期刊菜单

Reduced Order Estimation of Switching Complex Networks Based on Dwell Time

Abstract: In this paper, the l2-l reduced-order state estimation problem is investigated for a class of dis-crete time-delayed nonlinear switched complex networks. By introducing an auxiliary variable, a new model reduction method is proposed where the directly observed state can be represented by the measurement output and the unmeasured state is estimated by designing a reduced-order es-timator. With the help of the average dwell time method and the Lyapunov stability theory, a suffi-cient condition is presented to guarantee both the exponential stability of the resulting estimation error system and a prescribed l2-l performance level of the estimation error against exogenous disturbances. The desired reduced-order estimator gains are acquired in terms of the solution to a set of linear matrix inequalities. Finally, a numerical simulation is given to illustrate the usefulness of the proposed theoretical results and some comparative experiments are made to show the im-pact of the order of the designed estimator on the estimation accuracy.

1. 引言

1) 考虑的复杂网络模型综合了非线性、切换拓扑和时变时滞，更接近工程实际，并且考虑在很多工程应用中对滤波误差的峰值有度量要求，本文采用滤波，可以保证滤波误差的峰值低于一定水平；

2) 首次研究了复杂切换网络的降阶问题，通过引入辅助变量并进行线性变换，建立了构造降阶估计器的理论框架；

2. 问题描述

$\left\{\begin{array}{l}{x}_{i}\left(k+1\right)={A}_{i}{x}_{i}\left(k\right)+f\left({x}_{i}\left(k\right)\right)+{B}_{i}{\omega }_{i}\left(k\right)+g\left({x}_{i}\left(k-\tau \left(k\right)\right)\right)+\underset{j=1}{\overset{N}{\sum }}{l}_{ij}^{{\sigma }_{k}}\Gamma {x}_{j}\left(k\right)\\ {y}_{i}\left(k\right)={C}_{i}{x}_{i}\left(k\right)\\ {z}_{i}\left(k\right)={H}_{i}{x}_{i}\left(k\right)\end{array}$ (1)

${\sigma }_{k}:\left[0,\infty \right]\to \mathbb{M}\triangleq \left\{1,2,\cdots ,m\right\}$ 为切换信号。 ${L}^{{\sigma }_{k}}\triangleq {\left({l}_{ij}^{{\sigma }_{k}}\right)}_{N×N}$ 为表示网络耦合强度和拓扑结构的外耦合矩阵， ${l}_{ij}^{{\sigma }_{k}}={l}_{ji}^{{\sigma }_{k}}>0$ 如果节点i和节点j之间存在连接 $\left(i\ne j\right)$，则 ${l}_{ij}^{{\sigma }_{k}}={l}_{ji}^{{\sigma }_{k}}=0$。同时，矩阵 ${L}^{{\sigma }_{k}}$ 的对角元素满足

${l}_{ii}^{{\sigma }_{k}}=-\underset{j=1,j\ne i}{\overset{N}{\sum }}{l}_{ij}^{{\sigma }_{k}},i\in ℕ$$\Gamma$ 表示内部耦合矩阵。 $\tau \left(k\right)$ 表示满足 $0<{\tau }_{m}\le \tau \left(k\right)\le {\tau }_{M}$ 的时变时滞； ${\tau }_{m}$${\tau }_{M}$

$\begin{array}{l}‖f\left({x}_{1}\right)-f\left({x}_{2}\right)‖\le {\gamma }_{1}‖{x}_{1}-{x}_{2}‖\\ ‖g\left({x}_{1}\right)-g\left({x}_{2}\right)‖\le {\gamma }_{2}‖{x}_{1}-{x}_{2}‖\end{array}$ (2)

${C}_{i}=\left[\begin{array}{cc}{I}_{q}& 0\end{array}\right],i\in ℕ$ (3)

${A}_{i}\triangleq \left[\begin{array}{cc}{A}_{i11}& {A}_{i12}\\ {A}_{i21}& {A}_{i22}\end{array}\right],{B}_{i}\triangleq \left[\begin{array}{c}{B}_{i1}\\ {B}_{i2}\end{array}\right],\Gamma \triangleq \left[\begin{array}{cc}{\Gamma }_{11}& 0\\ 0& {\Gamma }_{22}\end{array}\right],$

${x}_{i}\left(k\right)\triangleq {\left[\begin{array}{cc}{x}_{i1}^{\text{T}}\left(k\right)& {x}_{i2}^{\text{T}}\left(k\right)\end{array}\right]}^{\text{T}},{H}_{i}\triangleq \left[\begin{array}{cc}{H}_{i11}& {H}_{i12}\end{array}\right]$

${x}_{i}\left(k-\tau \left(k\right)\right)\triangleq {\left[\begin{array}{cc}{x}_{i1}^{\text{T}}\left(k-\tau \left(k\right)\right)& {x}_{i2}^{\text{T}}\left(k-\tau \left(k\right)\right)\end{array}\right]}^{\text{T}}$

${\nu }_{i}={\stackrel{^}{C}}_{i}{x}_{i}\left(k\right)$。接下来，将 ${\stackrel{^}{C}}_{i}$ 展开为正交矩阵 ${M}_{i}=\left[\begin{array}{c}{\stackrel{^}{C}}_{i}\\ {N}_{i}\end{array}\right]\left({M}_{i}\in {ℝ}^{n×n}\right)$。然后，通过状态转换 ${\stackrel{˜}{x}}_{i}\left(k\right)={M}_{i}{x}_{i}\left(k\right)$

$\left\{\begin{array}{l}{\stackrel{¯}{x}}^{\left[1\right]}\left(k+1\right)=\left({A}_{11}+{L}^{{\sigma }_{k}}\otimes {\Gamma }_{11}\right){\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)+{A}_{12}{\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)+{\stackrel{¯}{B}}_{1}\omega \left(k\right)+{I}_{1}\stackrel{¯}{f}\left(\left[\begin{array}{c}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)\\ {\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)\end{array}\right]\right)\text{\hspace{0.17em}}+{I}_{1}\stackrel{¯}{g}\left(\left[\begin{array}{c}{\stackrel{¯}{x}}^{\left[1\right]}\left(k-\tau \left(k\right)\right)\\ {\stackrel{¯}{x}}^{\left[2\right]}\left(k-\tau \left(k\right)\right)\end{array}\right]\right)\\ {\stackrel{¯}{x}}^{\left[2\right]}\left(k+1\right)={A}_{21}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)+\left({A}_{22}+{L}^{{\sigma }_{k}}\otimes {\Gamma }_{22}\right){\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)+{\stackrel{¯}{B}}_{2}\omega \left(k\right)+{I}_{2}\stackrel{¯}{f}\left(\left[\begin{array}{c}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)\\ {\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)\end{array}\right]\right)+{I}_{2}\stackrel{¯}{g}\left(\left[\begin{array}{c}{\stackrel{¯}{x}}^{\left[1\right]}\left(k-\tau \left(k\right)\right)\\ {\stackrel{¯}{x}}^{\left[2\right]}\left(k-\tau \left(k\right)\right)\end{array}\right]\right)\\ z\left(k\right)={H}_{11}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)+{H}_{12}{\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)\\ y\left(k\right)={\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)\end{array}$ (4)

$\begin{array}{l}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)\triangleq {\left[{x}_{11}^{\text{T}}\left(k\right)\cdots {x}_{N1}^{\text{T}}\left(k\right)\right]}^{\text{T}},{\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)\triangleq {\left[{x}_{12}^{\text{T}}\left(k\right)\cdots {x}_{N2}^{\text{T}}\left(k\right)\right]}^{\text{T}},\omega \left(k\right)\triangleq {\left[{\omega }_{1}^{\text{T}}\left(k\right)\cdots {\omega }_{N}^{\text{T}}\left(k\right)\right]}^{\text{T}},\\ y\left(k\right)\triangleq {\left[{y}_{1}^{\text{T}}\left(k\right)\cdots {y}_{N}^{\text{T}}\left(k\right)\right]}^{\text{T}},z\left(k\right)\triangleq {\left[{z}_{1}^{\text{T}}\left(k\right)\cdots {z}_{N}^{\text{T}}\left(k\right)\right]}^{\text{T}},{H}_{11}\triangleq \text{diag}\left\{{H}_{111}\cdots {H}_{N11}\right\},\\ {\stackrel{¯}{B}}_{1}\triangleq \text{diag}\left\{{B}_{11}\cdots {B}_{N1}\right\},{\stackrel{¯}{B}}_{2}\triangleq \text{diag}\left\{{B}_{12}\cdots {B}_{N2}\right\},{I}_{1}\triangleq \text{diag}\left\{\left({I}_{q},0\right)\cdots \left({I}_{q},0\right)\right\},\\ {I}_{2}\triangleq \text{diag}\left\{\left(0,{I}_{n-q}\right)\cdots \left(0,{I}_{n-q}\right)\right\},{A}_{11}\triangleq \text{diag}\left\{{A}_{111}\cdots {A}_{N11}\right\},{A}_{12}\triangleq \text{diag}\left\{{A}_{112}\cdots {A}_{N12}\right\},\\ {A}_{21}\triangleq \text{diag}\left\{{A}_{121}\cdots {A}_{N21}\right\},{A}_{22}\triangleq \text{diag}\left\{{A}_{122}\cdots {A}_{N22}\right\},{H}_{12}\triangleq \text{diag}\left\{{H}_{112}\cdots {H}_{N12}\right\}.\end{array}$

$\stackrel{¯}{f}\left(\left[\begin{array}{c}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)\\ {\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)\end{array}\right]\right)\triangleq {\left[{f}^{\text{T}}\left({\Psi }_{1}\left[\begin{array}{c}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)\\ {\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)\end{array}\right]\right),{f}^{\text{T}}\left({\Psi }_{2}\left[\begin{array}{c}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)\\ {\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)\end{array}\right]\right),\cdots ,{f}^{\text{T}}\left({\Psi }_{N}\left[\begin{array}{c}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)\\ {\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)\end{array}\right]\right)\right]}^{\text{T}}$

$\stackrel{¯}{g}\left(\left[\begin{array}{c}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)\\ {\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)\end{array}\right]\right)\triangleq {\left[{g}^{\text{T}}\left({\Psi }_{1}\left[\begin{array}{c}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)\\ {\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)\end{array}\right]\right),{g}^{\text{T}}\left({\Psi }_{2}\left[\begin{array}{c}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)\\ {\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)\end{array}\right]\right),\cdots ,{g}^{\text{T}}\left({\Psi }_{N}\left[\begin{array}{c}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)\\ {\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)\end{array}\right]\right)\right]}^{\text{T}}$

$\begin{array}{l}{\Psi }_{1}\triangleq {\left[\begin{array}{cccccccc}{I}_{q}& {0}_{q×q}& \cdots & {0}_{q×q}& {0}_{q×\left(n-q\right)}& {0}_{q×\left(n-q\right)}& \cdots & {0}_{q×\left(n-q\right)}\\ {0}_{\left(n-q\right)×q}& {0}_{\left(n-q\right)×q}& \cdots & {0}_{\left(n-q\right)×q}& {I}_{n-q}& {0}_{\left(n-q\right)×\left(n-q\right)}& \cdots & {0}_{\left(n-q\right)×\left(n-q\right)}\end{array}\right]}_{n×Nn}\\ {\Psi }_{2}\triangleq {\left[\begin{array}{cccccccc}{0}_{q×q}& {I}_{q}& \cdots & {0}_{q×q}& {0}_{q×\left(n-q\right)}& {0}_{q×\left(n-q\right)}& \cdots & {0}_{q×\left(n-q\right)}\\ {0}_{\left(n-q\right)×q}& {0}_{\left(n-q\right)×q}& \cdots & {0}_{\left(n-q\right)×q}& {0}_{\left(n-q\right)×\left(n-q\right)}& {I}_{n-q}& \cdots & {0}_{\left(n-q\right)×\left(n-q\right)}\end{array}\right]}_{n×Nn}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⋮\\ {\Psi }_{N}\triangleq {\left[\begin{array}{cccccccc}{0}_{q×q}& {0}_{q×q}& \cdots & {I}_{q}& {0}_{q×\left(n-q\right)}& {0}_{q×\left(n-q\right)}& \cdots & {0}_{q×\left(n-q\right)}\\ {0}_{\left(n-q\right)×q}& {0}_{\left(n-q\right)×q}& \cdots & {0}_{\left(n-q\right)×q}& {0}_{\left(n-q\right)×\left(n-q\right)}& {0}_{\left(n-q\right)×\left(n-q\right)}& \cdots & {I}_{n-q}\end{array}\right]}_{n×Nn}.\end{array}$

$r\left(k\right)\triangleq \left[\begin{array}{c}{r}_{1}\left(k\right)\\ {r}_{2}\left(k\right)\end{array}\right],{G}^{{\sigma }_{k}}\triangleq \left[\begin{array}{cc}{I}_{Nq}& 0\\ {K}^{{\sigma }_{k}}& {I}_{N\left(n-q\right)}\end{array}\right]$ (6)

$r\left(k\right)={G}^{{\sigma }_{k}}x\left( k \right)$

$\left\{\begin{array}{l}{r}_{1}\left(k\right)={\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)\\ {r}_{2}\left(k\right)={K}^{{\sigma }_{k}}{\stackrel{¯}{x}}^{\left[1\right]}\left(k\right)+{\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)\end{array}$ (7)

$\left\{\begin{array}{l}{\stackrel{¯}{x}}^{\left[2\right]}\left(k\right)={r}_{2}\left(k\right)-{K}^{{\sigma }_{k}}y\left(k\right)\\ {r}_{2}\left(k+1\right)=\left({K}^{{\sigma }_{k}}{A}_{12}+{A}_{22}+{L}^{{\sigma }_{k}}\otimes {\Gamma }_{22}\right){r}_{2}\left(k\right)+\left({K}^{{\sigma }_{k}}{\stackrel{¯}{B}}_{1}+{\stackrel{¯}{B}}_{2}\right)\omega \left(k\right)+\left({K}^{{\sigma }_{k}}{A}_{11}+{A}_{21}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left({K}^{{\sigma }_{k}}{A}_{12}+{A}_{22}\right){K}^{{\sigma }_{k}}+{K}^{{\sigma }_{k}}{L}^{{\sigma }_{k}}\otimes {\Gamma }_{11}+{L}^{{\sigma }_{k}}\otimes {\Gamma }_{22}{K}^{{\sigma }_{k}}\right)y\left(k\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({I}_{2}+{K}^{{\sigma }_{k}}{I}_{1}\right)\stackrel{¯}{f}\left(\left[\begin{array}{c}y\left(k\right)\\ {r}_{2}\left(k\right)-{K}^{{\sigma }_{k}}y\left(k\right)\end{array}\right]\right)+\left({I}_{2}+{K}^{{\sigma }_{k}}{I}_{1}\right)×\stackrel{¯}{g}\left[\begin{array}{c}y\left(k-\tau \left(k\right)\right)\\ {r}_{2}\left(k-\tau \left(k\right)\right)-{K}^{{\sigma }_{k}}y\left(k-\tau \left(k\right)\right)\end{array}\right]\end{array}$ (8)

$\left\{\begin{array}{l}{\stackrel{^}{x}}_{2}^{\left[2\right]}\left(k\right)={\stackrel{^}{r}}_{2}\left(k\right)-{K}^{{\sigma }_{k}}y\left(k\right)\\ {\stackrel{^}{r}}_{2}\left(k+1\right)=\left({K}^{{\sigma }_{k}}{A}_{12}+{A}_{22}+{L}^{{\sigma }_{k}}\otimes {\Gamma }_{22}\right){\stackrel{^}{r}}_{2}\left(k\right)+\left({K}^{{\sigma }_{k}}{A}_{11}+{A}_{21}-\left({K}^{{\sigma }_{k}}{A}_{12}+{A}_{22}\right){K}^{{\sigma }_{k}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+{K}^{{\sigma }_{k}}{L}^{{\sigma }_{k}}\otimes {\Gamma }_{11}+{L}^{{\sigma }_{k}}\otimes {\Gamma }_{22}{K}^{{\sigma }_{k}}\right)y\left(k\right)+\left({I}_{2}+{K}^{{\sigma }_{k}}{I}_{1}\right)\stackrel{¯}{f}\left(\left[\begin{array}{c}y\left(k\right)\\ {\stackrel{^}{r}}_{2}\left(k\right)-{K}^{{\sigma }_{k}}y\left(k\right)\end{array}\right]\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({I}_{2}+{K}^{{\sigma }_{k}}{I}_{1}\right)×\stackrel{¯}{g}\left(\left[\begin{array}{c}y\left(k-\tau \left(k\right)\right)\\ {\stackrel{^}{r}}_{2}\left(k-\tau \left(k\right)\right)-{K}^{{\sigma }_{k}}y\left(k-\tau \left(k\right)\right)\end{array}\right]\right)\\ \stackrel{^}{z}\left(k\right)={H}_{11}y\left(k\right)+{H}_{12}{\stackrel{^}{x}}_{2}^{\left[2\right]}\left(k\right)\end{array}$ (9)

$\left\{\begin{array}{l}e\left(k+1\right)={\stackrel{¯}{x}}^{\left[2\right]}\left(k+1\right)-{\stackrel{^}{x}}^{\left[2\right]}\left(k+1\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={r}_{2}\left(k+1\right)-{\stackrel{^}{r}}_{2}\left(k+1\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left({K}^{{\sigma }_{k}}{A}_{12}+{A}_{22}+{L}^{{\sigma }_{k}}\otimes {\Gamma }_{22}\right)e\left(k\right)+\left({K}^{{\sigma }_{k}}{\stackrel{¯}{B}}_{1}+{\stackrel{¯}{B}}_{2}\right)\omega \left(k\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({I}_{2}+{K}^{{\sigma }_{k}}{I}_{1}\right)\left(\Delta f\left(k\right)+\Delta g\left(k-{\tau }_{k}\right)\right)\\ \stackrel{˜}{z}\left(k\right)={H}_{12}e\left(k\right)\end{array}$ (10)

$\begin{array}{l}\Delta g\left(k-{\tau }_{k}\right)\triangleq \stackrel{¯}{g}\left(\left[\begin{array}{c}y\left(k-\tau \left(k\right)\right)\\ {r}_{2}\left(k-\tau \left(k\right)\right)-{K}^{{\sigma }_{k}}y\left(k-\tau \left(k\right)\right)\end{array}\right]\right)-\stackrel{¯}{g}\left(\left[\begin{array}{c}y\left(k-\tau \left(k\right)\right)\\ {\stackrel{^}{r}}_{2}\left(k-\tau \left(k\right)\right)-{K}^{{\sigma }_{k}}y\left(k-\tau \left(k\right)\right)\end{array}\right]\right)\\ \Delta f\left(k\right)\triangleq \stackrel{¯}{f}\left(\left[\begin{array}{c}y\left(k\right)\\ {r}_{2}\left(k\right)-{K}^{{\sigma }_{k}}y\left(k\right)\end{array}\right]\right)-\stackrel{¯}{f}\left[\begin{array}{c}y\left(k\right)\\ {\stackrel{^}{r}}_{2}\left(k\right)-{K}^{{\sigma }_{k}}y\left(k\right)\end{array}\right]\end{array}$

$\left\{\begin{array}{l}e\left(k+1\right)=\left({\stackrel{¯}{A}}^{{\sigma }_{k}}+{K}^{{\sigma }_{k}}{A}_{12}\right)e\left(k\right)+{E}^{{\sigma }_{k}}\left(\Delta f\left(k\right)+\Delta g\left(k-{\tau }_{k}\right)\right)+{B}^{{\sigma }_{k}}\omega \left(k\right)\\ \stackrel{˜}{z}\left(k\right)={H}_{12}e\left(k\right)\end{array}$ (11)

${N}_{\sigma }\left({k}_{0},k\right)\le {N}_{0}+\frac{k-{k}_{0}}{{T}_{a}}$ (12)

${‖e\left(k\right)‖}^{2}\le \alpha {\xi }^{k-{k}_{0}}{‖e\left({k}_{0}\right)‖}^{2},\forall k\ge {k}_{0}$ (13)

${‖\stackrel{˜}{z}\left(k\right)‖}_{\infty }<\delta \sqrt{\underset{k=0}{\overset{\infty }{\sum }}{‖\omega \left(k\right)‖}^{2}}$ (14)

1) $\omega \left(k\right)=0$ 的增广误差系统(11)是指数稳定的。

2) 在零初始条件下，对于给定的扰动衰减水平 $\delta >0$ 和所有非零 $\omega \left(k\right)$，误差系统(11)满足 ${l}_{2}$ - ${l}_{\infty }$ 性能。

3. 主要结果

${T}_{a}>{T}_{a}^{\ast }=-\frac{\mathrm{ln}\epsilon }{\mathrm{ln}\mu }$ (15)

${\Pi }^{ϵ}\triangleq \left[\begin{array}{cc}{\Pi }_{3}^{ϵ}& {\Pi }_{2}^{ϵ}\\ \ast & -{P}^{ϵ}\end{array}\right]<0$ (16)

${P}^{\eta }\le \epsilon {P}^{\theta },{Q}^{\eta }\le \epsilon {Q}^{\theta },\forall \eta ,\theta \in \mathbb{M},\eta \ne \theta$ (17)

$\begin{array}{l}{\Pi }_{2}^{ϵ}\triangleq {\left[\begin{array}{cc}{\Pi }_{211}^{ϵ}& {\Pi }_{212}^{ϵ}\end{array}\right]}^{\text{T}},\rho \triangleq 1+{\tau }_{M}-{\tau }_{m}\\ {\Pi }_{211}^{ϵ}\triangleq \left[\begin{array}{cc}{P}^{ϵ}\left({\stackrel{¯}{A}}^{ϵ}+{K}^{ϵ}{A}_{12}\right)& 0\end{array}\right],\\ {\Pi }_{311}^{ϵ}\triangleq -\mu {P}^{ϵ}+\rho {Q}^{ϵ}+{\mu }_{1}{\gamma }_{1}^{2}\Psi ,\\ {\Pi }_{322}^{ϵ}\triangleq -{\mu }^{{\tau }_{M}}{Q}^{ϵ}+{\mu }_{2}{\gamma }_{2}^{2}\Psi ,\\ {\Pi }_{212}^{ϵ}\triangleq \left[\begin{array}{cc}{P}^{ϵ}\left({I}_{2}+{K}^{ϵ}{I}_{1}\right)& {P}^{ϵ}\left({I}_{2}+{K}^{ϵ}{I}_{1}\right)\end{array}\right],\\ {\Pi }_{3}^{ϵ}\triangleq \left[\begin{array}{cccc}{\Pi }_{311}^{ϵ}& 0& 0& 0\\ \ast & {\Pi }_{322}^{ϵ}& 0& 0\\ \ast & \ast & -{\mu }_{1}I& 0\\ \ast & \ast & \ast & -{\mu }_{2}I\end{array}\right].\end{array}$

${V}^{{\sigma }_{k}}\left(k\right)={V}_{1}^{{\sigma }_{k}}\left(k\right)+{V}_{2}^{{\sigma }_{k}}\left(k\right)+{V}_{3}^{{\sigma }_{k}}\left(k\right)$ (18)

${V}_{1}^{{\sigma }_{k}}\left(k\right)\triangleq {e}^{\text{T}}\left(k\right){P}^{{\sigma }_{k}}e\left( k \right)$

${V}_{2}^{{\sigma }_{k}}\left(k\right)\triangleq \underset{i=k-{\tau }_{k}}{\overset{k-1}{\sum }}{\mu }^{k-i-1}{e}^{\text{T}}\left(i\right){Q}^{{\sigma }_{k}}e\left( i \right)$

${V}_{3}^{{\sigma }_{k}}\left(k\right)\triangleq \underset{j=k-{\tau }_{M}+1}{\overset{k-{\tau }_{m}}{\sum }}\underset{i=j}{\overset{k-1}{\sum }}{\mu }^{k-i-1}{e}^{\text{T}}\left(i\right){Q}^{{\sigma }_{k}}e\left(i\right).$

$\begin{array}{c}\Delta {V}_{1}^{{\sigma }_{k}}\left(k\right)={V}_{1}^{{\sigma }_{k}}\left(k+1\right)-\mu {V}_{1}^{{\sigma }_{k}}\left(k\right)\\ ={e}^{\text{T}}\left(k+1\right){P}^{{\sigma }_{k}}e\left(k+1\right)-\mu {e}^{\text{T}}\left(k\right){P}^{{\sigma }_{k}}e\left(k\right)\end{array}$ (19)

$\begin{array}{c}\Delta {V}_{2}^{{\sigma }_{k}}\left(k\right)={V}_{2}^{{\sigma }_{k}}\left(k+1\right)-\mu {V}_{2}^{{\sigma }_{k}}\left(k\right)\\ =\underset{i=k+1-{\tau }_{k+1}}{\overset{k}{\sum }}{\mu }^{k-i}{e}^{\text{T}}\left(i\right){Q}^{{\sigma }_{k}}e\left(i\right)-\underset{i=k-{\tau }_{k}}{\overset{k-1}{\sum }}{\mu }^{k-i}{e}^{\text{T}}\left(i\right){Q}^{{\sigma }_{k}}e\left(i\right)\\ ={e}^{\text{T}}\left(k\right){Q}^{{\sigma }_{k}}e\left(k\right)-{\mu }^{{\tau }_{k}}{e}^{\text{T}}\left(k-{\tau }_{k}\right){Q}^{{\sigma }_{k}}e\left(k-{\tau }_{k}\right)+\underset{i=k-{\tau }_{m}+1}{\overset{k-1}{\sum }}{\mu }^{k-i}{e}^{\text{T}}\left(i\right){Q}^{{\sigma }_{k}}e\left(i\right)\\ \text{}+\underset{i=k-{\tau }_{k+1}+1}{\overset{k-{\tau }_{m}}{\sum }}{\mu }^{k-i}{e}^{\text{T}}\left(i\right){Q}^{{\sigma }_{k}}e\left(i\right)-\underset{i=k-{\tau }_{k}}{\overset{k-1}{\sum }}{\mu }^{k-i}{e}^{\text{T}}\left(i\right){Q}^{{\sigma }_{k}}e\left(i\right)\\ \le {e}^{\text{T}}\left(k\right){Q}^{{\sigma }_{k}}e\left(k\right)-{\mu }^{{\tau }_{M}}{e}^{\text{T}}\left(k-{\tau }_{k}\right){Q}^{{\sigma }_{k}}e\left(k-{\tau }_{k}\right)+\underset{i=k-{\tau }_{M}+1}{\overset{k-{\tau }_{m}}{\sum }}{\mu }^{k-i}{e}^{\text{T}}\left(i\right){Q}^{{\sigma }_{k}}e\left(i\right)\end{array}$ (20)

$\begin{array}{c}\Delta {V}_{3}^{{\sigma }_{k}}\left(k\right)={V}_{3}^{{\sigma }_{k}}\left(k+1\right)-\mu {V}_{3}^{{\sigma }_{k}}\left(k\right)\\ =\underset{j=k-{\tau }_{M}+2}{\overset{k+1-{\tau }_{m}}{\sum }}\underset{i=j}{\overset{k}{\sum }}{\mu }^{k-i}{e}^{\text{T}}\left(i\right){Q}^{{\sigma }_{k}}e\left(i\right)-\underset{j=k-{\tau }_{M}+1}{\overset{k-{\tau }_{m}}{\sum }}\underset{i=j}{\overset{k-1}{\sum }}{\mu }^{k-i}{e}^{\text{T}}\left(i\right){Q}^{{\sigma }_{k}}e\left(i\right)\\ =\underset{j=k-{\tau }_{M}+1}{\overset{k-{\tau }_{m}}{\sum }}\left({e}^{\text{T}}\left(k\right){Q}^{{\sigma }_{k}}e\left(k\right)-{\mu }^{k-j}{e}^{\text{T}}\left(j\right){Q}^{{\sigma }_{k}}e\left(j\right)\right)\\ =\left({\tau }_{M}-{\tau }_{m}\right){e}^{\text{T}}\left(k\right){Q}^{{\sigma }_{k}}e\left(k\right)-\underset{i=k-{\tau }_{M}+1}{\overset{k-{\tau }_{m}}{\sum }}{\mu }^{k-i}{e}^{\text{T}}\left(i\right){Q}^{{\sigma }_{k}}e\left(i\right).\end{array}$ (21)

$\begin{array}{c}\Delta {V}^{{\sigma }_{k}}\left(k\right)={V}^{{\sigma }_{k}}\left(k+1\right)-\mu {V}^{{\sigma }_{k}}\left(k\right)\\ =\underset{i=1}{\overset{3}{\sum }}\left({V}_{i}^{{\sigma }_{k}}\left(k+1\right)-\mu {V}_{i}^{{\sigma }_{k}}\left(k\right)\right)\\ \le {\xi }^{\text{T}}\left(k\right){\stackrel{¯}{\Pi }}_{1}^{{\sigma }_{k}}\xi \left(k\right)\end{array}$ (22)

$\xi \left(k\right)\triangleq {\left[\begin{array}{cccc}{e}^{\text{T}}\left(k\right)& {e}^{\text{T}}\left(k-{\tau }_{k}\right)& \Delta {f}^{\text{T}}\left(k\right)& \Delta {g}^{\text{T}}\left(k-{\tau }_{k}\right)\end{array}\right]}^{\text{T}}$

${\stackrel{¯}{\Pi }}_{1}^{{\sigma }_{k}}\triangleq \left[\begin{array}{cc}{\Pi }_{1}^{{\sigma }_{k}}& {\Pi }_{2}^{{\sigma }_{k}}\\ \ast & -{P}^{{\sigma }_{k}}\end{array}\right]$

${\Pi }_{1}^{{\sigma }_{k}}\triangleq \left[\begin{array}{cccc}-\mu {P}^{{\sigma }_{k}}+\rho {Q}^{{\sigma }_{k}}& 0& 0& 0\\ \ast & -{\mu }^{{\tau }_{M}}{Q}^{{\sigma }_{k}}& 0& 0\\ \ast & \ast & 0& 0\\ \ast & \ast & \ast & 0\end{array}\right]$

${\mu }_{1}\Delta {f}^{\text{T}}\left(k\right)\Delta f\left(k\right)\le {\mu }_{1}{\gamma }_{1}^{2}{e}^{\text{T}}\left(k\right)\Psi e\left(k\right)$ (23)

${\mu }_{2}\Delta {g}^{\text{T}}\left(k-{\tau }_{k}\right)\Delta g\left(k-{\tau }_{k}\right)\le {\mu }_{2}{\gamma }_{2}^{2}{e}^{\text{T}}\left(k-{\tau }_{k}\right)\Psi e\left(k-{\tau }_{k}\right)$ (24)

$\Psi \triangleq \underset{i=1}{\overset{N}{\sum }}{\stackrel{¯}{\Psi }}_{i}^{\text{T}}{\stackrel{¯}{\Psi }}_{i},{\stackrel{¯}{\Psi }}_{i}\triangleq {\Psi }_{i}\left[\begin{array}{c}{0}_{Nq×N\left(n-q\right)}\\ {I}_{N\left(n-q\right)}\end{array}\right],i\in ℕ$

$\Delta {V}^{{\sigma }_{k}}\left(k\right)\le {\xi }^{\text{T}}\left(k\right){\Pi }^{{\sigma }_{k}}\xi \left(k\right)$ (25)

${V}^{ϵ}\left(k+1\right)<\mu {V}^{ϵ}\left(k\right)$ (26)

${V}^{{\sigma }_{k}}\left(k\right)<{\mu }^{k-{k}_{l}}{V}^{{\sigma }_{{k}_{l}}}\left({k}_{l}\right)$ (27)

${V}^{{\sigma }_{k}}\left(k\right)<\epsilon {V}^{{\sigma }_{{k}_{l}-1}}\left({k}_{l}\right)$ (28)

${V}^{{\sigma }_{{k}_{l-1}}}\left({k}_{l-1}\right)<{\mu }^{{k}_{l}-{k}_{l-1}}{V}^{{\sigma }_{{k}_{l-1}}}\left({k}_{l-1}\right)$ (29)

$\begin{array}{c}{V}^{{\sigma }_{k}}\left(k\right)<{\mu }^{k-{k}_{l}}{V}^{{\sigma }_{{k}_{l}}}\left({k}_{l}\right)\\ <\epsilon {\mu }^{k-{k}_{l}}{V}^{{\sigma }_{{k}_{l-1}}}\left({k}_{l}\right)\\ <\epsilon {\mu }^{k-{k}_{l}-1}{V}^{{\sigma }_{{k}_{l-1}}}\left({k}_{l-1}\right)\\ <\cdots \\ <{\mu }^{k-{k}_{0}}{\epsilon }^{\frac{k-{k}_{0}}{{T}_{a}}}{V}^{{\sigma }_{{k}_{0}}}\left({k}_{0}\right)\end{array}$ (30)

${V}^{{\sigma }_{k}}\left(k\right)\ge a{‖e\left(k\right)‖}^{2}$ (31)

${V}^{{\sigma }_{{k}_{0}}}\left({k}_{0}\right)\le b{‖e\left({k}_{0}\right)‖}^{2}$ (32)

$\begin{array}{c}{‖e\left(k\right)‖}^{2}\le \frac{1}{a}{V}^{{\sigma }_{k}}\left(k\right)\\ \le \frac{1}{a}{\mu }^{k-{k}_{0}}{\epsilon }^{\frac{k-{k}_{0}}{{T}_{a}}}{V}^{{\sigma }_{{k}_{0}}}\left({k}_{0}\right)\\ \le \frac{b}{a}{\mu }^{k-{k}_{0}}{\epsilon }^{\frac{k-{k}_{0}}{{T}_{a}}}{‖e\left({k}_{0}\right)‖}^{2}\end{array}$ (33)

${‖e\left(k\right)‖}^{2}\le \frac{b}{a}{\beta }^{k-{k}_{0}}{‖e\left({k}_{0}\right)‖}^{2}$ (34)

${T}_{a}>{T}_{a}^{\ast }=-\frac{\mathrm{ln}\epsilon }{\mathrm{ln}\mu }$ (35)

${\stackrel{¯}{\Pi }}^{ϵ}\triangleq \left[\begin{array}{cc}{\stackrel{¯}{\Pi }}_{2}^{ϵ}& {\Pi }_{4}^{ϵ}\\ \ast & -{P}^{ϵ}\end{array}\right]<0$ (36)

${\Sigma }^{ϵ}\triangleq \left[\begin{array}{cc}-{P}^{ϵ}& {H}_{12}^{\text{T}}\\ \ast & -\mu {\epsilon }^{-{N}_{0}}I\end{array}\right]<0$ (37)

${P}^{\eta }\le \epsilon {P}^{\theta },{Q}^{\eta }\le \epsilon {Q}^{\theta },\forall \eta ,\theta \in \mathbb{M},\eta \ne \theta$ (38)

${\stackrel{¯}{\Pi }}_{2}^{ϵ}\triangleq \left[\begin{array}{cc}{\Pi }_{3}^{ϵ}& 0\\ \ast & -{\delta }^{2}I\end{array}\right]$

${\Pi }_{4}^{ϵ}\triangleq {\left[\begin{array}{cc}{\Pi }_{2}^{ϵ}& {P}^{ϵ}\left({K}^{ϵ}{\stackrel{¯}{B}}_{1}+{\stackrel{¯}{B}}_{2}\right)\end{array}\right]}^{\text{T}}$ (39)

$J\left(k\right)\triangleq \Delta {V}^{{\sigma }_{k}}\left(k\right)-{\delta }^{2}{\omega }^{\text{T}}\left(k\right)\omega \left(k\right)$ (40)

$J\left(k\right)<{\phi }^{\text{T}}\left(k\right){\stackrel{¯}{\Pi }}^{{\sigma }_{k}}\phi \left(k\right)<0$ (41)

$\begin{array}{c}{V}^{{\sigma }_{k}}\left(k\right)\le \mu {V}^{{\sigma }_{k-1}}\left(k-1\right)+{\delta }^{2}{\omega }^{\text{T}}\left(k-1\right)\omega \left(k-1\right)\\ \le {\mu }^{2}{V}^{{\sigma }_{k-2}}\left(k-2\right)+\underset{j=k-2}{\overset{k-1}{\sum }}{\mu }^{k-1-j}{\delta }^{2}{\omega }^{\text{T}}\left(j\right)\omega \left(j\right)\\ \le \cdots \\ \le {\mu }^{k-{k}_{l}}{V}^{{\sigma }_{{k}_{l}}}\left({k}_{l}\right)+\underset{j={k}_{l}}{\overset{k-1}{\sum }}{\mu }^{k-1-j}{\delta }^{2}{\omega }^{\text{T}}\left(j\right)\omega \left(j\right)\end{array}$ (42)

$\begin{array}{c}{V}^{{\sigma }_{k}}\left(k\right)\le {\mu }^{k-{k}_{l}}{V}^{{\sigma }_{{k}_{l}}}\left({k}_{l}\right)+\underset{j={k}_{l}}{\overset{k-1}{\sum }}{\mu }^{k-1-j}{\delta }^{2}{\omega }^{\text{T}}\left(j\right)\omega \left(j\right)\\ \le \epsilon {\mu }^{k-{k}_{l}}{V}^{{\sigma }_{{k}_{l-1}}}\left({k}_{l}\right)+\underset{j={k}_{l}}{\overset{k-1}{\sum }}{\mu }^{k-1-j}{\delta }^{2}{\omega }^{\text{T}}\left(j\right)\omega \left(j\right)\\ \le \epsilon {\mu }^{k-{k}_{l-1}}{V}^{{\sigma }_{{k}_{l-1}}}\left({k}_{l-1}\right)+\underset{j={k}_{l-1}}{\overset{k-1}{\sum }}{\epsilon }^{{N}_{\sigma }\left(j,k\right)}{\mu }^{k-1-j}{\delta }^{2}{\omega }^{\text{T}}\left(j\right)\omega \left(j\right)\\ \le \cdots \\ \le {\epsilon }^{{N}_{\sigma }\left({k}_{0},k\right)}{\mu }^{k-{k}_{0}}{V}^{{\sigma }_{{k}_{0}}}\left({k}_{0}\right)+\underset{j={k}_{0}}{\overset{k-1}{\sum }}{\epsilon }^{{N}_{\sigma }\left(j,k\right)}{\mu }^{k-1-j}{\delta }^{2}{\omega }^{\text{T}}\left(j\right)\omega \left(j\right)\end{array}$ (43)

${V}^{{\sigma }_{k}}\left(k\right)\le \underset{j={k}_{0}}{\overset{k-1}{\sum }}{\epsilon }^{{N}_{\sigma }\left(j,k\right)}{\mu }^{k-1-j}{\delta }^{2}{\omega }^{\text{T}}\left(j\right)\omega \left(j\right)$ (44)

$e{\left(k\right)}^{\text{T}}{P}^{{\sigma }_{k}}e\left(k\right)={V}_{1}^{{\sigma }_{k}}\left(k\right)\le {V}^{{\sigma }_{k}}\left(k\right)\le \underset{j={k}_{0}}{\overset{k-1}{\sum }}{\epsilon }^{{N}_{\sigma }\left(j,k\right)}{\mu }^{k-1-j}{\delta }^{2}{\omega }^{\text{T}}\left(j\right)\omega \left(j\right)$ (45)

${\stackrel{˜}{z}}^{\text{T}}\left(k\right)\stackrel{˜}{z}\left(k\right)={e}^{\text{T}}\left(k\right){H}_{12}^{\text{T}}{H}_{12}{e}_{2}\left(k\right)<\mu {\epsilon }^{-{N}_{0}}{e}^{\text{T}}\left(k\right){P}^{{\sigma }_{k}}e\left(k\right)$ (46)

${\stackrel{˜}{z}}^{\text{T}}\left(k\right)\stackrel{˜}{z}\left(k\right)<\underset{j={k}_{0}}{\overset{k-1}{\sum }}{\epsilon }^{{N}_{\sigma }\left(j,k\right)}{\epsilon }^{-{N}_{0}}{\mu }^{k-j}{\delta }^{2}{\omega }^{\text{T}}\left(j\right)\omega \left(j\right)$ (47)

${N}_{\sigma }\left(j,k\right)\le {N}_{0}+\frac{k-j}{{T}_{a}}\le {N}_{0}+\left(j-k\right)\frac{\mathrm{ln}\mu }{\mathrm{ln}\epsilon }$ (48)

${\epsilon }^{{N}_{\sigma }\left(j,k\right)}\le {\epsilon }^{{N}_{0}}{\mu }^{j-k}$ (49)

${\stackrel{˜}{z}}^{\text{T}}\left(k\right)\stackrel{˜}{z}\left(k\right)<{\delta }^{2}\underset{j={k}_{0}}{\overset{k-1}{\sum }}{\omega }^{\text{T}}\left(j\right)\omega \left(j\right)$ (50)

${\stackrel{˜}{z}}^{\text{T}}\left(k\right)\stackrel{˜}{z}\left(k\right)$ 的最大值 $\underset{j={k}_{0}}{\overset{k-1}{\sum }}{\omega }^{\text{T}}\left(j\right)\omega \left(j\right)$，可以得到

$\underset{k}{\mathrm{sup}}{\stackrel{˜}{z}}^{\text{T}}\left(k\right)\stackrel{˜}{z}\left(k\right)<{\delta }^{2}\underset{j=0}{\overset{\infty }{\sum }}{\omega }^{\text{T}}\left(j\right)\omega \left(j\right)$ (51)

${‖\stackrel{˜}{z}\left(k\right)‖}_{\infty }<\delta \sqrt{\underset{k=0}{\overset{\infty }{\sum }}{‖\omega \left(k\right)‖}^{2}}$ (52)

${T}_{a}>{T}_{a}^{\ast }=-\frac{\mathrm{ln}\epsilon }{\mathrm{ln}\mu }$ (53)

${\stackrel{¯}{\Pi }}^{ϵ}\triangleq \left[\begin{array}{cc}{\stackrel{¯}{\Pi }}_{2}^{ϵ}& {\stackrel{¯}{\Pi }}_{4}^{ϵ}\\ \ast & -{P}^{ϵ}\end{array}\right]<0$ (54)

${\Sigma }^{ϵ}\triangleq \left[\begin{array}{cc}-{P}^{ϵ}& {H}_{12}^{\text{T}}\\ \ast & -\mu {\epsilon }^{-{N}_{0}}I\end{array}\right]<0$ (55)

${P}^{\eta }\le \epsilon {P}^{\theta },{Q}^{\eta }\le \epsilon {Q}^{\theta },\forall \eta ,\theta \in \mathbb{M},\eta \ne \theta$ (56)

${\stackrel{¯}{\Pi }}_{4}^{ϵ}\triangleq {\left[\begin{array}{cc}{\stackrel{¯}{\Pi }}_{411}^{ϵ}& {\stackrel{¯}{\Pi }}_{412}^{ϵ}\end{array}\right]}^{\text{T}}$

$\begin{array}{l}{\stackrel{¯}{\Pi }}_{412}^{ϵ}\triangleq \left[\begin{array}{cc}{P}^{ϵ}{I}_{2}+{R}^{ϵ}{I}_{1}& {R}^{ϵ}{\stackrel{¯}{B}}_{1}+{P}^{ϵ}{\stackrel{¯}{B}}_{2}\end{array}\right]\\ {\stackrel{¯}{\Pi }}_{411}^{ϵ}\triangleq \left[\begin{array}{ccc}{P}^{ϵ}{\stackrel{¯}{A}}^{ϵ}+{R}^{ϵ}{A}_{12}& 0& {P}^{ϵ}{I}_{2}+{R}^{ϵ}{I}_{1}\end{array}\right]\end{array}$

4. 仿真实验

${A}_{1}=\left[\begin{array}{cccc}1& 0.5& 0.16& -0.21\\ -0.6& 0.6& 0.19& 0.14\\ 0.2& -0.1& 0.02& -0.11\\ 0.25& 0.3& 0.15& 0.02\end{array}\right],{A}_{2}=\left[\begin{array}{cccc}0.6& 0.1& 0.51& 0.1\\ -0.2& 0.01& 0& 0.18\\ 0.12& -0.2& 0.7& 0.01\\ 0.13& 0.1& 0.2& 0.4\end{array}\right]$

${A}_{3}=\left[\begin{array}{cccc}0.4& 0& -0.17& 0\\ -0.6& 0.25& 0& 0.19\\ 1.3& 0& 0.08& 0.5\\ 0.5& -0.16& 0.5& 0.19\end{array}\right],{B}_{1}={B}_{2}={B}_{3}={\left[\begin{array}{cccc}0& 0.1& 0& 0.1\end{array}\right]}^{\text{T}}$

${L}_{1}=\left[\begin{array}{ccc}-0.65& 0.1& 0.55\\ 0& -0.5& 0.05\\ 0.5& 0& -0.5\end{array}\right],{L}_{2}=\left[\begin{array}{ccc}-0.6& 0.1& 0.5\\ 0.15& -0.15& 0\\ 0.5& 0& -0.5\end{array}\right].$

$\begin{array}{l}f\left({x}_{i}\left(k\right)\right)={\left[\begin{array}{cccc}0& 0& f{\left({x}_{i}\left(k\right)\right)}_{11}& f{\left({x}_{i}\left(k\right)\right)}_{12}\end{array}\right]}^{\text{T}}\\ g\left({x}_{i}\left(k\right)\right)={\left[\begin{array}{cccc}0& 0& g{\left({x}_{i}\left(k\right)\right)}_{11}& g{\left({x}_{i}\left(k\right)\right)}_{12}\end{array}\right]}^{\text{T}}\end{array}$

$\begin{array}{l}f{\left({x}_{i}\left(k\right)\right)}_{11}=-0.01\mathrm{tanh}\left(0.2{x}_{i}^{1}\left(k\right)\right)+0.05\mathrm{tanh}\left({x}_{i}^{2}\left(k\right)\right)\\ f{\left({x}_{i}\left(k\right)\right)}_{12}=-0.01\mathrm{tanh}\left({x}_{i}^{1}\left(k\right)\right)+0.001\mathrm{tanh}\left(5{x}_{i}^{1}\left(k\right)\right)\\ g{\left({x}_{i}\left(k\right)\right)}_{11}=-0.01\mathrm{tanh}\left({x}_{i}^{1}\left(k\right)\right)+0.01\mathrm{tanh}\left(0.4{x}_{i}^{2}\left(k\right)\right)\\ g{\left({x}_{i}\left(k\right)\right)}_{12}=0.002\mathrm{tanh}\left(2{x}_{i}^{1}\left(k\right)\right)+0.001\mathrm{tanh}\left({x}_{i}^{2}\left( k \right)\right)\end{array}$

case 1：将输出矩阵设为 ${C}_{1}={C}_{2}={C}_{3}=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]$

${K}^{1}=\left[\begin{array}{ccc}-0.0182& 0.0082& 0.1022\\ -0.7372& -0.0105& 2.3262\\ -0.2056& -0.0010& -0.4152\\ -0.0231& 0.1628& 0.0098\\ -0.0045& 0.1122& -0.0281\\ 0.0171& -0.4396& 0.0486\\ -0.1242& -0.0326& -0.0093\\ -0.0118& -0.0551& -1.8976\\ 0.0852& -0.0186& 3.2210\end{array}\right]$

${K}^{2}=\left[\begin{array}{ccc}0.0449& 0.0449& 0.1748\\ -0.5722& -0.0495& 2.8434\\ -0.5116& 0.0031& -1.2074\\ -0.0471& 0.4881& 0.0089\\ -0.0144& 0.3069& -0.0701\\ 0.0492& -0.4963& 0.1133\\ -0.2119& -0.0836& -0.1427\\ -0.1347& -0.0273& -2.5961\\ 0.2748& -0.0293& 3.7211\end{array}\right]$

case 2：将输出矩阵设为 ${C}_{1}={C}_{2}={C}_{3}=\left[\begin{array}{cccc}1& 1& 0& 0\end{array}\right]$

${K}^{11}=\left[\begin{array}{ccc}0.3808& -0.0001& -0.2154\\ -0.0001& 0.3678& 0.0009\\ -0.2154& 0.0009& 1.7406\\ 0.0003& -0.2271& -0.0178\\ 0.0004& 0.0001& -0.0004\\ -0.0008& 0.0001& 0.0005\end{array}\right]$

${K}^{12}=\left[\begin{array}{ccc}0.0003& 0.0004& -0.0008\\ -0.2271& 0.0001& 0.0001\\ -0.0178& -0.0004& 0.0005\\ 2.6746& -0.0000& -0.0000\\ -0.0000& 0.1278& 0.0001\\ -0.0000& 0.0001& 0.1381\end{array}\right]$

${K}^{21}=\left[\begin{array}{ccc}0.2253& -0.0005& -0.1154\\ -0.0005& 0.3916& 0.0034\\ -0.1154& 0.0034& 1.1132\\ 0.0031& -0.2786& -0.0857\\ 0.0079& -0.0035& -0.0064\\ 0.0010& -0.0022& 0.0021\end{array}\right]$

${K}^{22}=\left[\begin{array}{ccc}0.0031& 0.0079& 0.0010\\ -0.2786& -0.0035& -0.0022\\ -0.0857& -0.0064& 0.0021\\ 2.0974& 0.0019& -0.0010\\ 0.0019& 0.0498& 0.0060\\ -0.0010& 0.0060& 0.0583\end{array}\right]$

Figure 1. Actual state and its estimate for ${x}_{i}^{2}\left(k\right)\left(i=1,2,3\right)$

Figure 2. Actual state and its estimate for ${x}_{i}^{3}\left(k\right)\left(i=1,2,3\right)$ in three-order system

Figure 3. Actual state and its estimate for ${x}_{i}^{4}\left(k\right)\left(i=1,2,3\right)$ in three-order system

Figure 4. Actual state and its estimate for ${x}_{i}^{3}\left(k\right)\left(i=1,2,3\right)$ in two-order system

Figure 5. Actual state and its estimate for ${x}_{i}^{4}\left(k\right)\left(i=1,2,3\right)$ in two-order system

Figure 6. Switching signal

5. 总结

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