We investigate the fundamental conditions on the sampling pattern, i.e., locations of the sampled entries, for finite completability of a low-rank tensor given some components of its Tucker rank. In order to find the deterministic necessary and sufficient conditions, we propose an algebraic geometric analysis on the Tucker manifold, which allows us to incorporate multiple rank components in the proposed analysis in contrast with the conventional geometric approaches on the Grassmannian manifold. This analysis characterizes the algebraic independence of a set of polynomials defined based on the sampling pattern, which is closely related to finite completion. Probabilistic conditions are then studied and a lower bound on the sampling probability is given, which guarantees that the proposed deterministic conditions on the sampling patterns for finite completability hold with high probability. Furthermore, using the proposed geometric approach for finite completability, we propose a sufficient condition on the sampling pattern that ensures there exists exactly one completion for the sampled tensor.

This paper presents a novel method to attenuate large horizontal wind disturbance for a small-scale unmanned autonomous helicopter combining wind tunnel-based experimental data and a backstepping algorithm. Large horizontal wind disturbance is harmful to autonomous helicopters, especially to small ones because of their low inertia and the high cross-coupling effects among the multiple inputs. In order to achieve more accurate and faster attenuation of large wind disturbance, a new hybrid control architecture is proposed to take advantage of the direct force/moment compensation based on the wind tunnel experimental data. In this architecture, large horizontal wind disturbance is treated as an additional input to the control system instead of a small perturbation around the equilibrium state. A backstepping algorithm is then designed to guarantee the stable convergence of the hilicopter to the desired position. The proposed technique is finally evaluated in simulation on the platform, HIROBO Eagle, compared with a traditional wind velocity compensation method.

Complex social systems are composed of interconnected individuals whose interactions result in group behaviors. Optimal control of a real-world complex system has many applications, including road traffic management, epidemic prevention, and information dissemination. However, such real-world complex system control is difficult to achieve because of high-dimensional and non-linear system dynamics, and the exploding state and action spaces for the decision maker. Prior methods can be divided into two categories: simulation-based and analytical approaches. Existing simulation approaches have high-variance in Monte Carlo integration, and the analytical approaches suffer from modeling inaccuracy. We adopted simulation modeling in specifying the complex dynamics of a complex system, and developed analytical solutions for searching optimal strategies in a complex network with high-dimensional state-action space. To capture the complex system dynamics, we formulate the complex social network decision making problem as a discrete event decision process. To address the curse of dimensionality and search in high-dimensional state action spaces in complex systems, we reduce control of a complex system to variational inference and parameter learning, introduce Bethe entropy approximation, and develop an expectation propagation algorithm. Our proposed algorithm leads to higher system expected rewards, faster convergence, and lower variance of value function in a real-world transportation scenario than state-of-the-art analytical and sampling approaches.

Robust cooperative formation control is investigated in this paper for fixed-wing unmanned aerial vehicles in close formation flight to save energy. A novel cooperative control method is developed. The concept of virtual structure is employed to resolve the difficulty in designing virtual leaders for a large number of UAVs in formation flight. To improve the transient performance, desired trajectories are passed through a group of cooperative filters to generate smooth reference signals, namely the states of the virtual leaders. Model uncertainties due to aerodynamic couplings among UAVs are estimated and compensated using uncertainty and disturbance observers. The entire design, therefore, contains three major components: cooperative filters for motion planning, baseline cooperative control, and uncertainty and disturbance observation. The proposed formation controller could at least secure ultimate bounded control performance for formation tracking. If certain conditions are satisfied, asymptotic formation tracking control could be obtained. Major contributions of this paper lie in two aspects: 1) the difficulty in designing virtual leaders is resolved in terms of the virtual structure concept; 2) a robust cooperative controller is proposed for close formation flight of a large number of UAVs suffering from aerodynamic couplings in between. The efficiency of the proposed design will be demonstrated using numerical simulations of five UAVs in close formation flight.

Our goal is to develop a principled and general algorithmic framework for task-driven estimation and control for robotic systems. State-of-the-art approaches for controlling robotic systems typically rely heavily on accurately estimating the full state of the robot (e.g., a running robot might estimate joint angles and velocities, torso state, and position relative to a goal). However, full state representations are often excessively rich for the specific task at hand and can lead to significant computational inefficiency and brittleness to errors in state estimation. In contrast, we present an approach that eschews such rich representations and seeks to create task-driven representations. The key technical insight is to leverage the theory of information bottlenecks}to formalize the notion of a "task-driven representation" in terms of information theoretic quantities that measure the minimality of a representation. We propose novel iterative algorithms for automatically synthesizing (offline) a task-driven representation (given in terms of a set of task-relevant variables (TRVs)) and a performant control policy that is a function of the TRVs. We present online algorithms for estimating the TRVs in order to apply the control policy. We demonstrate that our approach results in significant robustness to unmodeled measurement uncertainty both theoretically and via thorough simulation experiments including a spring-loaded inverted pendulum running to a goal location.

We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our results also extend to higher dimensional hypersurfaces. The resulting surface integrals are computable analytically on a triangulated mesh. This gives a simple computational algorithm for computing the spherical volume invariant for triangulated surfaces that does not involve discretizing the ambient space. We discuss potential applications to feature detection on broken bone fragments of interest in anthropology.

This paper presents the mathematical modeling, controller design, and flight-testing of an over-actuated Vertical Take-off and Landing (VTOL) tiltwing Unmanned Aerial Vehicle (UAV). Based on simplified aerodynamics and first-principles, a dynamical model of the UAV is developed which captures key aerodynamic effects including propeller slipstream on the wing and post-stall characteristics of the airfoils. The model-based steady-state flight envelope and the corresponding trim-actuation is analyzed and the overactuation of the UAV solved by optimizing for, e.g., power-optimal trims. The developed control system is composed of two controllers: First, a low-level attitude controller based on dynamic inversion and a daisy-chaining approach to handle allocation of redundant actuators. Secondly, a higher-level cruise controller to track a desired vertical velocity. It is based on a linearization of the system and look-up tables to determine the strong and nonlinear variation of the trims throughout the flight-envelope. We demonstrate the performance of the control-system for all flight phases (hover, transition, cruise) in extensive flight-tests.

Neural network models for dynamic systems can be trained either in parallel or in series-parallel configurations. Influenced by early arguments, several papers justify the choice of series-parallel rather than parallel configuration claiming it has a lower computational cost, better stability properties during training and provides more accurate results. Other published results, on the other hand, defend parallel training as being more robust and capable of yielding more accu- rate long-term predictions. The main contribution of this paper is to present a study comparing both methods under the same unified framework. We focus on three aspects: i) robustness of the estimation in the presence of noise; ii) computational cost; and, iii) convergence. A unifying mathematical framework and simulation studies show situations where each training method provides better validation results, being parallel training better in what is believed to be more realistic scenarios. An example using measured data seems to reinforce such claim. We also show, with a novel complexity analysis and numerical examples, that both methods have similar computational cost, being series series-parallel training, however, more amenable to parallelization. Some informal discussion about stability and convergence properties is presented and explored in the examples.

Popular deep neural networks (DNNs) spend the majority of their execution time computing convolutions. The Winograd family of algorithms can greatly reduce the number of arithmetic operations required and is present in many DNN software frameworks. However, the performance gain is at the expense of a reduction in floating point (FP) numerical accuracy. In this paper, we analyse the worst case FP error and prove the estimation of norm and conditioning of the algorithm. We show that the bound grows exponentially with the size of the convolution, but the error bound of the \textit{modified} algorithm is smaller than the original one. We propose several methods for reducing FP error. We propose a canonical evaluation ordering based on Huffman coding that reduces summation error. We study the selection of sampling "points" experimentally and find empirically good points for the most important sizes. We identify the main factors associated with good points. In addition, we explore other methods to reduce FP error, including mixed-precision convolution, and pairwise summation across DNN channels. Using our methods we can significantly reduce FP error for a given block size, which allows larger block sizes and reduced computation.

This paper discusses the stability of an equilibrium point of an ordinary differential equation (ODE) arising from a feed-forward position control for a musculoskeletal system. The studied system has a link, a joint and two muscles with routing points. The motion convergence of the system strongly depends on the muscular arrangement of the musculoskeletal system. In this paper, a sufficient condition for asymptotic stability is obtained. Furthermore, numerical simulations of the penalized ODE and experimental results are described.

Spectral clustering is a powerful method for finding structure in a dataset through the eigenvectors of a similarity matrix. It often outperforms traditional clustering algorithms such as $k$-means when the structure of the individual clusters is highly non-convex. Its accuracy depends on how the similarity between pairs of data points is defined. Two important items contribute to the construction of the similarity matrix: the sparsity of the underlying weighted graph, which depends mainly on the distances among data points, and the similarity function. When a Gaussian similarity function is used, the choice of the scale parameter $σ$ can be critical. In this paper we examine both items, the sparsity and the selection of suitable $σ$'s, based either directly on the graph associated to the dataset or on the minimal spanning tree (MST) of the graph. An extensive numerical experimentation on artificial and real-world datasets has been carried out to compare the performances of the methods.

Modern power grids are experiencing grand challenges caused by the stochastic and dynamic nature of growing renewable energy and demand response. Traditional theoretical assumptions and operational rules may be violated, which are difficult to be adapted by existing control systems due to the lack of computational power and accurate grid models for use in real time, leading to growing concerns in the secure and economic operation of the power grid. Existing operational control actions are typically determined offline, which are less optimized. This paper presents a novel paradigm, Grid Mind, for autonomous grid operational controls using deep reinforcement learning. The proposed AI agent for voltage control can learn its control policy through interactions with massive offline simulations, and adapts its behavior to new changes including not only load/generation variations but also topological changes. A properly trained agent is tested on the IEEE 14-bus system with tens of thousands of scenarios, and promising performance is demonstrated in applying autonomous voltage controls for secure grid operation.

This paper investigates the robust nonlinear close formation control problem. It aims to achieve precise position control at dynamic flight operation for a follower aircraft under the aerodynamic impact due to the trailing vortices generated by a leader aircraft. One crucial concern is the control robustness that ensures the boundedness of position error subject to uncertainties and disturbances to be regulated with accuracy. This paper develops a robust nonlinear formation control algorithm to fulfill precise close formation tracking control. The proposed control algorithm consists of baseline control laws and disturbance observers. The baseline control laws are employed to stabilize the nonlinear dynamics of close formation flight, while the disturbance observers are introduced to compensate system uncertainties and formation-related aerodynamic disturbances. The position control performance can be guaranteed within the desired boundedness to harvest enough drag reduction for a follower aircraft in close formation using the proposed design. The efficacy of the proposed design is demonstrated via numerical simulations of close formation flight of two aircraft.

This short note considers an efficient variant of the trust-region algorithm with dynamic accuracy proposed Carter (1993) and Conn, Gould and Toint (2000) as a tool for very high-performance computing, an area where it is critical to allow multi-precision computations for keeping the energy dissipation under control. Numerical experiments are presented indicating that the use of the considered method can bring substantial savings in objective function's and gradient's evaluation "energy costs" by efficiently exploiting multi-precision computations.

The Kaczmarz algorithm is an iterative method for solving systems of linear equations. We introduce a modified Kaczmarz algorithm for solving systems of linear equations in a distributed environment, i.e. the equations within the system are distributed over multiple nodes within a network. The modification we introduce is designed for a network with a tree structure that allows for passage of solution estimates between the nodes in the network. We prove that the modified algorithm converges under no additional assumptions on the equations. We demonstrate that the algorithm converges to the solution, or the solution of minimal norm, when the system is consistent. We also demonstrate that in the case of an inconsistent system of equations, the modified relaxed Kaczmarz algorithm converges to a weighted least squares solution as the relaxation parameter approaches $0$.

Motivated by the need to develop simulation tools for verification and validation of autonomous driving systems operating in traffic consisting of both autonomous and human-driven vehicles, we propose a framework for modeling vehicle interactions at uncontrolled intersections. The proposed interaction modeling approach is based on game theory with multiple concurrent leader-follower pairs, and accounts for common traffic rules. We parameterize the intersection layouts and geometries to model uncontrolled intersections with various configurations, and apply the proposed approach to model the interactive behavior of vehicles at these intersections. Based on simulation results in various traffic scenarios, we show that the model exhibits reasonable behavior expected in traffic, including the capability of reproducing scenarios extracted from real-world traffic data and reasonable performance in resolving traffic conflicts. The model is further validated based on the level-of-service traffic quality rating system and demonstrates manageable computational complexity compared to traditional multi-player game-theoretic models.

This paper is concerned with the approximation of the solution of partial differential equations by means of artificial neural networks. Here a feedforward neural network is used to approximate the solution of the partial differential equation. The learning problem is formulated as a least squares problem, choosing the residual of the partial differential equation as a loss function, whereas a multilevel Levenberg-Marquardt method is employed as a training method. This setting allows us to get further insight into the potential of multilevel methods. Indeed, when the least squares problem arises from the training of artificial neural networks, the variables subject to optimization are not related by any geometrical constraints and the standard interpolation and restriction operators cannot be employed any longer. A heuristic, inspired by algebraic multigrid methods, is then proposed to construct the multilevel transfer operators. Numerical experiments show encouraging results related to the efficiency of the new multilevel optimization method for the training of artificial neural networks, compared to the standard corresponding one-level procedure.

The complex tasks such as surveillance, construction, search and rescue can benefit of the maneuverability of multirotor Micro Aerial Vehicles (MAVs) to obtain robust, cooperative system behavior and formation control is a prominent component of the these complex tasks. This work focuses on the problem of three-dimensional formation control of multirotor MAVs by using exclusively relative sensory information. It proposes a centralized Nonlinear Model Predictive Control (NMPC) approach in a leader-follower scheme. A realistic six degrees of freedom mathematical model of a multirotor MAVs is introduced and leveraged in the control laws. The formulation of the problem is performed based on NMPC and relative sensing framework with respect to local coordinate frames of the robots. This type of formulation makes the formation independent of the full knowledge of global or common reference frames and the utilization of expensive global localization sensors. Real-time Iteration (RTI) based solution to optimal control problem (OCP) is proposed by taking the novel formulation into account. An extensive scenario is designed to test and validate the strategy. Evaluation of the results suggests that satisfactory robust performance is achieved and maintained under model uncertainty and noise in local sensors and even in cases where the dynamics of the formation suddenly changes.

While it has been repeatedly shown that learning-based controllers can provide superior performance, they often lack of safety guarantees. This paper aims at addressing this problem by introducing a model predictive safety certification (MPSC) scheme for polytopic linear systems with additive disturbances. The scheme verifies safety of a proposed learning-based input and modifies it as little as necessary in order to keep the system within a given set of constraints. Safety is thereby related to the existence of a model predictive controller (MPC) providing a feasible trajectory towards a safe target set. A robust MPC formulation accounts for the fact that the model is generally uncertain in the context of learning, which allows proving constraint satisfaction at all times under the proposed MPSC strategy. The MPSC scheme can be used in order to expand any potentially conservative set of safe states for learning and we prove an iterative technique for enlarging the safe set. Finally, a practical data-based design procedure for MPSC is proposed using scenario optimization.

In recent years, randomized methods for numerical linear algebra have received growing interest as a general approach to large-scale problems. Typically, the essential ingredient of these methods is some form of randomized dimension reduction, which accelerates computations, but also creates random approximation error. In this way, the dimension reduction step encodes a tradeoff between cost and accuracy. However, the exact numerical relationship between cost and accuracy is typically unknown, and consequently, it may be difficult for the user to precisely know (1) how accurate a given solution is, or (2) how much computation is needed to achieve a given level of accuracy. In the current paper, we study randomized matrix multiplication (sketching) as a prototype setting for addressing these general problems. As a solution, we develop a bootstrap method for \emph{directly estimating} the accuracy as a function of the reduced dimension (as opposed to deriving worst-case bounds on the accuracy in terms of the reduced dimension). From a computational standpoint, the proposed method does not substantially increase the cost of standard sketching methods, and this is made possible by an "extrapolation" technique. In addition, we provide both theoretical and empirical results to demonstrate the effectiveness of the proposed method.

Stochastic mirror descent (SMD) is a fairly new family of algorithms that has recently found a wide range of applications in optimization, machine learning, and control. It can be considered a generalization of the classical stochastic gradient algorithm (SGD), where instead of updating the weight vector along the negative direction of the stochastic gradient, the update is performed in a "mirror domain" defined by the gradient of a (strictly convex) potential function. This potential function, and the mirror domain it yields, provides considerable flexibility in the algorithm compared to SGD. While many properties of SMD have already been obtained in the literature, in this paper we exhibit a new interpretation of SMD, namely that it is a risk-sensitive optimal estimator when the unknown weight vector and additive noise are non-Gaussian and belong to the exponential family of distributions. The analysis also suggests a modified version of SMD, which we refer to as symmetric SMD (SSMD). The proofs rely on some simple properties of Bregman divergence, which allow us to extend results from quadratics and Gaussians to certain convex functions and exponential families in a rather seamless way.

This article discusses a useful tool in dimensionality reduction and low-rank matrix approximation called the CUR decomposition. Various viewpoints of this method in the literature are synergized and are compared and contrasted; included in this is a new characterization of exact CUR decompositions. A novel perturbation analysis is performed on CUR approximations of noisy versions of low-rank matrices, which compares them with the putative CUR decomposition of the underlying low-rank part. Additionally, we give new column and row sampling results which allow one to conclude that a CUR decomposition of a low-rank matrix is attained with high probability. We then illustrate the stability of these sampling methods under the perturbations studied before, and provide numerical illustrations of the methods and bounds discussed.

The performance of traditional graph Laplacian methods for semi-supervised learning degrades substantially as the ratio of labeled to unlabeled data decreases, due to a degeneracy in the graph Laplacian. Several approaches have been proposed recently to address this, however we show that some of them remain ill-posed in the large-data limit. In this paper, we show a way to correctly set the weights in Laplacian regularization so that the estimator remains well posed and stable in the large-sample limit. We prove that our semi-supervised learning algorithm converges, in the infinite sample size limit, to the smooth solution of a continuum variational problem that attains the labeled values continuously. Our method is fast and easy to implement.

Many processes in science and engineering can be described by partial differential equations (PDEs). Traditionally, PDEs are derived by considering first principles of physics to derive the relations between the involved physical quantities of interest. A different approach is to measure the quantities of interest and use deep learning to reverse engineer the PDEs which are describing the physical process. In this paper we use machine learning, and deep learning in particular, to discover PDEs hidden in complex data sets from measurement data. We include examples of data from a known model problem, and real data from weather station measurements. We show how necessary transformations of the input data amounts to coordinate transformations in the discovered PDE, and we elaborate on feature and model selection. It is shown that the dynamics of a non-linear, second order PDE can be accurately described by an ordinary differential equation which is automatically discovered by our deep learning algorithm. Even more interestingly, we show that similar results apply in the context of more complex simulations of the Swedish temperature distribution.

This paper proposes a novel neural network architecture inspired by the nonstandard form proposed by Beylkin, Coifman, and Rokhlin in [Communications on Pure and Applied Mathematics, 44(2), 141-183]. The nonstandard form is a highly effective wavelet-based compression scheme for linear integral operators. In this work, we first represent the matrix-vector product algorithm of the nonstandard form as a linear neural network where every scale of the multiresolution computation is carried out by a locally connected linear sub-network. In order to address nonlinear problems, we propose an extension, called BCR-Net, by replacing each linear sub-network with a deeper and more powerful nonlinear one. Numerical results demonstrate the efficiency of the new architecture by approximating nonlinear maps that arise in homogenization theory and stochastic computation.

Much effort has been devoted in the last two decades to characterize the situations in which a reservoir computing system exhibits the so-called echo state (ESP) and fading memory (FMP) properties. These important features amount, in mathematical terms, to the existence and continuity of global reservoir system solutions. That research is complemented in this paper with the characterization of the differentiability of reservoir filters for very general classes of discrete-time deterministic inputs. This constitutes a novel strong contribution to the long line of research on the ESP and the FMP and, in particular, links to existing research on the input-dependence of the ESP. Differentiability has been shown in the literature to be a key feature in the learning of attractors of chaotic dynamical systems. A Volterra-type series representation for reservoir filters with semi-infinite discrete-time inputs is constructed in the analytic case using Taylor's theorem and corresponding approximation bounds are provided. Finally, it is shown as a corollary of these results that any fading memory filter can be uniformly approximated by a finite Volterra series with finite memory.

Factor graphs have recently gained increasing attention as a unified framework for representing and constructing algorithms for signal processing, estimation, and control. One capability that does not seem to be well explored within the factor graph tool kit is the ability to handle deterministic nonlinear transformations, such as those occurring in nonlinear filtering and smoothing problems, using tabulated message passing rules. In this contribution, we provide general forward (filtering) and backward (smoothing) approximate Gaussian message passing rules for deterministic nonlinear transformation nodes in arbitrary factor graphs fulfilling a Markov property, based on numerical quadrature procedures for the forward pass and a Rauch-Tung-Striebel-type approximation of the backward pass. These message passing rules can be employed for deriving many algorithms for solving nonlinear problems using factor graphs, as is illustrated by the proposition of a nonlinear modified Bryson-Frazier (MBF) smoother based on the presented message passing rules.

Micro- and nanosatellites have become popular platforms for a variety of commercial and scientific applications, but today are considered suitable mainly for short and low-priority space missions due to their low reliability. In part, this can be attributed to their reliance upon cheap, low-feature size, COTS components originally designed for embedded and mobile-market applications, for which traditional hardware-voting concepts are ineffective. Software-fault-tolerance concepts have been shown effective for such systems, but have largely been ignored by the space industry due to low maturity, as most have only been researched in theory. In practice, designers of payload instruments and miniaturized satellites are usually forced to sacrifice reliability in favor deliver the level of performance necessary for cutting-edge science and innovative commercial applications. Thus, we developed a software-fault-tolerance-approach based upon thread-level coarse-grain lockstep, which was validated using fault-injection. To offer strong long-term fault coverage, our architecture is implemented as tiled MPSoC on an FPGA, utilizing partial reconfiguration, as well as mixed criticality. This architecture can satisfy the high performance requirements of current and future scientific and commercial space missions at very low cost, while offering the strong fault-coverage guarantees necessary for platform control even for missions with a long duration. This architecture was developed for a 4-year ESA project. Together with two industrial partners, we are developing a prototype to then undergo radiation testing.

Sensor-to-segment calibration is a crucial step in inertial motion tracking. When two segments are connected by a hinge joint, for example in human knee and finger joints as well as in many robotic limbs, then the joint axis vector must be identified in the intrinsic sensor coordinate systems. There exist methods that identify these coordinates by solving an optimization problem that is based on kinematic joint constraints, which involve either the measured accelerations or the measured angular rates. In the current paper we demonstrate that using only one of these constraints leads to inaccurate estimates at either fast or slow motions. We propose a novel method based on a cost function that combines both constraints. The restrictive assumption of a homogeneous magnetic field is avoided by using only accelerometer and gyroscope readings. To combine the advantages of both sensor types, the residual weights are adjusted automatically based on the estimated signal variances and a nonlinear weighting of the acceleration norm difference. The method is evaluated using real data from nine different motions of an upper limb exoskeleton. Results show that, unlike previous approaches, the proposed method yields accurate joint axis estimation after only five seconds for all fast and slow motions.

We study a setting where a group of agents, each receiving partially informative private observations, seek to collaboratively learn the true state (among a set of hypotheses) that explains their joint observation profiles over time. To solve this problem, we propose a distributed learning rule that differs fundamentally from existing approaches, in the sense, that it does not employ any form of "belief-averaging". Specifically, every agent maintains a local belief (on each hypothesis) that is updated in a Bayesian manner without any network influence, and an actual belief that is updated (up to normalization) as the minimum of its own local belief and the actual beliefs of its neighbors. Under minimal requirements on the signal structures of the agents and the underlying communication graph, we establish consistency of the proposed belief update rule, i.e., we show that the actual beliefs of the agents asymptotically concentrate on the true state almost surely. As one of the key benefits of our approach, we show that our learning rule can be extended to scenarios that capture misbehavior on the part of certain agents in the network, modeled via the Byzantine adversary model. In particular, we prove that each non-adversarial agent can asymptotically learn the true state of the world almost surely, under appropriate conditions on the observation model and the network topology.