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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 SystemEdit

Discrete-Time LTI System with state space realization

 

The DataEdit

The matrices: System  .

The Optimization ProblemEdit

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  

ImplementationEdit

A link to CodeOcean or other online implementation of the LMI
MATLAB Code

Related LMIsEdit

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

External LinksEdit

A list of references documenting and validating the LMI.

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