Page - 91 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

4.1. AdaptiveControl QDMCcontainstwomainsteps. Thefirststepistoapplythestandard DMCtothecontroltaskandobtainacontrolsolution. Ifthecontrolso- lutionfulfills the inputconstraint (asgivenbyequation 4.4), itwillbe directlyusedasthetruecontrolsolutionU(k). Otherwiseif itdoesnot fulfill the input constraint, a new quadratic programming (QP) prob- lem [GM86] has to be formulated and another control solution will be calculated. ProceduresofQDMCaregiven in the following. Instandardtime-invariantDMC,thelineardiscrete-timeequation3.41 isfirstlyre-written inacompletestate-space formas[Wan09] X(k+1) = [A(k)]X(k)+[B(k)]U(k), Y(k) = X(k), (4.7) where X(k) is the state variable vector. Defining the difference state, inputandoutputvectorsas ∆X(k) =X(k)−X(k−1), ∆U(k) =U(k)−U(k−1), ∆Y(k) =Y(k)−Y(k−1), there is ∆X(k+1) = X(k+1)−X(k) = [A(k)]∆X(k)+[B(k)]∆U(k). (4.8) In order to simplify notations during following derivations, new aug- mentedmatricesaredefinedas [Wan09] Xd(k) = [ ∆X(k) Y(k) ] , [Ad(k)] = [ [A(k)] [ON] [A(k)] [IN] ] , [Bd(k)] = [ [B(k)] [B(k)] ] , [Cd(k)] = [ [ON] [IN] ] , where [ON]and [IN]arezeroandidentitymatrices (N×N). Substituting above new defined matrices into equation 4.8, a new state-space formedsystemmodel isgeneratedas Xd(k+1) = [Ad(k)]Xd(k)+[Bd(k)]∆U(k), Y(k) = [Cd(k)]Xd(k) (4.9) 91