Natural consensus filters for second order infinite dimensional systems

Abstract

This work proposes consensus filters for a class of second order infinite dimensional systems. The proposed structure of the consensus filters is that of a local filter written in the natural setting of a second order formulation with the additional coupling that enforces consensus by minimizing the disagreement between all local filters. An advantage of the second order formulation imposed on the local filters is the natural interpretation that they retain, namely that the time derivative of the estimated position is equal to the estimated velocity. Stability analysis of the collective dynamics is achieved via the use of a parameter-dependent Lyapunov functional, and which guarantees that asymptotically, all filters agree with each other and that they all converge to the true state of the second order system.

Introduction

The advent of sensor technologies and ever increasing computing capabilities spurred the use of sensor networks, mobile or fixed, in improving the estimation of physical processes; see [1], [2], [3], [4] and the references therein. Employing an array of sensing devices, either homogeneous or heterogeneous, provides additional flexibility and improvement in filtering and smoothing of many processes. A dramatic increase in works dealing with sensor networks has been witnessed since the early 90s and the main recipients were systems governed by lumped parameter systems.

The control and estimation of infinite dimensional systems also enjoyed continuous attention of many researchers in the past 50 years. Works concentrated from system-theoretic properties to optimal and robust control design. Many aspects of control implementation were also considered, as for example finite dimensional approximation, optimal model reduction. Optimization of parameters (damping design) and of actuator/sensor shape and location were also considered. A special class of infinite dimensional systems, namely second order infinite dimensional systems, often describing structural systems and acoustic systems, was traditionally viewed in a first order formulation. However, as was pointed out in [5], such a framework will not lead to a physically meaningful state estimator. The reason is that the derivative of the first component of the estimated state is not equal to the second component of the estimated state. This means that the derivative of the estimated position is not equal to the estimated velocity. Such an equality may be achieved asymptotically, but when one desires a state observer with a physically meaningful estimate, one must then resort to the design in a second order formulation. In fact, such an observer was considered in [5]. The design proposed a Luenberger-type observer and a parameter-dependent Lyapunov functional produced the requisite stability, providing asymptotic convergence of the state position and state velocity errors to zero, while retaining at all times, the physical relation of the estimated position and velocity.

When moving to a second order system with multiple sensors, one approach is to consider a centralized observer where all measurements are used by a single filter. An alternative approach is to consider multiple filters, each producing its own state estimate by using a smaller number of sensor measurements. In order to ensure that all such state estimates agree and collectively produce the same estimate, a modification is included in order to induce and enforce consensus.

The use of sensor networks for the estimation of spatially distributed processes did not receive comparable attention as the finite dimensional case. Sporadic attempts to extend and apply the use of sensor networks for the estimation of spatially distributed processes is rather meager, concentrated in few researchers [6], [7], [8], [9], [10], [11]. Even further, the use of a sensor network in estimating spatially distributed processes that enforce dynamic agreement has not appeared in the literature. The latter constitutes the main thrust of this manuscript, which proposes: (i) the use of multiple local filters utilizing natural observers and obtaining measurements from a portion of the sensors within the sensor network, (ii) the enforcement of a dynamic agreement among all local filters via the inclusion of terms penalizing disagreement between different filters. The structure of the consensus filters that consist of the local filter coupled via the consensus step was inspired by the one presented for finite dimensional systems [12], [13]. It should be emphasized that due to the structure of the natural second order form that the local filters employ, two outcomes arise: (1) that the estimated velocity of each filter is equal to the time derivative of the estimated displacement produced by each filter, and (2) that the consensus terms penalize the disagreement of the filters over both the estimated velocity and the estimated displacement.

The manuscript is organized as follows: The next section provides the mathematical framework for which the class of systems under consideration can be viewed in. Additionally, the problem of consensus filters for a class of infinite dimensional systems is stated and possible approaches are suggested. The proposed natural Luenberger-consensus filters are introduced in Section 3 and both well-posedness and stability of the collective dynamics of the consensus filters are presented. An example with extensive numerical results of cable dynamics utilizing two consensus filters is presented in Section 4 to demonstrate the effectiveness of the dynamic agreement proposed for the consensus filters. Conclusions follow in Section 5.