Sum‐of‐squares–based consensus verification for directed networks with nonlinear protocols

Abstract

Summary In this paper, we investigate the consensus verification problem of nonlinear agents in a fixed directed network with a nonlinear protocol. Inspired by the classical Lipschitz‐like condition, we introduce a more relax condition for the dynamics of the nonlinear agents. This condition is motivated via the construction of general Lyapunov functions for achieving asymptotic consensus. Especially, for agents where dynamics are described by polynomial function of the states, our consensus criterion can be converted to a sum of squares (SOS) programming problem, solvable via semidefinite programming tools. Of interest is that the scale of the resulting SOS programming problem does not increase as the size of the network increases, and thus, the applicability to analyze consensus of large‐scale networks is promising. Finally, an example is given to illustrate the effectiveness of our theoretical results.

Uniform exponential stability criteria with verification for nonautonomous switched nonlinear systems with uncertainty

“…In our future works, we would like to construct a proper uncertainty‐independent Lyapunov function to gain a necessary and sufficient condition for the GES of UNSNS. Besides, extending our proposed results to discrete‐time UNSNS 40,48 and the multiagent systems with switching topology, 7‐11 and even designing the proportional‐integral‐derivative controller for UNSNS based on dwell time and scalar functions 49,50 are interesting. Moreover, the computation issue of more general UNSNS is also worthy to be investigated.…”

“…Remark Solving constraints (30)–(35) are inherently intractable since even testing non‐negativity of a polynomial is NP‐hard when the polynomial has degree 4 or higher 45 . Because of this, we solve them via sum of squares programming 7,33‐36 in Step 2. Moreover, considering the extra NP‐hardness of solving constraints (38) and (42), which are bilinear semidefinite constraints, 36,46,47 we in Step 3 experientially preset the rational function $$$ \mu (t) $$$, to transform the bilinear semidefinite programming problems into linear semidefinite programming problems.…”

“…where 0 < 𝜖 < min{2pL, 2p𝛼}, 𝛿 is big enough to meet 𝛽 2p e 𝜖−2p𝛼𝛿 − 1 < 0 and V u (t, x) was defined by Equation (7). Due to inequalities ( 12) and ( 13), we obtain that…”

“…Switched systems are composed of several subsystems along with a switching signal determining which subsystem is activated at each instant of time 1‐4 . As a special class of hybrid systems, 5,6 switched systems receive plenty of attentions since they can appropriately model many complex systems such as robotic systems, multiagent systems, signal processing, manufacturing systems, aircraft control, automotive engine control and so on 7‐11 . On the other hand, uncertainties are frequently encountered in complex systems 12 .…”

“…Another important problem arising from stability analysis is the computational problem of mechanically searching for the required Lyapunov functions. Fruitful advances in the areas of semidefinite programming and the use of the sum of squares decomposition to efficiently check stability of systems were proposed 7,33‐36 . Inspired by the aforementioned status of research, we in this paper are dedicated to develop several less conservative sufficient and necessary conditions in the framework of a nondecreasing condition and the corresponding mechanical verification for the global uniform exponential stability (GUES) of UNSNS.…”