Continuous- and discrete-time D-stability, joint D-stability, and their applications: μ theory and diagonal stability approaches

This paper studies relationships, implications, and applications of diagonal stability and D-stability. Necessary and sufficient conditions for continuous- and discrete-time D-stability are presented in terms of structured singular values of related matrices. It is shown that, for a certain class of interconnected systems, diagonal stability and D-stability are equivalent and the optimization of diagonal scaling gives a necessary and sufficient condition for stability of those systems. This paper also discusses several issues on diagonal stability and additive D-stability with their applications to robust optimal power distribution control and stability analysis of a certain class of reaction-diffusion systems with which the proposed robust stability and stabilizing conditions are illustrated. The resultant analysis and control design problems are formulated as linear or semidefinite programs.