Daniel Silvestre

Gossip average consensus in a Byzantine environment using stochastic Set-Valued Observers

We address the problem of a consensus system in the presence of Byzantine faults seen as an attacker injecting a perturbation in the state of the nodes. We propose the use of Set-Valued Observers to detect if the state observations are compatible with the system dynamics. The method is extended to the stochastic case by introducing a strategy to construct a set that is guaranteed to contain all possible states with, at least, a pre-specified desired probability. The proposed algorithm is stable in the sense that it requires a finite number of vertices to represent polytopic sets while also enabling the a priori computation of the largest magnitude of a disturbance that an attacker can inject without being detected.

Stochastic and deterministic fault detection for randomized gossip algorithms

This paper addresses the problem of detecting faults in linear randomized gossip algorithms, where the selection of the dynamics matrix is stochastic. A fault is a disturbance signal injected by an attacker to corrupt the states of the nodes. We propose the use of Set-Valued Observers (SVOs) to detect if the state observations are compatible with the system dynamics for the worst case in a deterministic setting. The concept of Stochastic Set-Valued Observers (SSVOs) is also introduced to construct a set that is guaranteed to contain all possible states with, at least, a pre-specified desired probability. The proposed algorithm is stable in the sense that it requires a finite number of vertices to represent polytopic sets and it allows for the computation of the largest magnitude of the disturbance that an attacker can inject in the network without being detected. Results are presented to reduce the computational cost of this approach and, in particular, by considering only local information and representing the remainder of the network as a disturbance. The case of a consensus algorithm is discussed leading to the conclusion that, by using the proposed SVOs, finite-time consensus is achieved in non-faulty environments. A novel algorithm is proposed that produces less conservative set-valued state estimates by having nodes exchanging local estimates. The algorithm inherits all the previous properties and also enables finite-time consensus computation regardless of the value of the horizon.

Finite-time average consensus in a Byzantine environment using Set-Valued Observers

This paper addresses the problem of consensus in the presence of Byzantine faults, modeled by an attacker injecting a perturbation in the state of the nodes of a network. It is firstly shown that Set-Valued Observers (SVOs) attain finite-time consensus, even in the case where the state estimates are not shared between nodes, at the expenses of requiring large horizons, thus rendering the computation problem intractable in the general case. A novel algorithm is therefore proposed that achieves finite-time consensus, even if the aforementioned requirement is dropped, by intersecting the set-valued state estimates of neighboring nodes, making it suitable for practical applications and enabling nodes to determine a stopping time. This is in contrast with the standard iterative solutions found in the literature, for which the algorithms typically converge asymptotically and without any guarantees regarding the maximum error of the final consensus value, under faulty environments. The algorithm suggested is evaluated in simulation, illustrating, in particular, the finite-time consensus property.

Self-Triggered and Event-Triggered Set-Valued Observers

Abstract This paper addresses the problem of reducing the required network load and computational power for the implementation of Set-Valued Observers (SVOs) in Networked Control System (NCS). Event- and self-triggered strategies for NCS, modeled as discrete-time Linear Parameter-Varying (LPV) systems, are studied by showing how the triggering condition can be selected. The methodology provided can be applied to determine when it is required to perform a full (“classical”) computation of the SVOs, while providing low-complexity state overbounds for the remaining time, at the expenses of temporarily reducing the estimation accuracy. As part of the procedure, an algorithm is provided to compute a suitable centrally symmetric polytope that allows to find hyper-parallelepiped and ellipsoidal overbounds to the exact set-valued state estimates calculated by the SVOs. By construction, the proposed triggering techniques do not influence the convergence of the SVOs, as at some subsequent time instants, set-valued estimates are computed using the conventional SVOs. Results are provided for the triggering frequency of the self-triggered strategy and two interesting cases: distributed systems when the dynamics of all nodes are equal up to a reordering of the matrix; and when the probability distribution of the parameters influencing the dynamics is known. The performance of the proposed algorithm is demonstrated in simulation by using a time-sensitive example.

Fault detection for LPV systems using Set-Valued Observers: A coprime factorization approach

Abstract This paper addresses the problem of fault detection for linear parameter-varying systems in the presence of measurement noise and exogenous disturbances using Set-Valued Observers (SVOs). The applicability of current methods is limited in the sense that, to increase accuracy, the detection requires a large number of past measurements and the boundedness of the set-valued estimates is only guaranteed for stable systems. In order to widen the class of systems to be modeled and also to reduce the associated computational cost, the aforementioned issues must be addressed. A solution involving left-coprime factorization and deadbeat observers is proposed that reduces the required number of past measurements without compromising accuracy and allowing the design of SVOs for fault detection of unstable systems by using the resulting coprime factorization stable subsystems. The algorithm is shown to produce bounded set-valued estimates and an example is provided. Performance is assessed through simulations, illustrating, in particular that small-magnitude faults (compared to exogenous disturbances) can be detected under mild assumptions.

Average consensus and gossip algorithms in networks with stochastic asymmetric communications

We consider that a set of distributed agents desire to reach consensus on the average of their initial state values, while communicating with neighboring agents through a shared medium. This communication medium allows only one agent to transmit unidirectionally at a given time, which is true, e.g., in wireless networks. We address scenarios where the choice of agents that transmit and receive messages at each transmission time follows a stochastic characterization, and we model the topology of allowable transmissions with asymmetric graphs. In particular, we consider: (i) randomized gossip algorithms in wireless networks, where each agent becomes active at randomly chosen times, transmitting its data to a single neighbor; (ii) broadcast wireless networks, where each agent transmits to all the other agents, and access to the network occurs with the same probability for every node. We propose a solution in terms of a linear distributed algorithm based on a state augmentation technique, and prove that this solution achieves average consensus in a stochastic sense, for the special cases (i) and (ii). Expressions for absolute time convergence rates at which average consensus is achieved are also given.

Self-Triggered Set-Valued Observers

This paper addresses the problem of high computational requirements in the implementation of Set-Valued Observers (SVOs), which places stringent constraints in terms of their use in applications where low computational power is available or the plant is sensitive to delay. It is firstly shown how to determine an overbound for the set-valued estimates, which reduces the overhead by limiting the number of inequalities defining those set-valued state estimates. In the particular setting of distributed gossip problems, the proposed algorithm is shown to have constant complexity. This algorithm is of prime importance to reduce the computational load and enable the use of such estimates for real-time applications. Results are also provided regarding the frequency of the triggers in the worst-case scenario. The performance of the proposed method is evaluated through simulation.

Finite-time convergence policies in state-dependent social networks

This paper addresses the problem of finite-time convergence in a social network for a political party or an association, modeled as a distributed iterative system with a graph dynamics chosen to mimic how people interact. It is firstly shown that, in this setting, finite-time convergence is achieved only when nodes form a complete network, and that contacting with agents with distinct opinions reduces to a half the required interconnections. Two novel strategies are presented that enable finite-time convergence, even for the case where each node only contacts the two closest neighbors. These strategies are of prime importance, for instance, in a company environment where agents can be motivated to reach faster conclusions. The performance of the proposed policies is assessed through simulation, illustrating, in particular the finite-time convergence property.