Fall 2001 16.318-6 State-Space Models to TFs Given the Linear Time-Invariant(LTI)state dynamics A r(t)+ Bu(t y t)= Ca(t)+ Du(t what is the corresponding transfer function? Start by taking the Laplace Transform of these equations Ci(t)=Ac(t)+Bu()h sX(s-T(0)=AX(s+BU(s) Cly(t)= C(t)+ Du(t) (s)=CX(s+DU( (sI-A)X(S)=BU(s)+c(0) (s)=(s-A)-BU(s)+(sI-A)-x(0-) and Y(s)=C(sI-A)-B+DU(s)+C(sI-A)-x(O By definition G(s)=C(sI- A)B+D is called the Transfer Function of the system And C(sI -A)-c0-)is the initial condition response. It is part of the response, but not part of the transfer functionFall 2001 16.31 8–6 State-Space Models to TF’s • Given the Linear Time-Invariant (LTI) state dynamics x˙(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) what is the corresponding transfer function? • Start by taking the Laplace Transform of these equations L{x˙(t) = Ax(t) + Bu(t)} sX(s) − x(0−) = AX(s) + BU(s) L{y(t) = Cx(t) + Du(t)} Y (s) = CX(s) + DU(s) which gives (sI − A)X(s) = BU(s) + x(0−) ⇒ X(s)=(sI − A) −1 BU(s)+(sI − A) −1 x(0−) and Y (s) =  C(sI − A) −1 B + D U(s) + C(sI − A) −1 x(0−) • By definition G(s) = C(sI − A) −1B + D is called the Transfer Function of the system. • And C(sI − A) −1x(0−) is the initial condition response. It is part of the response, but not part of the transfer function