Finite-time synchronization of hierarchical hybrid coupled neural networks with mismatched quantization

Abstract

A non-fragile memory feedback control methodology is precisely proposed in this study for synchronization of hierarchical hybrid coupled neural networks (HHCNNs) over finite-time domain with mismatched quantization channels and external disturbances. Specifically, the considered network model incorporates both higher level deterministic switching and lower level Markov switching. Moreover, an undirected communication topology is selected to project the addressed HHCNNs. The foremost intention of this study is to substantiate the synchronization criterion over finite interval of time with proposed $H_{\infty }$ disturbance attenuation. In consideration to this motive, by conferring Lyapunov stability theory in conjunction with average dwell-time technique, a collection of adequate conditions is established for assuring the exponential synchronization criterion through a set of linear matrix inequalities. Moreover, the desired the memory feedback controller with gain variations is computed based on the developed matrix inequalities. Finally, the developed theoretical results are validated through a numerical example, which showcases the significance and advantage of the developed control strategy.

Appendix I

Taking advantage of infinitesimal operator [18] indicated here as ${\mathcal {L}}$ on (18), we attained the following equations:

$$\begin{aligned} {\mathcal {L}}V_1(\varPsi (t),p,q,t)&=2\varPsi ^{T}(t)P_{p,q}{\dot{\varPsi }}(t)\nonumber \\&\quad +\sum _{n=1}^{{\mathcal {M}}}\varPi _{qn}^{p}\varPsi ^{T}(t)P_{p,n}\varPsi (t), \end{aligned}$$

(28)

$$\begin{aligned} {\mathcal {L}}V_2(\varPsi (t),p,q,t)&\le \varPsi ^{T}(t)Q_{1,p,q}{\varPsi }(t)-(1-\delta _{p,q})e^{\alpha \tau _{p,q}^{1}} \nonumber \\&\quad \times \varPsi (t-\tau _{p,q}^{T}(t))Q_{1,p,q}{\varPsi }(t-\tau _{p,q}(t))\nonumber \\&\quad +\sum _{n=1}^{{\mathcal {M}}}\varPi _{qn}^{p}\int _{t-\tau _{p,q}(t)}^{t}e^{\alpha (t-s)}\varPsi ^{T}(s)\nonumber \\&\quad \times Q_{1,p,n}\varPsi (s){\mathrm {d}}s +\varPsi ^{T}(t)R_{1,p,q}{\varPsi }(t) \nonumber \\&\quad -e^{\alpha \tau _{p,q}^{1}}\varPsi ^{T}(t-\tau _{p,q}^{1})\nonumber \\&\quad \times R_{1,p,q}{\varPsi }(t-\tau _{p,q}^{1})+\sum _{n=1}^{{\mathcal {M}}}\varPi _{qn}^{p}\nonumber \\&\quad \times \int _{t-\tau _{p,q}^{1}}^{t}e^{\alpha (t-s)}\varPsi ^{T}(s)R_{1,p,n}\varPsi (s){\mathrm {d}}s\nonumber \\&\quad +e^{\alpha \tau _{p,q}^{1}}\varPsi (t-\tau _{p,q}^{1})^{T}S_{1,p,q}{\varPsi }(t-\tau _{p,q}^{1})\nonumber \\&\quad -e^{\alpha \tau _{p,q}^{2}}\varPsi ^{T}(t-\tau _{p,q}^{2})S_{1,p,q}{\varPsi }(t-\tau _{p,q}^{2}) \nonumber \\&\quad +\sum _{n=1}^{{\mathcal {M}}}\varPi _{qn}^{p}\int _{t-\tau _{p,q}^{2}}^{t-\tau _{p,q}^{1}}e^{\alpha (t-s)}\varPsi ^{T}(s)S_{1,p,n}\nonumber \\&\quad \times \varPsi (s){\mathrm {d}}s+\alpha V_2(\varPsi (t),p,q,t), \end{aligned}$$

(29)

$$\begin{aligned} {\mathcal {L}}V_3(\varPsi (t),p,q,t)&=\varPsi ^{T}(t) [\tau ^2_p Q+\tau ^2_p R+(\tau ^2_p-\tau ^1_p) S] \varPsi (t)\nonumber \\&\quad -\int _{t-\tau ^2_p}^{t}e^{\alpha (t-s)}\psi ^{T}(s) Q \psi (s) {\mathrm {d}}s \nonumber \\&\quad -\int _{t-\tau ^2_p}^{t}e^{\alpha (t-s)}\psi ^{T}(s) R \psi (s) {\mathrm {d}}s \nonumber \\&\quad -\int _{t-\tau ^2_p}^{t-\tau ^1_p}e^{\alpha (t-s)}\psi ^{T}(s) S \psi (s){\mathrm {d}}s \nonumber \\&\quad +\alpha V_3(\varPsi (t),p,q,t), \end{aligned}$$

(30)

$$\begin{aligned} {\mathcal {L}}V_4(\varPsi (t),p,q,t)&={(\tau _{p,q}^{1})}^{2}{\dot{\varPsi }}^{T}(t)E_1{\dot{\varPsi }}(t) \nonumber \\&\quad +{(\tau _{p,q}^{2})}^{2}{\dot{\varPsi }}^{T}(t)E_2{\dot{\varPsi }}(t)\nonumber \\&\quad -\tau _{p,q}^{1}\int _{t-\tau _{p,q}^{1}}^{t}e^{\alpha (t-s)}{\dot{\varPsi }}^{T}(S)E_1{\dot{\varPsi }}(S){\mathrm {d}}s \nonumber \\&\quad -\tau _{p,q}^{2}\int _{t-\tau _{p,q}^{2}}^{t} e^{\alpha (t-s)}{\dot{\varPsi }}^{T}(S)E_2{\dot{\varPsi }}(S){\mathrm {d}}s\nonumber \\&\quad +\alpha V_4(\varPsi (t),p,q,t). \end{aligned}$$

(31)

Now, by the agency of Jensen’s inequality [16], the integral terms in (31) can be simplified as follows:

$$\begin{aligned}&-\tau ^{1}_{p,q}\int _{t-\tau _{p,q}^{1}}^{t}e^{\alpha (t-s)}{\dot{\varPsi }}^{T}(S)E_1{\dot{\varPsi }}(S){\mathrm {d}}s \nonumber \\&\quad \le -e^{\alpha \tau _{p,q}^{1}}\left[ \int _{t-\tau _{p,q}^{1}}^{t}{\dot{\varPsi }}^{T}(S)\right] E_1 \left[ \int _{t-\tau _{p,q}^{1}}^{t}{\dot{\varPsi }}(S)\right] \nonumber \\&\quad \le -e^{\alpha \tau _{p,q}^{1}}\left[ \varPsi (t)-\varPsi (t-\tau ^1_{p,q})\right] ^\mathrm{T} E_1 \left[ \varPsi (t)-\varPsi (t-\tau ^1_{p,q})\right] , \end{aligned}$$

(32)

$$\begin{aligned}&-\tau ^{2}_{p,q}\int _{t-\tau _{p,q}^{2}}^{t}e^{\alpha (t-s)}{\dot{\varPsi }}^{T}(S)E_2{\dot{\varPsi }}(S){\mathrm {d}}s\nonumber \\&\quad \le -e^{\alpha \tau _{p,q}^{2}}\left[ \int _{t-\tau _{p,q}^{2}}^{t}{\dot{\varPsi }}^{T}(S)\right] E_2 \left[ \int _{t-\tau _{p,q}^{2}}^{t}{\dot{\varPsi }}(S)\right] \nonumber \\&\quad \le -e^{\alpha \tau _{p,q}^{2}}\left[ \varPsi (t)-\varPsi (t-\tau ^2_{p,q})\right] ^\mathrm{T} E_2 \left[ \varPsi (t)-\varPsi (t-\tau ^2_{p,q})\right] . \end{aligned}$$

(33)

According to $\pi _{qn}\ge 0 (q\ne n),\ Q_{n}\ge 0$ and (13), we have

$$\begin{aligned}&\sum _{n=1}^{{\mathcal {M}}}\pi _{qn}^{p}\int _{t-\tau _{p,q}(t)}^{t}e^{\alpha (t-s)}\varPsi ^{T}(s)Q_{1,p,n}\varPsi (s){\mathrm {d}}s \nonumber \\&\quad \le \sum _{n=1,n\ne q}^{{\mathcal {M}}}\pi _{qn}^{p}\int _{t-\tau _{p,q}(t)}^{t}e^{\alpha (t-s)}\varPsi ^{T}(s)Q_{1,p,n}\varPsi (s){\mathrm {d}}s \nonumber \\&\quad \le \sum _{n=1,n\ne q}^{{\mathcal {M}}}\pi _{qn}^{p}\int _{t-\tau ^{2}_{p}}^{t}e^{\alpha (t-s)}\varPsi ^{T}(s)Q_{1,p,n}\varPsi (s){\mathrm {d}}s \nonumber \\&\quad \le \int _{t-\tau ^{2}_{p}}^{t}e^{\alpha (t-s)}\varPsi ^{T}(s)Q\varPsi (s){\mathrm {d}}s. \end{aligned}$$

(34)

Following the same methodology as above, we can get

$$\begin{aligned}&\sum _{n=1}^{{\mathcal {M}}}\pi _{qn}^{p}\int _{t-\tau ^{1}_{p,q}}^{t}e^{\alpha (t-s)}\varPsi ^{T}(s)R_{1,p,n}\varPsi (s){\mathrm {d}}s \nonumber \\&\quad \le \int _{t-\tau ^{2}_{p}}^{t}e^{\alpha (t-s)}\varPsi ^{T}(s)R\varPsi (s){\mathrm {d}}s, \end{aligned}$$

(35)

$$\begin{aligned}&\sum _{n=1}^{{\mathcal {M}}}\pi _{qn}^{p}\int _{t-\tau ^{2}_{p,q}}^{t-\tau ^{1}_{p,q}}e^{\alpha (t-s)}\varPsi ^{T}(s)S_{1,p,n}\varPsi (s){\mathrm {d}}s \nonumber \\&\quad \le \int _{t-\tau ^{2}_{p}}^{t-\tau ^{1}_{p}}e^{\alpha (t-s)}\varPsi ^{T}(s)S\varPsi (s){\mathrm {d}}s. \end{aligned}$$

(36)

Moreover, by utilizing Assumption 1, we can retrieve the following inequalities:

$$\begin{aligned}&\varPsi ^{T}(t){\bar{L}}{\varSigma }_1{\bar{L}}\varPsi (t)-g^{T}(\varPsi (t)){\varSigma }_1g(\varPsi (t))\ge 0, \end{aligned}$$

(37)

$$\begin{aligned}&\varPsi ^{T}(t-\tau _{p,q}(t)){\bar{L}}{\varSigma }_{2,p,q}{\bar{L}}\varPsi (t-\tau _{p,q}(t)) \nonumber \\&-g^{T}(\varPsi (t-\tau _{p,q}(t))){\varSigma }_{2,p,q}g(\varPsi (t-\tau _{p,q}(t)))\ge 0, \end{aligned}$$

(38)

where $\varSigma _1$ and $\varSigma _{2,p,q}$ are formerly described in theorem statement.

At the same instant, for any non-singular pertinent matrix $W_{p,q}$, the following equation holds:

$$\begin{aligned}&2\{\varPsi ^{T}(t)+{{\dot{\varPsi }}^{T}(t)} (I_{N} \otimes {\mathcal {W}}_{p,q}) [-{\bar{C}}_{p,q}\varPsi (t)+{\bar{A}}_{p,q}g(\varPsi (t))\nonumber \\&+{\bar{B}}_{p,q}g(\varPsi (t-\tau _{p,q}(t)))+{\bar{J}}_{p,q}+{\bar{H}}_{1,p,q}\psi (t)\nonumber \\&+{\bar{H}}_{2,p,q}\psi (t-\tau _{p,q}(t))+{\bar{D}}_{p,q}\hbar (t)K_{1,p,q}\varPsi {(t)}\nonumber \\&+{\bar{D}}_{p,q}\hbar (t)K_{2,p,q}\varPsi {(t-\tau ^{1}_{p})} +{\bar{D}}_{p,q}\hbar (t)K_{3,p,q}\varPsi {(t-\tau ^{2}_{p})}\nonumber \\&+{\bar{D}}_{p,q}\hbar (t)e_{\mu _c}(t)-{\bar{D}}_{p,q}\hbar (t) \{\phi /2+ \varepsilon _w/\lambda \}\nonumber \\&\times sign\{{\bar{D}}_{p,q}^{T}P_{p.q}\varPsi {(t)}\}-\dot{\psi (t)}]\}=0, \end{aligned}$$

(39)

Further, from (28)-(39) with addition of $z^{T}(s)z(s)-\gamma ^2 w^{T}(s)w(s)$ and taking mathematical expectations, we can easily fetch that

$$\begin{aligned}&{\mathbb {E}}\{{\mathcal {L}}V(\varPsi (t),p,q,t)\}- w^{T}(s)\alpha w(s)\le \ {\mathbb {E}}\big \{\varTheta ^{T}(t,p,q) \nonumber \\&\times [\varXi _{(v,w)pq}]_{(8\times 8)}\varTheta (t,p,q)+\alpha V(\varPsi (t),p,q,t)\big \}. \end{aligned}$$

(40)

Moreover, from relations (12) and (40), it is easy to obtain

$$\begin{aligned} {\mathbb {E}}\{{\mathcal {L}}V(\varPsi (t),p,q,t)\}<{\mathbb {E}}\{\alpha V(\varPsi (t),p,q,t)\}, \end{aligned}$$

(41)

where $\varTheta (t,p,q)=\bigg [\varPsi ^{T}(t) \ \ \varPsi ^{T}(t-\tau _{p,q}(t)) \ \ \varPsi ^{T}(t-\tau ^1_{p,q}(t)) \ \varPsi ^{T}(t-\tau ^2_{p,q}(t)) \ \ g^{T}(\varPsi (t)) \ \ g^{T}(\varPsi (t-\tau _{p,q}))\ \ {\dot{\varPsi }}^{T}(t)\ \ w^{T}(s) \bigg ]^\mathrm{T}.$

Further, multiplying $e^{-\alpha t}$ and integrating (41), we get

$$\begin{aligned}&{\mathbb {E}}\{V(\varPsi (t),p,q,t)\}- w_z(1-e^{-\alpha z})\nonumber \\&< e^{-\alpha (t-t_k)}{\mathbb {E}}\{V(\varPsi (t),p,q,t)\}. \end{aligned}$$

(42)

Moreover, in light of (15)-(17) and (40), we acquire

$$\begin{aligned} {\mathbb {E}}\{V(\varPsi (t),p,q,t)\}&-w_z(1-e^{-\alpha z})\nonumber \\&< \kappa {\mathbb {E}}\{V(\varPsi (t),p,q,t)\}, t \in [t,t_k). \end{aligned}$$

(43)

Based on the relations (42) and (43), and setting the recursion from $t_{k-1}$ to $t_k$, $t_{k-2}$ to $t_{k-1}$, $\ldots$ up to $t_0$, we bring off inequality by

$$\begin{aligned} {\mathbb {E}}\{V(\varPsi (t),p,q,t)\}< \kappa ^{{V}_p}e^{\alpha {z}}\{V(0)+w_z(1-e^{-\alpha z})\} \end{aligned}$$

(44)

Then, by letting ${\tilde{P}}_{p,q}={\mathfrak {D}}_{p,q}^{-\frac{1}{2}}P_{p,q}{\mathfrak {D}}_{p,q}^{-\frac{1}{2}}, {\tilde{Q}}_{1,p,q}={\mathfrak {D}}_{p,q}^{-\frac{1}{2}}Q_{1,p,q}{\mathfrak {D}}_{p,q}^{-\frac{1}{2}},\quad {\tilde{R}}_{1,p,q}={\mathfrak {D}}_{p,q}^{-\frac{1}{2}}R_{1,p,q}{\mathfrak {D}}_{p,q}^{-\frac{1}{2}}, {\tilde{S}}_{1,p,q}={\mathfrak {D}}_{p,q}^{-\frac{1}{2}}S_{1,p,q}{\mathfrak {D}}_{p,q}^{-\frac{1}{2}},\quad {\tilde{Q}}={\mathfrak {D}}^{-\frac{1}{2}}Q{\mathfrak {D}}^{-\frac{1}{2}}, {\tilde{R}}={\mathfrak {D}}^{-\frac{1}{2}}R{\mathfrak {D}}^{-\frac{1}{2}},\quad {\tilde{S}}={\mathfrak {D}}^{-\frac{1}{2}}S{\mathfrak {D}}^{-\frac{1}{2}},\ \tilde{E_1}={\mathfrak {D}}^{-\frac{1}{2}}E_1{\mathfrak {D}}^{-\frac{1}{2}}, \tilde{E_2}={\mathfrak {D}}^{-\frac{1}{2}}E_2{\mathfrak {D}}^{-\frac{1}{2}}$ and it is easy to obtain

$$\begin{aligned} {\mathbb {E}}V(\psi (t),p,q,t)&\ge \varPsi ^{T}(t)P_{p,q}\varPsi (t) \nonumber \\&\ge \varUpsilon _{min}({\tilde{P}}_{p,q})\varPsi ^{T}(t){\mathfrak {D}}_{p,q}\varPsi (t) \nonumber \\&=\varUpsilon _1 \varPsi ^{T}(t){\mathfrak {D}}_{p,q}\varPsi (t). \end{aligned}$$

(45)

Furthermore from (18), it follows that

$$\begin{aligned} {\mathbb {E}}V(0)&={\mathbb {E}}\{(\varPsi ^{T}(0)P_{p,q}\varPsi (0)\nonumber \\&\quad +\int _{-\tau _{p}^2}^{0}e^{-\alpha s}\varPsi ^{T}(s)Q_{1,p,q}\varPsi (s){\mathrm {d}}s \nonumber \\&\quad +\int _{-\tau ^{1}_{p,q}}^{0}e^{-\alpha s}\varPsi ^{T}(s)R_{1,p,q}\varPsi (s){\mathrm {d}}s\nonumber \\&\quad +\int _{-\tau ^{2}_{p,q}}^{-\tau ^{1}_{p,q}}e^{-\alpha s}\varPsi ^{T}(s)S_{1,p,q}\varPsi (s){\mathrm {d}}s\nonumber \\&\quad +\int _{-\tau ^2_p}^{0} \int _{\theta }^{0}e^{-\alpha s}\varPsi ^{T}(s)Q\varPsi (s){\mathrm {d}}s{\mathrm {d}}\theta \nonumber \\&\quad +\int _{-\tau ^2_p}^{0}\int _{\theta }^{0}e^{-\alpha s}\varPsi ^{T}(s)R\varPsi (s){\mathrm {d}}s{\mathrm {d}}\theta \nonumber \\&\quad +\int _{-\tau ^2_p}^{-\tau ^1_p} \int _{\theta }^{0}e^{-\alpha s}\varPsi ^{T}(s)S\varPsi (s){\mathrm {d}}s{\mathrm {d}}\theta \nonumber \\&\quad +\tau ^1_{p,q}\int _{-\tau ^{1}_{p,q}}^{0}\int _{\theta }^{0}e^{-\alpha s}{\dot{\varPsi }}^{T}(s)E_1{\dot{\varPsi }}(s){\mathrm {d}}s{\mathrm {d}}\theta \nonumber \\&\quad +\tau ^2_{p,q}\int _{-\tau ^{2}_{p,q}}^{0}\int _{\theta }^{0}e^{-\alpha s}{\dot{\varPsi }}^{T}(s)E_2{\dot{\varPsi }}(s){\mathrm {d}}s{\mathrm {d}}\theta \}, \nonumber \\&\le \bigg [\varUpsilon _{max} (P_{p,q})+e^{\alpha \tau _{p}^{2}} \tau _{p}^{2} \varUpsilon _{max} (Q_{1,p,q})\nonumber \\&\quad +e^{\alpha \tau _{p}^{2}} \tau _{p}^{2} \varUpsilon _{max} (R_{1,p,q})\nonumber \\&\quad +e^{\alpha (\tau _{p,q}^{2}-\tau _{p,q}^{1})} (\tau _{p,q}^{1}-\tau _{p,q}^{2}) \nonumber \\&\quad \times \varUpsilon _{max} (S_{1,p,q})+e^{\alpha \tau _{p}^{2}}\frac{(\tau _{p}^{2})^{2}}{2}\varUpsilon _{max}(Q)\nonumber \\&\quad +e^{\alpha \tau _{p}^{2}}\frac{(\tau _{p}^{2})^{2}}{2}\varUpsilon _{max}(R) \nonumber \\&\quad +e^{\alpha (\tau _{p,q}^{2}-\tau _{p,q}^{1})} \frac{(\tau _{p,q}^{1}-\tau _{p,q}^{2})^{2}}{2}\varUpsilon _{max}(S)\nonumber \\&\quad +e^{\alpha \tau _{p}^{2}} \frac{(\tau _{p}^{2})^{3}}{2} \varUpsilon _{max}(E_1)\nonumber \\&\quad +e^{\alpha \tau _{p}^{2}} \frac{(\tau _{p}^{2})^{3}}{2} \varUpsilon _{max}(E_2) \bigg ] \nonumber \\&\quad \times \sup _{{-\tau _{p^{2}}}\le s \le 0} \left\{ \psi ^{T}(s){\mathfrak {D}}\psi (s),{\dot{\psi }}^{T}(s){\mathfrak {D}}{\dot{\psi }}^{T}(s))\right\} , \nonumber \\&\le \bigg [\varUpsilon _2+e^{\alpha \tau _{p}^{2}} \tau _{p}^{2} \varUpsilon _3+e^{\alpha \tau _{p}^{2}} \tau _{p}^{2} \varUpsilon _4\nonumber \\&\quad +e^{\alpha (\tau _{p}^{2}-\tau _{p,q}^{1})} (\tau _{p,q}^{1}-\tau _{p,q}^{2}) \varUpsilon _5+e^{\alpha \tau _{p}^{2}}\frac{(\tau _{p}^{2})^{2}}{2}\varUpsilon _6 \nonumber \\&\quad +e^{\alpha \tau _{p}^{2}}\frac{(\tau _{p}^{2})^{2}}{2}\varUpsilon _7 +e^{\alpha (\tau _{p,q}^{2}-\tau _{p,q}^{1})} \frac{(\tau _{p,q}^{1}-\tau _{p,q}^{2})^{2}}{2}\varUpsilon _8\nonumber \\&\quad +e^{\alpha \tau _{p}^{2}} \frac{(\tau _{p}^{2})^{3}}{2} \varUpsilon _9+e^{\alpha \tau _{p}^{2}} \frac{(\tau _{p}^{2})^{3}}{2} \varUpsilon _{10}\bigg ]C_1. \end{aligned}$$

(46)

$$\begin{aligned}&=w_f C_1. \end{aligned}$$

(47)

From (44), we can get

$$\begin{aligned} \varUpsilon _1{\mathbb {E}}\{\varPsi (t){\mathfrak {D}}_{p,q}\varPsi (t)&\le {\mathbb {E}}\{V(\varPsi (t),p,q,t)\}\nonumber \\&<\kappa ^{{V}_p}e^{{z}\alpha }\{V(0)+w_z(1-e^{-\alpha z})\}. \end{aligned}$$

(48)

Then, in accordance with (46) and (48), we have

$$\begin{aligned} {\mathbb {E}}\{\varPsi ^{T}(t){\mathfrak {D}}_{p,q}\varPsi (t)\}&\le \frac{\kappa ^{{V}_p}e^{\alpha {z}}\{V(0)+w_z(1-e^{-\alpha z})\}}{\varUpsilon _1}\nonumber \\&\le \frac{\kappa ^{\frac{m}{\tau _{\alpha }}}e^{\alpha {z}}w_f C_1}{\varUpsilon _1}. \end{aligned}$$

(49)

From the relation (14), it is obvious that ${\mathbb {E}}\{\varPsi (t){\mathfrak {D}}_{p,q}\varPsi (t)\}<C_2 \ \forall t \in [0,{z}]$. Hence, finite-time exponential synchronization of HHCNNs (1) is achieved in accordance with Definition 2.1 of [30] under the control design (5). Thus, the proof of this theorem ends.