51 Solving MIMO

For any initial condition $X_0$ and any continuous input $U$, the solution of (A,B,C,D) is

$$Y(t)=C\exp (tA)X_0+{\int }_0^{t}C\exp ((t-s)A)BU(s)𝑑s+DU(t).$$

Proof. From the theorem, we take

$$X(t)=\exp (tA)X_0+{\int }_0^t\exp ((t-s)A)U(s)𝑑s$$

and then

$$Y(t)=CX(t)+DU(t).$$