Page - 88 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

4. ControlSystemDesign 2. MPCcontroller The MPC controller is responsible for calculating the new control input vector and giving it to the HEPHAISTOS oven, according to theestimatedmodelandtheMPCprinciple. In the MPC controller, the first task is to define the cost function, which is the objective to be solved in the controller. It is assumed that themodifiedtarget temperaturevectorat timek is Yt(k) =Tt(k)−Ta(k), (4.1) whereTt(k) is thereal target temperaturevector(N×1)andTa(k) isthetemperatureofthesurroundingair. Oneofthemostcommon cost functionsJ(k)usedin theMPCmethodis [Wan09] J(k) = p∑ l=1 [( Yt(k+ l)−Y(k+ l) )T Γ(l) ( Yt(k+ l)−Y(k+ l) ) +UT(k+ l−1)Λ(l)U(k+ l−1)], (4.2) with Y(l) =       Y1(l) Y2(l) ... YN(l)       , [Γ(l)] =       γ(l) 0 · ·· 0 0 γ(l) · ·· 0 ... ... ... ... 0 0 · ·· γ(l)       , U(l) =       u1(l) u2(l) ... uM(l)       , [Λ(l)] =       Λ(l) 0 · ·· 0 0 Λ(l) · ·· 0 ... ... ... ... 0 0 · ·· Λ(l)       , (4.3) where Γ(l)(N×N) and Λ(l)(M×M) are both positive definite tuning matrices used to adjust weights of future temperature and control input vectors, respectively. Thefuture temperaturevectors can be predicted using either the linear model 3.41 (linear MPC) or thenonlinearmodel 3.46 (nonlinearMPC). 88