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Saddle-point or primal-dual methods have recently attracted renewed interest as a systematic technique to design distributed algorithms, which solve convex optimization problems. When implemented online for streaming data or as dynamic feedback controllers, these algorithms become subject to disturbances and noise; convergence rates provide incomplete performance information, and quantifying input–output performance becomes more important. We analyze the input–output performance of the continuous-time saddle-point method applied to linearly constrained quadratic programs, providing explicit expressions for the saddle-point $\mathcal {H}_2$ norm under a relevant input–output configuration. We then proceed to derive analogous results for regularized and augmented versions of the saddle-point algorithm. We observe some rather peculiar effects—a modest amount of regularization significantly improves the transient performance, while augmentation does not necessarily offer improvement. We then propose a distributed dual version of the algorithm, which overcomes some of the performance limitations imposed by augmentation. Finally, we apply our results to a resource allocation problem to compare the input–output performance of various centralized and distributed saddle-point implementations and show that distributed algorithms may perform as well as their centralized counterparts. [ABSTRACT FROM AUTHOR]

Equilibrium-independent dissipativity (EID) is a recently introduced system property that requires a system to be dissipative with respect to any forced equilibrium configuration. This paper is a detailed examination of EID with quadratic supply rates for a common class of nonlinear control-affine systems. We provide an algebraic characterization of EID for such systems in the spirit of the Hill–Moylan lemma, where the usual stability condition is replaced by an incremental stability condition. Based on this characterization, we state results concerning internal stability, feedback stability, and absolute stability of EID systems. Finally, we study EID for discrete-time systems, providing the relevant definitions and an analogous Hill–Moylan-type characterization. Results for both continuous-time and discrete-time systems are illustrated through examples on physical systems and convex optimization algorithms. [ABSTRACT FROM AUTHOR]