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This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator The first result is the tunneling estimate The main result is a stability estimate for solutions to the hypoelliptic wave equation We then prove the approximate controllability of the hypoelliptic heat equation We also explain how the analyticity assumption can be relaxed, and a boundary Most results turn out to be optimal on a family of Grushin-type operators. The main proof relies on the general strategy to produce quantitative unique continuation estimates, developed by the authors in Laurent-Léautaud (2019).

The quantitative understanding and precise control of complex dynamical systems can only be achieved by observing their internal states via measurement and/or estimation. In large-scale dynamical networks, it is often difficult or physically impossible to have enough sensor nodes to make the system fully observable. Even if the system is in principle observable, high dimensionality poses fundamental limits on the computational tractability and performance of a full-state observer. To overcome the curse of dimensionality, we instead require the system to be functionally observable, meaning that a targeted subset of state variables can be reconstructed from the available measurements. Here, we develop a graph-based theory of functional observability, which leads to highly scalable algorithms to 1) determine the minimal set of required sensors and 2) design the corresponding state observer of minimum order. Compared with the full-state observer, the proposed functional observer achieves the same estimation quality with substantially less sensing and fewer computational resources, making it suitable for large-scale networks. We apply the proposed methods to the detection of cyberattacks in power grids from limited phase measurement data and the inference of the prevalence rate of infection during an epidemic under limited testing conditions. The applications demonstrate that the functional observer can significantly scale up our ability to explore otherwise inaccessible dynamical processes on complex networks.

A quantitative description of a complex system is inherently limited by our ability to estimate the system's internal state from experimentally accessible outputs. Although the simultaneous measurement of all internal variables, like all metabolite concentrations in a cell, offers a complete description of a system's state, in practice experimental access is limited to only a subset of variables, or sensors. A system is called observable if we can reconstruct the system's complete internal state from its outputs. Here, we adopt a graphical approach derived from the dynamical laws that govern a system to determine the sensors that are necessary to reconstruct the full internal state of a complex system. We apply this approach to biochemical reaction systems, finding that the identified sensors are not only necessary but also sufficient for observability. The developed approach can also identify the optimal sensors for target or partial observability, helping us reconstruct selected state variables from appropriately chosen outputs, a prerequisite for optimal biomarker design. Given the fundamental role observability plays in complex systems, these results offer avenues to systematically explore the dynamics of a wide range of natural, technological and socioeconomic systems.

Phasor measurement units (PMUs) are essential tools for monitoring, protection and control of power systems. The optimal PMU placement (OPP) problem refers to the determination of the minimal number of PMUs and their corresponding locations in order to achieve full network observability. This paper introduces a recursive Tabu search (RTS) method to solve the OPP problem. More specifically, the traditional Tabu search (TS) metaheuristic algorithm is executed multiple times, while in the initialisation of each TS the best solution found from all previous executions is used. The proposed RTS is found to be the best among three alternative TS initialisation schemes, in regard to the impact on the success rate of the algorithm. A numerical method is proposed for checking network observability, unlike most existing metaheuristic OPP methods, which are based on topological observability methods. The proposed RTS method is tested on the IEEE 14, 30, 57 and 118-bus test systems, on the New England 39-bus test system and on the 2383-bus power system. The obtained results are compared with other reported PMU placement methods. The simulation results show that the proposed RTS method finds the minimum number of PMUs, unlike earlier methods which may find either the same or even higher number of PMUs.

Phasor Measurement Units (PMUs) have enabled real-time power grid monitoring and control applications realizing an integrated power grid and communication system. The communication network formed by PMUs has strict latency requirements. If PMU measurements cannot reach the control centre within the latency bound, they will be invalid for calculation and may compromise the observability of the whole power grid as well as related applications. To address this issue, this study proposes a model to account for the power grid observability under communication constraints, where effective capacity is adopted to perform a cross-layer statistical analysis in the communication system. Based on this model, three algorithms are proposed for improving power grid observability, which are an observability redundancy algorithm, an observability sensitivity algorithm and an observability probability algorithm. These three algorithms aim at enhancing the power system observability via the optimal communication resource allocation for a given grid infrastructure. Case studies show that the proposed algorithms can improve the power system performance under constrained wireless communication resources.

SUMMARY The question of a priori observability of a dynamic system, that is, whether the states of a system can be identified given a particular set of measured quantities is of utmost importance in multiple disciplines including biology, economics, and engineering. More often than not, some of the parameters of the system need to be identified, and thus the issue of identifiability, that is, whether the measurements result in unique or finite solutions for the values of the parameters, is of interest. Identifiability arises in conjunction with the question of observability, when the notion of states may be augmented to include both the actual state variables of the dynamic system and its parameters. This results in the formulation of a nonlinear augmented system even though the dynamic equations of motion of the original system might be linear. In this work, three methods for the observability and identifiability of nonlinear dynamic systems are considered. More specifically, for a system whose state and measurement equations are analytic, the geometric Observability Rank Condition, which is based on Lie derivatives may be used. If the equations are rational, algebraic methods are also available. These include the algebraic observability methods and the algebraic identifiability algorithms which determine the finiteness or uniqueness of the solutions for the parameters. The aforementioned methods are used to study the observability and identifiability of suitable problems in civil engineering and highlight the connections between them and the corresponding concepts. Copyright © 2014 John Wiley & Sons, Ltd.

The characteristics of the goals and objectives of digitalization of pipeline systems (PS) are given, during which they acquire new properties of cyber-physical objects. A structure of tasks of general importance for PS of various types and purposes (heat, water, oil, gas supply, etc.) is proposed. The characteristic of probabilistic state models is given. On this basis, a system of indicators for analyzing the controllability, identifiability and observability of PS is proposed, and their relationship is disclosed. For the first time, analytical dependences of the parametric identifiability of PS on the composition of measurements have been obtained.

Phasor measurement unit (PMU) technology is a need of the power system due to its better resolution than conventional estimation devices used for wide-area monitoring. PMUs can provide synchronized phasor and magnitude of voltage and current measurements for state estimation of the power system to prevent blackouts. The drawbacks of a PMU are the high cost of the device and its installation. The main aspect of using PMUs in electrical networks is the property to observe the adjacent buses, thereby making it possible to observe the system with fewer PMUs than the number of buses through their optimal placement. In the last two decades, exhaustive research has been done on this issue. Considering the importance of this field, a comprehensive review of the progress achieved until now is carried out and the limitations of existing reviews in the literature are highlighted. This paper can be seen as a major attempt to provide an up-to-date review of the research work carried out in this all-important field of PMU placement and indicates that some perspectives of optimal PMU placement still need attention. Eventually, the work will open a new standpoint for the research community to fill the research gap.

Ecological systems can undergo sudden, catastrophic changes known as critical transitions. Anticipating these critical transitions remains challenging in systems with many species because the associated early warning signals can be weakly present or even absent in some species, depending on the system dynamics. Therefore, our limited knowledge of ecological dynamics may suggest that it is hard to identify those species in the system that display early warning signals. Here, we show that, in mutualistic ecological systems, it is possible to identify species that early anticipate critical transitions by knowing only the system structure—that is, the network topology of plant–animal interactions. Specifically, we leverage the mathematical theory of structural observability of dynamical systems to identify a minimum set of “sensor species,” whose measurement guarantees that we can infer changes in the abundance of all other species. Importantly, such a minimum set of sensor species can be identified by using the system structure only. We analyzed the performance of such minimum sets of sensor species for detecting early warnings using a large dataset of empirical plant–pollinator and seed-dispersal networks. We found that species that are more likely to be sensors tend to anticipate earlier critical transitions than other species. Our results underscore how knowing the structure of multispecies systems can improve our ability to anticipate critical transitions.

In this work, we consider the problem of localizing a team of robots, without access to direct pose measurements, under the influence of nonuniform environmental disturbances and measurement bias. Specifically, we are interested in the conditions under which teams remain range-only localizable when the environmental disturbances vary from robot to robot. We approach this problem through nonlinear observability and graph theory. After analyzing the system’s observability properties, we present theorems that identify the structural conditions under which the system maintains local weak observability. We demonstrate that rigid structures are important not only in defining multi-robot interactions, but also in characterizing the influence of nonuniform disturbances. We also give several example systems to cement intuition on the derived conditions. An observability-based planner is then presented that guides a subset of robots toward trajectories that are highly observable through finite-horizon optimization on robot headings. Simulations are then presented, along with an extended Kalman filter for state estimation, and a comparison to previous methods, to corroborate and demonstrate the results derived.