# LMIs in Control/pages/Multi-Criterion LQG

LMIs in Control/pages/Multi-Criterion LQG

This is a WIP based on the template

## Contents

**The System**[edit]

The system is a linear time-invariant system, that can be represented in state space as shown below:

where u is the input; y is the measured output; z is the output of interest; w and v are white noise signals with constant positive-definite spectral density matrices W and V, respectively. and , and the system is controllable and observable.

**The Data**[edit]

The matrices and the noise signals .

**The Optimization Problem**[edit]

In the Linear Quadratic Gaussian (LQG) control problem, the goal is to minimize a quadratic cost function while the plant has random initial conditions and suffers white noise disturbance on the input and measurement.

There are multiple outputs of interest for this problem. They are defined by

For each of these outputs of interest, we associate a cost function:

Additionally, the matrices and must be found as the solutions to the following Riccati equations:

The optimization problem is to minimize over u subject to the measurability condition and the constraints . This optimization problem can be formulated as:

over , with:

**The LMI:** Multi-Criterion LQG[edit]

over , subject to the following constraints:

**Conclusion:**[edit]

The result of this LMI is the optimal cost achievable, the result of the optimization.

**Implementation**[edit]

A link to CodeOcean or other online implementation of the LMI

**Related LMIs**[edit]

Links to other closely-related LMIs

## External Links[edit]

A 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.