STABILITY, FINITE-TIME STABILITY AND PASSIVITY CRITERIA FOR DISCRETE-TIME DELAYED NEURAL NETWORKS

In this paper, we present the problem of stability, finite-time stability and passivity for discrete-time neural networks (DNNs) with variable delays. For the purposes of stability analysis, an augmented Lyapunov-Krasovskii functional (LKF) with single and double summation terms and several augmented vectors is proposed by decomposing the time-delay interval into two non-equidistant subintervals. Then, by using the Wirtinger-based inequality, reciprocally and extended reciprocally convex combination lemmas, tight estimations for sum terms in the forward difference of LKF are given. In order to relax the existing results, several zero equalities are introduced and stability criteria are proposed in terms of linear matrix inequalities (LMIs). The main objective for the finite-time stability and passivity analysis is how to effectively evaluate the finite-time passivity conditions for DNNs. To achieve this, some weighted summation inequalities are proposed for application to a finite-sum term appearing in the forward difference of LKF, which helps to ensure that the considered delayed DNN is passive. The derived passivity criteria are presented in terms of linear matrix inequalities. Some numerical examples are presented to illustrate the proposed methodology

Consensus in multi-agent systems with time-delays

Different consensus problems in multi-agent systems have been addressed in this thesis. They represent improvements with respect to the state of the art. In the first part of the thesis in luding Chapters 2, 3, and 4, the state of the art of the representation and stability analysis of consensus problems, time-delay systems, and sampled-data systems have been presented. Novel contributions have been illustrated in Chapters 5-8. Particularly, in Chapter 5 we reported the results of Zareh et al. (2013b), where we investigated the consensus problem for networks of agents with double integrator dynamics affected by time-delay in their coupling. We provided a stability result based on the Lyapunov-Krasovskii functional method and a numerical proc edure based on an LMI condition which depends only on the algebraic connectivity of the considered network topologies, thus reducing greatly the computational complexity of the procedure. Obviously, this result implies the existence of a minimum dwell time such that the proposed consensus protocol is stable for slow swit things between network topologies with suffient algebraic connectivity. Future work will involve actually computing such a dwell time by adopting a multiple Lyapunov function method and evaluating the worst case sider only delayed relative measurements instead of delayed absolute values of the neighbors' state variables. The results of Zareh et al. (2013a) were addressed in Chapter 6, in which a on- tinuous time version of a consensus on the average protocol for arbitrary strongly connected directed graphs is proposed and its convergence properties with respect to time delays in the local state update are characterized. The convergenc e properties of this algorithm depend upon a tuning parameter that an be made arbitrary small to prove stability of the networked system. Simulations have been presented to corroborate the theoretical results and show that the existenc e of a small time delay an a tually improve the algorithm performance. Future work will include an extension of the mathematical characterization of the proposed algorithm to consider possibly heterogeneous or time-varying delays. In Chapter 7 we proposed a PD-like consensus algorithm for a second-order multi- agent system where, at non-periodic sampling times, agents transmit to their neighbors information about their position and veloc ity, while each agent has a perfect knowledge of its own state at any time instant. Conditions have been given to prove onsensus to a ommon xed point, based on LMIs verification. Moreover, we also show how it is possible to evaluate an upper bound on the de ay rate of exponential convergence of stable modes. In Chapter 8, mainly based on our paper Zareh et al. (2014b), we considered the same problem as in Chapter 7. The main contribution consists in proving consensus to a common fixed point, based on LMIs verification, under the assumption that the network topology is not known and the only information is an upper bound on the connectivity. Two are the main directions of our future research in this framework. First, we want to compute analytically an upper bound on the value of the second largest eigenvalue of the weighted adjacency matrix that guarantees consensus, as a function of the other design parameters. Second, we plan to study the case where agents do not have a perfect knowledge of their own state

Consensus in multi-agent systems with time-delays

Different consensus problems in multi-agent systems have been addressed in this thesis. They represent improvements with respect to the state of the art. In the first part of the thesis in luding Chapters 2, 3, and 4, the state of the art of the representation and stability analysis of consensus problems, time-delay systems, and sampled-data systems have been presented. Novel contributions have been illustrated in Chapters 5-8. Particularly, in Chapter 5 we reported the results of Zareh et al. (2013b), where we investigated the consensus problem for networks of agents with double integrator dynamics affected by time-delay in their coupling. We provided a stability result based on the Lyapunov-Krasovskii functional method and a numerical proc edure based on an LMI condition which depends only on the algebraic connectivity of the considered network topologies, thus reducing greatly the computational complexity of the procedure. Obviously, this result implies the existence of a minimum dwell time such that the proposed consensus protocol is stable for slow swit things between network topologies with suffient algebraic connectivity. Future work will involve actually computing such a dwell time by adopting a multiple Lyapunov function method and evaluating the worst case sider only delayed relative measurements instead of delayed absolute values of the neighbors' state variables. The results of Zareh et al. (2013a) were addressed in Chapter 6, in which a on- tinuous time version of a consensus on the average protocol for arbitrary strongly connected directed graphs is proposed and its convergence properties with respect to time delays in the local state update are characterized. The convergenc e properties of this algorithm depend upon a tuning parameter that an be made arbitrary small to prove stability of the networked system. Simulations have been presented to corroborate the theoretical results and show that the existenc e of a small time delay an a tually improve the algorithm performance. Future work will include an extension of the mathematical characterization of the proposed algorithm to consider possibly heterogeneous or time-varying delays. In Chapter 7 we proposed a PD-like consensus algorithm for a second-order multi- agent system where, at non-periodic sampling times, agents transmit to their neighbors information about their position and veloc ity, while each agent has a perfect knowledge of its own state at any time instant. Conditions have been given to prove onsensus to a ommon xed point, based on LMIs verification. Moreover, we also show how it is possible to evaluate an upper bound on the de ay rate of exponential convergence of stable modes. In Chapter 8, mainly based on our paper Zareh et al. (2014b), we considered the same problem as in Chapter 7. The main contribution consists in proving consensus to a common fixed point, based on LMIs verification, under the assumption that the network topology is not known and the only information is an upper bound on the connectivity. Two are the main directions of our future research in this framework. First, we want to compute analytically an upper bound on the value of the second largest eigenvalue of the weighted adjacency matrix that guarantees consensus, as a function of the other design parameters. Second, we plan to study the case where agents do not have a perfect knowledge of their own state

Stability of systems with fast-varying delay using improved Wirtinger's inequality

International audienceThis paper considers the stability of systems with fast-varying delay. The novelty of the paper comes from the consideration of a new integral inequality which is proved to be less conservative than the celebrated Jensen's inequality. Based on this new inequality, a dedicated construction of Lyapunov- Krasovskii functionals is proposed and is showed to have a great potential efficient in practice. The method is also combined with an efficient representation of the improved reciprocally convex combination inequality, recently provided in the literature, in order to reduce the conservatism induced by the LMIs optimization setup. The effectiveness of the proposed result is illustrated by some classical examples from the literature