A Test for Unconditional Stability Based on Polynomial Convexification

A theoretical analysis is carried out of the problem of checking the unconditional stability of a linear <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-port with arbitrary <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>, explaining at length its inherent difficulties. To overcome the algebraic roadblock, an optimization-based method is proposed; the main novelty is that notwithstanding the use of optimization, the problem formulation is specifically chosen so as to guarantee convergence to the global minimum. In particular, recent advancements in the field of convex optimization, mostly unknown to the high-frequency engineering community, are exploited to that end. The result is a method suitable for checking the unconditional stability of linear <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>-ports with <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> up to 4. Higher numbers of ports, although allowed in principle, are not viable with the current combination of hardware resources and software implementation. Several examples of applications are provided, both purely illustrative and from actual circuit designs.

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