Scheduler-based state estimation over multiple channels networks

Highlights

• The remote state estimation problem is studied for networked systems.

• Parallel noise-free communication channels are considered.

• Communication schedulers are implemented at the transmit side.

• Recursive approximate minimum mean-square error estimator is established.

• All sub-channels contribute to improve the estimation performance.

Abstract

We investigate the remote state estimation problem for networked systems over parallel noise-free communication channels. Due to limited network capabilities in practical network environments, communication schedulers are implemented at the transmit side of each subchannel to promote resource efficiency. Specifically, the processed signals are transmitted only when it is necessary to provide the real-time measurements to the remote estimator. The recursive approximate minimum mean-square error estimator is established to restore the state vector of a target plant by utilizing the scheduled transmission signals. All the information coming from the individual subchannels, even if no measurement is sent, will contribute to improve the estimation performance in an analytical form. Finally, a numerical example is given to illustrate the effectiveness of the main results.

Introduction

In the past decades, with the rapid development of sensing, computing and communication technologies, networked control has become a mainstream research topic receiving much attention from both the control and signal processing communities. A typical networked control system is composed of sensors, controllers, and actuators linked via a wired or wireless shared communication network [1], [2], [3]. To achieve high-quality control performance, state estimate is a necessary part for generating feedback control signals since the state vector of the target plant is extracted from the contaminated partial measurements [4], [5], [6], [7], [8], [9], [10]. The merits of network devices render the remote estimation possible and, in such scenarios, sensor measurements are transmitted to a central unit with sufficient computing resources for further processing [11], [12], [13], [14]. Since the networked environment greatly reduces the costs of installation and maintenance, the remote state estimation has been widely applied in engineering practice such as automated highway systems, battlefield surveillance, and environmental monitoring [15], [16], [17], [18].

Traditionally, the remote state estimation problems have mainly focused on the ideal channel settings, that is, energy supply and available bandwidth for communication networks are inexhaustible, and thus the remote estimator has access to all the raw measurements from the sensor, where a Kalman filter algorithm can be employed as an optimal estimator for linear systems with Gaussian noises. However, for some practical applications such as wireless sensor networks, the communication processes are inherently subject to limited bandwidth, and batteries of sensors are driven by restricted energy supply [19], [20], [21], [22]. These adverse factors limit the penetration of remote estimation because too frequent transmissions might not improve the estimation performance but, on the contrary, they could lead to some undesirable phenomena such as network congestion and lifespan reduction. A critical issue is how to utilize the available resources to achieve a satisfactory result efficiently. Notice that the communication process constitutes a major source of energy consumption. For the sake of preserving the bandwidth and prolonging the working hours simultaneously, a feasible scheme is to reduce the number of transmissions as much as possible on the premise of predetermined performance guarantee.

Up to now, a number of resource-efficient scheduling strategies have been extensively investigated, which include power scheduling [23], sensor selection [24], [25], event-based communication [26], [27], [28], [29], [30], [31], [32], [33], self-triggered communication [34], and compressed signals [35], etc. These strategies aim to preserve the system resources from various aspects. To be specific, for power scheduling problems, it is supposed that the transmit side can switch between two different transmission energy levels. A high energy level results in a high packet reception ratio while costing more resources, and vice versa. As a result, an optimal transmission power schedule is required for the remote estimator to achieve the optimal estimation performance under prescribed energy constraints. Moreover, for sensor networks with a large number of sensors, it is meaningful to employ an appropriate selection scheme by choosing reliable sensor signals among all the available sources, where the fundamental issues are to find out the optimal set of sensors and design the estimator so as to minimize the error covariance. As for event-based communication, it is essentially a controlled transmission scheduling strategy where the scheduler forwards signals to the remote estimator only when certain events happen. Different from the classical clock-driven mechanism that triggers a transmission at every sampling instant, in such a case, a batch of unnecessary signals can be removed from the transmission sequence to reduce resource consumption. Self-triggered communication can be regarded as an improved version of the event-based communication. In the mechanism of self-triggered communication, the next signal transmission instant is calculated by a triggering scheduling based on the previous transmitted data and the plant dynamics knowledge. Compared with the event-based communication, the main advantage of the self-triggered communication lies in the fact that mechanism of self-triggered is implemented based on certain “software” rather than the hardware (i.e. event-generator) adopted in event-based communication, and thereby reducing the hardware costs.

Due to its effectiveness in resource saving, the event-based mechanism has received increasing attention in recent years. Some initial works [36], [37] have considered an event-based rule called Send-on-Delta (or Lebesgue sampling) principle. By employing this principle the sensor data will be sent to the estimator when a certain specified threshold is reached. It can be further inferred that, when there is no transmission, the sensor data must lie in the given bound from the previously transmitted value. Therefore, one can utilize the previously transmitted value as the estimator input while keeping in mind that a bounded uncertainty exists. In this case, the exact optimal estimator is hard to obtain, but an alternative is to minimize the upper bound of the error covariance as [38]. Moreover, in [39], the Send-on-Delta principle has been extended to a more general one that is suitable for any type of sampling strategy. A sum of Gaussians approach has been employed to design the approximate optimal estimator for the sake of reducing computational complexity. On the other hand, another communication scheduling policy is based on the values of the real-time innovation as shown in [40], [41], [42]. Since innovations characterize the gap between the predicted and the current measurements, a small innovation implies that the estimator could utilize the predicted value as a quasi-optimal estimate and, in this case, the real-time transmissions are no longer necessary.

Following the existing works, the focus of this paper is on the remote state estimation problem under stringent energy and bandwidth constraints. By co-designing the scheduling policy and the state estimator, a balance between the estimation performance and available resources can be achieved. Furthermore, motivated by the multi-input-multi-output channel technique [43] developed in communication theory, we consider the communication channel to be composed of a set of parallel and independent subchannels, and each subchannel transmits the corresponding entry of the input vector. Since subchannels may own different available resources, the schedulers shall be specifically designed for the subchannels so that each subchannel can work at its desirable working condition. To the best of our knowledge, such a multiple channels setting has not yet been taken into account in the design of resource-efficient remote estimators.

The challenge for scheduler-based state estimation over multiple channels networks lies in the fact that the signals from subchannels are correlated and subject to the scheduling strategies. To achieve our objective, the channel input is first reconstructed by a dynamical linear transformation in order to eliminate the correlation between the components of the input vector. Therefore, the remote estimator can utilize the coming information from each subchannel to correct the one-step prediction independently. Furthermore, due to the scheduling process, it is almost impossible to give the exact minimum mean-square error (MMSE) when considering the amount of computation. An alternative way is to utilize a Gaussian assumption of the prior probability density function (PDF) at each step. Throughout this paper, we consider two scheduling strategies characterized by the signals injected to the channel. To be specific, when the pre-assigned conditions are fulfilled, the first one transmits the real-time signals, while the second one condenses the packet of the transmission signal by sending an indicator variable instead.

Summarizing the above discussion, the main contributions of our work can be highlighted as follows. 1) We investigate the remote state estimation problem over multiple communication channels. Under our framework, the average communication rate of each subchannel can be set specifically according to the channel condition; 2) the error covariance of the approximate MMSE estimator is obtained by a recursive algorithm. This covariance sequence turns out to be stochastic, but we can always find its tight upper and lower bounds at each step; 3) a bridge is established between the communication rate and the boundedness of the estimator, which works as a guideline to configure the schedulers.

The rest of this paper is organized as follows. In Section 2, the problem is formulated. Section 3 presents some preliminary knowledge for preparation. Section 4 computes the MMSE under the scheduler-based communication and gives the performance analysis. In Section 5, the result is extended to a more compressed scheduling policy. The results are illustrated by a numerical example in Section 6. Section 7 concludes this paper.

Notation

Throughout the paper, ${\mathbb{R}}^n$ denotes the $n$-dimensional Euclidean space. $\mathbb{E}\left[x\right]$ stands for the expectation of the stochastic variable $x$. When the expression for $x$ is long, $xW{x}^{\prime }$ is abbreviated as $xW{(\ast )}^{\prime }$. Let the cumulative distribution function of a standard normal distribution be $\Phi \left(x\right)={\int }_{-\infty }^x\frac{1}{\sqrt{2\pi }}exp(-\frac{x^2}{2})dx$. For any function $g\left(\cdot \right)$, its inverse function (if it exists) is denoted as $g^{-1}\left(\cdot \right)$.