This paper studies a large population dynamic game involving nonlinear stochastic dynamical systems with agents of the following mixed types: (i) a major agent, and (ii) a population of N minor agents where N is very large. The major and minor (MM) agents are coupled via both: (i) their individual nonlinear stochastic dynamics, and (ii) their individual finite time horizon nonlinear cost functions. This problem is approached by the so-called ǫ-Nash Mean Field Game (ǫ-NMFG) theory. A distinct feature of the mixed agent MFG problem is that even asymptotically (as the population size N 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, the overall asymptotic (N → ∞) mean field game problem is decomposed into: (i) two non-standard stochastic optimal control problems with random coefficient processes which yield forward adapted stochastic best response control processes determined from the solution of (backward in time) stochastic Hamilton-Jacobi-Bellman (SHJB) equations, and (ii) two stochastic coefficient McKean-Vlasov (SMV) equations which characterize the state of the major agent and the measure determining the mean field behaviour of the minor agents. This yields to a Stochastic Mean Field Game (SMFG) system which is in contrast to the deterministic mean field game system of the standard MFG problems with only minor agents. Existence and uniqueness of the solution to the SMFG system (SHJB and SMV equations) is established by a fixed point argument in the Wasserstein space of random probability measures. In the case that minor agents are coupled to the major agent only through their cost functions, the ǫ N -Nash equilibrium property of the SMFG best responses is shown for a finite N population system where ǫ N = O(1/ √ N ).
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 (infinitetime 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.Index Terms-Sensor networks, state estimation with packet dropouts, energy/power control, energy harvesting, Markov decision processes with imperfect state information, dynamic programming.
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. Index Terms-Wireless sensor networks, distributed estimation, best linear unbiased estimator (BLUE), energy/power control, energy harvesting, Markov decision processes, dynamic programming (DP), threshold policy. I. INTRODUCTIONA DVANCES in wireless communications and high-speed low-power electronics technologies have enabled various practical applications of inexpensive, compact and versatile wireless sensor networks (WSNs) in diverse areas such as Manuscript
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