I am trying to optimize the heat control of a test bench at our facility. The test bench is basically a pipe with an air flow that can be heated up to desired temperature. To heat the air flow up, three heat zones are available as shown in the sketch below. Measurements of the temperature are available for $T_1, T_2, T_3$ and $T_4$. The three zones are each equipped with there own PID controller, so it is possible to set target temperatures for each zone. However the single temperature states obviously affect the values of each other, especially in the direction of the flow, but also slightly in upstream direction via conduction in the pipe material.
The ultimate goal is to set a target temperature for $T_4$ and control the three heat zones accordingly.
Right now, basically, only the PID controller of the second heating zone is used for the temperature control. Ideally I would like to use all three heating zones. The PID controllers of the heating zones could either be used to do so or just set to a static gain and building a control on top of this.
What is the general strategy to get such a MISO control set up?