Extended Kalman Filter (EKF)

Introduction

So far, we have spent considerable time understanding the linear Kalman filter, its intuition, and practical examples.

However, most real-world systems are nonlinear.

This is where the standard Kalman filter reaches its limit.

To overcome this, engineers developed a method to apply Kalman filtering to nonlinear systems.
The earliest and most widely used solution is the Extended Kalman Filter (EKF), first applied in aerospace systems and famously used by NASA.

Core Idea of EKF

The idea behind EKF is extremely simple:

If the system is nonlinear, linearize it at every time step and apply the Kalman filter.

That is the entire concept.

From Linear KF to EKF

Recall the linear Kalman filter state prediction:

x_k = A x_{k-1}

In EKF, this becomes:

x_k = f(x_{k-1})

Similarly, the measurement model changes from:

z_k = H x_k

to:

z_k = h(x_k)

The only difference is that constant matrices $A$ and $H$ are replaced by nonlinear functions $f$ and $h$ .

Why This Works

Although the system is nonlinear, the Kalman filter itself is still linear.

EKF resolves this mismatch by:

Approximating nonlinear functions
Using a local linear model at each time step

This approximation is performed using first-order Taylor expansion.

Linearization via Jacobian

At each time step, the nonlinear functions are linearized around the current estimate.

Super wide

The Jacobian matrices are defined as:

F_k = \frac{\partial f}{\partial x} \Bigg|_{\hat{x}_{k-1}}

H_k = \frac{\partial h}{\partial x} \Bigg|_{\hat{x}_{k}^{-}}

These Jacobians replace $A$ and $H$ in the Kalman filter equations.

EKF Algorithm Structure

Once linearized, the EKF algorithm is identical to the linear Kalman filter:

Prediction
- Predict state using $f(x)$
- Predict covariance using Jacobian $F_k$
Update
- Compute Kalman gain using $H_k$
- Update state estimate
- Update covariance

Only the source of $A$ and $H$ changes.

What Is a Jacobian?

The Jacobian is a matrix of partial derivatives.

Suppose:

y = f(x_1, x_2, x_3) = x_1 + 2x_2 + 3x_3

Then:

\frac{\partial y}{\partial x_1} = 1,\quad \frac{\partial y}{\partial x_2} = 2,\quad \frac{\partial y}{\partial x_3} = 3

This vector forms one row of the Jacobian.

Why Partial Derivatives Are Necessary

In Kalman filtering, the state is typically a vector:

x = \begin{bmatrix} \text{position} \\ \text{velocity} \\ \vdots \end{bmatrix}

Each component affects the system differently.

Therefore, we must compute derivatives with respect to each state variable separately.

This is why Jacobians are essential in EKF.

Intuitive Interpretation

Nonlinear system → cannot apply Kalman filter directly
Linearize system locally → apply Kalman filter
Repeat at every time step

This produces a filter that works well as long as nonlinearities are not too severe.

Comparison Summary

Linear Kalman Filter

System is already linear
Directly apply Kalman filter

Extended Kalman Filter

System is nonlinear
Linearize using Jacobians
Apply Kalman filter to the linear approximation

Why EKF Looks Difficult

Most textbooks and papers present EKF using dense mathematical notation.

While the formulas are correct, they obscure the simplicity of the idea.

In reality:

EKF = Kalman Filter + Jacobian-based linearization

Nothing more.

Practical Notes

Jacobians can be derived analytically or computed numerically
Many tools (MATLAB, Python libraries) handle Jacobian computation automatically
EKF performance depends heavily on the quality of the linearization

Summary

Real-world systems are mostly nonlinear
EKF extends Kalman filtering to nonlinear systems
Nonlinear functions are linearized at every step
Jacobians replace constant matrices
The underlying Kalman filter structure remains unchanged

If you understand the Kalman filter, EKF is a natural and straightforward extension.