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We consider an optimization problem where the decision variable is a string of bounded length. For some time there has been an interest in bounding the performance of the greedy strategy for this problem. Here, we provide weakened sufficient conditions for the greedy strategy to be bounded by a factor of $(1-(1-1/K)^K)$, where $K$ is the optimization horizon length. Specifically, we introduce the notions of $K$-submodularity and $K$-GO-concavity, which together are sufficient for this bound to hold. By introducing a notion of \emph{curvature} $\eta\in(0,1]$, we prove an even tighter bound with the factor $(1/\eta)(1-e^{-\eta})$. Finally, we illustrate the strength of our results by considering two example applications. We show that our results provide weaker conditions on parameter values in these applications than in previous results., Comment: This paper has been accepted by 2015 IEEE CDC

A game-theoretic model for analysing the effects of privacy on strategic communication between agents is devised. In the model, a sender wishes to provide an accurate measurement of the state to a receiver while also protecting its private information (which is correlated with the state) private from a malicious agent that may eavesdrop on its communications with the receiver. A family of nontrivial equilibria, in which the communicated messages carry information, is constructed and its properties are studied., Accepted for Presentation at the 54th IEEE Conference on Decision and Control (CDC 2015)

This paper presents a first mathematical convergence analysis of a Fock states feedback stabilization scheme via single-photon corrections. This measurement-based feedback has been developed and experimentally tested in 2012 by the cavity quantum electrodynamics group of Serge Haroche and Jean-Michel Raimond. Here, we consider the infinite-dimensional Markov model corresponding to the ideal set-up where detection errors and feedback delays have been disregarded. In this ideal context, we show that any goal Fock state can be stabilized by a Lyapunov-based feedback for any initial quantum state belonging to the dense subset of finite rank density operators with support in a finite photon-number sub-space. Closed-loop simulations illustrate the performance of the feedback law., 2 figures, extended version of the IEEE CDC2015 conference paper

In this paper, we present a novel learning-aided energy management scheme ( $\mathtt {LEM}$ LEM ) for multihop energy harvesting networks. Different from prior works on this problem, our algorithm explicitly incorporates information learning into system control via a step called perturbed dual learning . $\mathtt {LEM}$ LEM does not require any statistical information of the system dynamics for implementation, and efficiently resolves the challenging energy outage problem. We show that $\mathtt {LEM}$ LEM achieves the near-optimal $[O(\epsilon), O(\log (1/\epsilon)^2)]$ [ O ( e ) , O ( log ( 1 / e ) 2 ) ] utility-delay tradeoff with an $O(1/\epsilon ^{1-c/2})$ O ( 1 / e 1 - c / 2 ) energy buffers ( $c\in (0,1)$ c ∈ ( 0 , 1 ) ). More interestingly, $\mathtt {LEM}$ LEM possesses a convergence time of $O(1/\epsilon ^{1-c/2} +1/\epsilon ^c)$ O ( 1 / e 1 - c / 2 + 1 / e c ) , which is much faster than the $\Theta (1/\epsilon)$ Θ ( 1 / e ) time of pure queue-based techniques or the $\Theta (1/\epsilon ^2)$ Θ ( 1 / e 2 ) time of approaches that rely purely on learning the system statistics. This fast convergence property makes $\mathtt {LEM}$ LEM more adaptive and efficient in resource allocation in dynamic environments. The design and analysis of $\mathtt {LEM}$ LEM demonstrate how system control algorithms can be augmented by learning and what the benefits are. The methodology and algorithm can also be applied to similar problems, e.g., processing networks, where nodes require nonzero contents to support their actions.

Closed-loop positivity of feedback interconnections of positive monotone nonlinear systems is investigated. It is shown that an instantaneous gain condition on the open-loop systems which implies feedback well-posedness also guarantees feedback positivity. Furthermore, the notion of integral linear constraints (ILC) is utilised as a tool to characterise uncertainty in positive feedback systems. Robustness analysis of positive linear time-varying and nonlinear feedback systems is studied using ILC, paralleling the well-known results based on integral quadratic constraints., 5 pages, 1 figure

In this paper, we propose a reachable set based collision avoidance algorithm for unmanned aerial vehicles (UAVs). UAVs have been deployed for agriculture research and management, surveillance and sensor coverage for threat detection and disaster search and rescue operations. It is essential for the aircraft to have on-board collision avoidance capability to guarantee safety. Instead of the traditional approach of collision avoidance between trajectories, we propose a collision avoidance scheme based on reachable sets and tubes. We then formulate the problem as a convex optimization problem seeking suitable control constraint sets for participating aircraft. We have applied the approach on a case study of two quadrotors and two fix-wing aircraft collision avoidance scenario., Comment: CDC 2015 fixed-wing nonlinear dynamics extension. CDC 2015 DOI: 10.1109/CDC.2015.7403154

A decentralized control system with linear dynamics, quadratic cost, and Gaussian disturbances is considered. The system consists of a finite number of subsystems whose dynamics and per-step cost function are coupled through their mean-field (empirical average). The system has mean-field sharing information structure, i.e., each controller observes the state of its local subsystem (either perfectly or with noise) and the mean-field. It is shown that the optimal control law is unique, linear, and identical across all subsystems. Moreover, the optimal gains are computed by solving two decoupled Riccati equations in the full observation model and by solving an additional filter Riccati equation in the noisy observation model. These Riccati equations do not depend on the number of subsystems. It is also shown that the optimal decentralized performance is the same as the optimal centralized performance. An example, motivated by smart grids, is presented to illustrate the result., Proceedings of IEEE Conference on Decision and Control, 2015

We address the optimal dynamic formation problem in mobile leader-follower networks where an optimal formation is generated to maximize a given objective function while continuously preserving connectivity. We show that in a convex mission space, the connectivity constraints can be satisfied by any feasible solution to a mixed integer nonlinear optimization problem. When the optimal formation objective is to maximize coverage in a mission space cluttered with obstacles, we separate the process into intervals with no obstacles detected and intervals where one or more obstacles are detected. In the latter case, we propose a minimum-effort reconfiguration approach for the formation which still optimizes the objective function while avoiding the obstacles and ensuring connectivity. We include simulation results illustrating this dynamic formation process.

State estimation is routinely being performed in high-voltage power transmission grids in order to assist in operation and to detect faulty equipment. In low- and medium-voltage power distribution grids, on the other hand, few real-time measurements are traditionally available, and operation is often conducted based on predicted and historical data. Today, in many parts of the world, smart meters have been deployed at many customers, and their measurements could in principle be shared with the operators in real time to enable improved state estimation. However, customers may feel reluctance in doing so due to privacy concerns. We therefore propose state estimation schemes for a distribution grid model, which ensure differential privacy to the customers. In particular, the state estimation schemes optimize different performance criteria, and a trade-off between a lower bound on the estimation performance versus the customers' differential privacy is derived. The proposed framework is general enough to be applicable also to other distribution networks, such as water and gas networks.

Most theoretical tools available for the analysis of spreading processes over networks assume exponentially distributed transmission and recovery times. In practice, the empirical distribution of transmission times for many real spreading processes, such as the spread of web content through the Internet, are far from exponential. To bridge this gap between theory and practice, we propose a methodology to model and analyze spreading processes with arbitrary transmission times using phase-type distributions. Phase-type distributions are a family of distributions that is dense in the set of positive-valued distributions and can be used to approximate any given distributions. To illustrate our methodology, we focus on a popular model of spreading over networks: the susceptible-infected-susceptible (SIS) networked model. In the standard version of this model, individuals informed about a piece of information transmit this piece to its neighbors at an exponential rate. In this paper, we extend this model to the case of transmission rates following a phase-type distribution. Using this extended model, we analyze the dynamics of the spread based on a vectorial representations of phase-type distributions. We illustrate our results by analyzing spreading processes over networks with transmission and recovery rates following a Weibull distribution.

Relative pose estimation between fixed-wing unmanned aerial vehicles (UAVs) is treated using a stable and robust estimation scheme. The motivating application of this scheme is that of "handoff" of an object being tracked from one fixed-wing UAV to another in a team of UAVs, using onboard sensors in a GPS-denied environment. This estimation scheme uses optical measurements from cameras onboard a vehicle, to estimate both the relative pose and relative velocities of another vehicle or target object. It is obtained by applying the Lagrange-d'Alembert principle to a Lagrangian constructed from measurement residuals using only the optical measurements. This nonlinear pose estimation scheme is discretized for computer implementation using the discrete Lagrange-d'Alembert principle, with a discrete-time linear filter for obtaining relative velocity estimates from optical measurements. Computer simulations depict the stability and robustness of this estimator to noisy measurements and uncertainties in initial relative pose and velocities., arXiv admin note: substantial text overlap with arXiv:1508.07671

In this paper we propose a general class of models for spreading processes we call the $SI^*V^*$ model. Unlike many works that consider a fixed number of compartmental states, we allow an arbitrary number of states on arbitrary graphs with heterogeneous parameters for all nodes and edges. As a result, this generalizes an extremely large number of models studied in the literature including the MSEIV, MSEIR, MSEIS, SEIV, SEIR, SEIS, SIV, SIRS, SIR, and SIS models. Furthermore, we show how the $SI^*V^*$ model allows us to model non-Poisson spreading processes letting us capture much more complicated dynamics than existing works such as information spreading through social networks or the delayed incubation period of a disease like Ebola. This is in contrast to the overwhelming majority of works in the literature that only consider spreading processes that can be captured by a Markov process. After developing the stochastic model, we analyze its deterministic mean-field approximation and provide conditions for when the disease-free equilibrium is stable. Simulations illustrate our results.

This paper is concerned with the mean square stabilization problem of discrete-time LTI systems over a power constrained fading channel. Different from existing research works, the channel considered in this paper suffers from both fading and additive noises. We allow any form of causal channel encoders/decoders, unlike linear encoders/decoders commonly studied in the literature. Sufficient conditions and necessary conditions for the mean square stabilizability are given in terms of channel parameters such as transmission power and fading and additive noise statistics in relation to the unstable eigenvalues of the open-loop system matrix. The corresponding mean square capacity of the power constrained fading channel under causal encoders/decoders is given. It is proved that this mean square capacity is smaller than the corresponding Shannon channel capacity. In the end, numerical examples are presented, which demonstrate that the causal encoders/decoders render less restrictive stabilizability conditions than those under linear encoders/decoders studied in the existing works., Accepted by the 54th IEEE Conference on Decision and Control

Convergence of the Kalman filter is best analyzed by studying the contraction of the Riccati map in the space of positive definite (covariance) matrices. In this paper, we explore how this contraction property relates to a more fundamental non-expansiveness property of filtering maps in the space of probability distributions endowed with the Hilbert metric. This is viewed as a preliminary step towards improving the convergence analysis of filtering algorithms over general graphical models., Comment: 6 pages

The identification of structured state-space model has been intensively studied for a long time but still has not been adequately addressed. The main challenge is that the involved estimation problem is a non-convex (or bilinear) optimization problem. This paper is devoted to developing an identification method which aims to find the global optimal solution under mild computational burden. Key to the developed identification algorithm is to transform a bilinear estimation to a rank constrained optimization problem and further a difference of convex programming (DCP) problem. The initial condition for the DCP problem is obtained by solving its convex part of the optimization problem which happens to be a nuclear norm regularized optimization problem. Since the nuclear norm regularized optimization is the closest convex form of the low-rank constrained estimation problem, the obtained initial condition is always of high quality which provides the DCP problem a good starting point. The DCP problem is then solved by the sequential convex programming method. Finally, numerical examples are included to show the effectiveness of the developed identification algorithm., Accepted to IEEE Conference on Decision and Control (CDC) 2015

Douglas-Rachford splitting and the alternating direction method of multipliers (ADMM) can be used to solve convex optimization problems that consist of a sum of two functions. Convergence rate estimates for these algorithms have received much attention lately. In particular, linear convergence rates have been shown by several authors under various assumptions. One such set of assumptions is strong convexity and smoothness of one of the functions in the minimization problem. The authors recently provided a linear convergence rate bound for such problems. In this paper, we show that this rate bound is tight for the class of problems under consideration.

Unlike the classical smoothing theory, it is well known that quantum smoothers are, in general, not well--defined by the quantum conditional expectation. The purpose of this paper is to propose a new quantum smoothing theory based on the least mean squared errors. The least mean square estimate of quantum physical quantity composes from symmetric part and skew part, and we developed the recursive equations, respectively., Comment: 6 pages. See also Ohki "An invitation to quantum filtering and smoothing theory based on two inner products," RIMS Kokyuroku, No. 2018, pp. 18-44 (2017), (http://hdl.handle.net/2433/231710), as a reference

Stable estimation of rigid body pose and velocities from noisy measurements, without any knowledge of the dynamics model, is treated using the Lagrange-d'Alembert principle from variational mechanics. With body-fixed optical and inertial sensor measurements, a Lagrangian is obtained as the difference between a kinetic energy-like term that is quadratic in velocity estimation error and the sum of two artificial potential functions; one obtained from a generalization of Wahba's function for attitude estimation and another which is quadratic in the position estimate error. An additional dissipation term that is linear in the velocity estimation error is introduced, and the Lagrange-d'Alembert principle is applied to the Lagrangian with this dissipation. This estimation scheme is discretized using discrete variational mechanics. The presented pose estimator requires optical measurements of at least three inertially fixed landmarks or beacons in order to estimate instantaneous pose. The discrete estimation scheme can also estimate velocities from such optical measurements. In the presence of bounded measurement noise in the vector measurements, numerical simulations show that the estimated states converge to a bounded neighborhood of the actual states., Comment: My earlier submitted manuscript (arXiv:1508.07671), is an extended version of this work, containing detailed proofs and more elaborated numerical simulations, currently under review in Automatica. This paper will be cited in the extended journal version (arXiv:1508.07671) upon publication

International audience; A nonlinear observer on the Special Euclidean group SE(3) for full pose estimation, that takes the system outputs on the real projective space directly as inputs, is proposed. The observer derivation is based on a recent advanced theory on nonlinear observer design. A key advantage with respect to existing pose observers on SE(3) is that we can now incorporate in a unique observer different types of measurements such as vectorial measurements of known inertial vectors and position measurements of known feature points. The proposed observer is extended allowing for the compensation of unknown constant bias present in the velocity measurements. Rigorous stability analyses are equally provided. Excellent performance of the proposed observers are shown by means of simulations.

We propose a parallel adaptive constraint-tightening approach to solve a linear model predictive control problem for discrete-time systems, based on inexact numerical optimization algorithms and operator splitting methods. The underlying algorithm first splits the original problem in as many independent subproblems as the length of the prediction horizon. Then, our algorithm computes a solution for these subproblems in parallel by exploiting auxiliary tightened subproblems in order to certify the control law in terms of suboptimality and recursive feasibility, along with closed-loop stability of the controlled system. Compared to prior approaches based on constraint tightening, our algorithm computes the tightening parameter for each subproblem to handle the propagation of errors introduced by the parallelization of the original problem. Our simulations show the computational benefits of the parallelization with positive impacts on performance and numerical conditioning when compared with a recent nonparallel adaptive tightening scheme., This technical report is an extended version of the paper "A Parallel Dual Fast Gradient Method for MPC Applications" by the same authors submitted to the 54th IEEE Conference on Decision and Control

A new challenge for learning algorithms in cyber-physical network systems is the distributed solution of big-data classification problems, i.e., problems in which both the number of training samples and their dimension is high. Motivated by several problem set-ups in Machine Learning, in this paper we consider a special class of quadratic optimization problems involving a "large" number of input data, whose dimension is "big". To solve these quadratic optimization problems over peer-to-peer networks, we propose an asynchronous, distributed algorithm that scales with both the number and the dimension of the input data (training samples in the classification problem). The proposed distributed optimization algorithm relies on the notion of "core-set" which is used in geometric optimization to approximate the value function associated to a given set of points with a smaller subset of points. By computing local core-sets on a smaller version of the global problem and exchanging them with neighbors, the nodes reach consensus on a set of active constraints representing an approximate solution for the global quadratic program.

The ergodic equation is a basic tool in the study of mean-payoff stochastic games. Its solvability entails that the mean payoff is independent of the initial state. Moreover, optimal stationary strategies are readily obtained from its solution. In this paper, we give a general sufficient condition for the solvability of the ergodic equation, for a game with finite state space but arbitrary action spaces. This condition involves a pair of directed hypergraphs depending only on the ``growth at infinity'' of the Shapley operator of the game. This refines a recent result of the authors which only applied to games with bounded payments, as well as earlier nonlinear fixed point results for order preserving maps, involving graph conditions., Comment: 6 pages, 1 figure, to appear in Proc. 54th IEEE Conference on Decision and Control (CDC 2015)

We outline a new control system model for the distributed dynamics of compressible gas flow through large-scale pipeline networks with time-varying injections, withdrawals, and control actions of compressors and regulators. The gas dynamics PDE equations over the pipelines, together with boundary conditions at junctions, are reduced using lumped elements to a sparse nonlinear ODE system expressed in vector-matrix form using graph theoretic notation. This system, which we call the reduced network flow (RNF) model, is a consistent discretization of the PDE equations for gas flow. The RNF forms the dynamic constraints for optimal control problems for pipeline systems with known time-varying withdrawals and injections and gas pressure limits throughout the network. The objectives include economic transient compression (ETC) and minimum load shedding (MLS), which involve minimizing compression costs or, if that is infeasible, minimizing the unfulfilled deliveries, respectively. These continuous functional optimization problems are approximated using the Legendre-Gauss-Lobatto (LGL) pseudospectral collocation scheme to yield a family of nonlinear programs, whose solutions approach the optima with finer discretization. Simulation and optimization of time-varying scenarios on an example natural gas transmission network demonstrate the gains in security and efficiency over methods that assume steady-state behavior.