2 added 4 characters in body edited Oct 31 '16 at 11:46 msm 3,57511 gold badge66 silver badges2121 bronze badges It is easier to work in the $$z$$$$s$$-domain: $$Z=YH_2$$ $$Y=(X-Z)H_1$$ Hence, $$Y=(X-YH_2)H_1=XH_1-YH_1H_2\Rightarrow Y(1+H_1H_2)=XH_1$$ Therefore, $$H(z)=\frac{Y(z)}{X(z)}=\frac{H_1(z)}{1+H_1(z)H_2(z)}$$$$H(s)=\frac{Y(s)}{X(s)}=\frac{H_1(s)}{1+H_1(s)H_2(s)}$$ which is called the closed-loop transfer function. The closed-loop impulse response can be found by inverse $$z$$-TransformLaplace transform. It is easier to work in the $$z$$-domain: $$Z=YH_2$$ $$Y=(X-Z)H_1$$ Hence, $$Y=(X-YH_2)H_1=XH_1-YH_1H_2\Rightarrow Y(1+H_1H_2)=XH_1$$ Therefore, $$H(z)=\frac{Y(z)}{X(z)}=\frac{H_1(z)}{1+H_1(z)H_2(z)}$$ which is called the closed-loop transfer function. The closed-loop impulse response can be found by inverse $$z$$-Transform. It is easier to work in the $$s$$-domain: $$Z=YH_2$$ $$Y=(X-Z)H_1$$ Hence, $$Y=(X-YH_2)H_1=XH_1-YH_1H_2\Rightarrow Y(1+H_1H_2)=XH_1$$ Therefore, $$H(s)=\frac{Y(s)}{X(s)}=\frac{H_1(s)}{1+H_1(s)H_2(s)}$$ which is called the closed-loop transfer function. The closed-loop impulse response can be found by inverse Laplace transform. 1 answered Oct 30 '16 at 22:05 msm 3,57511 gold badge66 silver badges2121 bronze badges It is easier to work in the $$z$$-domain: $$Z=YH_2$$ $$Y=(X-Z)H_1$$ Hence, $$Y=(X-YH_2)H_1=XH_1-YH_1H_2\Rightarrow Y(1+H_1H_2)=XH_1$$ Therefore, $$H(z)=\frac{Y(z)}{X(z)}=\frac{H_1(z)}{1+H_1(z)H_2(z)}$$ which is called the closed-loop transfer function. The closed-loop impulse response can be found by inverse $$z$$-Transform.