I am designing a PI controller for a heating coil. Air flowing around the heating coil is heated before being supplied to a heated space. Hot water flows through the heating coil. In the suggested control system, the controlled variable is the temperature of air and the manipulated variable is water flow rate through the coil. The equation describing the dynamics of the system is shown below:
Cah*DTair/dt=Mw(Twi-Two)+ Ma(Ta-Tair)
Cah= heating capacity ,which is contant
Tair= Temperature of air supplied to heating space, varies of time
Mw= Mass of water flowing through the coil, varies with time since its the manipulated variable
Twi= Temperature of water supplied to the coil, varies with time since it is part of another process
Two=Temperature of water exiting the coil, constant
Tao=Temperatue of air before approaching the coil, constant
In the above equation, Mw and Twi are both variables (function of time), and therefore the equation is nonlinear due to Mw*Twi; To design a controller, I should linearize around an operating point, which will restric the validity of the controller to this specific point (or around that point). Can anyone guide me regarding a better control design approach? should I check nonlinear control theory? multivariable control? etc..
Hi Mohamad Thabet,
Denoting x = Tair, and u = Mw, then
C·x' = (Tapp − x)·Mair + (Twi − Two)·u
From the above 1st-order ODE, the singularity occurs at Twi = Two. I'm unfamiliar with your heating process, but as long as Twi ≠ Two, then the following controller should be able to stabilize the heating system:
u = [C·(xref − Kp·x) − (Tapp − x)·Mair]/(Twi − Two)
where xref is the reference temperature to be followed. Since the condition requires that Twi ≠ Two, then you need to ensure that Twi in another process will never reach Two.
Dear Mohamed Thabet,
According to the equations and explanations that you gave, you have a nonlinear and variant system. First you can search to control's with appropriate controller . Second you can linearize this system around a function point and considered as variant linear system. Thirdly you can try LPV multimodel and advanced controller for your heating coil.
For more details and information about this subject, i suggest you to see links and attached file on topic.
https://www.google.com/search?q=Basic+and+Advanced+Regulatory+Control%3A+System+Design+and+Application&rlz=1C1AOHY_frDZ708DZ708&oq=Basic+and+Advanced+Regulatory+Control%3A+System+Design+and+Application&aqs=chrome..69i57.1599j0j7&sourceid=chrome&ie=UTF-8
https://www.wiley.com/en-us/Advanced+Building+Technologies+for+Sustainability-p-9780470546031
https://www.navigantresearch.com/reports/intelligent-building-technologies-for-sustainability
Article An experimental system for advanced heating, ventilating and...
Article Advanced building technologies for sustainability
Hi Mohamad Thabet
As Yew-Chung Chak said, you can set Tair as your state x, and Mw, as your system input u. Another approach is to consider Twi*u to be a process disturbance (since Twi is not controllable from your process). That way, the ODE describing your process will be:
x' = (−Mair/C)*x + (−Two/C)·u + (Tapp/C)·Mair + disturbance.
The integral action in the PI controller should take care of the disturbance and the constant addition of (Tapp/C)*Mair.
On the other hand, you can also try some advance controllers like MPC, where you can take into account your system constraints as well.
Best
Alejandro Astudillo Vigoya Yew-Chung Chak I am very thankful for your answer. I was not capable of fully understanding your answers because probably I am not very familiar with the theory of state space modelling; Kindly can you recommend a reference for me to check and familiarise myself with the topic?
Regards
Hi Mohamad Thabet,
Thank you for your honest reply. That's why I didn't recommend any advanced control strategies because I presumed that you lacked the knowledge in control design. On the references, any modern control textbooks {Ogata, Dorf & Bishop, Golnaraghi & Kuo, Nise, and Franklin et al.} cover the topics of state-space analysis and linear control design.
If you are good at ordinary differential equations and solving eigenvalue problems, then you can pick up the above topics with relative ease.
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