Network-based state estimation for neural networks using imperfect measurement
Introduction
The study on nonlinear systems have been becoming more and more important because nonlinear phenomena are ubiquitous [1]. As one of representative nonlinear systems, recently, neural networks have drawn much attention because they have been successfully applied in diverse research fields such as clouding computing, power network, associative memories, signal processing, pattern recognition, smart antenna arrays, and so on [2], [3], [4], [5], [6], [7], [8], [9]. To use neural networks in these applications, exact information of the neuron states is needed. In general, this goal can be achieved by designing a certain state estimator [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].
Communication technology is rapidly growing nowadays, so the sensed neuron states are usually transmitted through networks, so a necessity for research on network-based state estimation naturally arises. Network-based state estimation offers several benefits compared with traditional state estimation, such as flexible architecture, low cost, simple installation and maintenance. However, it also has several undesired factors caused by the networks, such as external disturbance, transmission delay, and packet dropout. In practice, the network is not always ideal due to the limited bit rate by busy channels and abrupt changes of the network circumstances or customer requests. These disadvantages are unavoidable and bring some significant problems such as the instability, poor performance of the system, and so on.
When we exchange data through network channels, digital sensors are commonly used for sensing exact values of targets. If the targets are continuous-time systems, then continuous-time signals are captured by digital sensors at only the sampling instants tk. After passing through a sensor, the continuous-time signals become discrete-time signals with a sampling period . The combination of a sampling effect and network-induced undesired factors (external disturbance, transmission delay, packet dropout) leads to other problems. Because of the sampling effect, sensed data can be transmitted at only the sampling instants, so the network-induced external disturbance and transmission delay occur at only sampling instants too. For example, for network-induced external disturbance and transmission delay w(t) and τ(t), w(tk) and τ(tk) are affected to the transmitting data.
Packet dropout is a more important factor for the sampled signal because if the packet dropout occurs at the present sampling instant, then packets will disappear until the next sampling instant. For example, if a packet dropout occurs during where and is a sampling instant, then the packet dropout will occur during in the case of a continuous-time signal, but in the case of a sampled signal, packet dropout will occur during [t1 t2). Because of these issues, the final transmitted signals will be quite different from the original one and contain a large amount of network uncertainty. Thus, it is clear that solving a state estimation problem using network-constrained measurements is very difficult.
Most research has handled these network constraints individually [14], [15], [16], [17], [18], [19], [20], [21]. Zhu et al. [14] investigated the state estimation problem for discrete-time switching neural networks in which mode-dependent time-varying delays and a new concept, persistent dwell time switching regularities, were considered. The measurements included random packet dropout and disturbance. Huang et al. [16] used measurements that contain delay and disturbance and designed the Arcak-type guaranteed state estimator for neural networks with a time-delay. In [19], based on the discontinuous Lyapunov approach, the stochastic sampled-data controller was designed for estimating states of neural networks with a time-varying delay. To the best of our knowledge, there is no report on the state estimation of neural networks via network-constrained measurements that simultaneously includes the sampling effect, external disturbance, transmission delay, and packet dropout.
From above discussion, it is clear that there is a strong need to figure out original signals, because the final measurements would be quite different with original signals when original signals are sensed and transmitted through a network. For example, if we construct a model (or state estimator) from these measurements to copy the original neural networks and install it to an industrial plant, then it cannot provide expected control performance or sometimes cause serious problems because the constructed neural network model cannot copy the same outputs with original one. As these reasons, it is very important to estimate actual state information of neural networks with high accuracy. For this situation, this paper addresses the followings:
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Consider imperfect measurements
This paper designs a state estimator by considering four issues, sampling effect, external disturbance, transmission delay, and packet dropout, simultaneously. To design a state estimator in the existence of these four issues simultaneously is one of the main contributions of this paper because most of design problems of the state estimator have been solved by considering these items individually.
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Design a compensator
Usually, to reduce the effects of external disturbance under a certain acceptable value, the control method is adopted. However, this paper considered four issues, so only using technique could not provide aimed performance. As a solution to improve the estimation accuracy, we design a compensator to compensate network uncertainties, and then, a state estimator is designed using the output of the compensator.
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When signals transmitted through a network, it firstly sensed by sensor and send in digital format. For this case, if sampling information of sensor is accessible, then sampling issue can be easily handled by using the information. However, if we cannot know the sampling information due to aging of devices, abrupt changes of circumstances, the sampling is an issue. Therefore, in this paper, we consider both cases and derived robust criteria for each cased.
Throughout this paper, when packet dropouts happen, the last received data will be used in the estimator because of the existence of the Zero-Order-Hold (ZOH). Both design problems of the network-based state estimator and compensator are solved by deriving new sufficient conditions in terms of a linear matrix inequality (LMI) based on Lyapunov approach.
is the n-dimensional Euclidean space, X > 0 is a real symmetric and positive definite. I means the identity matrix. diag{⋅⋅⋅} denotes block diagonal matrix. ⋆ represents the symmetric block in a symmetric matrix.
Section snippets
Preliminaries
Consider the following neural networks:
where is the neuron state vectors of the neural networks (1), n is the number of neurons, is the neuron activation function, is a positive diagonal matrix, is an interconnection matrix, and is an external input vector.
Assumption 1 The neuron activation function f( · ) in (1) satisfies the following Lipschitz condition:
where
Main results
In this section, the design problem of the network-based state estimator for neural networks is considered. Measurement signals are limited by the network and contain many uncertain factors, so to achieve the main goal of this study, a network uncertainty compensator is established for the imperfect signal. The theory is developed for two cases, one for accessible sampling information, and one for inaccessible sampling information.
Numerical example
An example is presented for illustrating the effectiveness of our method. Let us take the following parameters for neural networks (1): The network-induced external disturbance and transmission delay are chosen as and respectively. In this example, the sampling interval and maximum number of accumulated packet dropouts are taken as and respectively.
Conclusions
This study has investigated the network-based state estimation problem of neural networks. The sampling effect, external disturbance, transmission delay, and packet dropout were included as network constraints in measurements of neural networks, and a network uncertainty compensator was designed for the difference between the original and measured signals. Two theorems were developed for designing the network-based state estimator and network uncertainty compensator for the case of
Acknowledgments
This work was supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant no. NRF-2017R1A2B2004671). Also, this work was supported in part by the Natural Science Foundation of CQ under Grant CSTC 2014J-CYJA40004, and the Natural Science Foundation of China under Grant 11471061.
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