Distributed event-triggered hybrid wired-wireless networked control with ${H_2}/{H_\infty }$ filtering

Abstract

Aiming at the fact that distributed multi-channel hybrid network-induced delays and noise interference may deteriorate the control performance of hybrid networked control systems, distributed event-triggered hybrid wired-wireless networked control with ${H_2}/{H_\infty }$ filtering is proposed. A distributed event-triggered mechanism is firstly employed to reduce communication burden, and two Markov chains are used to respectively describe different characters of network-induced delays of hybrid wired-wireless networks. Then, a ${H_2}/{H_\infty }$ filter is employed to improve the input signal precision of the controller, where a general closed-feedback filtering and control system model with distributed event-triggered parameters and network-induced delays of hybrid wired-wireless networks is proposed. Furthermore, the designed filter and controller enable the closed-feedback filtering and control system to be stochastic stability and to achieve a prescribed ${H_2}/{H_\infty }$ performance, and the relationships between the stability criteria and the maximum network-induced delays, distributed event-triggered parameters, the filter and controller parameters and the system performance parameter are established. Finally, simulation results confirm the effectiveness of the proposed method.

Appendix

1.1 Proof of Theorem 1

Proof

Construct a Lyapunov function as

$$\begin{aligned} V(\tilde{\xi } (k),k) = {\tilde{\xi } ^T}(k)P(\iota ,r)\tilde{\xi } (k) \end{aligned}$$

where $k \in \left[ {S_k^i + d_{S_k^i}^j + {\tau _{S_k^i + d_{S_k^i}^j}},} \right. \left. {S_{k + 1}^i + d_{S_{k + 1}^i}^j + {\tau _{S_{k + 1}^i + d_{S_{k + 1}^i}^j}}} \right) , i,j \in \upsilon , \tilde{\xi } (k) = dia{g_N}\left\{ {\xi (k)} \right\}$.

1. Supposed that $\omega (k) = 0$, it follows that

where ${\Theta _{1r}} = \left[ {\begin{array}{cc} {{\chi _{11}} - P(\iota ,r)}&{}{{\chi _{12}}}\\ * &{}{{\chi _{22}}} \end{array}} \right] ,$.

Moreover, for $k \in \varOmega$, the proposed distributed event-triggered mechanism (5) ensures that

$$\begin{aligned} \begin{array}{l} {e^T}(k)e(k) \le {y^T}({S_k})\wp y({S_k})\\ = {\left[ {y(k - d(k)) - e(k)} \right] ^T}\wp \left[ {y(k - d(k)) - e(k)} \right] \\ = {{\tilde{\xi } }^T}(k){\varOmega _1}\tilde{\xi } (k) - {{\tilde{\xi } }^T}(k){\varOmega _2}e(k) - {e^T}(k)\varOmega _2^T\tilde{\xi } (k)\\ \quad + {e^T}(k)\wp e(k) \end{array} \end{aligned}$$

Therefore,

where ${\Theta _{2r}} = \left[ {\begin{array}{c@{\quad }c@{\quad }c} {{\chi _{11}} - P(\iota ,r) + {\varOmega _1}}&{}{{\chi _{12}}}&{}{ - {\varOmega _2}}\\ * &{}{{\chi _{22}}}&{}0\\ * &{} * &{}{\wp - I} \end{array}} \right]$.

If ${\Theta _{2r}} < 0$, then

where $\beta = \inf \left\{ {{\lambda _{\min }}( - {\Theta _{2r}})} \right\} .$

Since $EV(\tilde{\xi } (k + 1),k + 1) - V(\tilde{\xi } (k),k) \le - \beta x{(k)^T}x(k)$, we let the inequality overlay both sides from 0 to $\ell (\ell \rightarrow \infty )$, we have

$$\begin{aligned}&\mathop {\lim }\limits _{\ell \rightarrow \infty } EV(\tilde{\xi } (\ell + 1),\ell + 1) - V({\varphi _0},{s_0}) \le \\&\quad - \beta \mathop {\lim }\limits _{\ell \rightarrow \infty } \sum \limits _{k = 0}^\ell {x{{(k)}^T}x(k)} \\&\Rightarrow \beta \mathop {\lim }\limits _{\ell \rightarrow \infty } \sum \limits _{k = 0}^\ell {x{{(k)}^T}x(k)} \le V({\varphi _0},{s_0}) \\&\quad - \mathop {\lim }\limits _{\ell \rightarrow \infty } EV(\xi (\ell + 1),\ell + 1)\\&{ \le V({\varphi _0},{s_0})}\\&\Rightarrow \mathop {\lim }\limits _{\ell \rightarrow \infty } \sum \limits _{k = 0}^\ell {x{{(k)}^T}x(k)} \\&\quad \le \dfrac{1}{\beta }V({\varphi _0},{s_0}) < \infty \end{aligned}$$

where ${\varphi _0}$ and ${s_0}$ are the initial conditions.

Therefore, the closed-feedback filtering and control system is stochastically stable.

2. Supposed that $\omega (k) \ne 0$, when the initial condition is zero, We have

where ${\Theta _{3r}} = \left[ {\begin{array}{cccc} {{\chi _{11}} - P(\iota ,r) + {\varOmega _1}}&{}{{\chi _{12}}}&{}{{\chi _{13}} + {\varOmega _3}}&{}{ - {\varOmega _2}}\\ * &{}{{\chi _{22}}}&{}{{\chi _{23}}}&{}0\\ * &{} * &{}{{\chi _{33}} + {{\tilde{D}}^T}\varOmega \tilde{D}}&{}{ - {{\tilde{D}}^T}\wp }\\ * &{} * &{} * &{}{\wp - I} \end{array}} \right]$.

Define $J = E(V(\tilde{\xi } (k),k) - E(\sum \limits _{h = 0}^{k - 1} {{\omega ^T}(h)\omega (h)} )$. When the initial condition is zero, the value of E(V(0), 0) is also zero. Then, we have

where ${\Theta _{4r}} = \left[ {\begin{array}{cccc} {{\chi _{11}} - P(\iota ,r) + {\varOmega _1}}&{}{{\chi _{12}}}&{}{{\chi _{13}} + {\varOmega _3}}&{}{ - {\varOmega _2}}\\ * &{}{{\chi _{22}}}&{}{{\chi _{23}}}&{}0\\ * &{} * &{}{{\chi _{33}} + {{\tilde{D}}^T}\varOmega \tilde{D} - I}&{}{ - {{\tilde{D}}^T}\wp }\\ * &{} * &{} * &{}{\wp - I} \end{array}} \right]$.

If ${\Theta _{4r}} < 0$, we have $J < 0$, i.e.,

$$\begin{aligned} E({\tilde{\xi } ^T}(k)P(\iota ,r)\tilde{\xi } (k)) < E(\sum \limits _{h = 0}^{k - 1} {{\omega ^T}(h)\omega (h)} ). \end{aligned}$$

Using Schur complement Lemma, (21) can be re-written as

Then, we have

Therefore, the system is stochastically stable with an ${H_2}/{H_\infty }$ noise attenuation level $\gamma$. This completes the proof. $\square$

1.2 Proof of Theorem 2

Proof

From Theorem 1, (20) can be re-written as

Using Lemma 1, the above inequality can be written as

(25)

where $\mathrm{T} = \left[ {\begin{array}{c@{\quad }c@{\quad }c@{\quad }c} I&I&\ldots&I \end{array}} \right]$.

Substituting $,$ and into (25), we can get the inequality (23). This completes the proof. $\square$