Persistent inputs and the standard H₂ multivariable control problem

Abstract

The H ₂ control problem is formulated with exogenous inputs having unstable shape-deterministic components. Such inputs are called persistent. Some issues in connection with past solutions of the H ₂ control problem in this case are carefully described and addressed. In particular, it is established that two- and three-degree-of freedom (2DOF and 3DOF) systems with persistent inputs cannot be treated within the framework of the standard configuration. In addition, past treatments of persistent inputs within the framework of the generalised 2DOF configuration focused on solutions which yielded stable error transforms without explicitly requiring the same for controller outputs. In this article, a more general configuration is treated and physical considerations are invoked to justify imposition of the requirement that both the regulated variable z(s) and the controller output u(s) be stable. A persistent input model for which there exists a stabilising controller that makes both z(s) and u(s) stable is called acceptable and the necessary and sufficient condition for such acceptability is determined. Also considered is a persistent input model for which there exists a stabilising controller that makes the controller output, the measured output and the regulated variable stable. Such a model is called strictly acceptable and the necessary and sufficient condition for strict acceptability is given. The subset of all stabilising controllers associated with an acceptable persistent input model is parameterised and this parameterisation is used to formulate an H ₂ optimisation problem with persistent inputs which can then be solved using standard procedures.

Keywords:

• optimal H ₂ design
• persistent inputs
• acceptable persistent inputs
• generalised plant
• standard configuration

Acknowledgements

The authors are grateful to Professor D.C. Youla who supplied significant insights in connection with Theorem 1 and other helpful comments.