Page - 89 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

4.1. AdaptiveControl In the above cost function p denotes the prediction and control horizon length. In general, the prediction horizon length is not equivalent to the control horizon length. But here in order to sim- plifythederivation, it isassumedtheyareidentical,whichwillnot affect thecontrolperformance. Basedonthecost function, theformalrepresentationof thecontrol taskcanbeexpressedas min J(k) subject to 0≤um(k+ l−1)≤1, for 1≤m≤M, 1≤ l≤p, (4.4) where the constraints in equation 4.4 guarantees that the mi- crowave feeding power is always between 0 and 100%. This con- trol taskconsistsof twoaspects. Thefirstaspect is to trackthepre- defined target temperature and minimize the quadratic difference between the measured temperatures and the target temperature (the first term in the cost functionJ(k)). The second aspect refers to the minimization of the control power and save the heating en- ergy(thesecondterminthecost functionJ(k)). Definingacontrol sequence (as illustrated infigure 4.1) Uset(k) = { U(k),U(k+1), . . .,U(k+p−1)}, (4.5) the control solutionU(k) can be obtained by searching for the op- timalsequenceU∗set(k) fulfilling U∗set(k) = argUminJ(k), (4.6) andthenimplementingthefirstcontrol inputvector fromtheopti- malsequence. Whenthecontrolledsystemmodel is linear(suchas equation 3.41), normally an analytical expression can be derived for U∗set(k) as well as U(k). Otherwise when the system model is nonlinear (such as equation 3.46), an analytical control solution is not feasible. In this case, the control task 4.4 has to be solved with anumerical solution. Inthefollowing, the linearMPCandthenonlinearMPCmethodswill be introducedindetail. 89