Page - (000104) - in Control Theory Tutorial - Basic Concepts Illustrated by Software Examples

13.5 SmithPredictor 101 To obtain robust stability, we can design a controller,C, under the assumption that themodelingerror iszero, M =0.Forexample,wecanusethemethodsfrom theearlierchapterStabilization toobtainagoodstabilitymarginforC relativetoP∗. Thenwecanexplicitlyanalyzethesetofmodelingerrors, M , forwhichourrobust controllerwill remain stable. A designwith a good stabilitymargin also typically providesgoodperformance. 13.6 Derivationof theSmithPredictor ThederivationofEq.13.5beginswith the transfer functionsobtaineddirectly from Fig.13.4 forvariousoutputs Y =ECPe−δs Y∗ =ECP∗ =Y P ∗ Pe−δs Ym =ECP∗e−δ∗s =Y P ∗e−δ∗s Pe−δs witherror input E =R−Y −Y∗+Ym =R−Y(1+ P ∗ Pe−δs − P ∗e−δ∗s Pe−δs ) =R−Y 1 Pe−δs ( P∗+ M) with M =Pe−δs −P∗e−δ∗s. Substituting theexpression forE into theexpression forY yields Y =CPe−δs[R−Y 1 Pe−δs ( P∗+ M)]. The system response,Y, to an input,R, isG =Y/R, whichweobtain bydividing both sidesof theprior equationbyR, yielding G =CPe−δs −GC(P∗+ M), fromwhichweobtain G = ( CP 1+C (P∗+ M) ) e−δs,