PIDAnalytix.COM
Universal Control Law E=PC**2
Since its first use in 1911, PID controller has survived in its original form. However, PID has been loved and despised by many. It survived this long is the testimony of its enduring power for being a simple and intuitively easy to understand formulation. There has been many attempts to replace it with other forms of control, however, most of them being more sophisticated but not able to dethrone PID. PID still constitutes 90% of all control loops. However, presently in terms of quality of control, performance of PID loops has remained the same, unreliable and majority of time oscillatory. There are roughly more than 100 methods of PID tuning, none of these tuning methods have the degree of robustness and adaptability to satisfy the real world process variations. They all fail miserably when the realities hit them.
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In order to create yet another PID-Tuning method, the founder of PIDAnalytix devised a control law that would encapsulate the realities of real world process and control in general. This control law is termed as universal control law. This law is a "metaphorical", and not literal. However its application in conjunction with an adaptive intelligent method of re-tuning PID has led to creating what is termed herein as AI-PID. This universal control law was conceived to form the basis of controller tuning adaptation to variation of process P to ensure consistent minimal value in error of control, E. Typically E would be an integrated value error over a time period arising from the actions of control on P. The effect of controller C is viewed being of magnitude of square. The reason for this is that any mismatch of C or its deviation from being the most efficient with respect to P (known or Unknown) would result in E being affected in the manner of square magnitude, irrespectively of deviation being positive or negative.
Strict adherence to this universal control law and discovery of the innate properties of PID tuning allowed formulation of various adaptation strategies for the entire range of variation in P, first at the level of individual processes and then at the level of a network of processes. Therefore, it is envisaged that as individual PID loops are tuned in accordance with universal control law (UCL), when interconnected each of them will adapt individually without negatively impacting neighboring loops.
The power of Adaptive Intelligent PID (AI-PID) tuning is demonstrated by two sets of 256 random process variations, namely of process gain and what is generally associated with controller OP, valve stiction and in particular valve deadband.
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These 512 random variations performance comparison with non AI-PID tuning, clearly provide a totally unrivalled benchmark for PID tuning hereto not known in the literature or history of PID tuning. In house at PIDAnalytix, a much wider range of comparison up to 10000+ random variations of P to automatically tune PID was done. We will release snippets of these comparison in due time.
E=P*C**2 (1)
where E being the error of control, typically of small magnitude and definitely devoid of oscillations.
P being of dynamic process of any type, first order, second order or higher of linear gain or nonlinear gain including time variant.
The term C**2 in the above formulation can be viewed as being C1*C2, where in
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C1 relates to regulatory and disturbance rejection by PID,
C2 relates to various Meta manipulation of PID tuning to deal with Non-linear Gain change, Process Type variations
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This may seem outlandish, but it captures what is incorporated in the adaptive intelligent (AI) algorithm.
There are various types of adaptation of PID tuning based on certain characteristics of E for various forms of Process Type and Parameter variations.
E Optimal Error of Control when AI-PID Tuned
E is a measure of performance of a controller such as PID, to be reduced to a minimum value. It is both a primary driver of controller actions and yet it can also be increased by the controller actions. In other words, this is a catch 22 situation more often than not. The challenge of PID controller tuning is to be such that it reduces E without causing it to be increased during the period of control. This delicate balance of control error E decreasing and possibly increasing what makes tuning of PID controller such a challenging and unforgiving task. This can happen with the simplest of Process P or the hardest of Process P (unstable).
After many years of researching this subject and experimenting with the various tuning methods of PID, PIDAnalytix has devised what is called AI-PID tuning method that eliminates the unforgiving part of control E increasing with control actions. This has resulted in a unique capability of PID-Tuning which has eluded most practitioners in the field. Therefore, AI-PID tuning is formulated to avoid this self created problem of increasing E. In fact this is the underlying issue with any form of feedback controller, not particularly confined to PID type controller.
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The 512 cases of random variations in P and their AI-PID tuning clearly establishes the consistent pattern of E value as follows.
This can be described as being:
Linearized Optimal Error Control
E=(Td+1)*Delta(SP) + (Td+1)*Delta(PVL) + e(PVNoise) (2)
Td in eqn 2 is dead time of PV change to a change in controller OP. Delta(PVL) relates to what is generally considered as step change in PV due to external process parameters, like load change. For example when PV is at SP, but due to load change PV changes from its SP resulting in Error which is followed by the controller actions to reduce E. With AI-PID Tuning, the response of the controller would be a mirror image to a SP change. AI-PID Tuning is such that it reduces E in much the same way albeit in the opposite direction to an equivalent opposite SP change. Typically e(PVNoise) would be a small number which can only be reduced by troubleshooting instruments and sensors. Therefore it can be assumed to be small and hereon assumed to be zero.
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In most non AI-PID tuned controller, E will be significantly greater than E of eqn 2 above. Whereas in AI-PID tuned controller, E of 2 eqn would be a normal value expected. Therefore any deviation of E from this value would be indicative of change in P or C not being optimally tuned. Thus eqn (2) serves as a benchmark for monitoring of P and C in closed loop and hence updating of C in real time. This behavior of E with AI-PID tuning will greatly facilitate Plant Wide Optimization in that it will allow overall optimal controller to perform allowing each of AI-PID loop to respond. By allowing each loop to respond in accordance with its own E, the plant wide optimal control can play the orchestra while each control loop play their individual "tunes".