2.1a.1 Discrete-Time LTI System

State Trajectory In Discrete-Time LTI Model

The solution to the discrete-time LTI model
$\vec{x}[i+1]=A\vec{x}[i]+B\vec{u}[i]$
$\vec{x}[0]={\vec{x}}_0,$
for the state vector as a function of timestep $\vec{x}:\mathbb{N}\to {\mathbb{R}}^n$, the control inputs as a function of timestep $\vec{u}:\mathbb{N}\to {\mathbb{R}}^m$ and matrices $A\in {\mathbb{R}}^{n\times n},B\in {\mathbb{R}}^{n\times m}$ is
$\vec{x}[i]=A^i{\vec{x}}_0+{\sum }_{k=0}^{i-1}{A}^{i-k-1}B\vec{u}[k].$

Proof

The proof to this solution is very straightforward. We will use the principle of mathematical induction to figure it out. We will start with the base case,
$\vec{x}[0]={\vec{x}}_0.$
Knowing this, we can assume the LTI model is true for timestep $i$:
$\vec{x}[i]=A\vec{x}[i-1]+B\vec{u}[i-1].$
We will then prove it for $i+1$ as follows:

$$\begin{array}{c}\begin{array}{rl}\vec{x}[i+1]& =A\vec{x}[i]+B\vec{u}[i]\\ & =A^2\vec{x}[i-1]+AB\vec{u}[i-1]+B\vec{u}[i]\\ & =A^3\vec{x}[i-2]+A^{2}B\vec{u}[i-2]+AB\vec{u}[i-1]+B\vec{u}[i]\\ & =\text{...}\\ & =A^{i+1}{\vec{x}}_0+\sum _{k=0}^i{A}^{i-k}B\vec{u}[k].\end{array}\end{array}$$

Therefore, for timestep $i$, the solution is
$\vec{x}[i]=A^i{\vec{x}}_0+{\sum }_{k=0}^{i-1}{A}^{i-k-1}B\vec{u}[k].$