# System control of an induction heated system

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

Problem
Create a control system that can precisely control the temperature of a container heated by an induction system.
where:
$$Overshoot_{max} = \begin{cases} 1°C & \text{if}~ T_{final}\,\lt\,100°C \\ 2.5°C & \text{if}~ T_{final}\,\gt\,100°C \end{cases}$$ $$Error_{max} = \begin{cases} \pm0.5°C & \text{if}~T_{final}\,\lt\,100°C \\ \pm1°C & \text{if}~ T_{final}\,\gt\,100°C. \end{cases}$$
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

Question
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?

Goals
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.

Assumption
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. $$\left[cm\Delta T\right]-\left[heat\;lost\right]=\left[heat\;accumulated\right]$$
where:
- $c$ 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.

• 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. – Ian Jun 6 '16 at 20:50
• @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? – Johan Zwarteveld Jun 7 '16 at 6:10