Ion Matei

Convergence results for the linear consensus problem under Markovian random graphs

This paper discusses the linear asymptotic consensus problem for a network of dynamic agents whose communication network is modeled by a randomly switching graph. The switching is determined by a finite state Markov process, each topology corresponding to a state of the process. We address the cases where the dynamics of the agents is expressed both in continuous time and in discrete time. We show that, if the consensus matrices are doubly stochastic, average consensus is achieved in the mean square sense and the almost sure sense if and only if the graph resulting from the union of graphs corresponding to the states of the Markov process is strongly connected. The aim of this paper is to show how techniques from the theory of Markovian jump linear systems, in conjunction with results inspired by matrix and graph theory, can be used to prove convergence results for stochastic consensus problems.

Consensus-based linear distributed filtering

We address the consensus-based distributed linear filtering problem, where a discrete time, linear stochastic process is observed by a network of sensors. We assume that the consensus weights are known and we first provide sufficient conditions under which the stochastic process is detectable, i.e. for a specific choice of consensus weights there exists a set of filtering gains such that the dynamics of the estimation errors (without noise) is asymptotically stable. Next, we develop a distributed, sub-optimal filtering scheme based on minimizing an upper bound on a quadratic filtering cost. In the stationary case, we provide sufficient conditions under which this scheme converges; conditions expressed in terms of the convergence properties of a set of coupled Riccati equations.

Almost sure convergence to consensus in Markovian random graphs

In this paper we discuss the consensus problem for a network of dynamic agents with undirected information flow and random switching topologies. The switching is determined by a Markov chain, each topology corresponding to a state of the Markov chain. We show that in order to achieve consensus almost surely and from any initial state the sets of graphs corresponding to the closed positive recurrent sets of the Markov chain must be jointly connected. The analysis relies on tools from matrix theory, Markovian jump linear systems theory and random processes theory. The distinctive feature of this work is addressing the consensus problem with ¿Markovian switching¿ topologies.

Consensus problems with directed Markovian communication patterns

This paper is a continuation of our previous work and discusses the consensus problem for a network of dynamic agents with directed information flows and random switching topologies. The switching is determined by a Markov chain, each topology corresponding to a state of the Markov chain. We show that in order to achieve consensus almost surely and from any initial state, each union of graphs from the sets of graphs corresponding to the closed positive recurrent sets of states of the Markov chain must have a spanning tree. The analysis relies on tools from matrix theory, Markovian jump linear systems theory and random process theory. The distinctive feature of this work is addressing the consensus problem with “Markovian switching” topologies.

Optimal State Estimation for Discrete-Time Markovian Jump Linear Systems, in the Presence of Delayed Output Observations

In this paper, we investigate an optimal state estimation problem for Markovian Jump Linear Systems. We consider that the state has two components: the first component of the state is finite valued and is denoted as mode, while the second (continuous) component is in a finite dimensional Euclidean space. The continuous state is driven by a deterministic control input and a zero mean, white and Gaussian process noise. The observable output has two components: the first is the mode delayed by a fixed amount and the second is a linear combination of the continuous state observed in zero mean white Gaussian noise. Our paradigm is to design optimal estimators for the current state, given the current output observation. We provide a solution to this paradigm by giving a recursive estimator of the continuous state, in the minimum mean square sense, and a finitely parameterized recursive scheme for computing the probability mass function of the current mode conditional on the observed output. We show that the optimal estimator is nonlinear on the observed output and on the control input. In addition, we show that the computation complexity of our recursive schemes is polynomial in the number of modes and exponential in the mode observation delay.

Technical communique: A linear distributed filter inspired by the Markovian jump linear system filtering problem

In this paper we introduce a consensus-based distributed filter, executed by a sensor network, inspired by the Markovian jump linear system filtering theory. We show that the optimal filtering gains of the Markovian jump linear system can be used as an approximate solution of the optimal distributed filtering problem. This parallel allows us to interpret each filtering gain corresponding to a mode of operation of the Markovian jump linear system as a filtering gain corresponding to a sensor in the network. The approximate solution can be implemented distributively and guarantees a quantifiable level of performance.

Trust-based multi-agent filtering for increased Smart Grid security

We address the problem of state estimation of the power system for the Smart Grid. We assume that the monitoring of the electrical grid is done by a network of agents with both computing and communication capabilities. We propose a security mechanism aimed at protecting the state estimation process against false data injections originating from faulty equipment or cyber-attacks. Our approach is based on a multi-agent filtering scheme, where in addition to taking measurements, the agents are also computing local estimates based on their own measurements and on the estimates of the neighboring agents. We combine the multi-agent filtering scheme with a trust-based mechanism under which each agent associates a trust metric to each of its neighbors. These trust metrics are taken into account in the filtering scheme so that information transmitted from agents with low trust is disregarded. In addition, a mechanism for the trust metric update is also introduced, which ensures that agents that diverge considerably from their expected behavior have their trust values lowered.

Consensus-based distributed linear filtering

We address the consensus-based distributed linear filtering problem, where a discrete time, linear stochastic process is observed by a network of sensors. We assume that the consensus weights are known and we first provide sufficient conditions under which the stochastic process is detectable, i.e. for a specific choice of consensus weights there exists a set of filtering gains such that the dynamics of the estimation errors (without noise) is asymptotically stable. Next, we provide a distributed, sub-optimal filtering scheme based on minimizing an upper bound on a quadratic filtering cost. In the stationary case, we provide sufficient conditions under which this scheme converges; conditions expressed in terms of the convergence properties of a set of coupled Riccati equations. We continue with presenting a connection between the consensus-based distributed linear filter and the optimal linear filter of a Markovian jump linear system, appropriately defined. More specifically, we show that if the Markovian jump linear system is (mean square) detectable, then the stochastic process is detectable under the consensus-based distributed linear filtering scheme. We also show that the optimal gains of a linear filter for estimating the state of a Markovian jump linear system appropriately defined can be used to approximate the optimal gains of the consensus-based linear filter.

Distributed algorithms for optimization problems with equality constraints

In this paper we introduce two discrete-time, distributed optimization algorithms executed by a set of agents whose interactions are subject to a communication graph. The algorithms can be applied to optimization problems where the cost function is expressed as a sum of functions, and where each function is associated to an agent. In addition, the agents can have equality constraints as well. The algorithms are not consensus-based and can be applied to non-convex optimization problems with equality constraints. We demonstrate that the first distributed algorithm results naturally from applying a first order method to solve the first order necessary conditions for a lifted optimization problem with equality constraints; the solution of our original problem is embedded in the solution of this lifted optimization problem. Using an augmented Lagrangian idea, we derive a second distributed algorithm that requires weaker conditions for local convergence compared to the first algorithm. For both algorithms we address the local convergence properties.

Optimal Linear Quadratic Regulator for Markovian Jump Linear Systems, in the presence of one time-step delayed mode observations

Abstract In this paper, we provide the solution to the optimal Linear Quadratic Regulator (LQR) paradigm for Markovian Jump linear Systems, when the continuous state is available at the controller instantaneously, but the mode is available only after a delay of one time step. This paper is the first to investigate the LQR paradigm in the presence of such mismatch between the delay in observing the mode and the continuous state at the controller. We show that the optimal LQR policy is a time-varying matrix gain multiplied by the continuous component of the state, where the gain is indexed in time by the one-step delayed mode. The solution to the LQR is expressed as a collection of coupled Riccati iterations and equations, for the finite and the infinite horizon cases respectively. In the infinite horizon case the solution of the coupled Riccati equations or a certificate of infeasibility is obtained by solving a set of linear matrix inequalities. We also explain the difficulties of solving the LQR problem when the mode is observed by the controller with a delay of more than one step. We show that, with delays of more than one time-step, the optimal control will be a time-varying nonlinear function of the continuous state and of the control input, without presenting an exact solution.