Robust Control: The Parametric Approach
Robust Control: The Parametric Approach
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This paper revisits the problem of checking the robust stability of matrix familes generated by interval parameters in a matrix. Previous research on this topic (including that of this author) erroneously assumed that this family can be represented as a standard convex combination of vertex matrices (matrices evaluated at the end points of the interval parameters). Solutions offered to this important problem with this erroneous assumption suffered various set backs in the form of counterexamples which caused considerable disillusionment in the research community, especially for this author, warranting continued research on this problem. As a result of this new research, for the first time in the literature, in this paper, explicit expressions for the convex combination coefficients in terms of the interval parameters are derived. These expressions help to clarify and explain the misconceptions that currently exist in the research community about the nature of the convex combination coefficients induced by the interval parameters and shed significant insight into the 'correct' scenario for this case. A previously presented 'vertex algorithm' by the author for this tough problem was derived under the misunderstood mapping of the parameter space to the matrix element space that currently exists in the literature (in the absence of the explicit expressions derived in this paper). Based on the correct mapping presented in this paper, a thorough and correct vertex solution is offered for this tough problem. Several examples are given which clearly demonstrate effectiveness of the new, corrected algorithm.