Multivariable State Vector

In many dynamic processes, the state has to be described with multiple variables. For example, the state vector that describes airplane position and velocity is 6-dimensional:

$$\left[\begin{array}{c}x\\ y\\ z\\ \dot{x}\\ \dot{y}\\ \dot{z}\end{array}\right]$$

Including velocity requires a 9-dimensional vector:

$$\left[\begin{array}{c}x\\ y\\ z\\ \dot{x}\\ \dot{y}\\ \dot{z}\\ \ddot{x}\\ \ddot{y}\\ \ddot{z}\end{array}\right]$$

Assuming a constant acceleration dynamic model, we can describe the airplane state at time $n$ by motion equations:

$$\{\begin{array}{l}{x}_n=x_{n-1}+{\dot{x}}_{n-1}\Delta t+\frac{1}{2}{\ddot{x}}_{n-1}\Delta t^2\\ y_n=y_{n-1}+{\dot{y}}_{n-1}\Delta t+\frac{1}{2}{\ddot{y}}_{n-1}\Delta t^2\\ z_n=z_{n-1}+{\dot{z}}_{n-1}\Delta t+\frac{1}{2}{\ddot{z}}_{n-1}\Delta t^2\\ {\dot{x}}_n={\dot{x}}_{n-1}+{\ddot{x}}_{n-1}\Delta t\\ {\dot{y}}_n={\dot{y}}_{n-1}+{\ddot{y}}_{n-1}\Delta t\\ {\dot{z}}_n={\dot{z}}_{n-1}+{\ddot{z}}_{n-1}\Delta t\\ {\ddot{x}}_n={\ddot{x}}_{n-1}\\ {\ddot{y}}_n={\ddot{y}}_{n-1}\\ {\ddot{z}}_n={\ddot{z}}_{n-1}\end{array}$$

It is common practice to describe a multidimensional process with a single equation in matrix form. Represent them in matrix notation is much shorter and more elegant.