Problem Definition

Consider the following scalar first order differential equation: \[ \begin{equation} \dot{x} = ax + bu \end{equation} \] where the notations are self-explanatory [REF]. By using various methods, the solution of this equation \(x(t)\) is: \[ \begin{equation*} x(t) = \exp(at)x(0) + \int_{0}^{t} \exp(a(t-\tau))bu(\tau)d\tau \end{equation*} \] The system response varies with respect to input \(u(t)\).

x = rnorm(10)
print(x)
##  [1]  0.53927228  0.59404045  0.50220726 -0.05374421  0.55276639 -0.88438989
##  [7] -0.16640982 -0.84669756 -0.34759300 -1.18285991
hist(x)