Optimal Sensor Networks Scheduling in Identification of Distributed Parameter Systems

Systems with spatio-temporal dynamics, commonly known as Distributed- Parameter Systems (DPSs), constitute one of the most general and important classes of systems which are widely used in modelling for a great variety of real-world engineering problems. As a matter of fact, rapidly growing demands of recent control and quality monitoring systems stimulate engineers to search for more precise mathematical models of the phenomena considered. This leads directly to the description of the system at hand using Partial Differential Equations (PDEs), usually expressing physical laws governing the behavior of the system. The reason is that lumped descriptions of system dynamics basically become unsatisfactory as they cannot provide a sufficient approximation of the distributed nature of the investigated process. Despite the more sophisticated formulation in terms of PDEs, such models achieve high quality and efficiency of simulations and control techniques [19, 47, 87, 145].

Prior to the analysis of more complex situations which are of interest in the context of the present work, it is useful to investigate first the less sophisticated case of a lumped dynamic system. To achieve a possibly high level of generality, we assume that the system under consideration is of the multipleinput and multiple-output (MIMO) type and evolves in a continuous time domain (results for a simpler case of observations in discrete time may be derived in much the same way).

In recent years, rapid advances in ad-hoc networking systems and sensor technology have begun a strong trend of measurement devices to become miniaturized and cheap, at the same time making the networks of sensors increasingly powerful. The reliability and robustness of DSNs have improved to the point where their wide applicability is unquestionable [29, 37, 98, 108, 110, 245, 309, 315]. This is especially important in the context of spatiotemporal systems where a large number of sensors can be used for the task of monitoring the dynamics of a system.

In modern observation systems, sensors can be located on various platforms which can be highly dynamic in motion. Each sensor node has a sensing capability, as well as limited energy supply, computing power, memory and communication ability. Endowing nodes in a sensor network with mobility drastically expands the spectrum of the network’s capabilities. Moreover, assuming that each mobile node possesses a certain amount of decision making autonomy gives rise to a dynamic system with a considerable amount of flexibility, depending on the extent to which the nodes can cooperate in order to perform a mission. This flexibility, for example, allows us to handle a large number of data source targets with a much smaller number of nodes that can move comparing to the stationary or scanning sensing strategies. What is more, technological advances in communication systems and the growing ease in making small, low power and inexpensive mobile systems now make it feasible to deploy a group of networked vehicles in a number of environments, see [35, 37, 42, 163, 176, 255].

All the approaches to sensor scheduling discussed so far have relied on centralized techniques, which assume the existence of some superior entity to maintain the whole network and responsible for global optimization of the observation strategy. The distributed nature of the design problem is taken into account very occasionally. But recent advancements in sensor networks necessitate effective, distributed and fault-tolerant algorithms for computation and information exchange. The purpose of the investigations undertaken in this chapter was to establish a practical approach to properly formulate and solve the sensor scheduling problem in a decentralized manner allowing simultaneous parallelization and robustness of the design problem.

As was elucidated in Chapter 3, although laborious research on the development of strategies for efficient sensor placement has been conducted with numerous contributions and the need for systematic methods was widely recognized, most techniques communicated by various authors usually rely on exhaustive search over a predefined set of candidate solutions and the combinatorial nature of the design problem is taken into account very occasionally [305]. Obviously, such an approach is feasible for a relatively small number of possible sensor locations, and becomes useless as the number of possible candidate locations increases.

In Section 2.2.5.4 it was indicated that, in general, for nonlinear parametrization of the system responses, optimum experimental conditions strongly depend on the unknown parameter values which only have to be estimated. This causes one of the main complications related to the determination of optimal experimental conditions. A common approach is then to design the experiment for some reasonable nominal parameter values whose knowledge is a prerequisite for applying the locally optimal sensor location methods described in the previous chapters. Since the uncertainty of those nominal values is not taken into account, practical application of such procedures is limited to situations when system responses change slowly in the set of admissible parameters.

Recently, one can observe an extremely fast development of methods of Fault Detection and Isolation (FDI) for dynamical systems. A wide variety of techniques with many potential applications is described in the rich literature. For surveys, the interested reader is referred to [17, 39, 41, 60, 124, 125, 209]. Nevertheless, a great majority of contributions focus on the methodology dedicated to lumped systems, and there are no effective methods tailored to spatio-temporal systems. Proper recognition of an abnormal behavior of the examined process leads to the necessity of very precise fitting of a nominal model corresponding to the conditions of normal work of real physical phenomena associated with it, as well as the need for appropriate models of abnormal work.

So far, numerous application-driven developments for the experimental design for DPSs related to the sensor scheduling in monitoring networks have been carefully studied. Nevertheless, all problems considered by no means exhaust all potential situations motivated by practical settings, and many difficult issues still remain open, posing challenges to researchers concerned with distributed measurement systems. In this chapter, two important extensions of the foregoing results to other experimental settings related to difficult design problems encountered in identification of real-world processes are discussed. The first one is the sensor scheduling problem for observations collected in a series for different realizations of processes with random parameters. Both the theoretical background and an algorithm for calculating optimum group experimental designs are provided to address this issue. The theory is applicable to those practical situations in which a dynamic system is sensitive to sampling or gives a different response at each run of the experiment. Together with the definition of group designs that is also introduced, this structure leads to a practical and numerically tractable representation of optimum designs for estimation of the mean values of the parameters. The second setting of great practical relevance which is investigated in this chapter is the problem of realization of the observational process under the presence of spatially correlated measurements. The task is extremely difficult, since information from different sensor nodes cannot be separated during the data fusion, leading to a far higher level of complexity compared to the uncorrelated setting.

Undoubtedly, the optimization and control of DPSs are intensively expanding research areas with a high number of applications. The process of data acquisition, being an integral part of control design, is fundamental since in distributed systems it exerts a strong influence on the accuracy of estimation, the quality of control and prediction of system behavior. Therefore, sensor networks, being, in fact, modern observational systems which have recently emerged in the context of monitoring distributed processes, are becoming a very important field of research.