Page - 94 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

4. ControlSystemDesign [Λ] :=       [Λ(1)] [OM] · ·· [OM] [OM] [Λ(2)] · ·· [OM] ... ... ... ... [OM] [OM] · ·· [Λ(p)]       , andYd,t(k) is themodifiedtarget temperaturevectoras Yd,t(k) = [ Yt(k) T Yt(k+1) T . .. Yt(k+p−1)T ]T . Both [Γ] (Np×Np)and [Λ] (Mp×Mp)arepositivedefinitediagonal matrices thatareusedtoadjustweightsof∆U(k)andYd,t(k) respec- tively. Substituting the equations 4.12 and 4.13 into the above cost function and differentiating both sides with respect to ∆Ud(k), the analytical formedsolution isgivenas [Wan09] ∆Ud(k) = ( [Ξd(k)] T [Γ] T [Γ] [Ξd(k)]+[Λ] )−1 [Ξd(k)] T [Γ] T [Γ]( Yd,t(k)− [Fd(k)]Xd(k) ) , U(k) = U(k−1)+∆U(k), (4.16) where∆U(k) is thefirstvector takenfrom∆Ud(k). Equation 4.16 is the final control solution of the standard DMC algo- rithm. For non constrained system, it can be directly implemented to calculateU(k). However, for systems with control constraints, the re- sulted control solutionU(k) does not automatically fulfill the desired control constraints (such as equation 4.4). If the resulted U(k) does not fulfill the input constraint, an additional quadratic programming step is needed. Due to the modified system model 4.12, the form of inputconstraintalsohas tobeadjustedas [Wan09] Ud(k) = [Id] U(k−1)+[Πd] ∆Ud(k), with Ud(k) =       U(k) U(k+1) ... U(k+p−1)       , Id=       [IM] [IM] ... [IM]       , 94