Laura Giarré

Non-linear protocols for optimal distributed consensus in networks of dynamic agents

We consider stationary consensus protocols for networks of dynamic agents with fixed topologies. At each time instant, each agent knows only its and its neighbors’ state, but must reach consensus on a group decision value that is function of all the agents’ initial state. We show that the agents can reach consensus if the value of such a function is time-invariant when computed over the agents’ state trajectories. We use this basic result to introduce a non-linear protocol design rule allowing consensus on a quite general set of values. Such a set includes, e.g., any generalized mean of order p of the agents’ initial states. As a second contribution we show that our protocol design is the solution of individual optimizations performed by the agents. This notion suggests a game theoretic interpretation of consensus problems as mechanism design problems. Under this perspective a supervisor entails the agents to reach a consensus by imposing individual objectives. We prove that such objectives can be chosen so that rational agents have a unique optimal protocol, and asymptotically reach consensus on a desired group decision value. We use a Lyapunov approach to prove that the asymptotical consensus can be reached when the communication links between nearby agents define a time-invariant undirected network. Finally we perform a simulation study concerning the vertical alignment maneuver of a team of unmanned air vehicles.

Identification of linear parameter varying models

We consider the problem of identifying discrete-time linear parameter varying models of nonlinear or time-varying systems. We assume that inputs, outputs and the scheduling parameters are measured, and a form of the functional dependence of the coefficients on the parameters. We show how the identification problem can be reduced to a linear regression, and we give conditions on persistency of excitation in terms of the inputs and parameter trajectories.

NARX models of an industrial power plant gas turbine

This brief reports the experience with the identification of a nonlinear autoregressive with exogenous inputs (NARX) model for the PGT10B1 power plant gas turbine manufactured by General Electric-Nuovo Pignone. Two operating conditions of the turbine are considered: isolated mode and nonisolated mode. The NARX model parameters are estimated iteratively with a Gram-Schmidt procedure, exploiting both forward and stepwise regression. Many indexes have been evaluated and compared in order to perform subset selection in the functional basis set and determine the structure of the nonlinear model. Various input signals (from narrow to broadband) for identification and validation have been considered.

Complex Dynamics in a Harmonically Excited Lennard-Jones Oscillator: Microcantilever-Sample Interaction in Scanning Probe Microscopes

In this paper we model the microcantilever-sample interaction in an atomic force microscope (AFM) via a Lennard-Jones potential and consider the dynamical behavior of a harmonically forced system. Using nonlinear analysis techniques on attracting limit sets, we numerically verify the presence of chaotic invariant sets. The chaotic behavior appears to be generated via a cascade of period doubling, whose occurrence has been studied as a function of the system parameters. As expected, the chaotic attractors are obtained for values of parameters predicted by Melnikov theory. Moreover, the numerical analysis can be fruitfully employed to analyze the region of the parameter space where no theoretical information on the presence of a chaotic invariant set is available. In addition to explaining the experimentally observed chaotic behavior, this analysis can be useful in finding a controller that stabilizes the system on a nonchaotic trajectory. The analysis can also be used to change the AFM operating conditions to a region of the parameter space where regular motion is ensured.

Model quality evaluation in set membership identification

Abstract Identification from corrupted input-output measurements of systems that do not necessarily belong to the model class used is investigated. This leads to a nonstandard set membership (SM) identification problem. The ‘goodness’ of different model classes is measured by the conditional radius of information, a generalization of the radius in standard SM identification theory, giving a measure of the minimal worst-case modeling error. Upper and lower bounds on the radius are derived for linearly parameterized model classes. Specific formulas for the upper and lower bounds are given for the case of H 2 identification of exponentially stable systems in the presence of powerbounded noise. The bounds are shown to coincide with the conditional radius when the model space dimension is equal to the number of output measurements. An almost-optimal identification algorithm is derived, giving identification error within the range of the derived bounds.

Numerical analysis of complex dynamics in atomic force microscopes

We model the microcantilever-sample interaction in an atomic force microscope (AFM) via a Lennard-Jones potential, and consider the dynamical behavior of the corresponding harmonically forced system. Existence of chaotic invariant sets predicted by Melnikov theory is then numerically verified. Such dynamics appears to be generated via a cascade of period doubling bifurcations, whose occurrence has been studied as a function of the system parameters. The importance of this analysis is twofold: it can be used to change the AFM operating conditions to a region of the parameter space where regular motion is ensured or it can be useful in designing a controller that stabilizes the system on a non-chaotic trajectory.

Consensus in Noncooperative Dynamic Games: A Multiretailer Inventory Application

We focus on Nash equilibria and Pareto optimal Nash equilibria for a finite horizon noncooperative dynamic game with a special structure of the stage cost. We study the existence of these solutions by proving that the game is a potential game. For the single-stage version of the game, we characterize the aforementioned solutions and derive a consensus protocol that makes the players converge to the unique Pareto optimal Nash equilibrium. Such an equilibrium guarantees the interests of the players and is also social optimal in the set of Nash equilibria. For the multistage version of the game, we present an algorithm that converges to Nash equilibria, unfortunately, not necessarily Pareto optimal. The algorithm returns a sequence of joint decisions, each one obtained from the previous one by an unilateral improvement on the part of a single player. We also specialize the game to a multiretailer inventory system.

Identification of approximated hammerstein models in a worst-case setting

The identification of Hammerstein models for nonlinear systems in considered in a worst-case setting, assuming unknown-but-bounded measurement noise. A new approach is proposed in which the identification of a low-complexity Hammerstein model amounts to the computation of the Chebichev center of a set of matrices conditioned to the manifold of rank-one matrices. An identification algorithm, based on a relaxation technique, is proposed and its consistency is proven. The algorithm is computationally attractive in two cases: noise bounded either in 2 or in ∞ norm. The effectiveness of the proposed central algorithm and the comparison with the corresponding projection algorithm, which is based on the singular-value decomposition, are investigated both analytically and through numerical examples. In particular, tight error bounds are obtained for the projection algorithm.

Distributed consensus protocols for coordinating buyers

In this paper, we introduce a distributed consensus protocol for coordinating orders of a network of buyers also called agents/decision makers. Each buyer chooses a different threshold strategy, defining its intention to place an order only if at least other l buyers will do the same. We prove that consensus is reached asymptotically globally and coordination is the same that if the decision making process would be centralized, namely, any decision maker (DM) has access to the thresholds of all other DMs and chooses to order or not. The proposed distributed protocol has the advantage that buyers do not have to communicate their threshold strategy in advance, and consensus is reached without exploring all the possible threshold values.

SM identification of approximating models for H, robust control

Set Membership (SM) W, identification of mixed parametric and nonparametric models is investigated, aimed to estimate a low order approximating model and an identification error, giving a measure of the unmodeled dynamics in a form well suited for H, control methodologies. In particular, the problem of estimating the parameters of the parametric part and the H, bound on the modeling error is solved using frequency domain data, supposing lbo bounded measurement errors and exponentially stable unmodeled dynamics. The effectiveness of the proposed procedure is tested on some numerical examples, showing the advantages of the proposed methods over the existing nonparametric H, identification approaches are shown, in terms of lower model order and of tightness in the modeling error bounds.