A structure-preserving algorithm for the minimum H_∞ norm computation of finite-time state feedback control problem

Abstract

The algorithm being developed here is based on the generating function approach for finite-time H _∞ control and application of canonical transformation of linear Hamiltonian system. First, an equivalent finite-time H _∞ control law in terms of the third-type generating function is derived. Then, by using symplectic structure of the Hamiltonian system's state transition matrix, a group of matrix recursive formulae are deduced for the evaluation of the finite-time H _∞ control law. Combining with a matrix singularity testing procedure, this recursive algorithm verifies the existence condition of sub-optimal H _∞ controllers and gives the minimum H _∞ norm of finite-time control systems. Inherited from the canonical transformation, the matrix recursive formulae have a standard symplectic form; this structure-preserving property helps facilitate reliable and effective computation. Numerical results show the effectiveness of the proposed algorithm.

Keywords:

• H _∞ control
• H _∞ norm
• generating function
• canonical transformation
• Hamiltonian system
• differential Riccati equations

Acknowledgements

The authors would like to thank the reviewers for their valuable comments and suggestions on this article. This work was supported by National Natural Science Foundation of China (No.10632030) and Research Fund for the Doctoral Program of Higher Education (No.20070141067).