Page - 95 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

4.1. AdaptiveControl Πd=       [IM] [OM] . .. [OM] [IM] [IM] . .. [OM] ... ... ... ... [IM] [IM] . .. [IM]       , where [IM]and [OM]are identicalandzeromatrices (M×M), respec- tively. Thedimensionsof [Id] isMp×M andthedimensionof [Πd] is Mp×Mp. Using the above notations, the control constraint in equa- tion 4.4 isconverted into −[Πd]−1 [Id]U(k−1)≤ ∆Ud(k) ≤ [Πd]−1 ( 1− [Id]U(k−1) ) , (4.17) where1 isanall-onesvector (Mp×1) as 1= [ 1, 1, 1, . . ., 1︸ ︷︷ ︸ Mp ]T . Theoriginalcontrol task(equation 4.15)canbedescribedbyaregular quadraticprogramming(QP)problem[GM86] minJLM(k) = min [ ∆UTd(k) ( [Ξd(k)] T [Γ] [Ξd(k)]+[Λ] ) ∆Ud(k) −2∆UTd(k)[Ξd(k)]T [Γ] ( Yd,t(k)− [Fd(k)]Xd(k) )) + ( Yd,t(k)− [Fd(k)]Xd(k) )T Γ ( Yd,t(k)− [Fd(k)]Xd(k) )] , (4.18) subject to −[Πd]−1 [Id]U(k−1)≤ ∆Ud(k) ≤ [Πd]−1 ( 1− [Id]U(k−1) ) , The above QP problem can be solved using different numerical meth- ods [Wan09]. In the practical MPC system of HEPHAISTOS, the con- trolsolutionisfirstlycalculatedusingthestandardDMCalgorithm. If theresultedcontrolvectorU(k) iswithinthecontrolconstraint, itwill be directly implemented. Otherwise, the control task will be trans- ferred into the QP problem (equation 4.18) and solved using the nu- mericalmethod. 95