2.1b.2 Diagonal State Matrix Model

State Trajectory in Diagonal Continuous-Time LTI Differential Equation Model

The solution to the diagonal continuous-time LTI model
$\frac{d}{dt}\vec{x}(t)=\text{Uplambda}\vec{x}(t)+B\vec{u}(t)$
$\vec{x}(0)={\vec{x}}_0,$
for the state vector as a function of time $\vec{x}:{\mathbb{R}}_{+}\to {\mathbb{R}}^n$, the control inputs as a function of time $\vec{u}:{\mathbb{R}}_{+}\to {\mathbb{R}}^m$, matrix $B\in {\mathbb{R}}^{n\times m}$, and square diagonal matrix $\text{Uplambda}:=diag({\lambda }_{11},{\lambda }_{22},...,{\lambda }_{nn})$ is
$\vec{x}(t)=e^{\text{Uplambda}t}{\vec{x}}_0+{\int }_0^t{e}^{\text{Uplambda}(t-\tau )}B\vec{u}(\tau )d\tau .$

Proof

Left for later (Note 9)