Stability analysis of neural network controller based on event triggering

doi:10.1016/j.jfranklin.2020.07.040

Journal of the Franklin Institute

Volume 357, Issue 14, September 2020, Pages 9960-9975

https://doi.org/10.1016/j.jfranklin.2020.07.040 Get rights and content

Abstract

In this paper, we investigated the stability of continuous-time nonlinear systems and designed a Neural-Network-Controller based event-triggered. First, we designed a system delay model for the transmission time-delay contained in the system. Based on the designed time-delay model, the input delay method was used to convert the synchronous controller solution problem into a stability problem for the corresponding time-delay system. Second, by using a piecewise Lyapunov-Krasovskii functional combined with the Jensen's inequality, both the stability condition of the continuous-time nonlinear systems and the solution method of sampling controller were given. The designed sampling controller is able to update the control parameters only when the triggering rules are fulfilled, thus reducing the transmission rate of the network. Finally, the effectiveness of the proposed method was verified by numerical simulation.

Introduction

In the field of modern advanced control, Neural networks are used for system parameter identification [1], [2], [3] and state estimation [4], [5]–6], as well as image recognition and classification [7], [8], [9]. In [10,11], the author studies the stability design of nonlinear systems, the asymptotic stability analysis and state-feedback control design are investigated for a class of discrete-time switched nonlinear systems via the smooth approximation technique. And in [11], each nonlinear subsystem of the presented switched system is modeled as a piecewise affine (PWA) one by splitting the state space into polyhedron regions. However, we focus on intelligent controllers to solve the stability problems of nonlinear systems. It has been shown [12,13] that the Three-layer Fully Connected Feed-forward Neural Network (TLFCFFNN) is a powerful method superior to the fuzzy system in terms of high approximation precision and non-linear function generalization ability. At the same time, with the rapid development of digital computer technology, digital devices with low installation cost, good reliability, and easy maintenance have gradually been applied in industrial applications. In [12], the current state of the controlled system is acquired by a highly sensitive sensor, then passed to the TLFCFFNN controller, and finally to the zero-order holder (ZOH) controls the continuous-time controlled system. This method greatly improves the bandwidth usage efficiency and reduces the system information transmission.

The control signal between any two consecutive sampling moments remains unchanged (it changes only at each sampling time). As a consequence, the analysis of the sampling data system is usually quite difficult and problematic. So that, in recent years, the analysis of sampled data control systems became a hot topic, and several methods have been proposed [14,15]. In particular, the input delay method has been intensively used [16,17]. Through the input delay method, the system can be considered as a continuous-time system with time-varying delay produced by ZOH. Then, the stability condition in LMI (Linear Matrix Inequality) is established by the Lyapunov-Krasovskii-Functional (LKF) method. In [18], a controller based on TLFCFFNN is proposed. It consists of a sampler, a neural network controller and a zero-order keeper. It is used to control continuous-time nonlinear systems. This paper is based on the Lyapunov stability theory. Considering the influence of time delay, the conservative condition as given in [18] is lower than [12]. Finally, the numerical simulation is given to prove the feasibility of the controller. From the above article, we can see that the above methods are based on time-triggered methods to study the continuous system information transmission, which however is leading to network information redundancy, resulting in the waste of broadband resources, while increasing the pressure on the network load. The event trigger mechanism only transmits data through the network when the current state of the system satisfies the given trigger condition, which can greatly reduce the resource consumption of the network communication bandwidth. In [19], the author discusses in detail the stability conditions of nonlinear systems based on event triggering mechanism, and the stability conditions of the system under the presence of quantization error. In [20], the Authors are studying the active and passive hybrid robust H_∞ fault-tolerant control design methods for networked control systems (NCS) under the condition of discrete event triggering (DETCS). For the uncertain linear NCS with time-varying delay, under the influence of external finite energy disturbances, based on the discrete event triggering mechanism, a normal controller and a passive robust H_∞ fault-tolerant controller are designed respectively, which are eliminated by adaptive compensation control. However the unknown fault is affecting the system, thus reducing the accuracy. In [21] the Authors introduced the influence of quantization error caused by the networked control system. They also introduced the event trigger mechanism, and gave the L₂ stability condition of the networked system under the quantitative control. But, in [22,23], the authors considered the introduction of intelligent algorithms into the control system, through the application of ant colony algorithm, the system tends to stabilize.

Based on these results obtained by several authors, consider the increasing controller accuracy as one of the main goal of research in this topics. In this paper we propose an event-triggered Four-layer Fully Connected Feed-forward Neural Network (FLFCFFNN) sampling data controller, which triggers event triggering on the premise of sampling feedback control. The main contributions of this paper are summarized as follow:

(1)
The mechanism changes the real-time transmission of the signal. When the current sampling value exceeds the set threshold, transmission is performed, which greatly reduces the amount of information transmitted and reduces the bandwidth occupation.
(2)
When solving the controller, the genetic algorithm (GA) is used to optimize the initial weights and thresholds of the neural network to reduce the controller's error.
(3)
The article adopts a piecewise LKF, which fully considers the influence of different time lags and reduces the conservativeness of the designed controller.

The organization of this paper is as follows: in Section 2 we give the formulation of the problem with some prelim nary remarks on this topics; the stability analysis is discussed in Section 3; a computer-numerical simulation is given in Section 4, while in the conclusion (Section 5) some future perspectives are suggested.

In this paper, we will use the following notations: Rⁿ denotes the n-dimensional Euclidean space, R^m × n is the set of all m × n real matrices. The superscript “ T ” and “ −1 ” represent the transpose and inverse of a matrix respectively, and “*” denotes the term that is induced by symmetry. The term He(H) denotes $H + H^{T}$ , n denotes the nonnegative integers.

Section snippets

Problem formulation

Let us consider the event trigger data sampling based on nonlinear control system, as shown in Fig. 1:

In Fig. 1, the high-sensitivity sensor monitors the controlled object in real time, and the signal output by the sensor are the input to the event detector via the sampler. Only when the event detection requirement is satisfied, the sampled discrete signal x(t_k) is transmitted to the controller based on FLFCFFNN, and then it goes through ZOH. For continuous signals, feedback to the controlled

Stability analysis

This part mainly analyzes the stability of FLFCFFNN controller based on event triggering. At the same time, the stability condition of closed-loop system is given. Based on the stability condition, a new piecewise LKF is constructed, and the stability of closed-loop system is proved.

Theorem 1

Consider the closed-loop system (12) and cost function (13), for given scalars $h > 0, δ_{i} > 0 (i = 1, 2, \dots, n)$ and matrices $G_{j} (j = 1, 2, \dots, n h)$ , then there exists a positive definite symmetric matrix with proper dimension P > 0, Q

Simulation example

In this section, an inverted pendulum is studied in order to show the effectiveness of the proposed design method. The system dynamics can be described as follows: $\begin{matrix} \ddot{x} (t) & = & \frac{F - b \dot{x} (t) + m l \dot{θ} {(t)}^{2} sin (θ (t)) + \frac{1}{2} m g \sin (2 θ (t))}{- m co s^{2} (θ (t)) + M + m}, \\ \ddot{θ} (t) & = & \frac{- \ddot{x} cos (θ (t)) - g sin (θ (t))}{l}, \end{matrix}$ where x(t) is the displacement of the cart, θ(t) is the angular displacement of the pendulum, $M = 1 kg$ is the mass of the cart, $m = 0.2 kg$ is the mass of the pendulum, $l = 0.5 m$ is the distance between the axis of rotation of the pendulum rod and the

Conclusion

In this paper, a FLFCFFNN controller based on event trigger data sampling is designed for the stability of continuous-time nonlinear systems. Based on Lyapunov-Krasovskii stability theory and Jensen inequality, the stability condition of FLFCFFNN controller based on event triggering is given. The designed controller can not only make the nonlinear system asymptotically stable, but also set the event trigger conditions in the text, reducing the sampling frequency of the system, reducing the

Declaration of Competing Interest

None.

References (26)

W.Q. Song et al.
Multi fractional brownian motion and quantum-behaved particle swarm optimization for short term power load forecasting: an integrated approach
Energy
(2020)
D.Y. Meng et al.
Robust cooperative learning control for directed networks with nonlinear dynamics
Automatica.
(2017)
H. Liu et al.
Fractional Lévy stable motion: finite difference iterative forecasting model. Chaos
Solitons Fractals
(2020)
E. Fridman et al.
Robust sampled-data stabilization of linear systems: an input delay approach
Automatica
(2004)
E. Fridman
A refined input delay approach to sampled-data control
Automatica
(2010)
X.F. Jiang et al.
On H_∞ control for linear systems with interval time-varying delay
Automatica
(2005)
Y.X. Song et al.
Exponential synchronization of coupled neutral-type neural networks with mixed delays via quantized output control
J. Frankl. Inst.
(2019)
G. Nagamani et al.
Robust state estimation for fractional-order delayed BAM neural networks via LMI approach
J. Frankl. Inst.
(2020)
L.Y. Jun et al.
H_∞ state estimation for stochastic markovian jumping neural network with time-varying delay and leakage delay
Int. J. Automat. Comput.
(2019)
G. Fei et al.
A new algorithm for SAR image target recognition based on an improved deep convolutional neural network
Cognit. Comput
(2019)

W. Shan et al.

Multimodal computer image recognition based on depth neural network

Cluster Comput.

(2019)

B.E. Amgalan et al.

Breast tumor classification using fast convergence recurrent wavelet elman neural networks

Neural Process. Lett.

(2019)

Z.Y. Zheng et al.

Multiple Lyapunov Functions Analysis approach for discrete-time switched piecewise-affine systems under dwell-time constraints

IEEE Trans. Autom. Control.

(2019)

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Stability analysis of neural network controller based on event triggering

Abstract

Introduction

Section snippets

Problem formulation

Stability analysis

Simulation example

Conclusion

Declaration of Competing Interest

Energy

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Solitons Fractals

Automatica

Automatica

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J. Frankl. Inst.

Robust state estimation for fractional-order delayed BAM neural networks via LMI approach

J. Frankl. Inst.

H∞ state estimation for stochastic markovian jumping neural network with time-varying delay and leakage delay

Int. J. Automat. Comput.

A new algorithm for SAR image target recognition based on an improved deep convolutional neural network

Cognit. Comput

Multimodal computer image recognition based on depth neural network

Cluster Comput.

Breast tumor classification using fast convergence recurrent wavelet elman neural networks

Neural Process. Lett.

Multiple Lyapunov Functions Analysis approach for discrete-time switched piecewise-affine systems under dwell-time constraints

IEEE Trans. Autom. Control.

H_∞ state estimation for stochastic markovian jumping neural network with time-varying delay and leakage delay