Determine gain, damping ratio, overshoot and first crossing…

The heat treatment furnace operates like a subcritical damped dual$–$capacity process. The furnace is heated electrically and the heating power is infinitely adjustable between $0$ and $3000$W$.$ Correspondingly, the furnace temperature varies between $0$ and $1200$C$.$ When heated with a power of $1500$W$,$ the furnace temperature remains at $520$C$.$ When the heating power is increased stepwise from $1500$W to $2000$W$,$ the furnace temperature stabilizes over time at $720$C$.$ The natural frequency of the process had previously been determined to be $1/61/$min$,$ but the damping coefficient had not been determined. In the step test, the first stage, when the furnace temperature rose and exceeded $720$C$,$ was not observed, as was the highest temperature of the furnace. As the experiment continued, the following passes to $720$C were recorded:

$$ second pass $($temperature dropped below $720$C$)(1)/(2)$ hour after the start of the experiment,

$$ third pass $($temperature rose above $720$C$)48$ minutes later.

Calculate based on the measurement results of the experiment:

$$ Process amplification

$$ Process damping coefficient

$$ Highest temperature in the furnace during the experiment

$$ How long after the start of the experiment did the furnace temperature exceed $720$C for the first time.

Note! Answers alone are not enough, they must show the intermediate steps and explanations of how you arrived at the solution. If you make assumptions, please justify them.

The necessary graphs can be found in the pictures.

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