Abstract
This chapter introduces a class of heterogeneous delayed impulsive neural networks with memristors and focuses on their collective evolution for multisynchronization. The multisynchronization represents a diversified collective behavior that is inspired by multitasking as well as observations of heterogeneity and hybridity arising from system models. In view of memristor, the memristor-based impulsive neural network is first represented by an impulsive differential inclusion. According to the memristive and impulsive mechanism, a fuzzy logic rule is introduced, and then a new fuzzy hybrid impulsive and switching control method is presented correspondingly. It is shown that using the proposed fuzzy hybrid control scheme, multisynchronization of interconnected memristor-based impulsive neural networks can be guaranteed with a positive exponential convergence rate. The heterogeneity and hybridity in system models thus can be indicated by the obtained error thresholds that contribute to the multisynchronization. Numerical examples are presented and compared to demonstrate the effectiveness of the developed theoretical results.
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References
J. Pu, H. Gong, X. Li, and Q. Luo, “Developing neuronal networks: Self-organized criticality predicts the future,” Sci. Reports, vol. 3, no. 1081, pp. 1–6, 2013.
F. Varela, J. P. Lachaux, E. Rodriguez, and J. Martinerie, “The brainweb: phase synchronization and large-scale integration,” Nature Reviews Neuroscience, vol. 2, pp. 229–239, 2001.
H. Modares, I. Ranatunga, F. L. Lewis, and D. Popa, “Optimized assistive human-robot interaction using reinforcement learning,” IEEE Trans. Cybern., vol. 46, no. 3, pp. 655–667, 2016.
J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl Acad. Sci., vol. 79, no. 8, pp. 2554–2558, 1982.
Z.-H. Guan and G. Chen, “On delayed impulsive Hopfield neural networks,” Neural Netw., vol. 12, pp. 273–280, 1999.
A. Thomas, “Memristor-based neural networks,” J. Phys. D: Appl. Phys., vol. 46, no. 9, pp. 093001(1–12), 2013.
L. Chua, “Memristor-The missing circuit element,” IEEE Trans. Circuit Theory, vol. 18, no. 5, pp. 507–519, 1971.
D. B. Strukov, G. S. Snider, D. R. Stewart, and R. S. Williams, “The missing memristor found,” Nature, vol. 453, no. 7191, pp. 80–83, 2008.
X. Liu, T. Chen, J. Cao, and W. Lu, “Dissipativity and quasi-synchronization for neural networks with discontinuous activations and parameter mismatches,” Neural Netw., vol. 24, no. 10, pp. 1013–1021, 2011.
B. Hu, D.-X. He, Z.-H. Guan, D.-X. Zhang, and X.-H. Zhang, “Hybrid subgroup coordination of multi-agent systems via nonidentical information exchange,” Neurocomputing, vol. 168, pp. 646–654, 2015.
D.-X. He, G. Ling, Z.-H. Guan, B. Hu, and R.-Q. Liao, “Multisynchronization of coupled heterogeneous genetic oscillator networks via partial impulsive control,” IEEE Trans. Neural Netw. Learn. Syst., vol. 29, no. 2, pp. 335–342, 2018.
Z.-H. Guan, Z.-W. Liu, G. Feng, and Y.-W. Wang, “Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control,” IEEE Trans. Circuits Syst. I, vol. 57, no. 8, pp. 2182–2195, 2010.
W. Zhang, C. Li, T. Huang, and X. He, “Synchronization of memristor-based coupling recurrent neural networks with time-varying delays and impulses,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 12, pp. 3308–3313, 2015.
X. Yang and D. W. C. Ho, “Synchronization of delayed memristive neural networks: robust analysis approach,” IEEE Trans. Cybern., vol. 46, no. 12, pp. 3377–3387, 2016.
Z. Guo, S. Yang, and J. Wang, “Global exponential synchronization of multiple memristive neural networks with time delay via nonlinear coupling,” IEEE Trans. Neural Netw. Learn. Syst., vol. 26, no. 6, pp. 1300–1311, 2015.
S. Wen, Z. Zeng, T. Huang, and Y. Zhang, “Exponential adaptive lag synchronization of memristive neural networks via fuzzy method and applications in pseudorandom number generators,” IEEE Trans. Fuzzy Syst., vol. 22, no, 6, pp. 1704–1713, 2014.
A. Wu, S. Wen, and Z. Zeng, “Synchronization control of a class of memristor-based recurrent neural networks,” Inform. Sci., vol. 183, no. 1, pp. 106–116, 2012.
F. Sorrentino, L. M. Pecora, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Complete characterization of the stability of cluster synchronization in complex dynamical networks,” Sci. Advances, vol. 2, no. 4, pp. e1501737, 2016.
J. Cao and L. Li, “Cluster synchronization in an array of hybrid coupled neural networks with delay,” Neural Netw., vol. 22, no. 4, pp. 335–342, 2009.
L. Li, D. W. C. Ho, J. Cao, and J. Lu, “Pinning cluster synchronization in an array of coupled neural networks under event-based mechanism,” Neural Netw., vol. 76, pp. 1–12, 2016.
W. Wu, W. Zhou, and T. Chen, “Cluster synchronization of linearly coupled complex networks under pinning control,” IEEE Trans. Circuits Syst. I, vol. 56, no. 4, pp. 829–839, 2009.
Q. Gao, G. Feng, D. Dong, and L. Liu, “Universal fuzzy models and universal fuzzy controllers for discrete-time nonlinear systems,” IEEE Trans. Cybern., vol. 45, no. 5, pp. 880–887, 2015.
Y. Li and S. Tong, “Hybrid adaptive fuzzy control for uncertain MIMO nonlinear systems with unknown dead-zone,” Inform. Sci., vol. 328, pp. 97–114, 2016.
X. Yang, D. W. C. Ho, J. Lu, and Q. Song, “Finite-time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays,” IEEE Trans. Fuzzy Syst., vol. 23, no. 6, pp. 2302–2316, 2015.
Y. Liu, S. Tong, D. Li, and Y. Gao, “Fuzzy adaptive control with state observer for a class of nonlinear discrete-time systems with input constraint,” IEEE Trans. Fuzzy Syst., vol. 24, no. 5, pp. 1147–1158, 2016.
S. Zhang, Z. Wang, D. Ding, H. Dong, F. E. Alsaadi, and T. Hayat, “Nonfragile H∞ fuzzy filtering with randomly occurring gain variations and channel fadings,” IEEE Trans. Fuzzy Syst., vol. 24, no. 3, pp. 505–518, 2016.
T. Wang, Y. Zhang, J. Qiu, and H. Gao, “Adaptive fuzzy backstepping control for a class of nonlinear systems with sampled and delayed measurements,” IEEE Trans. Fuzzy Syst., vol. 23, no. 2, pp. 302–312, 2015.
W.-H. Chen, D. Wei, and W. X. Zheng, “Delayed impulsive control of Takagi-Sugeno fuzzy delay systems,” IEEE Trans. Fuzzy Syst., vol. 21, no. 3, pp. 516–526, 2013.
H. Zhang, H. Yan, T. Liu, and Q. Chen, “Fuzzy controller design for nonlinear impulsive fuzzy systems with time delay,” IEEE Trans. Fuzzy Syst., vol. 19, no. 5, pp. 844–856, 2011.
H. Dong, Z. Wang, D. W. C. Ho, and H. Gao, “Robust H∞ fuzzy output feedback control with multiple probabilistic delays and multiple missing measurements,” IEEE Trans. Fuzzy Syst., vol. 18, no. 4, pp. 712–725, 2010.
G. Feng, “A survey on analysis and design of model-based fuzzy control systems,” IEEE Trans. Fuzzy Syst., vol. 14, no. 5, pp. 676–697, 2006.
T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. SMC-15, no. 1, pp. 116–132, 1985.
I. Saboori and K. Khorasani, “H∞ consensus achievement of multi-agent systems with directed and switching topology networks,” IEEE Trans. Autom. Contr., vol. 59, no. 11, pp. 3104–3109, 2014.
Z.-H. Guan, B. Hu, M. Chi, D.-X. He, and X.-M. Cheng, “Guaranteed performance consensus in second-order multi-agent systems with hybrid impulsive control,” Automatica, vol. 50, no. 9, pp. 2415–2418, 2014.
Z.-H. Guan, D. J. Hill, and J. Yao, “A hybrid impulsive and switching control strategy for synchronizaiton of nonlinear systems and application to Chua’s chaotic circuit,” Int. J. Bifurcation and Chaos, vol. 16, no. 1, pp. 229–238, 2006.
Z.-H. Guan, D. J. Hill, and X. Shen, “On hybrid impulsive and switching systems and application to nonlinear control,” IEEE Trans. Autom. Contr., vol. 50, no. 7, pp. 1058–1062, 2005.
J. P. Hespanha and A. S. Morse, “Switching between stabilizing controllers,” Automatica, vol. 38, no. 11, pp. 1905–1917, 2002.
Z. Yang and D. Xu, “Stability analysis and design of impulsive control systems with time delay,” IEEE Trans. Autom. Contr., vol. 52, no. 8, pp. 1448–1454, 2007.
M. Forti and P. Nistri, “Global convergence of neural networks with discontinuous neuron activations,” IEEE Trans. Circuits Syst. I, vol. 50, no. 11, pp. 1421–1435, 2003.
B. Hu, Z.-H. Guan, X.-W. Jiang, M. Chi, R.-Q. Liao, “Event-driven multi-consensus of multi-agent networks with repulsive links,” Inform. Sci., vol. 373, pp. 110–123, 2016.
F. Yaghmaie, R. Su, F. Lewis, and L. Xie, “Multi-party consensus of linear heterogeneous multi-agent systems,” IEEE Trans. Autom. Contr., vol. 62, no. 11, pp. 5578–5589, 2017.
S. Jafarzadeh, M. Fadali, and A. Sonbol, “Stability analysis and control of discrete type-1 and type-2 TSK fuzzy systems: Part I. Stability analysis,” IEEE Trans. Fuzzy Syst., vol. 19, no. 6, pp. 989–1000, 2011.
S. Jafarzadeh, M. Fadali, and A. Sonbol, “Stability analysis and control of discrete type-1 and type-2 TSK fuzzy systems: Part II. Control design,” IEEE Trans. Fuzzy Syst., vol. 19, no. 6, pp. 1001–1013, 2011.
T. M. Guerra and L. Vermeiren, “LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi-Sugeno’s form,” Automatica, vol. 40, no. 5, pp. 823–829, 2004.
H. Ying, “Structure and stability analysis of general Mamdani fuzzy dynamic models,” Int. J. Intelligent Systems, vol. 20, no. 1, pp. 103–125, 2005.
R.-E. Precup, M.-L. Tomescu, and St. Preitl, “Fuzzy logic control system stability analysis based on Lyapunov’s direct method,” Int. J. Computers, Communication & Contr., vol. 4, no. 4, pp. 415–426, 2009.
A. F. Filippov, “Differential equations with discontinuous right-hand side,” Norwell, MA, USA: Kluwer, 1988.
M. Benchohra, J. Henderson, and S. K. Ntouyas, “Impulsive differential equations and inclusions,” vol. 2, New York: Hindawi Publishing Corporation, Chap. 3, 2006.
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Appendix
Appendix
Proof of Theorem 6.1
Proof
Construct a collection of Lyapunov functions as
First, consider the flow part of (6.11). Let \(\widetilde {\mathcal {D}} W_i(t)\) be the Lie derivative of W i, and ∂W i(t) the generalized gradient of W i at e i, as given by Definition 6.1. Based on (6.11), for any \(\mu _i\in \widetilde {\mathcal {D}}W_i(t)\), t ∈ (t k−1, t k], there exists \(\varrho _i\in \mathcal {K}[\dot {e}_i]\) such that \(\mu _i = \nu _i^\top \varrho _i\), with ν i = ∂W i(t). Note that, for any \(\varrho _i\in \mathcal {K}[\dot {e}_i]\), there also exist △B ∈co[−B (1), B (1)], △C ∈co[−C (1), C (1)], and \(\triangle \tilde {C}\in \mathrm {co}[-\tilde {C}^{(1)},\tilde {C}^{(1)}]\) such that
where \(\tilde {\varXi }(z_q(t),z_q(t-\tau (t)))=-\big (B^{(0)}-B+\triangle B\big ) z_q(t) +\big (C^{(0)}-C+\triangle C\big ) g(z_q(t)) +\big (\tilde {C}^{(0)}-\tilde {C}+\triangle \tilde {C} \big ) g(z_q(t-\tau (t)))\).
Taking \(\nu _i=\dot {e}_i^\top \varPsi (e_i,\dot {e}_i)\), equation \(\mu _i = \nu _i^\top \varrho _i\) gives
Next, the right-hand side of Eq. (6.21) is discussed.
By Assumption 6.1, one has
With \(\sum _{i=1}^N l_{ij}=0\), the coupling term satisfies
This gives
Contrarily, if
then
The above hypothesis implies
which contradicts with the fact that \(\sum _{j=1}^N l_{ij}=0\) with l ii > 0 and l ij = l ji ≤ 0 (j ≠ i), i, j = 1, 2, ⋯ , N.
Under Assumptions 6.1 and 6.2, it also follows that
Moreover, one has
Hence, Eq. (6.21) together with the preceding relationships gives
where t ∈ (t k−1, t k], α i, β i and \(d_i^q\) are as given by (6.13).
In the following, the impulse effect in (6.11) is analyzed. Similarly, there also exists △C ∈co[−C (1), C (1)] such that
Then with Assumption 6.2, for any \(\varOmega _i\in \mathcal {K}[W_i(t_k^+)]\), one can write
where \(f_i^q=\big \|C^{(0)}+\triangle C \big \| \varepsilon _q\).
With \(\sigma _i^q=1-\min \limits _r\min \{\nu _{r}^1,\nu _{r}^2\}+l_g \big \|C^{(0)}+\triangle C \big \|\), it follows
Take the delayed impulsive differential equation
as a comparison equation, k = 1, 2, ⋯, i = 1, 2, ⋯ , N. According to the comparison principle for delayed impulsive differential equations [5, 12, 38], it follows that W i(t) ≤ V i(t) for all t ≥ t 0.
Then, the solution V i(t) of dynamical system (6.24) is discussed. For t ∈ (t k−1, t k], one has
That is,
For k = 1, with \(V_i(t_0^+)=V_i(t_0)\), it follows
For k = 2, one has
For k = 3, it follows
Overall, it can be verified that for any t ∈ (t k−1, t k],
where \(\varPhi (t,t_{0})=(\sigma _i^q)^{\mathbb {N}(t,t_{0})} e^{-\alpha _i (t-t_{0})}\), \(\mathbb {N}(t,t_0)\) denotes the number of impulses during time interval (t 0, t].
According to Definition 6.2, with the average impulsive intermittence τ a > 0, α i > 0, and \(0<\sigma _i^q<1\), the following three relationships hold.
and
where the last inequality is due to \(\ln \sigma _i^q<0\).
Thus for any t ∈ (t k−1, t k], Eq. (6.26) gives
where \(\tilde {f}_i^q=\frac {1-(\sigma _i^q)^{\mathbb {N}_1}}{1-\sigma _i^q} f_i^q+\frac {(\sigma _i^q)^{-\mathbb {N}_0} }{\alpha _i}d_i^q\).
Hence, for all t ≥ t 0, from (6.27) it follows that
where \(\tilde {\psi }_i=\sup _{t_0-\bar {\tau }\leq \vartheta \leq t_0} \|\psi _i(\vartheta )-z_q(\vartheta )\|\), and \(\tilde {\alpha }_i\) is the unique solution of algebra equation
It can be verified that \(\tilde {\alpha }_i>0\) since one has \(\big (\alpha _i-\frac {\ln \sigma _i^q}{\tau _a}\big )- (\sigma _i^q)^{-\mathbb {N}_0} \beta _i>0\) based on condition (6.13).
It is now to verify the relationship (6.28) by a contradiction [12, 38].
If (6.28) does not hold, then there exists T 1 > 0 such that
for all t < T 1.
Based on inequality (6.27), one has
Due to the fact that \(\alpha _i-\frac {\ln \sigma _i^q}{\tau _a}= \tilde {\alpha }_i+ (\sigma _i^q)^{-\mathbb {N}_0} \beta _i e^{\tilde {\alpha }_i\kappa }>(\sigma _i^q)^{-\mathbb {N}_0} \beta _i\), one gets
Substituting the above inequality into inequality (6.30) gives
which leads to the contradiction.
Therefore, with \(\tilde {\alpha }_i >0\) and W i(t) ≤ V i(t) (t ≥ t 0), for any ε > 0, there exists a positive constant \(\mathcal {T}_i\) such that W i(t) ≤ ε + ξ q, \(\forall t>\mathcal {T}_i\), where ξ q is given by (6.12), i ∈ гq, \(q=1,\cdots ,\mathcal {Q}\). That is, multisynchronization of MINNs (6.3) is achieved asymptotically in regard to master system (6.6). This completes the proof. □
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Guan, ZH., Hu, B., Shen, X.(. (2019). Hybrid Memristor-Based Impulsive Neural Networks. In: Introduction to Hybrid Intelligent Networks. Springer, Cham. https://doi.org/10.1007/978-3-030-02161-0_6
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