We consider a joint sensor and controller design problem for linear Gaussian stochastic systems in which a weighted sum of quadratic control cost and the amount of information acquired by the sensor is minimized. This problem formulation is motivated by situations where a control law must be designed in the presence of sensing, communication, and privacy constraints. We show that the optimal joint sensor-controller design is relatively easy when the sensing policy is restricted to be linear. Namely, an explicit form of the optimal linear sensor equation, the Kalman filter, and the certainty equivalence controller that jointly solves the problem can be efficiently found by semidefinite programming (SDP). Whether the linearity assumption in our design is restrictive or not is currently an open problem.
However, the majority of the existing work in this context assume sensor models and or channel models , and are different from ). A few exceptions include @cite_13 and @cite_26 , where sensor-controller joint design problems are considered. However, these works are concerned with sensor power constraints rather than information constraints, and are different from ). Our problem formulation ) falls into a general class of sensor optimization problems considered in @cite_17 , where several results are derived regarding the convexity of the problem and the existence of an optimal solution under different choices of topologies in the space of sensors. However, no structural results on specific problems appear there.
@cite_26 @cite_13 @cite_17
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We consider a Wireless Sensor Network where each sensor device is able to harvest energy from the environment, and randomly accesses the channel to transmit packets of random importance to a fusion center. If a collision occurs, the transmission fails and all packets involved are discarded. We design distributed transmission schemes where each sensor node, based on its own energy level and the importance of its own data packet, decides whether to transmit the packet or remain idle, so as to maximize the network utility, defined as the average long-term aggregate network importance of the data packets successfully reported to the fusion center. Due to the generally non-convex structure of the optimization problem, we resort to approximate solutions. In particular, we use a mathematical artifice based on a game theoretic formulation of the multiaccess problem, where each sensor node is a player that attempts to selfishly maximize the network utility. We characterize the Symmetric Nash Equilibrium (SNE) of this game, where all the sensor nodes employ the same policy. We prove the existence and uniqueness of the SNE, and we show that it is a local maximum of the original optimization problem. Moreover, we derive an algorithm to compute it. In this paper, we study the problem of tracking an object moving randomly through a network of wireless sensors. Our objective is to devise strategies for scheduling the sensors to optimize the tradeoff between tracking performance and energy consumption. We cast the scheduling problem as a partially observable Markov decision process (POMDP), where the control actions correspond to the set of sensors to activate at each time step. Using a bottom-up approach, we consider different sensing, motion and cost models with increasing levels of difficulty. At the first level, the sensing regions of the different sensors do not overlap and the target is only observed within the sensing range of an active sensor. Then, we consider sensors with overlapping sensing range such that the tracking error, and hence the actions of the different sensors, are tightly coupled. Finally, we consider scenarios wherein the target locations and sensors' observations assume values on continuous spaces. Exact solutions are generally intractable even for the simplest models due to the dimensionality of the information and action spaces. Hence, we devise approximate solution techniques, and in some cases derive lower bounds on the optimal tradeoff curves. The generated scheduling policies, albeit suboptimal, often provide close-to-optimal energy-tracking tradeoffs. In a sensor network, in practice, the communication among sensors is subject to: 1) errors that can cause failures of links among sensors at random times; 2) costs; and 3) constraints, such as power, data rate, or communication, since sensors and networks operate under scarce resources. The paper studies the problem of designing the topology, i.e., assigning the probabilities of reliable communication among sensors (or of link failures) to maximize the rate of convergence of average consensus, when the link communication costs are taken into account, and there is an overall communication budget constraint. We model the network as a Bernoulli random topology and establish necessary and sufficient conditions for mean square sense (mss) and almost sure (a.s.) convergence of average consensus when network links fail. In particular, a necessary and sufficient condition is for the algebraic connectivity of the mean graph topology to be strictly positive. With these results, we show that the topology design with random link failures, link communication costs, and a communication cost constraint is a constrained convex optimization problem that can be efficiently solved for large networks by semidefinite programming techniques. Simulations demonstrate that the optimal design improves significantly the convergence speed of the consensus algorithm and can achieve the performance of a non-random network at a fraction of the communication cost.