After reading the "Kalman Filter in one dimension" section, you shall be familiar with the concepts of the Kalman Filter. In this section we will derive equations for the multidimensional Kalman Filter.

Until now, we've dealt with one dimensional processes, like estimating the liquid temperature. But many dynamic processes have two, three or even more dimensions.

For instance, the state vector that describes the airplane position in the space is three-dimensional:

\[ \left[ \begin{matrix} x\\ y\\ z\\ \end{matrix} \right] \]

The state vector that describes the airplane position and velocity is six-dimensional:

\[ \left[ \begin{matrix} x\\ y\\ z\\ \dot{x}\\ \dot{y}\\ \dot{z}\\ \end{matrix} \right] \]

The state vector that describes the airplane position, velocity and acceleration is nine-dimensional:

\[ \left[ \begin{matrix} x\\ y\\ z\\ \dot{x}\\ \dot{y}\\ \dot{z}\\ \ddot{x}\\ \ddot{y}\\ \ddot{z}\\ \end{matrix} \right] \]

Assuming constant acceleration model, the extrapolated airplane state at time \( n \) can be described by nine equations:

\[ \begin{cases} x_{n} = x_{n-1} + \dot{x}_{n-1} \Delta t+ \frac{1}{2}\ddot{x}_{n-1} \Delta t^{2}\\ y_{n} = y_{n-1} + \dot{y}_{n-1} \Delta t+ \frac{1}{2}\ddot{y}_{n-1} \Delta t^{2}\\ z_{n} = z_{n-1} + \dot{z}_{n-1} \Delta t+ \frac{1}{2}\ddot{z}_{n-1} \Delta t^{2}\\ \dot{x}_{n} = \dot{x}_{n-1} + \ddot{x}_{n-1} \Delta t\\ \dot{y}_{n} = \dot{y}_{n-1} + \ddot{y}_{n-1} \Delta t\\ \dot{z}_{n} = \dot{z}_{n-1} + \ddot{z}_{n-1} \Delta t\\ \ddot{x}_{n} = \ddot{x}_{n-1}\\ \ddot{y}_{n} = \ddot{y}_{n-1}\\ \ddot{z}_{n} = \ddot{z}_{n-1}\\ \end{cases} \]

It is common practice to describe multidimensional process with a single equation in a matrix form.

First, it is very exhausting to write all these equations, representation in matrix notation is much shorter and elegant.

Second, the computers are extremely efficient in matrix calculations. Implementing the Kalman Filter in matrix form yields faster computation run time.

The following chapters describe the Kalman Filter equations in the matrix form. And, of course, I will solve numerical examples.