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Quadratic Lyapunov function has been widely used in the analysis of linear time invariant (LTI) systems ever since it has shown that the existence of such quadratic Lyapunov function certifies the stability of the LTI system. In this work, the problem of finding upper and lower bounds for the $L_{\infty}$-induced norm of the LTI system is considered. Quadratic Lyapunov functions are used to find the star norm, the best upper on the $L_{\infty}$-induced norm, by bounding the unit peak input reachable sets by inescapable ellipsoids. Instead, a more general class of homogeneous Lyapunov functions is used to get less conservative upper bounds on the $L_{\infty}$-induced norm and better conservative approximations for the reachable sets than those obtained using standard quadratic Lyapunov functions. The homogeneous Lyapunov function for the LTI system is considered to be a quadratic Lyapunov function for a higher-order system obtained by Lifting the LTI system via Kronecker product. Different examples are provided to show the significant improvements on the bounds obtained by using Homogeneous Lyapunov functions.

Uncertainty quantification of predictive models is crucial in decision-making problems. Conformal prediction is a general and theoretically sound answer. However, it requires exchangeable data, excluding time series. While recent works tackled this issue, we argue that Adaptive Conformal Inference (ACI, Gibbs and Cand{\`e}s, 2021), developed for distribution-shift time series, is a good procedure for time series with general dependency. We theoretically analyse the impact of the learning rate on its efficiency in the exchangeable and auto-regressive case. We propose a parameter-free method, AgACI, that adaptively builds upon ACI based on online expert aggregation. We lead extensive fair simulations against competing methods that advocate for ACI's use in time series. We conduct a real case study: electricity price forecasting. The proposed aggregation algorithm provides efficient prediction intervals for day-ahead forecasting. All the code and data to reproduce the experiments is made available.

Run Time Assurance (RTA) Systems are online verification mechanisms that filter an unverified primary controller output to ensure system safety. The primary control may come from a human operator, an advanced control approach, or an autonomous control approach that cannot be verified to the same level as simpler control systems designs. The critical feature of RTA systems is their ability to alter unsafe control inputs explicitly to assure safety. In many cases, RTA systems can functionally be described as containing a monitor that watches the state of the system and output of a primary controller, and a backup controller that replaces or modifies control input when necessary to assure safety. An important quality of an RTA system is that the assurance mechanism is constructed in a way that is entirely agnostic to the underlying structure of the primary controller. By effectively decoupling the enforcement of safety constraints from performance-related objectives, RTA offers a number of useful advantages over traditional (offline) verification. This article provides a tutorial on developing RTA systems.

In this note, a novel methodology that can extract a number of analysis results for linear time-invariant systems (LTI) given only a single trajectory of the considered system is proposed. The superiority of the proposed technique relies on the fact that it provides an automatic and formal way to obtain valuable information about the controlled system by only having access to a single trajectory over a finite period of time (i.e., the system dynamics is assumed to be unknown). At first, we characterize the stability region of LTI systems given only a single trajectory dataset by constructing the associated Lyapunov function of the system. The Lyapunov function is found by formulating and solving a linear programming (LP) problem. Then, we extend the same methodology to a variety of essential analysis results for LTI systems such as deriving bounds on the output energy, deriving bounds on output peak, deriving $\mathbf{L}_2$ and RMS gains. To illustrate the efficacy of the proposed data-driven paradigm, a comparison analysis between the learned LTI system metrics and the true ones is provided.

We developed a new class of mono- or few-layered two-dimensional polymers based on dinuclear (arene)ruthenium nodes, obtained by combining the imine condensation with an interfacial chemistry process, and use a modified Langmuir-Schaefer method to transfer them onto solid surfaces. Robust nano-sheets of 2D polymers including dinuclear complexes of heavy ruthenium atoms as nodes were synthesised. These nano-sheets, whose thickness is of a few tens of nanometers, were suspended onto solid porous membranes. Then, they were thoroughly characterised with a combination of local probes, including Raman scattering, Fourier transform infrared spectroscopy and transmission electron microscopy in imaging and diffraction mode., Comment: 9 pages, 2 schemes, 4 figures

This review presents the set of electricity price models proposed in the literature since the opening of power markets. We focus on price models applied to financial pricing and risk management. We classify these models according to their ability to represent the random behavior of prices and some of their characteristics. In particular, this classification helps users to choose among the most suitable models for their risk management problems.

Improving the passenger air travel experience is one of the explicit goals set by the Next Generation Air Transportation System in the United States and by the Advisory Council for Aeronautics Research in Europe FlightPath 2050. Both suggest door-to-door travel times as a potential metric for these objectives. In this paper, we propose a data-driven model to estimate door-to-door travel times and compare the reach and performance of different access modes to a city, as well as conduct segment analysis of full door-to-door trips. This model can also be used to compare cities with respect to the integration of their airport within their road structure. We showcase multiple applications of this full door-to-door travel time model to demonstrate how the model can be used to locate where progress can be made., Comment: 11 pages, 15 figures and 6 tables

In natural and artificial light-harvesting complexes (LHC) the resonant energy transfer (RET) between chromophores enables an efficient and directional transport of solar energy between collection and reaction centers. The detailed mechanisms involved in this energy funneling are intensely debated, essentially because they rely on a succession of individual RET steps that can hardly be addressed separately. Here, we developed a scanning tunnelling microscopy-induced luminescence (STML) approach allowing visualizing, addressing and manipulating energy funneling within multi-chromophoric structures with sub-molecular precision. We first rationalize the efficiency of the RET process at the level of chromophore dimers. We then use highly resolved fluorescence microscopy (HRFM) maps to follow energy transfer paths along an artificial trimer of descending excitonic energies which reveals a cascaded RET from high- to low-energy gap molecules. Mimicking strategies developed by photosynthetic systems, this experiment demonstrates that intermediate gap molecules can be used as efficient ancillary units to convey energy between distant donor and acceptor chromophores. Eventually, we demonstrate that the RET between donors and acceptors is enhanced by the insertion of passive molecules acting as non-covalent RET bridges. This mechanism, that occurs in experiments performed in inhomogeneous media and which plays a decisive role in fastening RET in photosynthetic systems, is reported at the level of individual chromophores with atomic-scale resolution. As it relies on organic chromophores as elementary components, our approach constitutes a powerful model to address fundamental physical processes at play in natural LHC.

Stability margins for linear time-varying (LTV) and switched-linear systems are traditionally computed via quadratic Lyapunov functions, and these functions certify the stability of the system under study. In this work, we show how the more general class of homogeneous polynomial Lyapunov functions is used to compute stability margins with reduced conservatism, and we show how these Lyapunov functions aid in the search for periodic trajectories for marginally stable LTV systems. Our work is premised on the recent observation that the search for a homogeneous polynomial Lyapunov function for some LTV systems is easily encoded as the search for a quadratic Lyapunov function for a related LTV system, and our main contribution is an intuitive algorithm for generating upper and lower bounds on the system's stability margin. We show also how the worst-case switching scheme - which draws an LTV system closest to a periodic orbit - is generated. Three numerical examples are provided to aid the reader and demonstrate the theoretical contributions of the work.

We study price formation in intraday electricity markets in the presence of intermittent renewable generation. We consider the setting where a major producer may interact strategically with a large number of small producers. Using stochastic control theory we identify the optimal strategies of agents with market impact and exhibit the Nash equilibrium in closed form in the asymptotic framework of mean field games with a major player. This is a companion paper to [F\'eron, Tankov, and Tinsi, Price formation and optimal trading in intraday electricity markets, arXiv:2009.04786, 2020], where a similar model is developed in the setting of identical agents.

We develop a tractable equilibrium model for price formation in intraday electricity markets in the presence of intermittent renewable generation. Using stochastic control theory, we identify the optimal strategies of agents with market impact and exhibit the Nash equilibrium in closed form for a finite number of agents as well as in the asymptotic framework of mean field games. Our model reproduces the empirical features of intraday market prices, such as increasing price volatility at the approach of the delivery date and the correlation between price and renewable infeed forecasts, and relates these features with market characteristics like liquidity, number of agents, and imbalance penalty.

Performance analysis for linear time-invariant (LTI) systems has been closely tied to quadratic Lyapunov functions ever since it was shown that LTI system stability is equivalent to the existence of such a Lyapunov function. Some metrics for LTI systems, however, have resisted treatment via means of quadratic Lyapunov functions. Among these, point-wise-in-time metrics, such as peak norms, are not captured accurately using these techniques, and this shortcoming has prevented the development of tools to analyze system behavior by means other than e.g. time-domain simulations. This work demonstrates how the more general class of homogeneous polynomial Lyapunov functions can be used to approximate point-wise-in-time behavior for LTI systems with greater accuracy, and we extend this to the case of linear time-varying (LTV) systems as well. Our findings rely on the recent observation that the search for homogeneous polynomial Lyapunov functions for LTV systems can be recast as a search for quadratic Lyapunov functions for a related hierarchy of time-varying Lyapunov differential equations; thus, performance guarantees for LTV systems are attainable without heavy computation. Numerous examples are provided to demonstrate the findings of this work., Comment: 8 Pages, 12 Figures, Submitted to American Controls Conference 2020

Advanced embedded algorithms are growing in complexity and they are an essential contributor to the growth of autonomy in many areas. However, the promise held by these algorithms cannot be kept without proper attention to the considerably stronger design constraints that arise when the applications of interest, such as aerospace systems, are safety-critical. Formal verification is the process of proving or disproving the ''correctness'' of an algorithm with respect to a certain mathematical description of it by means of a computer. This article discusses the formal verification of the Ellipsoid method, a convex optimization algorithm, and its code implementation as it applies to receding horizon control. Options for encoding code properties and their proofs are detailed. The applicability and limitations of those code properties and proofs are presented as well. Finally, floating-point errors are taken into account in a numerical analysis of the Ellipsoid algorithm. Modifications to the algorithm are presented which can be used to control its numerical stability.

This paper aims at analyzing the effect on the US air transportation system of the travel restriction measures implemented during the COVID-19 pandemic from a passenger perspective. Flight centric data are not already publicly and widely available therefore the traditional metrics used to measure the state of this system are not yet available. Seven metrics based on three different passenger-generated datasets are proposed here. They aim to measure in close to real-time how the travel restriction measures impacted the relation between major stakeholders of the US air transportation system, namely passengers, airports and airlines., Comment: 22 pages, 12 figures, 6 tables, pre-print version

This paper presents the foundation of a modular robotic system comprised of several novel modules in the shape of a tetrahedron. Four single-propeller submodules are assembled to create the Tetracopter, a tetrahedron-shaped quad-rotorcraft used as the elementary module of a modular flying system. This modular flying system is built by assembling the different elementary modules in a fractal shape. The fractal tetrahedron structure of the modular flying assembly grants the vehicle more rigidity than a conventional two-dimensional modular robotic flight system while maintaining the relative efficiency of a two-dimensional modular robotic flight system. A prototype of the Tetracopter has been modeled, fabricated, and successfully flight-tested by the Decision and Control Laboratory at the Georgia Institute of Technology. The results of this research set the foundation for the development of Tetrahedron rotorcraft that can maintain controllable flight and assemble in flight to create a Fractal Tetrahedron Assembly., Comment: Presented as a conference paper at the 2020 IEEE Aerospace Conference

This work presents a decentralized allocation algorithm of safety-critical application on parallel computing architectures, where individual Computational Units can be affected by faults. The described method consists in representing the architecture by an abstract graph where each node represents a Computational Unit. Applications are also represented by the graph of Computational Units they require for execution. The problem is then to decide how to allocate Computational Units to applications to guarantee execution of the safety-critical application. The problem is formulated as an optimization problem, with the form of an Integer Linear Program. A state-of-the-art solver is then used to solve the problem. Decentralizing the allocation process is achieved through redundancy of the allocator executed on the architecture. No centralized element decides on the allocation of the entire architecture, thus improving the reliability of the system. Experimental reproduction of a multi-core architecture is also presented. It is used to demonstrate the capabilities of the proposed allocation process to maintain the operation of a physical system in a decentralized way while individual component fails.

The field of bio-inspired robotics seeks to create mechanical systems that mimic the designs and concepts used by biological systems. One of the more challenging biological concepts to imitate in mechanical systems is the ability to create an internal environment that can foster homeostasis through the use of a membrane similar to skin. A robot with this ability would be able to regulate internal parameters, repair itself using the internal sub-environment, and defend its internal parts from the surrounding environment. This paper presents the internal structure of a non-holonomic wheeled system that enables homeostasis via a fully connected interior, protected from the outside environment by a flexible membrane. The three objectives of this paper are to: 1) Explore the idea of nature creating higher-order life forms with wheeled limbs given the correct intermediate steps. 2) Characterize a robot that uses a homeostasis enabling wheel. 3) Determine the feasibility of using a homeostasis enabling wheel as a mode of locomotion.

We propose to develop a full-accuracy flight test environment for the Mars helicopter and related Mars-atmospheric vehicles. The experiment would use reduced-g atmospheric flights with an aircraft that houses a properly sized vacuum chamber., Comment: 8 pages

This paper introduces the safety controller architecture as a runtime assurance mechanism for system specifications expressed as safety properties in Linear Temporal Logic (LTL). The safety controller has three fundamental components: a performance controller, a backup controller, and an assurance mechanism. The assurance mechanism uses a monitor, constructed as a finite state machine (FSM), to analyze a suggested performance control input and search for system trajectories that are bad prefixes of the system specification. A fault flag from the assurance mechanism denotes a potentially dangerous future system state and triggers a sequence of inputs that is guaranteed to keep the system safe for all time. A case study is presented which details the construction and implementation of a safety controller on a non-deterministic cyber-physical system., Comment: 8 Pages, 4 Figures. IEEE Conference on Decision and Control, 2019, to appear

This work studies the problem of searching for homogeneous polynomial Lyapunov functions for stable switched linear systems. Specifically, we show an equivalence between polynomial Lyapunov functions for systems of this class and quadratic Lyapunov functions for a related hierarchy of Lyapunov differential equations. This creates an intuitive procedure for checking the stability properties of switched linear systems and a computationally competitive algorithm is presented for generating high-order homogeneous polynomial Lyapunov functions in this manner. Additionally, we provide a comparison between polynomial Lyapunov functions generated with our proposed approach and polynomial Lyapunov functions generated with a more traditional sum-of-squares based approach., Comment: 8 pages, 2 figures

This paper discusses the possibility of making an object that precisely meets global structural requirements using additive manufacturing and feedback control. An experimental validation is presented by printing a cantilever beam with a prescribed stiffness requirement. The printing process is formalized as a model-based finite-horizon discrete control problem, where the control variables are the widths of the successive layers. Sensing is performed by making {\em in situ} intermediate stiffness measurements on the partially built part. The hypothesis that feedback control is effective in enabling the 3D-printed beam to meet precise stiffness requirements is validated experimentally., Comment: Accepted for publication as a brief paper in IEEE Transactions on Control Systems Technology

The propagation of a state vector is governed by a set of time-invariant state transition matrices that switch arbitrarily between two values. The evolution of the state is also perturbed by white Gaussian noise with a variance that switches randomly with the state transition relation. The behavior of this system can be characterized by the covariance matrix of the state vector, which is time varying. However, we can bound the set of covariances by comparing the switching system to an augmented system derived with Kronecker algebra. We formulate a matrix optimization problem to compute an ellipsoid that bounds the covariance dynamics, which in turn bounds the state covariance of the set of switching systems subject to white noise. In developing this approach, an invariant ellipsoid for a linear switching affine system is computed along the way.

We report on an experimental study of laser regime in erbium-doped whispering gallery mode (WGM) microspheres under modal-coupling between the co- and counter-propagating modes. The evidence of modal coupling has been observed in the relative intensity noise spectrum of several WGM lasers. Cross-correlation measurements have been carried out in order to determine precisely the emission regimes. It is shown that depending on the material constituting the WGM resonator, frequency locked bidirectional emission or self-modulated regime can be reached. The control of the laser emission regime of WGM micro-lasers is of great importance in the aim of applications in microwave optics or optical sensor miniaturization.