Abstract
In this article, we study the global exponential stability of the equilibrium point for a class of memristor-based recurrent neural networks (MRNNs). The MRNNs are based on a realistic memristor model and can be implemented by a very large scale of integration circuits. By introducing a proper Lyapunov functional, it is proved that the equilibrium point of the MRNN is globally exponentially stable under two less conservative assumptions. Furthermore, an algorithm is proposed for the design of MRNN-based circuits with stable voltages. Finally, an illustration example is performed to show the validation of the proposed theoretical results; an MRNN-based circuit with stable voltages is designed according to the proposed algorithm.
1 Introduction
Recurrent networks have been one of the necessary tools to character system states since their wide applications in optimization (; ), games (, ; ), control (; ; ), and so on (; ; ). In recent years, a new type of recurrent network was proposed based on a new two-terminal circuit element called the memristor (; ). Note that a memristor works like a biological synapse (; ) and has the ability of automatic information storage. Thus, memristors replaced resistors as synapses in recurrent neural networks, that is, memristor-based recurrent neural networks (MRNNs) (; ; ). In recent years, the stability and stabilization of Boolean networks have been extensively investigated (; , ).
MRNNs have been a promising architecture in neuromorphic systems by virtue of their non-volatility, high-density, and physical storable feature. According to the realistic structure of MRNNs, several different mathematical models for MRNNs were proposed (; ; ; ; ). Meanwhile, notice that the MRNN, a special recurrent network, depends on the stability of its equilibrium points in application scenarios. Therefore, many interesting works were addressed to analyze the stability for the MRNNs (; ; ; ; ). A mathematical model of MRNN was proposed, and its global uniform asymptotic stability was investigated in a Lyapunov sense (). A simple model of MRNN was introduced by ) by means of the typical current–voltage characteristics of memristors. A stochastic MRNN was proposed by ) based on the work by ), in which therewas some unavoidable noise in real networks. Furthermore, the global exponential stability for the stochastic MRNN was studied under the framework of Filppov’s solution; three sufficient conditions with the form of linear inequalities were provided to determine the global exponential stability of the stochastic MRNN. The global asymptotic stability and synchronization of a class of fractional-order memristor-based delayed neural networks were investigated by ). The existence and global exponential stability were discussed by ) for an uncertain MRNN with mixed time delay under two assumptions.
Motivated by the aforementioned works, the global exponential stability of the equilibrium point is investigated for a class of MRNNs with time-varying delay, and its application to stabilize the voltage in a circuit network is carried out in this study. A sufficient condition is obtained for the global exponential stability of MRNNs. Based on this condition, an algorithm is proposed to stabilize the voltage of the MRNN-based circuit. The time-varying delay was considered in the activation functions of MRNN in this study. In addition, the activation functions in the MRNN are not necessarily non-decreasing, while the activation functions are non-decreasing in the works by ); ); ); ); ). Thus, the MRNN considered in this study is the extension from the view of activation functions compared with those in the works by ); ); ); ); ). Meanwhile, the stable voltage is a necessary prerequisite for obtaining high-quality electric energy in power systems, such as wind power converters (). Consequently, the obtained theretical results are successfully applied to design the MRNN-based circuit system with global exponential stability, which makes it possible to apply the MRNN to power converters.
The structure of this article is given as follows: an MRNN with time-varying delay and some notations is introduced in Section 2. In Section 3, the global exponential stability of the equilibrium point for the MRNN is obtained, and an example is given to show the effectiveness of the obtained results. Then, an algorithm to design the MRNN-based circuit with stable voltage is proposed, and a simple application is carried out in Section 4. Finally, the main conclusions are given in Section 5.
2 Memristor-Based Recurrent Neural Network
In this section, some notations are introduced, and an MRNN is described under two assumptions based on the mathematical models by ); ).
Notation: denotes the set of real numbers. is an m − dimensional column, and the superscript T stands for the transpose operator. . is a matrix. , where λM(A) represents the maximum eigenvalue of A. stands for an identity matrix. For a real symmetric matrix A, A > 0 (A < 0) means that A is positive (negative) definite.
Consider the following MRNN, which was originated from ),
Here, fj (⋅) is the activation function, τj (⋅) is the time-varying delay, Ci is the capacitance of the capacitor, and xi(t) is the voltage of the capacitor. is the resistor between the feedback function fj (xj(t)) and the state xi(t), and is the resistor between the feedback function gj (xj (t − τj(t))) and the state xi(t). signij is defined as
Wi[xi(t)] is the memductance of the i − th memristor satisfying
Ii is an external input or bias and i, j = 1, 2, …, n. Let
From ), the MRNN (Eq. 1) is transformed into:
Here,
Next, let D = diag{d1, d2, …, dn}, , , , , , and Then, Eq. 5 is rewritten as:
In addition, there are two assumptions and one lemma, which will be needed in the sequel, for the MRNN (Eq. 7). The first assumption about the activation function fi is from ). The second assumption about the time-varying delay τj is from ).
S1. For i ∈ {1, 2, …, n}, the activation function fi is bounded continuous, and , there exists real number li > 0 such that
Here, we set Lf = diag{l1, l2, …, ln}.
For i ∈ {1, 2, …, n}, the activation function gi is bounded continuous, gi (0) = 0, and , there exists real number such that
Here, we set .
S2. For i ∈ {1, 2, …, n}, τi(t) satisfies
Here, we let , and μ = max{μ1, …, μn}.
Remark 1. From Eq. 9, the activation functions gi [xi(t)] are non-monotonic in this study. On the other hand, we notice that the activation functions of MRNNs in the works by ; ); ); ); ) are non-decreasing. Thus, Eq. 1 is the extension from the view of activation functions compared with those references.
3 Globally Exponential Stability
In this section, we will prove that the MRNN (Eq. 1) is globally exponentially stable under the assumptions S1 and S2. A sufficient condition with the form of linear matrix inequalities can be obtained for globally exponential stability of MRNN by constructing a suitable Lyapunov functional.
Theorem 1. Assume that S1 and S2 hold. If there exist a matrix P = diag{p1, p2, …, pn} > 0, a constant k > 0, and small enough constants ξ > 0 and ϑ > 0 such thatThen, the equilibrium point of the MRNN (Eq. 1) is globally exponentially stable.
ProofTo simplify the proof, we make the following transformation:where is the equilibrium point of the MRNN (Eq. 1). Then, the MRNN (Eq. 1) can be rewritten equivalently aswhere (z(t)) = f(z(t) + x*) − f (x*) and (z(t − τ(t))) = g(z(t − τ(t)) + x*) − g (x*). It is obvious that i (0) = 0 and i (0) = 0. By the assumption S1, we getWe define a Lyapunov functional as follows:whereHere, ξ, ϑ are small positive constants, and η is a positive constant to be determined.First, calculating the time derivative of along the trajectories of the MRNN (Eq. 13), we haveIn addition,Here, the parameter k is a positive constant. Substituting Eqs 18–20 into Eq. 17, we obtainSecond, by calculating the time derivative of along the trajectories of the MRNN (Eq. 13), it followsBy Eq. 14, we haveNotice thatSubstituting Eqs 23–25 into Eq. 22, we haveThird, calculating the time derivative of along the trajectories of the MRNN (Eq. 13), we haveHence, by Eqs 21, 26, 27, we haveLet in Eq. 28. It means thatSince Φ < 0, Ψ < 0, and by Eq. 29, we havewhich means that . More precisely,where and p = max{pi: i = 1, …, n}, that is, the unique equilibrium point of the MRNN (Eq. 1) is globally exponentially stable.Remark 2. Motivated by the representation of the Lyapunov functional in the work by ), we construct a new Lyapunov functional , in order to overcome the difficulty brought by the nonmonotone activation functions in MRNN (Eq. 1) in the proof of Theorem 1.Now, we give an example to illustrate that the equilibrium point of the MRNN is globally exponentially stable when the conditions in Theorem 1 are satisfied.Example 1. Consider an MRNN (Eq. 1) with four state voltages, for which the parameter values of MRNN (Eq. 1) are originated from the work by ), especially the capacitors C1 = 2, C2 = 3, C3 = 2, and C4 = 7; the external inputs I1 = 9, I2 = 3, I3 = 9.5, and I4 = 6; the memductances ,,, and for xi(t) ≤ 0; the memductances ,,, and for xi(t) ≥ 0; and the resistors and are given as follows:Next, by the aforementioned parameters and Eqs 2–6, it follows that D, , U, A, and B. Let the activation functionsand the time-varying delaysfor i = 1, 2, 3, 4. It is obvious that the assumptions S1 and S2 are satisfied. Then, by assumptions S1 and S2, we have , and μ = 0.5.Now, by fixing the parameters k = 1000,ξ = 0.001, and ϑ = 0.001 in Theorem 1 and substituting the matrices into the linear matrix inequalities (Eq. 11), we get a positive definite diagonal matrixnamely, by Theorem 1, the equilibrium point of the MRNN (Eq. 1) is globally exponentially stable.The initial values of the neural network (Eq. 1) are set at (0.1,0.1,0.1,0.1)T, (0.5,0.5,0.5,0.5)T, and (0.9,0.9,0.9,0.9)T. The solution trajectories of Eq. 1 are illustrated in Figure 1. From Figure 1, we see that the equilibrium point of the MRNN is globally exponentially stable, which shows the validation of the obtained result from Theorem 1.
FIGURE 1
4 An Algorithm to Design the MRNN-Based Circuit With Stable Voltages
Note that the stable voltage is a necessary prerequisite for obtaining high-quality electric energy in power systems. In this section, the two linear inequalities in Theorem 1 are used to design the MRNN-based circuit with globally exponentially stable voltages, which make it possible to apply the MRNN to power converters. The design process is described by the following four steps:
Step 1Fix the values of capacitor Ci, external input Ii, and the resistors and in Eq. 1 for i, j = 1, 2, …, n.
Step 2For the given time-varying delay τi(t) and the activation functions fi, gi, calculate the matrices Lf, Lg in the assumption S1 and the parameters and μ in the assumption S2.
Step 3
Determine the parameters
and
in the memductance
Wi(
xi(
t)) of the
i− th memristor in
Eq. 1for
i= 1, 2, …,
n.
• Fix a matrix P > 0 and the parameters k, ξ, and ϑ in Theorem 1.
• Substitute Ci, Ii, , and into aij, bij, and Ui in Eq. 6 to obtain matrices A, B, D, and U.
• Substitute the matrices P, D, A, B, and U into the linear matrix inequalities (11).
• Solve Eq. 11 to obtain the matrix .
• Calculate and by the di and in Eq. 5.
Step 4By substituting Ci, Ii, , , , and into Eq. 1, the MRNN-based circuit with stable voltages is obtained.
Remark 3. From Step 3, the parameters and in the MRNN (Eq. 1) can be determined at the same time by the parameter di and for i = 1, 2, …, n in (6). Consequently, we can select or make the memristor guarantee the MRNN-based circuit with stable voltage when the other elements are given beforehand by means of the proposed algorithm.Next, we will design an MRNN-based circuit with four stable voltages by the proposed algorithm, where the activation functions and some of the parameters in the MRNN-based circuit in this example are the same as those in the first example.
Example 2. It is declared that the activation functions
fi(xi(t)),
gi(xi(t)), the time-varying delay
τi(t), and the values of parameters C
i, I
i,
, and
for the MRNN-based circuit are the same as those in the first example. Next, by Step 3, we determine the values of
and
in the memducta
nceWi[
xi(
t)]
of thei− th memristor in
Eq. 1fo
ri,
j= 1, 2, 3, 4
.• Fix the values of parameters k = 1000, ξ = 0.001, and ϑ = 0.001 in Theorem 1 and a matrix P = diag{5, 5, 5, 5} > 0 and D = diag{30, 20, 25, 10}. Substitute the matrices P, D, B, and U into the linear matrix inequalities (11). Then, solve Eq. 11 to obtain the matrix :
• Calculate and by di and in Eq. 6, especially ,,,; ,,, and .
FIGURE 2
5 Conclusion
In this study, the global exponential stability of the equilibrium point of the MRNN is investigated for a class of general activation functions. A sufficient condition with the form of linear matrix inequalities is obtained for the global exponential stability. Furthermore, the proposed results are applied to design the MRNN-based circuits with stable voltages. From the view of the MRNN-based circuit, some elements of the MRNN-based circuit with stable voltages can be determined by the proposed algorithm. Note that the earth’s environmental pollution and the lack of energy restrict the survival and development of the human society. Wind energy, an environment-friendly renewable resource, has become one of the effective ways to solve these two difficulties. The conversion of wind energy into electric energy can rely on wind power converters. The mathematical model of the power system of new wind turbines was described by a recurrent network. Thus, further research will focus on transforming the output voltage of the wind power converter to ensure the stable amplitude of its output voltage based on MRNN with stability.
Statements
Data availability statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
Author contributions
ZY contributed to the globally exponential stability of MRNN by considering the proper assumptions and constructing a suitable Lyapunov functional. YL drafted the manuscript and contributed to the algorithm of design of the MRNN-based circuit with stable voltages, experiments, and conclusions. All authors agree to be accountable for the content of the work.
Funding
This work was financially supported by the China Postdoctoral Science Foundation (Grant No. 2020M670785).
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
1
AnthesG. (2011). Memristors. Commun. ACM54, 22–24. 10.1145/1897852.1897859
2
ChenL.WuR.CaoJ.LiuJ.-B. (2015). Stability and Synchronization of Memristor-Based Fractional-Order Delayed Neural Networks. Neural Networks71, 37–44. 10.1016/j.neunet.2015.07.012
3
ChenS.WuY.MacauleyM.SunX.-M. (2018). Monostability and Bistability of Boolean Networks Using Semitensor Products. IEEE Trans. Control. Netw. Syst.6, 1379.
4
ChengD.WuY.ZhaoG.FuS. (2021). A Comprehensive Survey on STP Approach to Finite Games. J. Syst. Sci. Complex34, 1666–1680. 10.1007/s11424-021-1232-8
5
ChuaL. (1971). Memristor-the Missing Circuit Element. IEEE Trans. Circuit Theor.18, 507–519. 10.1109/tct.1971.1083337
6
GuoY.WuY.GuiW. (2021). Stability of Discrete-Time Systems under Restricted Switching via Logic Dynamical Generator and Stp-Based Mergence of Hybrid States. IEEE Trans. Automatic Control.10.1109/tac.2021.3105319
7
GuoY.ZhouR.WuY.GuiW.YangC. (2019). Stability and Set Stability in Distribution of Probabilistic Boolean Networks. IEEE Trans. Automatic Control.64, 736–742. 10.1109/TAC.2018.2833170
8
HuJ.WangJ. (2010). “Global Uniform Asymptotic Stability of Memristor-Based Recurrent Neural Networks with Time Delays,” in IEEE Congress on Cumputational Intelligence (Spain): Barcelona), 2127–2134. 10.1109/ijcnn.2010.5596359
9
JianminW.FengqiuL.SitianQ. (2021). Exponential Stabilization of Memristor-Based Recurrent Neural Networks with Disturbance and Mixed Time Delays via Periodically Intermittent Control. Int. J. Control Automation Syst.19, 2284–2296. 10.1007/s12555-020-0083-8
10
JianminW.FengqiuL.SitianQ. (2019). Global Exponential Stability of Uncertain Memristor-Based Recurrent Neural Networks with Mixed Time Delays. Int. J. Machine Learn. Cybernetics10, 743–755. 10.1007/s13042-017-0759-4
11
KobraviK.KinsnerW.FilizadehS. (2007). Analysis of Bifurcation and Stability in a Simple Power System Using Matcont. Can. Conf. Electr. Comput. Eng., 1150–1154. 10.1109/ccece.2007.292
12
LiH.ShaoS.QinS.YangY. (2021). Neural Networks with Finite-Time Convergence for Solving Time-Varying Linear Complementarity Problem. Neurocomputing439, 146–158. 10.1016/j.neucom.2021.01.015
13
LiJ.HuM.GuoL. (2014). Exponential Stability of Stochastic Memristor-Based Recurrent Neural Networks with Time-Varying Delays. Neurocomputing138, 92–98. 10.1016/j.neucom.2014.02.042
14
MaL.BianW. (2021). A Novel Multiagent Neurodynamic Approach to Constrained Distributed Convex Optimization. IEEE Trans. Cybern.51, 1322–1333. 10.1109/TCYB.2019.2895885
15
QinS.WangJ.XueX. (2015). Convergence and Attractivity of Memristor-Based Cellular Neural Networks with Time Delays. Neural Networks63, 223–233. 10.1016/j.neunet.2014.12.002
16
ShenX.RaksincharoensakP. (2021). Pedestrian-aware Statistical Risk Assessment. IEEE Trans. Intell. Transport. Syst., 1–9. 10.1109/TITS.2021.3074522
17
ShenX.ZhangX.RaksincharoensakP. (2020). Probabilistic Bounds on Vehicle Trajectory Prediction Using Scenario Approach. IFAC-PapersOnLine53, 2385–2390. 10.1016/j.ifacol.2020.12.038
18
StrukovD. B.SniderG. S.StewartD. R.WilliamsR. S. (2008). The Missing Memristor Found. Nature453, 80–83. 10.1038/nature06932
19
ToyodaM.WuY. (2021). Mayer-type Optimal Control of Probabilistic Boolean Control Network with Uncertain Selection Probabilities. IEEE Trans. Cybern.51, 3079–3092. 10.1109/tcyb.2019.2954849
20
WangZ.LauriaS.FangJ. a.LiuX. (2007). Exponential Stability of Uncertain Stochastic Neural Networks with Mixed Time-Delays. Chaos, Solitons & Fractals32, 62–72. 10.1016/j.chaos.2005.10.061
21
WenS.BaoG.ZengZ.ChenY.HuangT. (2013). Global Exponential Synchronization of Memristor-Based Recurrent Neural Networks with Time-Varying Delays. Neural Networks48, 195–203. 10.1016/j.neunet.2013.10.001
22
WuA.ZengZ.ZhuX.ZhangJ. (2011). Exponential Synchronization of Memristor-Based Recurrent Neural Networks with Time Delays. Neurocomputing74, 3043–3050. 10.1016/j.neucom.2011.04.016
23
WuY.ChengD.GhoshB. K.ShenT. (2019). Recent Advances in Optimization and Game Theoretic Control for Networked Systems. Asian J. Control.21, 2493–2512. 10.1002/asjc.2303
24
WuY.GuoY.ToyodaM. (2021). Policy Iteration Approach to the Infinite Horizon Average Optimal Control of Probabilistic Boolean Networks. IEEE Trans. Neural Netw. Learn. Syst.32, 2910–2924. 10.1109/tnnls.2020.3008960
25
YangS.GuoZ.WangJ. (2015). Robust Synchronization of Multiple Memristive Neural Networks with Uncertain Parameters via Nonlinear Coupling. IEEE Trans. Syst. Man. Cybern, Syst.45, 1077–1086. 10.1109/tsmc.2014.2388199
26
ZhangG.ShenY.YinQ.SunJ. (2013). Global Exponential Periodicity and Stability of a Class of Memristor-Based Recurrent Neural Networks with Multiple Delays. Inf. Sci.232, 386–396. 10.1016/j.ins.2012.11.023
Summary
Keywords
memristor, voltage, circuit, recurrent neural network, stability
Citation
Yao Z and Li Y (2022) Global Exponential Stability of a Class of Memristor-Based RNN and Its Application to Design Stable Voltage Circuits. Front. Energy Res. 10:887769. doi: 10.3389/fenrg.2022.887769
Received
02 March 2022
Accepted
17 March 2022
Published
27 April 2022
Volume
10 - 2022
Edited by
Xun Shen, Tokyo Institute of Technology, Japan
Updates
Copyright
© 2022 Yao and Li.
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*Correspondence: Yingshun Li, leeys@dlut.edu.cn
This article was submitted to Smart Grids, a section of the journal Frontiers in Energy Research
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.