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 SystemEdit

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 DataEdit

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 FilterEdit

The Filter and Estimator equations can be written as:

Continuous Time

 

Discrete Time

 

The ErrorEdit

The error dynamics evolve according to the following expression

Continuous Time

 

Discrete Time

 

The Optimization ProblemEdit

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 FilterEdit

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.

ImplementationEdit

A link to CodeOcean or other online implementation of the LMI

Related LMIsEdit

Links to other closely-related LMIs

External LinksEdit

A list of references documenting and validating the LMI.

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