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This paper provides finite dimensional convex conditions to construct homogeneous polynomially parameter-dependent Lur'e functions which ensure the stability of nonlinear systems with state-dependent nonlinearities lying in general sectors and which are affected by uncertain parameters belonging to the unit simplex. The proposed conditions are written as linear matrix inequalities parametrized in terms of the degree g of the parameterdependent solution and in terms of the relaxation level d of the inequality constraints, based on algebraic properties of positive matrix polynomials with parameters in the unit simplex. As g and d increase, progressively less conservative solutions are obtained. The results in the paper include as special cases existing conditions for robust stability and for absolute stability analysis. A convex solution suitable for the design of robust nonlinear state feedback stabilizing controllers is also provided. Numerical examples illustrate the efficiency of the proposed conditions.

This paper focuses on the problem of designing a decentralized output feedback control strategy for synchronization of homogeneous multi-agent systems with global performance guarantees. The agents under investigation are described as linear singularly perturbed dynamics representing a wide class of physical systems characterized by processes evolving on two time-scales. The collaborative decentralized control is achieved using only output information from neighboring agents and considering that the only available graph information consists in its connectivity, that is, there is no centralized information related to the interconnection network structure. As methodology, the synchronization problem is rewritten as a dynamic output feedback robust stabilization of a singularly perturbed uncertain linear system with guaranteed cost. We show that these problems can be solved by using convex conditions expressed by LMIs and by decoupling the slow and fast dynamics. As an advantage, the fast dynamic matrix can be singular (nonstandard systems) and unstable. The proposed conditions circumvent some drawbacks of the existing works on this topic by providing a dynamic controller that does not depend on the singular parameter or by allowing the design of slow controllers when the fast system is stable. Numerical examples are presented to demonstrate the effectiveness of the proposed protocol and design method.

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