2.1b.3 Diagonizable State Matrix Model

State Trajectory in Diagonal Continuous-Time LTI Differential Equation Model

The solution to the diagonal continuous-time LTI model
$\frac{d}{dt}\vec{x}(t)=A\vec{x}(t)+B\vec{u}(t)$
$\vec{x}(0)={\vec{x}}_0,$
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 a diagonizable matrix with diagonalization $A=V\text{Uplambda}{V}^{-1}$ is
$\vec{x}(t)=V{e}^{\text{Uplambda}t}{V}^{-1}{\vec{x}}_0+{\int }_0^{t}V{e}^{\text{Uplambda}(t-\tau )}{V}^{-1}B\vec{u}(\tau )d\tau .$

Proof

Left for later (Note 9)