Achieving diagonal interactor matrix for multivariable linear systems with uncertain parameters

• Published in: Automatica (Oxford) Vol. 29; no. 6; pp. 1547 - 1550
• Main Authors: Gibbens, Peter W., Schwartz, Carla A., Fu, Minyue
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.11.1993; Elsevier
• Subjects: Adaptive control; Applied sciences; Computer science; control theory; systems; Control theory. Systems; Exact sciences and technology; interactor matrix; linear systems; robustness; System theory; vector relative degree; vector relative degree; robustness; linear systems; Adaptive control; interactor matrix; Uncertain system; Decoupling; Linear system; Matrix calculus; Robustness; Multivariable system
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/0005-1098(93)90019-P

The notion of interactor matrix or equivalently the Hermite normal form, is a generalization of relative degree to multivariable systems, and is crucial in problems such as decoupling, inverse dynamics, and adaptive control. In order for a system to be input-output decoupled using static state feedback, the existence of a diagonal interactor matrix must first be established. For a multivariable linear system which does not have a diagonal interactor matrix, dynamic precompensation or dynamic state feedback is required for achieving a diagonal interactor matrix for the compensated system. Such precompensation often depends on the parameters of system, and is thus difficult to implement with accuracy when the system is subject to parameter uncertainty. In this paper we characterize a class of linear systems which can be precompensated to achieve a diagonal interactor matrix without the exact knowledge of the system parameters. More precisely, we present necessary and sufficient conditions on the transfer matrix of the system under which there exists a diagonal dynamic precompensator such that the compensated system has a diagonal interactor matrix. These conditions are associated with the so-called (non)generic singularity of certain matrix related to the system structure but independent of the system parameters. The result of this paper is expected to be useful in robust and adaptive designs.