Page - 53 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3.2. Grey-boxModeling Thefinal lineardiscrete-timeMIMOheat transfermodel is Y(k+1) = ( [IN]+∆t [Ac(k)] ) Y(k)+∆t [Bc(k)] U(k), = [A(k)] Y(k)+[B(k)] U(k) (3.38) with [A(k)] := [IN]+∆t [Ac(k)] =      A1(k) 0 · ·· 0 0 A2(k) · ·· 0 ... ... ... ... 0 0 · ·· AN(k)      , [B(k)] :=∆t · [Bc(k)] =       B1,1(k) B1,2(k) · ·· B1,M(k) B2,1(k) B2,2(k) · ·· B2,M(k) ... ... ... ... BN,1(k) BN,2(k) · ·· BN,M(k)       . (3.39) and An(k) = 1+∆t ·Anc(k), Bn,m(k) = ∆t ·Bn,mc (k). (3.40) The matrix [IN] is an identity matrix with the dimensionN×N. The equation 3.38 indicates that thefuture temperaturevectorY(k+1)at timek+1 can be determined by the current temperature vectorY(k) and the control input vector U(k) at time k. Therefore the current temperaturevectorY(k) (equation 3.38) iswrittenas Y(k) = [A(k−1)] Y(k−1)+[B(k−1)] U(k−1), (3.41) which will be used later (section 3.2.4) for the linear system identifi- cationderivation. Following the similar derivation process, a nonlinear discrete-time MIMOmodelcanalsobeconstructedsuchas Y(k+1) = [A(k)] Y(k)+Ψ(k), (3.42) 53