Extended Full Block S-Procedure for Distributed Control of Interconnected Systems

Abstract

This paper proposes a novel method for distributed controller synthesis of homogeneous interconnected systems consisting of identical subsystems. The objective of the designed controller is to minimize the L 2 -gain of the performance channel. The proposed method is an extended formulation of the Full Block S-Procedure (FBSP) where we introduce an additional set of variables. This allows to relax the block-diagonal structural assumptions on the Lyapunov and multiplier matrices required for distributed control design, which reduces conservatism w.r.t. most existing approaches. We show how to decompose the proposed extended FBSP into small synthesis conditions, of the size of one individual subsystem.

Self‐triggered consensus of multi‐agent systems with quantized relative state measurements

“…Inspired by the norm used in the theory of operator semigroups (see, e.g. the proof of Theorem 5.2 in Chapter 1 of [60]), we introduce a new semi‐norm on $$\mathbb {R}^N$$, which will lead to the semi‐contractivity property [58, 59] of the matrix exponential of the negative Laplacian matrix. Lemma Let $$\Vert \cdot \Vert$$ be an arbitrary norm on $$\mathbb {R}^N$$, and let $$L,F \in \mathbb {R}^{N\times N}$$.…”

“…The semi‐norm is constructed from the maximum norm and is suitable for handling errors of individual agents due to quantization and self‐triggered sampling. Moreover, the Laplacian matrix $$L\in \mathbb {R}^{N \times N}$$ of the multi‐agent system has the following semi‐contractivity property: There exists a constant $$\gamma >0$$ such that $$$$\begin{equation*} {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert e^{-Lt} v \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }_{\infty } \le e^{-\gamma t} {\vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert v \vert \hspace{-1.0625pt}\vert \hspace{-1.0625pt}\vert }_{\infty } \end{equation*}$$$$for all $$v \in \mathbb {R}^N$$ and $$t \ge 0$$; see [58, 59] for the semi‐contraction theory. The semi‐contractivity property facilitates the analysis of state trajectories under self‐triggered sampling and consequently leads to a simple design of the scaling parameter for finite‐level dynamic quantization.…”