While I was completing my bachelor’s degree in electrical engineering, I was, among other things, studying “automatic process control.” I was using a couple of texts that elucidated terms and concepts such as “observability,” “transfer function”, and “sensitivity.” I recall a huge hit by The Rolling Stones and I’m struck with its pertinence to relating process observations with process control.
In many situations it’s hard to get every observation you need when, where, and how you need it. Often, some assessment of importance is useful in deciding what, how, and where to measure something. Mick Jagger provided the insight:
“You can’t always get what you want, but if you try sometimes, you might find, you get what you need.”
There are things you need to know in an application and there are things you can measure. The characteristics of a measurement include how it was measured, where it was measured, and when it was measured. What you are trying to assess is how well the outcome of the underlying process meets expectations. If the results are disappointing, we wonder about and investigate why. This can be a tedious process, especially with new ideas based on weak or untested science (or creative conjecture). Around a third of the way through his book Automatic Control Systems, Dr. Benjamin Kuo introduces the idea of sensitivity and use of the “sensitivity function” to assess it.
Sensitivity addresses the question, “How does this parameter I’m investigating affect the intended outcome?” This helps answer some profound questions, including:
- Do I really need to know this?
- How accurately do I need to know this?
- Does how I measure it matter?
- Does where I measure it matter?
This thinking can be applied to all processes. A good friend was recently wondering how his pension for a lifetime of municipal work could dissolve into a tableful of new job applications, suddenly necessary for his future.
Clearly, the “output” from that system, his pension, could be assessed as “non-conforming to expectations” and “inadequate.” The “inputs,” the payroll deductions, work concessions, etc. over the years, were certainly observable. The “process” depended on investment practices, economic factors, and assumptions; not just hopes and expectations. In his case, lifelong plans changed. In the case of a process automation engineer optimizing a control system, similar manifestations of the same root concepts can profoundly affect his company’s future as well as his own.
A transfer function describes the connection between the inputs and related factors with the outcome we desire. Fundamentally, an equation is written in which the outcome is set equal to the mathematical collection of process observations that science says will produce or influence it. The transfer function describes in explicit or characteristic terms how the inputs become the output. Just because it’s easy and familiar, let’s consider the pressure lost in moving a fluid in a full pipe from one place to another. The most common methodology is the Darcy equation which, for this simple example, can be written as:
Δ ρ = ƒ L v2 / 2D
- p is the fluid density
- ƒ is the friction between the fluid and the pipe
- L is the length of the pipe
- v is the flow velocity of the fluid
- D is the pipe diameter in which flow occurs
Essentially, the sensitivity function relates a change in the output, in this case, to the change in one of the parameters on which it depends. This relationship could be explored several ways but the fast way through the concepts is to observe that the sensitivity of to velocity (for example) can be found by taking the partial derivative of with respect to v. The operator symbol ¶ is usually used to denote partial differentiation. In this case the result is:
∂ΔP/∂v = ρ ƒ L v / D
From this we deduce that how sensitive ΔP is to velocity depends linearly on the velocity – changes at higher velocity produce more effect than changes at lower velocity. Thinking about it, that seems “normal” considering v is “squared” in the transfer function, accentuating its effect as velocity becomes larger and becoming less and less significant as it become small. Remember, we are investigating the sensitivity of the change in pressure drop to the change in velocity, not the value of the pressure decrease resulting from a particular velocity. This calculation can be repeated for each parameter of interest in the equation.
Having assessed the importance of v we can move onto assessing the methodology, placement and timing of the measurements available to the system for use in appropriate process observation and effective control. If sensitivity to a parameter is low or non-existent the care involved with observing it is less important. If the sensitivity is very high then obtaining an appropriate, accurate, and timely observation can be crucial.
What about the sensitivity to the friction factor,ƒ? That seems simple to evaluate until we realize that the calculation of it depends on several other process parameters, such as the viscosity of the fluid, which depends on its temperature. In some cases, it might depend on the pressure, or pipe condition, or even the exact composition of the actual fluid – there are equations pertaining to friction factor for specific fluids over a variety of conditions. Including all that complicates the transfer function but improves the visibility of the issues involved. In some cases, a broadly applicable and accurate transfer function can be very hard to develop or to solve in a manner consistent with actual process behavior. Sometimes the cost of knowing these things can exceed their value. Conversely, sometimes the value is realizing what things are important and how they are changing.
Sometimes sensitivity takes us past the limits of what we can reliably or economically observe. Exploring it, though, discloses a lot about reasonable expectations and divergences from them. Future discussions about some observability issues will come back to this idea of sensitivity.
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