HTML

ePub

PDF

1 Introduction

In some practical applications of delayed neural networks (DNNs), one needs to estimate the neuron states by available measurements and then employ the estimated values to achieve a certain desired performance [28]. Thus effective algorithms capable of estimat- ing the state of neurons are of both theoretical and practical significance. As we know, the Kalman filtering method is one of the most efficient ways to handle state estimation problems. However, when the external interferences do not have stationary Gaussian noise properties, this scheme will no longer be valid, which limits its application in DNNs and leads to the development of the passive filtering methods. A passive filter consists of resistors, capacitors, and inductors. It has been shown that passive filters are not only suitable for the situation of large current or voltage levels, but also can work well at very high frequencies [14, 17]. In 2010, the problem of passive filtering for DNNs was studied in [1], and a delay-dependent passive filter was proposed for ensuring that the filtering error system is stable as well as passive. Later, robust passive filtering for DNNs with uncertain parameters was considered in [38], where a cone complementarity linearization algorithm was used to calculate the desired filter gain.

Over the last two decades, switched systems, as a particular class of hybrid dy- namical systems, have attracted enormous attention owing to their potential applications in the vestibulo–ocular reflex [13], automotive roll dynamics control [25], image en- cryption [30], and other fields [6, 20, 31]. Generally, switched systems are made up of a set of continuous-time (or discrete-time) subsystems defined by differential (or dif- ference) equations as well as a switching rule that supervises the switching among the subsystems. The switching specifying which subsystem is activated every instant can be either arbitrary or restricted (e.g., obeying a certain probability distribution constraint) [40]. In recent years, by combining the theory of switched systems with DNNs, various mathematical models of switched DNNs have been introduced and a number of theoretical achievements have been reported; see, for instance, [22, 23, 33, 42]. Particularly, in the context of passive filtering for switched delayed neural networks (SDNNs), an error passivation method was put forward in [18], where it was ensured that the corresponding filtering error system is passive and asymptotically stable; the exponential passive filtering for SDNNs was addressed in [2], where a sufficient condition for the needed filter was derived in the form of linear matrix inequalities (LMIs).

In traditional communications, one often assumes that data is transmitted through per- fect communication network channels. In practice, however, as a result of the limitation of storage and digital communication bandwidth among nodes, the original data needs to be quantized before transmission. Quantization can be regarded as a map from continuous signals to discrete finite sets [32]. The quantized control strategy is able to save channel resources and cut down both the amount of transmitted data and channel blocking [36]. During the past few years, the design of quantized filters has been a hot topic and a variety of outstanding results have been acquired. To name a few, in [8], a quantized H∞ filter for time-varying switched systems was designed via employing the gridding method. In [9], based on a sector bound method, both H∞ and l2 l∞ filtering designs for a class of discrete switched system with quantized measurements were investigated. To the best of our knowledge, nevertheless, there is no relevant report on quantized passive filtering for SDNNs in the open literature, which inspires our current research.

From the above discussions this paper addresses the quantized passive filtering for SDNNs with noise interference. Both arbitrary switching and semi-Markov switching are taken into account. By choosing Lyapunov functionals and applying several inequality techniques, sufficient conditions are proposed to ensure the filtering error system to be not only exponentially stable, but also exponentially passive from the noise interference to the output error. The gain matrix for the proposed quantized passive filter is able to be determined through the feasible solution of LMIs, which are computationally tractable with the help of popular convex optimization tools. The remainder of this paper is as follows: in Section 2, we give the SDNN model, the quantized filter, as well as two types of switching rules under consideration. In Section 3, we propose quantized passive filter design methods for SDNNs under arbitrary switching and semi-Markov switching, respectively. In Section 4, we provide two numerical examples to illustrate the usefulness of the quantized passive filter design methods. Section 5 summarizes our conclusions

Notations. Throughout the present study, we apply Rn to represent a n-dimensional Euclidean space with norm , Rn×m to represent the set of all n m real matrices, and Z+ (respectively, R+) to stand for the set of positive integer numbers (respectively, non- negative real numbers). For any matrix X Rn×m, XT denotes its transpose, λmin(X) denotes its smallest eigenvalue, and X > 0 means that it is symmetric positive definite. In the case when n = m, let us define by the symmetric blocks in X and by S (X) the sum of X and XT. Moreover, we denote by the expectation operator, by by diag a diagonal matrix, and by I (respectively, 0) the identity (respectively, zero) matrix with appropriate dimension

2 Preliminaries

Consider a switched system composed of multiple DNNs given by

where x(t) Rn, y(t) Rm, and ω(t) Rm represent the state, output, and noise inter- ference, respectively; θ > 0 stands for the time-delay; (γ(t)) = diag a1(γ(t)), . . . , an(γ(t)) Rn×n (ak(γ(t)) > 0, k = 1, . . . , n) and (γ(t))(t) Rn×n are th self-feedback matrix and the delayed connection weight matrix, respectively; (γ(t)) Rn×m, (γ(t)) Rm×n, (γ(t)) Rm×n, and (γ(t)) Rm×m are known constant matrices; (γ(t))(t) Rn is an external input vector; γ(t) is the switching signal which chooses its values in Γ = (1, . . . , N ), N Z+; ψ(x(t)) denotes the activation function, which is assumed to be global Lipschitz continuous with Lipschitz constant Lψ > 0 [11, 12], i.e.

For more general assumptions on the activation function, one may refer to [26, 27, 34].

A quantizer q(•) : Rm → Φm is defined as q(ν) = [q1(ν1), . . . , qm(νm)]T, where Φ = {±φl, φl = χlφ0, l = 0, ±1, ±2, . . . } ∪ {0} with φ0 > 0 and 0 < χ < 1 [5, 19]. For any νj ∈ R (j = 1, . . . , m), quantizer qj(ν) is given by

where δ =(1−χ)/(1+χ). Note that q(ν) can be expressed by the sector bound method [10]:

Remark 1.

In networked control practice, owing to the limited transmission capacity of the network, signals need to be quantized before transmission for acquiring better control results. The quantizer can be seen as a coder that transforms the continuous signal into a piecewise continuous one [15].

Considering quantization effect (3), we propose the following filter

where x˘(t) Rn, y˘(t) Rm, and L(γ(t)) Rn×m are the filter state vector, the filter output vector, and the filter gain matrix, respectively. If we define by z(t) = x(t) x˘(t) the filtering error and by y¯(t) = y(t) y˘(t) the output error, then the filtering error system is able to be represented as follows

where ψ¯(z(t θ)) = ψ(x(t- θ))- ψ(x˘(t-θ)). In this paper, the following two types of switching rules are considered:

Case 1. γ(t) is a arbitrary switching signal

Case 2. γ(t) is a semi-Markov switching signal; i.e., (γ(t), h ≥ 0)t≥0 =(γn, hn)n∈Z+ represents a continuous-time and discrete-state semi-Markov process, where (γn)n∈Z+ is the index of system mode at nth transition selecting values in Γ , and (hn)n∈Z+ is the sojourn time of mode γn−1 between the (n−1)th transition and nth transition selecting val- ues in R+. The entries of transition probability matrix Π(h) = {πuv(h)} is determined by

where h > 0 denotes the sojourn time, limα→0o(α)/α = 0, πuv(h) ≥ 0 is the transitiΣon rate from mode u at time t to mode v at time t + α for u /= v, and πuu(h) =—v=1,vu πuv(h).

Remark 2.

The semi-Markov switching is a switching process that can be applied to describe sudden structure changes as well as abrupt component errors. Compared with the usual Markov switching [21, 24], the semi-Markov switching is more general since its sojourn-time can follow a nonexponential distribution that results in time-varying transition rates.

3 Main results

In this section, we propose design methods for quantized passive filtering of SDNN (1) under arbitrary switching and semi-Markov switching, respectively.

3.1Quantized passive filtering under arbitrary switching

The issue of quantized passive filtering under arbitrary switching to be addressed can be formulated explicitly as follows: for the switching rule in Case 1, design a quantized passive filter having the form in (4) such as the filtering error system in (5):

(i) is exponentially stable when ω(t) = 0;

(ii) is exponentially passive for ω(t) =/ 0 [2]; i.e., for a given scalar β > 0

holds, in which yˆ(s) = eκsy¯(s), κ > 0 is a real scalar, and H(z(s)) is a positive semi-definite storage function.

Define the indicator function as ζ(t) = [ζ1(t), . . . , ζN (t)]T, where

with u ∈ Γ . Then the filtering error system can be rewritten as

Note that ΣN

For the arbitrary switching rule, one can obtain the following result

Theorem 1.

If there exist matrices Pu > 0, Ru > 0, S> 0, Mu, and scalar ε > 0 such that

holds true for any u ∈ Γ, where

then the issue of quantized passive filtering under arbitrary switching is solvable, and the needed gain matrix can be chosen as

Proof. Define

Then, in view of the well-known inequality XY T+Y XT ≤ (1/ε)XXT+εY Y T (ε > 0), one can write

It follows by ∆2 ≤ δ2 that

Where

By Schur’s complement, (7) is equivalent to Ω˜u < 0, which together with (9) ensures that

Now, construct multiple Lyapunov functionals as follows

Then, along the trajectories of system (6), it can be calculated that

By adding and subtracting eκtωT(t)[Cuz(t) + Duz(t − θ) + Fuω(t)], one gets

Using (2), one has

By Mu = PuLu and (12), one can write

Where

and the second inequality follows from (10).

When ω(t) = 0, from (13) one can get V˙u(z(t), t) ≤ eκtzT(t)Sz(t). Owing to the fact that S > 0, V˙u(z(t), t) < 0 for any z(t) = 0. Thus, for any t > 0, it can be obtaine that

In addition, (11) gives

From (14) and (15) one has

Thus, the exponential stability of the filtering error system in (6) is guaranteed.

Next, one focuses on the passivity of system (6) with ω(t) = 0. Integrating both sides of (13) from t to 0 gives

Let β = maxu∈Γ Vu(z(0), 0). Then one has

which implies that filtering error system (6) is ensured to be exponentially passive from noise interference ω(t) to output error y¯(t) under the arbitrary switching rule. This com- pletes the proof When there is no quantization, the passive filter to be applied becomes

In the case, the filtering error system is represented by

which corresponds to (5) with ∆ = 0. Thus, one can write the following result.

Corollary 1.

If there exist matrices Pu > 0, Ru > 0, S > 0, and Mu such that

holds true for any u Γ, where Σ1u = S (Pu u Mu u) + κPu + S + Ru, then the issue of passive filtering under arbitrary switching is solvable, and the needed gain matrix of the passive filter can be chosen as (8)

Remark 3

Corollary 1 gives a novel existence criterion for the passive filtering of SDNN (1) without quantization. As going to be shown in Example 1, the criterion in Corollary 1, which is based on multiple Lyapunov functionals, is less conservative than the main result of [2].

3.2 Quantized passive filtering under semi-Markov switching

The issue of quantized passive filtering under semi-Markov switching to be addressed can be formulated explicitly as follows: for the switching rule in Case 2, design a quantized passive filter having the form in (4) such as the filtering error system in (5) is both exponentially stable and exponentially passive in the mean square sense

Set A(γ(t)) = Au, W(γ(t)) = Wu, J (γ(t)) = Ju, G(γ(t)) = Gu, C(γ(t)) = Cu, D(γ(t)) = Du, and F(γ(t)) = Fu. Then the filtering error system changes into

For the quantized passive filtering under semi-Markov switching, one can give the following result

Theorem 2.

Suppose that there are matrices Pu > 0, R > 0, S > 0, Mu, and scalar ε > 0 such that

holds for any u ∈ Γ, where

with gu(h) being the probability density function of sojourn time h at mode u. Then the issue of quantized passive filtering under semi-Markov switching is solvable, and the needed gain matrix can be chosen as (8).

Proof. Define

Along the same line as the proof in Theorem 1, we can write

Where

By Schur’s complement, the LMI in (18) ensures Ω˜u < 0. Then from (19) we have

Now, choose a mode-dependent Lyapunov functional as

Where

Define by L the infinitesimal generator [29], i.e.,

Then, for (t) = u, we can write

It follows that

Similarly, it can be obtained that

Thus, for γ(t) = u, by (12), (22), (23), and Mu = PuLu, we have

where the inequality follows by (20).

When ω(t) = 0, noting S > 0, from (24) we get E{LV (z(t), γ(t), t)} < 0 for all z(t) /= 0, which, together with Dynkin’s formula, yields

On the other hand, (21) implies

where

According to (25) and (26), we have

which means that the filtering error system is exponentially stable in the mean square sense.

When ω(t) /= 0, by (24) and Dynkin’s formula we can get that

Let β = maxu∈Γ {λmax(Pu)}E{W (z(0), 0)}. Then we can write

Thus, the filtering error system is ensured to be exponentially passive in the mean square sense from noise interference .(.) to output error .¯(.) under the semi-Markov switching rule. This completes the proof.

When there is no quantization, we can write the following result:

Corollary 2.

Suppose that there exist matrices P. > ., R > ., S > ., and M. such that

holds for any u . Γ, where

with g.(.) being the probability density function of sojourn time h at mode u. Then the issue of passive filtering under the semi-Markov switching is solvable, and the needed gain matrix of the passive filter can be chosen as (8).

Remark 4.

With the aid of multiple Lyapunov functionals and several inequality tech- niques, design methods for the quantized passive filtering under arbitrary switching and semi-Markov switching are proposed in Theorems 1 and 2, respectively. It is shown that the needed gain matrix is able to be obtained through the feasible solution of LMIs,which are known to be computationally tractable using some popular convex optimization tools. In addition, from the proofs of Theorems 1 and 2 it can be seen that κ/2 corresponds to the decay rate. Thus, the larger the scalar ., the faster the filtering error system converges.

Remark 5.

Over the past few decades, there has been an increasing interest in time-delay systems and a great number of research results have been achieved; see., e.g., [16, 35, 37, 41]. To our knowledge, most of the results are based on the Lyapunov functional method. It is worth pointing out that the choice of suitable Lyapunov functionals is of considerable significance. By extending the Lyapunov functionals in Theorems 1 and 2 as [3, 4], it is expected to obtain less conservative conditions. However, this may lead to increases in the dimension of LMIs and the number of decision variables, which in turn will result in higher computational costs.

4 Numerical examples

In this section, we give two numerical examples to show the usefulness of the proposed quantized passive filter design methods for SDNNs under arbitrary switching and semi- Markov switching, respectively.

Example 1. Consider SDNN (1) under arbitrary switching with

Notice that the activation function satisfies (2) with L. = 1 [7].

First, let us consider the case that there is no quantization. When ..(2) = 3.9, it can be verified that the LMIs in (17) are feasible for any . ≤ 0.75, while the condition in Theorem 2 of [2] fails for . ≥ 0.48. This means that, for . [0.48, 0.75], Corollary 1 of this paper can be applied for designing passive filter (16) while Theorem 2 of [2] is unavailable. When ..(2) = 3.6, it is found that the maximum allowed values of . are 0.66 by Corollary 1 and 0.38 by Theorem 2 of [2], respectively. A more detailed comparison of the maximum allowed . obtained by Corollary 1 of this paper and Theorem 2 of [2] for different choices of ..(2) is given in Table 1, it can be inferred that the present design method is always less conservative.

Next, we consider the passive filtering with quantization. Set . = 0.74 and . = 0.6.

Then, by solving the LMIs in (7), the filter gains can be obtained as follows:

Let we set .(.) = 1 when . [1,2] and .(.) = 2 otherwise, the initial condition to be .(.) = [ 3 1.5]., .˘(.) = [2 2]. (. [ θ, 0]), and .(.) to be a Gaussian noise subject to mean 0 and variance 1. Then the trajectories of state .(.) and its estimate .˘(.), output error .¯(.) and quantized measurement .(.¯(.)) are displayed in Fig. 1. The simulation results show that the quantized passive filter reduces the impact of noise interference .(.) on the filtering error system.

Example 2. Consider SDNN (1) under semi-Markov switching with

Suppose that sojourn time . obeys the Weibull distribution [39]. Specifically, assume that . ∼ Weibull(1, 2) (i.e., ..(.) = 2. exp( ..)) for . = 1 and . ∼ Weibull(1, 3) (i.e., ..(.) = 3.. exp( ..)) for . = 2. Then the transition probability matrix is given by.

Consequently, the mathematical expectation of .(.) is able to be acquired as

Choose . = 0.63 and . = 0.7. Then, by solving the LMI in (18), the corresponding gains can be obtained as follows:

Let the initial condition be .(.) = [ 0.5 1.5]., .˘(.) = [1 2]. (. [ θ, 0]), and .(.) be a Gaussian noise subject to mean 0 and variance 1. Then the trajectories of state .(.) and its estimate .˘(.), output error .¯(.) and quantized measurement .(.¯(.)) are shown in Fig. 2. The simulation results show the usefulness of the proposed quantized passive filter method in Theorem 2.

5 Conclusion

The issue of quantized passive filtering for SDNNs with noise interference has been addressed in this paper. Both arbitrary and semi-Markov switching rules have been dis- cussed. By choosing Lyapunov functionals and applying several inequality techniques, sufficient conditions have been established to ensure the filtering error systems to be not only exponentially stable, but also exponentially passive from the noise interference to the output error. It has been shown that the needed gain matrix for the proposed quantized passive filter can be constructed through the feasible solution of LMIs, which are compu- tationally tractable using some popular convex optimization tools. Finally, two numerical examples have been given to illustrate the usefulness of the present quantized passive filter design methods. It is worth mentioning that the quantizer under consideration is mode- independent. Over the past decade, robust filtering under mode-dependent quantization has received increasing attention. The robust passive filtering for SDNNs with mode- dependent quantization will be considered in our future work.

References

1. C.K. Ahn, Linear matrix inequality approach to passive filtering for delayed neural networks, Proc. Inst. Mech. Eng., I. J. Syst. Control Eng., 224(8):1040–1047, 2010, https://doi. org/10.1243/09596518JSCE1056.

2. C.K. Ahn, An error passivation approach to filtering for switched neural networks with noise disturbance, Neural Comput. Appl., 21(5):853–861, 2012, https://doi.org/10.1007/ s00521-010-0474-5.

3. Y. Cao, R. Samidurai, R. Sriraman, Robust passivity analysis for uncertain neural networks with leakage delay and additive time-varying delays by using general activation function, Math. Comput. Simul.,155:57–77, 2019, https://doi.org/10.1016/j.matcom.2017. 10.016.

4. Y. Cao, R. Samidurai, R. Sriraman, Stability and dissipativity analysis for neutral type stochastic Markovian jump static neural networks with time delays, J. Artif. Intell. Soft Comput. Res., .(3):189–204, 2019, https://doi.org/10.2478/jaiscr-2019-0003.

5. X. Chang, Y. Wang, Peak-to-peak filtering for networked nonlinear DC motor systems with quantization, IEEE Trans. Ind. Inf., 14(12):5378–5388, 2018, https://doi.org/10. 1109/TII.2018.2805707.

6. Y. Chen, S. Fei, K. Zhang, L. Yu, Control of switched linear systems with actuator saturation and its applications, Math. Comput. Modelling, 56(1–2):14–26, 2012, https://doi.org/ 10.1016/j.mcm.2012.01.002.

7. Y. Chen, Z. Zhang, Y. Liu, J. Zhou, Z. Wang, Resilient input-to-state stable filter design for nonlinear time-delay systems, Commun. Nonlinear Sci. Numer. Simul.,89:105335, 2020, https://doi.org/10.1016/j.cnsns.2020.105335.

8. J. Cheng, J.H. Park, J. Cao, D. Zhang, Quantized .∞ filtering for switched linear parameter- varying systems with sojourn probabilities and unreliable communication channels, Inf. Sci., 466:289–302, 2018, https://doi.org/10.1016/j.ins.2018.07.048.

9. S. Dong, H. Su, P. Shi, R. Lu, Z. Wu, Filtering for discrete-time switched fuzzy systems with quantization, IEEE Trans. Fuzzy Syst., 25(6):1616–1628, 2017, https://doi.org/10. 1109/TFUZZ.2016.2612699.

10. M. Fu, L. Xie, The sector bound approach to quantized feedback control, IEEE Trans. Autom. Control, 50(11):1698–1711, 2005, https://doi.org/10.1109/TAC.2005. 858689.

11. B. Hu, Q. Song, Z. Zhao, Robust state estimation for fractional-order complex-valued delayed neural networks with interval parameter uncertainties: LMI approach, Appl. Math. Comput.,373:125033, 2020, https://doi.org/10.1016/j.amc.2020.125033.

12. J. Hu, G. Sui, X. Lv, X. Li, Fixed-time control of delayed neural networks with impulsive perturbations, Nonlinear Anal., Model. Control, 23:904–920, 2018, https://doi.org/ 10.15388/NA.2018.6.6.

13. S. L. Kukreja, R. E. Kearney, H. L. Galiana, A least-squares parameter estimation algorithm for switched Hammerstein systems with applications to the VOR, IEEE Trans. Biomed. Eng., 52(3):431–444, 2005, https://doi.org/10.1109/TBME.2004.843286.

14. K. Lacanette, A basic introduction to filters-active, passive, and switched-capacitor, Application Note 779, National Semiconductor, Santa Clara, USA, 2010.

15. F. Li, X. Wang, P. Shi, Robust quantized .∞ control for network control systems with Markovian jumps and time delays, IEEE Trans. Biomed. Eng., .(12):4889–4902, 2013.

16. X. Li, X. Yang, T. Huang, Persistence of delayed cooperative models: impulsive control method, Appl. Math. Comput.,342:130–146, 2019, https://doi.org/10.1016/j. amc.2018.09.003.

17. X. Lin, X. Zhang, Y. Wang, Robust passive filtering for neutral-type neural networks with time- varying discrete and unbounded distributed delays, J. Franklin Inst., 350(5):966–989, 2013, https://doi.org/10.1016/j.jfranklin.2013.01.021.

18. J. Liu, J. Huang, Error passivation to filtering for a general class of switched recurrent neural networks with noise disturbance, in International Conference on Information Science and Technology Proceedings of the International, Nanjing, China, May 26–28, 2011, IEEE, Piscataway, NJ, 2011, pp. 839–842, https://doi.org/10.1109/ICIST.2011. 5765110.

19. Y. Liu, F. Fang, J. H. Park, H. Kim, X. Yi, Asynchronous output feedback dissipative control of Markovian jump systems with input time delay and quantized measurements, Nonlinear Anal., Hybrid Syst., 31:109–122, 2019, https://doi.org/10.1016/j.nahs.2018. 08.006.

20. L. Long, J. Zhao, A small-gain theorem for switched interconnected nonlinear systems and its applications, IEEE Trans. Autom. Control, 59(4):1082–1088, 2013, https://doi.org/ 10.1109/TAC.2013.2286898.

21. L. Pan, J. Cao, A. Alsaedi, Stability of reaction–diffusion systems with stochastic switching, Nonlinear Anal. Model. Control, 24(3):315–331, 2019, https://doi.org/0.15388/ NA.2019.3.1.

22. J. Qi, C. Li, T. Huang, W. Zhang, Exponential stability of switched time-varying delayed neural networks with all modes being unstable, Neural Process. Lett.,43(2):553–565, 2016, https://doi.org/10.1007/s11063-015-9428-3.

23. R. Rakkiyappan, K. Maheswari, G. Velmurugan, J.H. Park, Event-triggered .∞ state estima- tion for semi-Markov jumping discrete-time neural networks with quantization, Neural Netw., 105:236–248, 2018, https://doi.org/10.1016/j.neunet.2018.05.007.

24. R. Samidurai, R. Sriraman, S. Zhu, Stability and dissipativity analysis for uncertain Markovian jump systems with random delays via new approach, Int. J. Syst. Sci., 50(8):1609–1625, 2019, https://doi.org/10.1080/00207721.2019.1618942.

25. S. Solmaz, R. Shorten, K. Wulff, F.O. Cairbre, A design methodology for switched discrete time linear systems with applications to automotive roll dynamics control, Automatica, 44(9): 2358–2363, 2008, https://doi.org/10.1016/j.automatica.2008.01.014.

26. W. Tai, Q. Teng, Y. Zhou, J. Zhou, Z. Wang, Chaos synchronization of stochastic reaction- diffusion time-delay neural networks via non-fragile output-feedback control, Appl. Math. Comput.,354:115–127, 2019, https://doi.org/10.1016/j.amc.2019.02.028.

27. X. Wang, Z. Wang, Q. Song, H. Shen, X. Huang, A waiting-time-based event-triggered scheme for stabilization of complex-valued neural networks, Neural Netw., 121:329–338, 2020, https://doi.org/10.1016/j.neunet.2019.09.032.

28. Z. Wang, D.W.C. Ho, X. Liu, State estimation for delayed neural networks, IEEE Trans. Neural Networks, 16(1):279–284, 2005, https://doi.org/10.1109/TNN.2004.841813.

29. Y. Wei, J. Qiu, H.R. Karimi, W. Ji, A novel memory filtering design for semi-Markovian jump time-delay systems, IEEE Trans. Syst. Man Cybern., Syst., 48(12):2229–2241, 2018, https://doi.org/10.1109/TSMC.2017.2759900.

30. S. Wen, Z. Zeng, T. Huang, Q. Meng, W. Yao, Lag synchronization of switched neural networks via neural activation function and applications in image encryption, IEEE Trans. Neural Networks Learn. Syst., 26(7):1493–1502, 2015, https://doi.org/10.1109/ TNNLS.2014.2387355.

31. D. Xie, Y. Wu, Stabilisation of time-delay switched systems with constrained switching signals and its applications in networked control systems, IET Control Theory Appl., .(10):2120–2128, 2010, https://doi.org/10.1049/iet-cta.2009.0016.

32. L. Xing, C. Wen, Y. Zhu, H. Su, Z. Liu, Output feedback control for uncertain nonlinear systems with input quantization, Automatica, 65:191–202, 2016, https://doi.org/10. 1016/j.automatica.2015.11.028.

33. Z. Yan, X. Huang, J. Cao, Variable-sampling-period dependent global stabilization of delayed memristive neural networks via refined switching event-triggered control, Sci. China, Inf. Sci., 2020, https://doi.org/10.1007/s11432-019-2664-7.

34. Z. Yan, X. Huang, Y. Fan, J. Xia, H. Shen, Threshold-function-dependent quasi-synchroniza- tion of delayed memristive neural networks via hybrid event-triggered control, IEEE Trans. Syst. Man Cybern., Syst., 2020, https://doi:10.1109/TSMC.2020.2964605.

35. D. Yang, X. Li, J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Anal., Hybrid Syst., 32:294–305, 2019, https://doi.org/10.1016/j.nahs.2019.01.006.

36. X. Yang, Q. Song, J. Cao, J. Lu, Synchronization of coupled Markovian reaction–diffusion neural networks with proportional delays via quantized control, IEEE Trans. Neural Networks Learn. Syst., 30(3):951–958, 2019, https://doi.org/10.1109/TNNLS. 2018.2853650.

37. Y. Yang, J. Xia, J. Zhao, X. Li, Z. Wang, Multiobjective nonfragile fuzzy control for nonlinear stochastic financial systems with mixed time delays, Nonlinear Anal. Model. Control, 24(5): 696–717, 2019, https://doi.org/10.15388/NA.2019.5.2.

38. X. Zhang, X. Fan, Y. Xue, Y. Wang, W. Cai, Robust exponential passive filtering for uncertain neutral-type neural networks with time-varying mixed delays via Wirtinger-based integral inequality, Int. J. Control Autom. Syst., 15(2):585–594, 2017, https://doi.org/10. 1007/s12555-015-0441-0.

39. J. Zhou, Y. Liu, J. Xia, Z. Wang, S. Arik, Resilient fault-tolerant anti-synchronization for stochastic delayed reaction–diffusion neural networks with semi-Markov jump parameters, Neural Netw., 125:194–204, 2020, https://doi.org/10.1016/j.neunet.2020. 02.015.

40. J. Zhou, Y. Wang, X. Zheng, Z. Wang, H. Shen, Weighted .∞ consensus design for stochastic multi-agent systems subject to external disturbances and ADT switching topologies, Nonlinear Dyn., 96(2):853–868, 2019, https://doi.org/10.1007/s11071-019-04826-9.

41. Y. Zhou, J. Xia, H. Shen, J. Zhou, Z. Wang, Extended dissipative learning of time-delay recurrent neural networks, J. Franklin Inst., 356(15):8745–8769, 2019, https://doi. org/10.1016/j.jfranklin.2019.08.003.

42. S. Zhu, Y. Shen, Robustness analysis of global exponential stability of neural networks with Markovian switching in the presence of time-varying delays or noises, Neural Comput. Appl., 23(6):1563–1571, 2013, https://doi.org/10.1007/s00521-012-1105-0.

IR