A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> C. Power Schedule Control of Sensor Networks <s> We consider finite time-horizon sensor data scheduling under limited communication resource. A sensor can only send d of its measurement data to a remote estimator within a time-horizon T >;>; d. When use the terminal estimation error covariance of the estimator as a performance metric, we provide an explicit form of the optimal data schedule; when use the average estimation error covariance as a performance metric, we provide a necessary condition for a schedule to be optimal for a general T . When T has a special form, the necessary condition allows us to construct an explicit optimal data schedule. <s> BIB001 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> C. Power Schedule Control of Sensor Networks <s> We consider sensor power scheduling for estimating the state of a general high-order Gauss-Markov system. A sensor decides whether to use a high or low transmission power to communicate its local state estimate or raw measurement data with a remote estimator over a packet-dropping network. We construct the optimal sensor power schedule which minimizes the expected terminal estimation error covariance at the remote estimator under the constraint that the high transmission power can only be used m <; T + 1 times, given the time-horizon from k = 0 to k = T. We also discuss how to extend the result to cases involving multiple power levels scheduling. Simulation examples are the provided to demonstrate the results. <s> BIB002 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> C. Power Schedule Control of Sensor Networks <s> We consider the problem of finding optimal time-periodic sensor schedules for estimating the state of discrete-time dynamical systems. We assume that {multiple} sensors have been deployed and that the sensors are subject to resource constraints, which limits the number of times each can be activated over one period of the periodic schedule. We seek an algorithm that strikes a balance between estimation accuracy and total sensor activations over one period. We make a correspondence between active sensors and the nonzero columns of estimator gain. We formulate an optimization problem in which we minimize the trace of the error covariance with respect to the estimator gain while simultaneously penalizing the number of nonzero columns of the estimator gain. This optimization problem is combinatorial in nature, and we employ the alternating direction method of multipliers (ADMM) to find its locally optimal solutions. Numerical results and comparisons with other sensor scheduling algorithms in the literature are provided to illustrate the effectiveness of our proposed method. <s> BIB003 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> C. Power Schedule Control of Sensor Networks <s> This paper studies the infinite-horizon sensor scheduling problem for linear Gaussian processes with linear measurement functions. Several important properties of the optimal infinite-horizon schedules are derived. In particular, it is proved that under some mild conditions, both the optimal infinite-horizon average-per-stage cost and the corresponding optimal sensor schedules are independent of the covariance matrix of the initial state. It is also proved that the optimal estimation cost can be approximated arbitrarily closely by a periodic schedule with a finite period. Moreover, it is shown that the sequence of the average-per-stage costs of the optimal schedule must converge. These theoretical results provide valuable insights into the design and analysis of various infinite-horizon sensor scheduling algorithms. <s> BIB004 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> C. Power Schedule Control of Sensor Networks <s> In this paper, we propose a stochastic scheduling strategy for estimating the states of N discrete-time linear time invariant (DTLTI) dynamic systems, where only one system can be observed by the sensor at each time instant due to practical resource constraints. The idea of our stochastic strategy is that a system is randomly selected for observation at each time instant according to a pre-assigned probability distribution. We aim to find the optimal pre-assigned probability in order to minimize the maximal estimate error covariance among dynamic systems. We first show that under mild conditions, the stochastic scheduling problem gives an upper bound on the performance of the optimal sensor selection problem, notoriously difficult to solve. We next relax the stochastic scheduling problem into a tractable suboptimal quasi-convex form. We then show that the new problem can be decomposed into coupled small convex optimization problems, and it can be solved in a distributed fashion. Finally, for scheduling implementation, we propose centralized and distributed deterministic scheduling strategies based on the optimal stochastic solution and provide simulation examples. <s> BIB005 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> C. Power Schedule Control of Sensor Networks <s> Abstract In this paper, we consider the problem of sensor transmission power scheduling for remote state estimation. We assume that the sensor has two transmission energy levels, where the high level corresponds to a high packet reception ratio. By exploiting the feedback information from the remote estimator, we aim to find an optimal transmission power schedule. We formulate the problem as a Markov decision process, and analytically develop a simple and optimal dynamic schedule which minimizes the average estimation error under the energy constraint. Furthermore, we derive the necessary and sufficient condition under which the remote state estimator is stable. It is shown that the estimation stability only depends on the high-energy packet reception ratio and the spectral radius of the system dynamic matrix. <s> BIB006 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> C. Power Schedule Control of Sensor Networks <s> This paper considers a remote state estimation problem, where a sensor measures the state of a linear discrete-time system. The sensor has computational capability to implement a local Kalman filter. The sensor-to-estimator communications are scheduled intentionally over a finite time horizon to obtain a desirable tradeoff between the state estimation quality and the limited communication resources. Compared with the literature, we adopt a Gaussianity-preserving event-based sensor schedule bypassing the nonlinearity problem met in threshold event-based polices. We derive the closed-form of minimum mean-square error (MMSE) estimator and show that, if communication is triggered, the estimator cannot do better than the local Kalman filter, otherwise, the associated error covariance, is simply a sum of the estimation error of the local Kalman filter and the performance loss due to the absence of communication. We further design the scheduler's parameters by solving a dynamic programming (DP) problem. The computational overhead of the DP problem is less sensitive to the system dimension compared with that of existing algorithms in the literature. <s> BIB007 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> C. Power Schedule Control of Sensor Networks <s> We consider a sensor scheduling problem where the sensors have multiple choices of communication channel to send their local measurements to a remote state estimator for state estimation. Specifically, the sensors can transmit high-precision data packets over an expensive channel or low-precision data packets, which are quantized in several bits, over some cheap channels. The expensive channel, though being able to deliver more accurate data which leads to good estimation quality at the remote estimator, can only be used scarcely due to its high cost (e.g., high energy consumption). On the other hand, the cheap channel, though having a small cost, delivers less accurate data which inevitably deteriorates the remote estimation quality. In this work we propose a new framework in which the sensors switch between the two channels to achieve a better tradeoff among the communication cost, the estimation performance and the computational complexity, where the two-channel case can be easily extended to a multiple-channel case. We propose an opportunistic sensor schedule which reduces the communication cost by randomly switching among the expensive and cheap channels, and in the meantime maintains low computational complexity while introducing data quantization into the estimation problem. We present a minimum mean square error (MMSE) estimator in a closed-form under the proposed opportunistic sensor schedule. We also formulate an optimization problem to search the best opportunistic schedule with a linear quantizer. Furthermore, we show that the MMSE estimator in the limiting case becomes the standard Kalman filter. <s> BIB008 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> C. Power Schedule Control of Sensor Networks <s> Wireless charging is a promising way to solve the energy constraint problem in sensor networks. While extensive efforts have been made to improve the performance of charging and communication in wireless rechargeable sensor networks (WRSNs), little has been done to address the operation scheduling problem. To fill this void, we propose a joint energy replenishment and scheduling mechanism so as to maximize the network lifetime while making strict sensing guarantees in the WRSN. We first formulate the problem in a general 2-D space and prove its NP-completeness. We then devise an f -approximate scheduling mechanism by transforming the classical minimum set cover problem and develop an optimal energy-replenish strategy based on the energy consumption of nodes returned by the scheduling mechanism. Large-scale simulation results validate our design and show a $39.2\%$ improvement of network lifetime over a baseline method. <s> BIB009 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> C. Power Schedule Control of Sensor Networks <s> We consider sensor data scheduling with communication energy constraint for remote state estimation over a time-varying fading channel. Given that the fade channel state with temporal correlated variation influences the estimation accuracy, the goal is to obtain a schedule for a sensor to decide whether using high energy to send its local estimate to a remote estimator in order to minimize the expected average estimation error covariance. By exploiting the feedback information, we propose an effective closed-form dynamic stationary scheduling scheme. We prove that such simple schedule is optimal in terms of the estimation performance by comparing with a general stationary schedule under energy constraint. Furthermore, we derive the sufficient and necessary condition for estimation stability when the sensor can only use low energy at each time. Numerical examples are provided to illustrate the effectiveness of the proposed scheme. <s> BIB010 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> C. Power Schedule Control of Sensor Networks <s> This paper considers a remote state estimation problem with multiple sensors observing a dynamical process, where sensors transmit local state estimates over an independent and identically distributed (i.i.d.) packet dropping channel to a remote estimator. At every discrete time instant, the remote estimator decides whether each sensor should transmit or not, with each sensor transmission incurring a fixed energy cost. The channel is shared such that collisions will occur if more than one sensor transmits at a time. Performance is quantified via an optimization problem that minimizes a convex combination of the expected estimation error covariance at the remote estimator and expected energy usage across the sensors. For transmission schedules dependent only on the estimation error covariance at the remote estimator, this work establishes structural results on the optimal scheduling which show that 1) for unstable systems, if the error covariance is large then a sensor will always be scheduled to transmit, and 2) there is a threshold-type behaviour in switching from one sensor transmitting to another. Specializing to the single sensor case, these structural results demonstrate that a threshold policy (i.e. transmit if the error covariance exceeds a certain threshold and don't transmit otherwise) is optimal. We also consider the situation where sensors transmit measurements instead of state estimates, and establish structural results including the optimality of threshold policies for the single sensor, scalar case. These results provide a theoretical justification for the use of such threshold policies in variance based event triggered estimation. Numerical studies confirm the qualitative behaviour predicted by our structural results. An extension of the structural results to Markovian packet drops is also outlined. <s> BIB011 </s> A Survey on Model-Based Distributed Control and Filtering for Industrial Cyber-Physical Systems <s> C. Power Schedule Control of Sensor Networks <s> We consider the problem of multiple sensor scheduling for remote state estimation of multiple process over a shared link. In this problem, a set of sensors monitor mutually independent dynamical systems in parallel but only one sensor can access the shared channel at each time to transmit the data packet to the estimator. We propose a stochastic event-based sensor scheduling in which each sensor makes transmission decisions based on both channel accessibility and distributed event-triggering conditions. The corresponding minimum mean squared error (MMSE) estimator is explicitly given. Considering information patterns accessed by sensor schedulers, time-based ones can be treated as a special case of the proposed one. By ultilizing realtime information, the proposed schedule outperforms the time-based ones in terms of the estimation quality. Resorting to solving an Markov decision process (MDP) problem with average cost criterion, we can find optimal parameters for the proposed schedule. As for practical use, a greedy algorithm is devised for parameter design, which has rather low computational complexity. We also provide a method to quantify the performance gap between the schedule optimized via MDP and any other schedules. <s> BIB012
In parallel with distributed filtering, significant efforts are devoted to the energy consumption minimization issues of sensor networks with the purpose of extending the lifespan via media access control protocols or sensor scheduling. Sensor scheduling in general refers to reducing the communication rate or energy consumption intentionally in order to achieve a desirable tradeoff between the filtering quality and the limited resource utilization BIB009 , BIB007 . In other words, a suitable scheduling strategy of sensors should be adopted to effectively decrease the negative impact that forward error correction (FEC) is usually enlarged along with the reduced communication energy or communication rate. In the framework of Kalman filtering, various strategies are developed recently and can be roughly categorized as follows: 1) the FEC optimization under given scheduling constraints BIB002 , BIB005 ; 2) the FEC optimization under given constraints on average energy usage BIB006 , BIB010 ; 3) the optimization of energy usage under given FEC constraints BIB008 ; and 4) the combination optimization between FEC and energy usage BIB011 , BIB003 . For the purpose of analysis, a set of determinate or random binary variables is usually employed to indicate scheduled sensors, and the evolution of FEC is highly dependent on these binary variables. For issues with predetermined scheduling limitation, the sum of binary variables is subject to a given constraint describing the number of authorized sensors BIB005 or senors with the high transmission power BIB002 . For finite-horizon scheduling, the characteristic of optimal policy is to schedule all transmission in a certain part of the horizon BIB002 or the limited number of transmission as uniformly as possible BIB001 . For an infinite horizon case, a metric of averaged FEC is generally adopted and some interesting algorithms, such as greedy algorithms and convex optimization approaches BIB005 , BIB012 , are developed to find optimal policies themselves or corresponding parameters therein (e.g., the ideal scheduling probability) in the limit case. Furthermore, the optimal policy can be approximated closely by a periodic scheduling with finite length BIB004 . On the other hand, considering the constraint on energy usage, a Markov decision process is utilized to reveal the structure of optimal scheduling policies and the derived sufficient and necessary condition of filtering stability depends on the high-energy packet reception ratio in BIB006 or the channel recovery rate in BIB010 . Unfortunately, it is indispensable for the feedback information from the remote estimator.