H₂ optimal control for a wide class of discrete-time linear stochastic systems

Abstract

In this article, the problem of H ₂-control of a discrete-time linear system subject to Markovian jumping and independent random perturbations is considered. Different H ₂ performance criteria (often called H ₂-norms) are introduced and characterised via solutions of some suitable linear equations on certain spaces of symmetric matrices. Some aspects specific to the discrete-time framework are revealed. The problem of optimisation of H ₂-norms is solved under the assumption that full state vector is available for measurements. One shows that among all stabilising controllers of higher dimension, the best performance is achieved by a zero-order controller. The corresponding feedback gain of the optimal controller is constructed based on the stabilising solution of a system of discrete-time generalised Riccati equations.

Keywords:

• discrete-time linear systems
• Markov chains
• independent random perturbations
• H ₂-norms
• H ₂-optimal control
• robust control
• Markov processes
• stochastic systems

AMS Subject Classifications:

• Primary: 93E20
• 39A11
• Secondary: 93C55
• 93C15

Acknowledgement

This work was partially supported by Grant no. 2-CEx06-11-18/2006 of the Romanian Ministry of Education and Research.