Optimal Energy Allocation for Kalman Filtering over Packet Dropping Links with Imperfect Acknowledgments and Energy Harvesting Constraints

This paper presents a design methodology for optimal transmission energy allocation at a sensor equipped with energy harvesting technology for remote state estimation of linear stochastic dynamical systems. In this framework, the sensor measurements as noisy versions of the system states are sent to the receiver over a packet dropping communication channel. The packet dropout probabilities of the channel depend on both the sensor's transmission energies and time varying wireless fading channel gains. The sensor has access to an energy harvesting source which is an everlasting but unreliable energy source compared to conventional batteries with fixed energy storages. The receiver performs optimal state estimation with random packet dropouts to minimize the estimation error covariances based on received measurements. The receiver also sends packet receipt acknowledgments to the sensor via an erroneous feedback communication channel which is itself packet dropping. The objective is to design optimal transmission energy allocation at the energy harvesting sensor to minimize either a finite-time horizon sum or a long term average (infinite-time horizon) of the trace of the expected estimation error covariance of the receiver's Kalman filter. These problems are formulated as Markov decision processes with imperfect state information. The optimal transmission energy allocation policies are obtained by the use of dynamic programming techniques. Using the concept of submodularity, the structure of the optimal transmission energy policies are studied. Suboptimal solutions are also discussed which are far less computationally intensive than optimal solutions. Numerical simulation results are presented illustrating the performance of the energy allocation algorithms.Comment: Submitted to IEEE Transactions on Automatic Control. arXiv admin note: text overlap with arXiv:1402.663

An Optimal Transmission Strategy for Kalman Filtering over Packet Dropping Links with Imperfect Acknowledgements

This paper presents a novel design methodology for optimal transmission policies at a smart sensor to remotely estimate the state of a stable linear stochastic dynamical system. The sensor makes measurements of the process and forms estimates of the state using a local Kalman filter. The sensor transmits quantized information over a packet dropping link to the remote receiver. The receiver sends packet receipt acknowledgments back to the sensor via an erroneous feedback communication channel which is itself packet dropping. The key novelty of this formulation is that the smart sensor decides, at each discrete time instant, whether to transmit a quantized version of either its local state estimate or its local innovation. The objective is to design optimal transmission policies in order to minimize a long term average cost function as a convex combination of the receiver's expected estimation error covariance and the energy needed to transmit the packets. The optimal transmission policy is obtained by the use of dynamic programming techniques. Using the concept of submodularity, the optimality of a threshold policy in the case of scalar systems with perfect packet receipt acknowledgments is proved. Suboptimal solutions and their structural results are also discussed. Numerical results are presented illustrating the performance of the optimal and suboptimal transmission policies.Comment: Conditionally accepted in IEEE Transactions on Control of Network System

Comparing the Diagnostic Precision of Clinical Examination and MRI with Findings from Arthroscopy in Traumatic Knee Injuries with Femur or Tibia Shaft Fracture

Background: Diagnosis of knee injuries following trauma to the lower extremity is very important and needs to be carefully examined. This study aimed at comparing the diagnostic precision of clinical examination (CE) and MRI with findings from arthroscopy in traumatic knee injuries with femur or tibia shaft fracture.Methods: A cross-sectional study was conducted on 164 patients with traumatic knee injuries with femur or tibia shaft fracture who had been referred to Imam Hossein Hospital, Shahroud, between March 2014 and February 2015. We compared CE and MRI with arthroscopic findings (gold standard) to determine the concordance, accuracy, sensitivity, and specificity of injuries to the meniscus and knee ligaments.Results: The results showed that internal mucus rupture was the most common trauma, noted in 83 cases (50.6%), followed by anterior corrosion rupture, noted in 65 cases (39.6%). CE sensitivity was 68.4% and specificity was 96.2% for medial meniscal (MM) injuries, while sensitivity was 53.6% and specificity was 96.4% for lateral meniscal (LM) injuries. For anterior cruciate ligament (ACL) injuries, CE showed sensitivity of 77.2% and specificity of 91.8%. For posterior cruciate ligament (PCL) injuries, CE showed sensitivity of 52.6% and specificity of 98.6%. For MM injuries, MRI showed sensitivity of 92.5% and specificity of 86.5%, while for LM injuries, it showed sensitivity of 85.00% and specificity of 98.6%. For ACL injuries, MRI showed sensitivity of 86.7% and specificity of 93.8%, and for PCL injuries, MRI showed sensitivity of 84.5% and specificity of 98.8. For ACL injuries, the best concordance was with CE, while for MM and LM injuries, it was with MRI (P<0.001).Conclusions: Meniscal and ligament injuries in traumatic knee injury can be diagnosed through careful clinical examination, while requests for MRI can be reserved for complex or doubtful cases. CE and MRI used together have high sensitivity for ACL, PCL, and MM lesions, while for LM lesions, the specificity is higher

Comparing the Diagnostic Precision of Clinical Examination and MRI with Findings from Arthroscopy in Traumatic Knee Injuries with Femur or Tibia Shaft Fracture

Background: Diagnosis of knee injuries following trauma to the lower extremity is very important and needs to be carefully examined. This study aimed at comparing the diagnostic precision of clinical examination (CE) and MRI with findings from arthroscopy in traumatic knee injuries with femur or tibia shaft fracture.Methods: A cross-sectional study was conducted on 164 patients with traumatic knee injuries with femur or tibia shaft fracture who had been referred to Imam Hossein Hospital, Shahroud, between March 2014 and February 2015. We compared CE and MRI with arthroscopic findings (gold standard) to determine the concordance, accuracy, sensitivity, and specificity of injuries to the meniscus and knee ligaments.Results: The results showed that internal mucus rupture was the most common trauma, noted in 83 cases (50.6%), followed by anterior corrosion rupture, noted in 65 cases (39.6%). CE sensitivity was 68.4% and specificity was 96.2% for medial meniscal (MM) injuries, while sensitivity was 53.6% and specificity was 96.4% for lateral meniscal (LM) injuries. For anterior cruciate ligament (ACL) injuries, CE showed sensitivity of 77.2% and specificity of 91.8%. For posterior cruciate ligament (PCL) injuries, CE showed sensitivity of 52.6% and specificity of 98.6%. For MM injuries, MRI showed sensitivity of 92.5% and specificity of 86.5%, while for LM injuries, it showed sensitivity of 85.00% and specificity of 98.6%. For ACL injuries, MRI showed sensitivity of 86.7% and specificity of 93.8%, and for PCL injuries, MRI showed sensitivity of 84.5% and specificity of 98.8. For ACL injuries, the best concordance was with CE, while for MM and LM injuries, it was with MRI (P<0.001).Conclusions: Meniscal and ligament injuries in traumatic knee injury can be diagnosed through careful clinical examination, while requests for MRI can be reserved for complex or doubtful cases. CE and MRI used together have high sensitivity for ACL, PCL, and MM lesions, while for LM lesions, the specificity is higher

Mean field game theory: consensus, leader-follower and major-minor agent systems

This thesis focuses on Mean Field Game (MFG) theory with applications to consensus, flocking, leader-follower and major-minor agent systems. The MFG methodology addresses a class of dynamic games with a large number of minor agents in which each agent interacts with the average or so-called mean field effect of other agents via couplings in their individual dynamics and cost functions. A minor agent is an agent which, asymptotically as the population size goes to infinity, has a negligible influence on the overall system while the overall population's effect on it is significant. The thesis is presented in three main parts.The first part consists of applications of the MFG methodology to large population consensus and flocking behaviour. In these formulations each agent seeks to minimize its individual quadratic discounted or long time average (i.e., ergodic) cost functions involving the mean of the states of all other agents. The resulting MFG control strategies steer each agent's state toward the initial state population mean, and by applying these decentralized strategies, the system reaches mean-consensus asymptotically as time and population size go to infinity.The second part is concerned with the extension of the mean field linear-quadratic-Gaussian (MF LQG) framework so as to model the collective system dynamics which include large population of leaders and followers, and an unknown (to the followers) reference trajectory for the leaders. The cost of each leader is based on a trade-off between moving toward the reference trajectory and staying near leaders' own centroid. On the other hand, followers react by tracking a convex combination of their own centroid and the centroid of the leaders. The MF LQG equations characterizing the Nash equilibrium for infinite population systems are derived, and under appropriate conditions, they have a unique solution leading to decentralized control laws. The computation of the followers' mean field control laws requires knowledge of the complete reference trajectory which is in general not known to the followers but is estimated by a likelihood ratio based adaptation scheme based on noisy observations taken by the followers on a random sample of leaders. The final part focuses on large population dynamic games with nonlinear stochastic dynamical systems involving agents of the following mixed types: (i) a major agent, and (ii) a large population of minor agents. The major and minor agents are coupled via both: (i) their individual nonlinear stochastic dynamics of controlled McKean-Vlasov type, and (ii) their individual finite time horizon nonlinear cost functions. A distinct feature of MFG problems with mixed agents is that even asymptotically (as the population size approaches infinity) the noise process of the major agent causes random fluctuation of the mean field behaviour of the minor agents. To deal with this, a stochastic mean field system is formulated in contrast to the deterministic mean field system employed in standard MFG problems.Cette thèse se concentre sur la théorie des jeux à population importante (en Anglais, Mean Field Games (MFG)) avec des applications aux systèms de consensus, flocage, chef-suiveur et aux systèmes d'agents majeure-mineure. La méthodologie MFG aborde une classe de jeux dynamiques avec un grand nombre d'agents mineures dans laquelle chaque agent interagit avec l'effet du champ moyen des autres agents par l'intermédiaire d'accouplements dans leurs dynamiques individuelles et des fonctions de coût. Un agent mineur est un agent qui a une influence négligeable sur l'ensemble du système, mais sur lequel la population globale a un effet significatif. Cette thèse est présentée en trois parties principales. La première partie developpe des applications de la méthodologie MFG au consensus d'une population importante et le comportement de flocage. Dans ces formulations, chaque agent cherche à minimiser ses coûts quadratiques individuels, soit escomptés, soit moyennés en temps (c'est-à-dire ergodique), impliquant la moyenne des états de tous les autres agents. Les stratégies résultant de contrôle MFG orientent l'état de chaque agent vers la moyenne de la population initiale, et en appliquant ces stratégies décentraliseés, le systéme atteint un consensus moyen asymptotiquement en temps et en population. La deuxième partie s'intéresse à l'extension du cadre des jeux à population importante linéaire-quadratique-Gaussienne (MF LQG) pour modéliser la dynamique du système collective qui comprennent une grande population de chefs et de suiveurs, et une trajectoire de référence pour les chefs qui est inconnue aux suiveurs. Le coût de chaque chef est basé sur un compromis entre le déplacement vers la trajectoire de référence et de rester près du centre de gravité propre des chefs. D'autre part, les suiveurs réagissent en faisant le suivi d'une combinaison convexe de leur centre de gravité propre et celui des chefs. Les équations MF LQG qui caractérisent l'équilibre de Nash pour les systèmes de population infinie sont dérivées, et, étant donnédes conditions appropriées, ils ont des solutions uniques qui menent aux lois de contrôle décentralisées. Les calculs des lois de contrôle MFG des suiveurs nécessitent la connaissance complète de la trajectoire de référence qui n'est pas généralement connue aux suiveurs, mais qui est estimée par un rapport de vraisemblance, basé sur des observations bruitées d'un échantillon aléatoire des chefs. La dernière partie se concentre sur les jeux dynamiques des populations importantes avec des systèmes dynamiques stochastiques non-linéaires impliquant des agents mixtes suivants: (i) un agent majeur, et (ii) une grande population d'agents mineurs. Les agents majeurs et mineurs sont couplés par ces deux: (i) leurs propres dynamiques stochastiques non-linéaires et contrôlées de type McKean-Vlasov, et (ii) leurs fonctions de coûts individuelles non-linéaires à horizon de temps fini. Une caractéristique distincte des problèmes MFG avec des agents mixtes est que, même asymptotiquement (lorsque la taille de la population tend vers l'infini), le processus de bruit de l'agent majeur provoque une fluctuation aléatoire du comportement du champ moyen des agents mineurs. Pour faire face à cela, un système stochastique à champ moyen est introduite comme extension du système déterministe de champ moyen des problèmes de MFG standard

Distortion Minimization in Multi-Sensor Estimation With Energy Harvesting

This paper presents a design methodology for optimal energy allocation to estimate a random source using multiple wireless sensors equipped with energy harvesting technology. In this framework, multiple sensors observe a random process and then transmit an amplified uncoded analog version of the observed signal through Markovian fading wireless channels to a remote station. The sensors have access to an energy harvesting source, which is an everlasting but unreliable random energy source compared to conventional batteries with fixed energy storage. The remote station or so-called fusion centre estimates the realization of the random process by using a best linear unbiased estimator. The objective is to design optimal energy allocation policies at the sensor transmitters for minimizing total distortion over a finite-time horizon or a long term average distortion over an infinite-time horizon subject to energy harvesting constraints. This problem is formulated as a Markov decision process (MDP) based stochastic control problem and the optimal energy allocation policies are obtained by the use of dynamic programming techniques. Using the concept of submodularity, the structure of the optimal energy allocation policies is studied, which leads to an optimal threshold policy for binary energy allocation levels. Motivated by the excessive communication burden for the optimal control solutions where each sensor needs to know the channel gains and harvested energies of all other sensors, suboptimal decentralized strategies are developed where only statistical information about all other sensors' channel gains and harvested energies is required. Numerical simulation results are presented illustrating the performance of the optimal and suboptimal algorithms