they are doing. The parameters set in this form can be made equivalent (except for the treatment of gain-limiting on derivative) to those in the first form in this way:

Kc' = ((Ti +Td)/Ti))Kc, "effective" gain. Ti' = Ti + Td, "effective" integral or reset time Td' = TiTd/(Ti + Td), "effective" derivative time

These conversions are made by equating the coefficients of s. Conversions in the reverse direction are:

Kc = FKc' Ti = FTi' Td = Td'/F

where

F =0.5 + sqrt(0.25 - Td'/Ti')

Typically Ti is set about 4 to 8 times Td, so the conversion factor is not huge, but it is important to not loose sight of the correction. With this algorithm it is possible to have very troublesome combinations of Ti' and Td'. If Ti'<4Td' then the reset and derivative times, as differentiated from settings, become complex numbers, which can confuse tuning. Don't slip into these settings inadvertently! A very knowledgeable tuner may be able to take advantage of that characteristic in very special cases, but it is not for everyone, every day. Some companies advise to use the interacting form if available, simply to avoid that potential pitfall.

This algorithm also has no provision for limiting high frequency gain from derivative action, a virtually essential feature. In the first algorithm Kd is typically fixed at 10, or if adjustable, should typically be set somewhere in the range of 6 to 10. This desirable limiting of the derivative component is sometimes accomplished in this second form by writing it as:

Kc'(1 + 1/Ti's + Td's)/(1 + Td's/Kd)

or

Kc'(1 + 1/Ti's + Td's/(1 + Td's/Kd))

There are likely many variations on the theme.

The variables Kc', Ti' and Td' have been called "effective". In the Bode plot, IF Ti'>4Td', THEN Kc' is the minimum frequency-dependent gain (Kc is a frequency-independent gain). This is at a frequency which is midway between the "corners" defined by Ti and Td, which is also midway between the "effective " corners associated with Ti' and Td'. Ti' is always larger than Ti and Td' is always smaller than Td, which recognizes the slight spreading of the "effective" corners of the Bode plot as they approach each other.

This algorithm is also called the "ISA" algorithm. The ISA has no association with this algorithm. Apparently this attribution got started when someone working on the Fieldbus thought it would become "THE" algorithm. It didn't. Or hasn't. ANSI/ISA-S51.1-1979 (Rev. 1993) is a standard on Process Instrumentation Terminology. While this is a standard on terminology, not algorithms, it uses the first form of the algorithm for examples and in its Bode plot for a PID controller. Another term used to identify this algorithm is "ideal". Think of this word as one to identify the algorithm, not describe it. It is true that it can do everything the first form can do, and more, provided the gain for derivative is handled appropriately. But settings which produce complex roots should be used only by the very knowledgeable.

THE FIRST FORM OF THE PID ALGORITHM

This first form is called "series" or "interacting" or "analog" or "classical". The variables are:

Kc = controller gain = 100/proportional band Ti = Integral or reset time = 1/reset rate in repeats/time Td = derivative time Kd = derivative gain

Early pneumatic controllers were probably designed more to meet mechanical and patent constraints than by a zeal to achieve a certain algorithm. Later pneumatic controllers tended to have an algorithm close to this first form. Electronic controllers of major vendors tended to use this algorithm. It is what process industry control users were used to at the time. If you are unsure what algorithm is being used for the controller you are tuning, find out what it is before you start to tune.

I did not follow closely the evolution of algorithms as digital controllers were introduced. It is my understanding that most major vendors of digital controllers provide this algorithm as basic, and many provide the second form as well. Also, many provide several variations (I'm told Allen-Bradley has 10, and that other manufacturers are adding variations continually).

The choice of the word interacting is interesting. At least one author says that it is interacting in the time domain and noninteracting in the frequency domain. Another author disagrees with this distinction. This really becomes a discussion of what interacts with what. To be safe, think of the word interacting as one to identify the algorithm, not to describe it.