# LMIs in Control/Applications/An LMI for the Kalman Filter

LMIs in Control/Applications/An LMI for the Kalman Filter

This is a **An LMI for the Kalman Filter**. The Kalman Filter is one of the most widely used state-estimation techniques. It has applications in multiple aspects of navigation (inertial, terrain-aided, stellar.)

**The System**

edit
Continuous Time:

The process and sensor noises are given by and respectively.

Discrete Time:

The process and sensor noises are given by and respectively.

**The Data**

edit
The data required for the Kalman Filter include a model of the system that the states are trying to be output and a measurement that is the output of the system dynamics being estimated.

**The Filter**

edit
The Filter and Estimator equations can be written as:

Continuous Time

Discrete Time

**The Error**

edit
The error dynamics evolve according to the following expression

Continuous Time

Discrete Time

**The Optimization Problem**

edit
The Kalman Filtering (or LQE) problem is a Dual to the LQR problem. Replace the matrices from LQR with

The Kalman Filter chooses to minimize the cost This cost can be thought of as the covariance of the state error between the actual and estimated state. When the state error covariance is low the filter has converged and the estimate is good.

The Luenberger or Kalman gain can be computed from

The process and measurement noise covariances for the Kalman filter are given by

The matrix satisfies the following equality

We also cover the discrete Kalman Filter formulation which is more useful for real-life computer implementations.

The discrete Kalman filter chooses the gain where the PSDs of the process and sensor noises are given by

The steady-state covariance of the error in the estimated state is given by and satisfies the following Riccati equation.

- Objective: State Estimate Error Covariance
- Variables: Observer Gains
- Constraints: Dynamics of System to be Estimated

**The LMI:** H2-Optimal Control Full-State Feedback to LQR to Kalman Filter

edit
The Kalman Filter is a dual to the LQR problem which has been shown to be equivalent to a special case of H2-static state feedback.

Start with the H2-Optimal Control Full-State Feedback.

The following are equivalent

To solve the LQR problem using H2 optimal state-feedback control the following variable substitutions are required.

Then

This results in the following LMI.

To solve the Kalman Filtering problem using the LQR LMI replace with and This results in the following LMI.

The discrete-time Kalman Filtering LMI is saved for another page as it requires derivation of the Discrete-Time LQR LMI problem which was not covered in class.

**Conclusion:**

edit
The LMI for the Kalman Filter allows us to calculate the optimal gain for state estimation. It is shown that it can be found as a special case of the H2-optimal state feedback with the appropriate substitution of matrices. The LMI gives us a different way of computing the optimal Kalman gain.

**Implementation**

edit
A link to CodeOcean or other online implementation of the LMI

**Related LMIs**

edit
Links to other closely-related LMIs

## External Links

editA list of references documenting and validating the LMI.

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd
- [1] - Applied Optimal Estimation, Arthur Gelb (Kalman Filtering and Optimal Estimation Classic)
- [2] - All Source Position, Navigation and Timing. New textbook that has a good development and description of the Kalman Filter and some of its very practical uses and implementations by Boeing Senior Technical Fellow Ken Li.