Consider the problem based on first-order linear system

Question

Consider the first-order linear system

x^· (t) = A [x (t) - u (t)]

with A < 0. Note that tyhe system is stable, the time constant is τ_c = 1/(-A), and the steady-sate gain from u to x is 1. Suppose the input (for t ≥ 0) is a ramp input, that is

u(t) = βt

(where β is a known constant) starting with an initial condition x(0) = x₀. Show that the solution for t ≥ 0 is

x(t)= β (t - τ_c)+(x₀ + βτ_c)e^At

Make an accurate sketch of u(t) and x(t) (versus t) on the same graph, for x₀ = 0, β = 3 and A = 0.5.