Previous Question
i understand i asked my previous question regarding this topic the wrong way. thou before i had time to respond the question was closed. On the other side i couldn't rephrase my question without completely rewriting it. so there for my new question

Create a control system that can precisely control the temperature of a container heated by an induction system.
$$Overshoot_{max} = \begin{cases} 1°C & \text{if}~ T_{final}\,\lt\,100°C \\ 2.5°C & \text{if}~ T_{final}\,\gt\,100°C \end{cases}$$ $$Error_{max} = \begin{cases} \pm0.5°C & \text{if}~T_{final}\,\lt\,100°C \\ \pm1°C & \text{if}~ T_{final}\,\gt\,100°C. \end{cases}$$
Rise time should be as fast a possible without violating the above.

Known Values
Control value: $30°C\,\lt\, T_{final}\lt\,250°C$

Of a given induction heating plate the following is known:
- The nominal watt of the induction coil.
- Control variable which is a percentage of the nominal watt.

Of a given container the following is know:
- Coating of the container (ferromagnetic)
- The material of the container (aluminum)
- The weight of the container
- The size / shape of the container
- Bottom thickness $\neq$ wall thickness

The temperature of the container is measure every second.

Unknown Values
- Contents of the container

1. Is it possible to create a control loop as described?
2. What would be the appropriate control loop (PID, State Space, ect)
3. Any pointers on how to create / describe the transfer function of the system?
4. Any pointer on how to setup the control system?

To begin with all goals are theoretical, in the future the model will be tested in a real life environment.
1. Create a generic control system for the system described above. Supported with the appropriate formula's.
2. Create a estimation if there is a fluid is present in the container, what type of fluid (ea. specific heat) and the amount of fluid. Given the parameters of the system designed in goal 1.

as Ian noted in my previous question:

It seems that a significant portion of this problem would depend on what assumptions you may make. For example, may you assume that the temperature distribution in the container is homogeneous? (In reality the answer is definitely "no", but the diffusion constant could be so large that this is a reasonable approximation).

  1. The temperature is homogeneous for the container.

Current Progress
Currently this problem is implemented as black box with a PID control and it works ok, but only in the current situation. I know in the future there will be different kinds of containers and induction heating plates. So i want to try to create a control system where i can put in the parameters of the container and induction plate. (Thou these are fixed for the duration of the control)

Below is described how far i currently am with the system description.
I assume that the basic description of the system state is: $$\left[rate\;of\;heat\;entering\right] - \left[rate\;of\;heat\;leaving\right] = \left[rate\;of\;heat\;accumulated\right]$$

for $\left[rate\;of\;heat\;entering\right]$ i think i need to solve the specific heat equation for the ferromagnetic coating, so basically for iron.
but i'm stuck on how to create the equation for $\left[rate\;of\;heat\;leaving\right]$. i probably need to use a combination of the equations for heat conduction, heat convection and heat radiation. $$\left[cm\Delta T\right]-\left[heat\;lost\right]=\left[heat\;accumulated\right]$$
- $c$ is the specific heat capacity
- $m$ is the mass of the object
- $\Delta T$ is the change in temperature

i know the system is nonlinear because i can only heat and not cool.

Notes I have a basic knowledge of control theory, and zero knowledge of thermodynamics. so please support your answers with an explanation.

  • $\begingroup$ Again, as I said on the other version of the question, without some specific model for the heat conduction, all you're going to be able to do is a black box that you can't possibly prove will actually work to within the specifications. We as mathematicians cannot help you with that. Talk to engineers or physicists about that. $\endgroup$ – Ian Jun 6 '16 at 20:50
  • $\begingroup$ @Ian what more can i add to further specify the model? is more information needed of the container? like dimensions and such? or am i missing something completely? $\endgroup$ – Johan Zwarteveld Jun 7 '16 at 6:10


It's really tough to give you good advice with such a loose mathematical model, which is what it sounds like needs to be tackled before you go further with theoretical calculations. Since you say you know controls, you should know that the only theoretical modeling which is possible is possible when you have ordinary differential equations which model the process. In this instance you probably want to look into the physics of your induction setup including: (1) the physics of the induction heating mechanism and what its power characteristic looks like, (2) the modes of heat transfer you expect to dominate and/or affect the setup appreciably, and (3) other sources or sinks of heat in the environment. Once you have looked at these carefully and figured out which physics to include, you can come up with a mathematical model and apply typical control theory to it as is done in any textbook. From the statement of your problem, this sounds like it will ultimately be a SISO controller, so you should not worry about state space methods.

Alternatively you can build your setup, connect a PID, and tune a series of gains for the cases you are expecting (in this case two). You can buy tunable general purpose analog PIDs or even build your own out of op amps and basic components (the place to discuss this more would be the Electrical Engineering stack exchange, which I also frequent and where I would be glad to help you should you decide to go this route). Implementation of the system, assuming you know how to setup the plant, is the same as any other control system: you need sensors--probably thermocouples if your response variable is temp, you will probably actuate voltage, not power, and from my experience with regulating inductors, you will probably want an inner loop which stabilizes the current feed forward into the coils, since inductors tend to create a lot of back-emf.

  • $\begingroup$ The system is any existing system where different modules control the user interface or the induction coil. The behavior of these modules can't be changes as the source code is unknown. therefor i can only change the % of the nominal power of the induction modules. The temperature of the container / container content is indeed my response variable. The current PID controller is implemented in software. as the system is really slow i found it hard to get the appropriate value for my PID controller, aside from that i wanted to make a more generic controller. $\endgroup$ – Johan Zwarteveld Jun 8 '16 at 6:53
  • $\begingroup$ is this still the appropriate place for this question as i'm really stuck getting the differential equations for the system. $\endgroup$ – Johan Zwarteveld Jun 8 '16 at 6:58
  • $\begingroup$ @JohanZwarteveld I think physics.stackexchange.com would be more appropriate. $\endgroup$ – WG- Jun 8 '16 at 8:59

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