# LMIs in Control/pages/Discrete Time Mixed H2-H∞ Optimal Full State Feedback Control

**Discrete-Time Mixed H2-H∞-Optimal Full-State Feedback Control**

A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.

A full-state feedback controller (i.e., ) is to be designed to minimize the H2 norm of the closed loop transfer matrix from the exogenous input to the performance output while ensuring the H∞ norm of the closed-loop transfer matrix from the exogenous input to the performance output is less than .

**The System**Edit

Discrete-Time LTI System with state space realization

**The Data**Edit

The matrices: System .

**The Optimization Problem**Edit

The following feasibility problem should be optimized:

Minimize the H2 norm of the closed loop transfer matrix , while ensuring the H∞ norm of the closed-loop transfer matrix is less than , while obeying the LMI constraints.

**The LMI:**Edit

Discrete-Time Mixed H2-H∞-Optimal Full-State Feedback Controller is synthesized by solving for , and that minimize subject to

The LMI formulation

H∞ norm <

H2 norm <

**Conclusion:**Edit

The H2-optimal full-state feedback controller gain is recovered by

**Implementation**Edit

A link to CodeOcean or other online implementation of the LMI

MATLAB Code

**Related LMIs**Edit

[1] - Continuous Time Mixed H2-H∞ Optimal Full State Feedback Control

**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.