Page - 52 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3. ModelingMicrowaveHeating thenequation 3.32 is further transferred into dYn dt = Anc(t) ·Yn+Bnc(t)U(t). (3.34) Basedonthissingleoutputmodel,acontinuous-timeMIMOmodel is builtas dY dt = [Ac(t)]Y+[Bc(t)]U(t), (3.35) whereY is theoutputvectordefinedas Y = [ Y1,Y2, .. . ,YN ]T , (3.36) andthestatematrixandtheoutputmatrixaredefinedas [Ac(t)] :=      A1c(t) 0 · ·· 0 0 A2c(t) · ·· 0 ... ... ... ... 0 0 · ·· ANc (t)      , [Bc(t)] :=       B1,1c (t) B 1,2 c (t) · ·· B1,Mc (t) B2,1c (t) B 2,2 c (t) · ·· B2,Mc (t) ... ... ... ... BN,1c (t) B N,2 c (t) · ·· BN,Mc (t)       . (3.37) Beforeabovemodelscanbeusedinrealcontrollerdesign, thelaststep is discretization, because in practice the controller is always operated indiscrete-time. Differentdiscretizationmethodsexistandinthisdis- sertationtheEulermethod[Lju98]isapplied. Whenthesamplingtime ∆t is a sufficiently small constant (1.5 s or even less), the temperature changing rate on the left-hand side of equation 3.35 can be approxi- matedby dY dt ≈ 1 ∆t ( Y(k+1)−Y(k) ) , wherek indicates thediscrete timeinterval. 52