# PID controller - How is plant equation derived?

During our control systems course we often work on control techniques where the plant equation/state space model is given .However , how are these derived? For example, if I wanted to build a soldering iron or just an ordinary iron . How should I go about modelling it as a plant and applying what I learnt in theory?

• Why don't you start with something simple like an RC low pass filter and move up in complication as your analytical skills improve. And please don't say that an RC FILTER doesn't constitute a plant. – Andy aka Jul 8 '17 at 11:52
• @Andyaka I know how to do it for electrical systems consisting of discrete components and for mechanical systems but heat/temperature is something that I haven't yet encountered. However , i'm curious to know how people model complicated systems into equations . – Aakusti Jul 8 '17 at 11:55
• All systems are the same, just the letters change. You need to learn the underpinning science - thermodynamics. Analysis is then maths, which is common to all fields. – Chu Jul 8 '17 at 12:02
• You construct a graph of input vs output as a function of frequency, so input current vs temperature. Then visually deduce a simple 1st or second order transfer function – sstobbe Jul 8 '17 at 15:29
• System identification – Rodrigo de Azevedo Jul 9 '17 at 7:44

PID really isn't appropriate for soldering irons. A simple proportional loop with high gain is a much better investment. The big problem is that the iron works in two regimes, resting and loaded, and the loaded regime is very badly defined.

Starting with the available heater power, the mass of the coil and tip, and the surface area of the hot section of the iron, you can get an idea of the temperature sensitivity of the iron. Thermal impedance from coil to air is the important factor, with the heat capacity providing another constraint on time to reach desired temperature.

A PID can be used to reach operating temperature in minimum time and maximum accuracy, but the working temperature is typically not extremely precise (a degree or so for extremely demanding applications, or 10 degrees for a "real life" soldering iron.

The thing is, once you go to actually soldering, the amount of heat you need depends greatly on exactly what you're soldering. A big, heavy wire or buss bar will require much more heat than an IC lead, and the thermal lag in the iron tip will mean that response times of a few seconds can be expected, and will depend on the target.

So, depending on what you're soldering, the plant model changes and you cannot optimally tune a PID. A simple proportional loop will work adequately for any likely real-world soldering iron.

• would newton's law of cooling be a good place to start thinking about characterizing the plant? – Aakusti Jul 8 '17 at 12:09
• @Aakusti - Only as a very preliminary start. You'll also need the Stefan-Boltzmann equation, not to mention the equations dealing with convection. – WhatRoughBeast Jul 8 '17 at 12:15

Knowing the transfer function of the plant is truly the starting point before attempting to close the loop. There are several ways to characterize a plant:

1. Experimental data: build a prototype and collect data on the bench. If I speak for a switching converter, assemble selected components and characterize its control-to-output dynamic response. You must identify what is the control variable (the duty ratio $D$ in this example) and what is the output variable ($V_{out}$ the output voltage). The dynamic analysis is usually done with a frequency-response analyser (FRA) which plots the transfer function by exciting the plant control variable $D$ at various frequency points while computing and storing the magnitude and phase observed in the output variable $V_{out}$: you obtain a Bode plot of the plant, $H(s)=\frac{V_{out}(s)}{D(s)}$.

2. Simulation: you have a model of your system and you enrich it with parasitic elements to faithfully reproduce its operating environment. For a switching converter, it means taking a model of the switching core and adding parasitic elements such as capacitor and inductor equivalent resistances (ESR, respectively labelled $r_C$ and $r_L$) or MOSFET and diode resistive drops etc. to produce a computer model of the plant. You start with the simplest model and increase complexity depending on the amount of details you need. This is usually an iterative process: you must compare your model's results to the bench data before validating the model. Once this is done, you can carry the analysis in the computer (characterize the control-to-output plant transfer function) and even close the loop before reproducing the circuit on the bench.

3. Analytical analysis: this is a pure mathematical approach in which you start from a non-linear equation describing the control-to-output transfer function (perhaps obtained from bench data also) and you linearise it around an operating point. From there, you can obtain its small-signal dynamic response and infer the right compensation strategy. The important point here is to organize the transfer function in a low-entropy format meaning that you factor the equation so that poles, zeros and gains (or attenuations) are clearly organized. That way, you can identify what element in the plant contributes a pole or a zero and understand how this element is going to move in temperature, production etc. and see how the whole response is affected. Then, you will design the compensator so that these offenders ($r_C$ and $r_L$ for instance) do not compromise the integrity of the closed-loop system once released to production. This is a very important aspect that must not be overlooked and is often ignored when people compensate by trial and error.

As you can read, there is no single answer to your question. My experience taught me however that combining the 3 above points is the recipe to successfully compensating a given system.