Question

Consider the following difference equation that represents the dynamics of a system: (y = output of the system, u = system input): Vk -0.5yₖ₋₁ +0.5 yₖ₋₂ + Uₖ₋₃ + Uₖ₋₄. Find the discrete transfer function of the system.

Answer 1

To find the discrete transfer function of the system, rearrange the given difference equation and express it in terms of z-transforms. Factor out the Y(z) and U(z) terms. Divide both sides by (1 - 0.5z^-1 + 0.5z^-2) to get the transfer function.

Step-by-step explanation:

To find the discrete transfer function of the system, we can rearrange the given difference equation to solve for y:

yk = 0.5yk-1 - 0.5yk-2 + Uk-3 + Uk-4

Now, we can express the equation in terms of z-transforms:

Y(z) = 0.5z-1Y(z) - 0.5z-2Y(z) + z-3U(z) + z-4U(z)

Next, we can factor out the Y(z) and U(z) terms:

Y(z)(1 - 0.5z-1 + 0.5z-2) = U(z)(z-3 + z-4)

Finally, we can divide both sides by (1 - 0.5z-1 + 0.5z-2) to get the transfer function:

H(z) = Y(z)/U(z) = (z-3 + z-4)/(1 - 0.5z-1 + 0.5z-2)