﻿ 非线性布尔网络系统模糊建模与动态性能分析
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 智能系统学报  2018, Vol. 13 Issue (5): 707-715  DOI: 10.11992/tis.201704023 0

引用本文

LYU Hongli, SONG Yujing, DUAN Peiyong. Fuzzy modeling and dynamic analysis of nonlinear Boolean networks systems[J]. CAAI Transactions on Intelligent Systems, 2018, 13(5): 707-715. DOI: 10.11992/tis.201704023.

文章历史

1. 山东建筑大学 信息与电气工程学院，山东 济南 250101;
2. 山东师范大学 信息科学与工程学院，山东 济南 250014

Fuzzy modeling and dynamic analysis of nonlinear Boolean networks systems
LYU Hongli1, SONG Yujing1, DUAN Peiyong2
1. School of Information and Electrical Engineering, Shandong Jianzhu University, Ji’nan 250101, China;
2. School of Information Science and Engineering, Shandong Normal University, Ji’nan 250014, China
Abstract: Considering the difficulty in accurately modeling nonlinear systems and analyzing their dynamic properties, the Boolean network system theories are extended to the nonlinear Boolean network system based on a fuzzy dynamic model, establishing a model of fuzzy dynamic Boolean network control systems. The fuzzy dynamic model is introduced to build a fuzzy model of nonlinear Boolean network, establishing the local and global models of the nonlinear Boolean network systems. The dynamic properties, which include controllability, observability, and stability of the system, are analyzed from the local and global meanings of the system. Finally, a multi-input multi-output nonlinear Boolean network system is taken as a numerical example, and the local and global models of the system are established. The dynamic properties are simulated and analyzed, and the experimental results are obtained. The results show that the fuzzy dynamic Boolean network control system is effective in modeling nonlinear Boolean network systems and reasonably analyzes the dynamic properties, which is of great significance for further analysis of the fuzzy dynamic Boolean network control systems.
Key words: fuzzy dynamical model    Boolean network    semi-tensor product    local model    global model    controllability    observability    stability

1969年，Kauffman[11]首先提出布尔网络模型，布尔网络是关于布尔状态变量的一种简单的逻辑动力系统，是当前学者专家们共同关心的热点问题。针对布尔网络研究缺少有效的数学工具问题，程代展教授在文献[12]中首次提出矩阵半张量积方法。这种方法将逻辑运算转换成代数运算，使得许多经典的处理量变过程的数学工具可直接用来分析逻辑动态系统。在文献[13]中，程代展教授将这种方法应用于布尔网络，将逻辑动态控制系统转化为普通离散时间系统，提出了一系列关于布尔网络的新理论。随后在文献[14-17]中研究了布尔网络系统的能控能观性等性质，形成了布尔控制网络分析设计的完整理论框架。之后，学者们在控制理论方面对线性布尔网络系统做了大量的深入研究[18-23]，但是没有针对非线性布尔网络系统进行分析和研究。

1 预备知识 1.1 数学符号说明

1.2 模糊动态模型

 $\begin{array}{c}{R_k}:\;{\rm{if}}\;{x_1}\;{\rm{is}}\;{A_{1k}}, {x_2}\;{\rm{is}}\;{A_{2k}} ,\cdots ,{x_m}\;{\rm{is}}\;{A_{mk}}\\{\rm{Then}}\left\{ \begin{array}{c}{y_{1k}} = p_{0k}^1 + p_{1k}^1{x_1} + \cdots + p_{ik}^1{x_i} + \cdots p_{mk}^1{x_m}\\ \vdots \\{y_{jk}} = p_{0k}^j + p_{1k}^j{x_1} + \cdots + p_{ik}^j{x_i} + \cdots p_{mk}^j{x_m}\\ \vdots \\{y_{nk}} = p_{0k}^n + p_{1k}^n{x_1} + \cdots + p_{ik}^n{x_i} + \cdots p_{mk}^n{x_m}\end{array} \right.\end{array}$ (1)

 $y_{jk}^* = p_{1k}^jx_1^* + p_{2k}^jx_2^* + \cdots + p_{mk}^jx_m^*$ (2)

 $\left\{ \begin{gathered} {\mu _1} = {\mu _{{A_{11}}}}(x_{_1}^*) \wedge {\mu _{{A_{21}}}}(x_{_2}^*) \wedge \cdots \wedge {\mu _{{A_{m1}}}}(x_{_m}^*) \\ \vdots \\ {\mu _k} = {\mu _{{A_{1k}}}}(x_{_1}^*) \wedge {\mu _{{A_{2k}}}}(x_{_2}^*) \wedge \cdots \wedge {\mu _{{A_{mk}}}}(x_{_m}^*) \\ \vdots \\ {\mu _N} = {\mu _{{A_{1N}}}}(x_{_1}^*) \wedge {\mu _{{A_{2N}}}}(x_{_2}^*) \wedge \cdots \wedge {\mu _{{A_{mN}}}}(x_{_m}^*) \\ \end{gathered} \right.$ (3)

 ${Y_j} = \frac{{\sum\limits_{k = 1}^N {{\mu _k}} \cdot y_{_{jk}}^*}}{{\sum\limits_{k = 1}^N {{\mu _k}} }}$ (4)

1.3 逻辑的矩阵表示

1)如果 $n = p$ ,则称AB满足等维数关系；

2)果 $n = tp$ (记为 ${{A}}{ \succ _t}{{B}}$ )，或者 $nt = p$ (记 ${{A}}{ \prec _t}{{B}}$ )，则称 ${{A}}$ ${{B}}$ 满足倍维数关系，否则称一般维数关系。矩阵乘积在倍维数关系下的一种推广如定义1。

 ${{X}} \triangleright {{Y}} = \sum\limits_{i = 1}^p {{{{X}}^i}{{{y}}_i}} \in {{\bf R}^q}$ (5)

 ${{{Y}}^{\rm T}} \triangleright {{{X}}^{\rm T}} = \sum\limits_{i = 1}^p {{{{y}}_i}({{\bf X}^i}} {)^{\rm T}} \in {{\bf R}^q}$ (6)

 ${{f}}({{{x}}_1},{{{x}}_2}, \cdots ,{{{x}}_n}) = {{{M}}_f} \triangleright {{x}}$ (7)
 $式中${{x}} = \triangleright _{i = 1}^n{{{x}}_i}$。常用的逻辑算子及其结构矩阵分别为$\begin{array}{c}{{{M}}_ \wedge } = {\delta _2}[1\;\;2\;\;2\;\;2],\quad{{{M}}_ \vee } = {\delta _2}[1\;\;1\;\;1\;\;2],\\{{{M}}_ \to } = {\delta _2}[1\;\;2\;\;1\;\;1],\quad{{{M}}_{\overline \vee }} = {\delta _2}[2\;\;1\;\;1\;\;2].\end{array}
1.4 布尔(控制)网络

 $\left\{ \begin{gathered} {x_1}(t + 1) = {f_1}({x_1}(t),{x_2}(t), \cdots ,{x_n}(t)) \\ {x_2}(t + 1) = {f_2}({x_1}(t),{x_2}(t), \cdots ,{x_n}(t)) \\ \vdots \\ {x_n}(t + 1) = {f_n}({x_1}(t),{x_2}(t), \cdots ,{x_n}(t)) \\ \end{gathered} \right.{\kern 1pt}$ (8)

2)布尔控制网络是指一个含有输入输出的布尔网络，其动态方程为

 $\left\{ \begin{gathered} {x_1}(t + 1) = {f_1}({x_1}(t),{x_2}(t), \cdots ,{x_n}(t),{u_1}(t),\; \cdots ,{u_m}(t)) \\ {x_2}(t + 1) = {f_2}({x_1}(t),{x_2}(t), \cdots ,{x_n}(t),{u_1}(t),\; \cdots ,{u_m}(t)) \\ \vdots \\ {x_n}(t + 1) = {f_n}({x_1}(t),{x_2}(t), \cdots ,{x_n}(t),{u_1}(t),\; \cdots ,{u_m}(t)) \\\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{y_j}(t) = {h_j}({x_1}(t),{x_2}(t), \cdots ,{x_n}(t) \\ \end{gathered} \right.$ (9)

1)布尔网络的动态方程式(8)可表示为

 ${{X}}(t + 1) = {{LX}}(t),\;\;\;\;{{L}} \in {{\cal{L}}_{{2^n} \times {2^n}}}$ (10)

2)布尔控制网络的动态方程式(9)可表示为

 \left\{ \begin{aligned}& {{X}}(t + 1) = {{LU}}(t){{X}}(t) \\&{{Y}}(t) = {{HX}}(t) \\ \end{aligned} \right. (11)

2 非线性布尔网络系统的基本概念

 \begin{aligned}& {R^k}:\;{\rm{if}}\;{z_1}\;{\rm{is}}\;F_1^k\;{\rm{and}}\; \cdots \;{z_i}\;{\rm{is}}\;F_i^k\; \cdots \;{z_n}\;{\rm{is}}\;F_n^k\\& {\rm{Then}}\;\left\{ \begin{aligned}& {{X}}(t + 1) = {{{L}}_k}{{U}}(t){{X}}(t)\\& {{Y}}(t) = {{{H}}_k}{{X}}(t)\end{aligned} \right.,\quad k = 1,\;2, \cdots ,{\kern 1pt} \;n\end{aligned} (12)

 \begin{aligned}& {R^k}:\;{\rm{if}}\;{z_1}\;{\rm{is}}\;F_1^k\;{\rm{and}}\; \cdots \;{z_i}\;{\rm{is}}\;F_i^k\; \cdots \;{z_n}\;{\rm{is}}\;F_n^k\\& {\rm{Then}}\;\;{\rm{L}}{{\rm{M}}_k} = [\;{\mu _k}(z),{\kern 1pt} ({{{L}}_k},\;{{{H}}_k})],\quad k = 1,\;2, \cdots \;,\;{\kern 1pt} n\end{aligned} (13)

2)第 $k$ 个FDBNM为

 ${\rm{FDBNML}}{{\rm{M}}_k} = [{\;\mu _k}(z),({{{L}}_k},{\kern 1pt} {\kern 1pt} {{{H}}_k})]$ (14)

 $\left\{ \begin{gathered} {\mu _1}\left( z \right) = {\mu _{F_1^1}}(z_{_1}^*) \wedge {\mu _{F_2^1}}(z_{_2}^*) \wedge \cdots \wedge {\mu _{F_n^1}}(z_{_m}^*) \\ \vdots \\ {\mu _k}\left( z \right) = {\mu _{F_1^k}}(z_{_1}^*) \wedge {\mu _{F_2^k}}(z_{_2}^*) \wedge \cdots \wedge {\mu _{F_n^k}}(z_{_m}^*) \\ \vdots \\ {\mu _N}\left( z \right) = {\mu _{F_1^N}}(z_{_1}^*) \wedge {\mu _{F_2^N}}(z_{_2}^*) \wedge \cdots \wedge {\mu _{F_n^N}}(z_{_m}^*) \\ \end{gathered} \right.$

3)使用加权平均法解模糊，可得FDBNM的全局模型为

 \left\{ \begin{aligned} & {{X}}(t + 1) = {{LU}}(t){{X}}(t) \\ &{{Y}}(t) = {{HX}}(t) \end{aligned} \right. (15)

 $GM = [\;\mu (z),({{L}},{\kern 1pt} {\kern 1pt} {{H}})]$

2)对于模糊动态布尔网络控制系统的全局模型式(15)，如果存在控制变量 ${U_0}$ ，能使方程式(15)从初始状态 $X(U,0) = {X_0}$ 到达终端状态 $X(U,t) =$ ${X_d}$ ,则称 ${X_d}$ ${X_0}$ 经过 $t$ 步是能控的。如果式(15)能从任意初始状态 ${X_0}$ 到达 $X(T) = {X_d}$ ，则称模糊动态布尔网络控制系统的全局模型是能控的。

2)对于模糊动态布尔网络控制系统的全局模型式(15)，对任意给定的初始状态如果至少存在一个布尔控制序列，使初始状态能由输出序列唯一地确定，则称全部模型是状态能观测的。

2)对于模糊动态布尔网络控制系统的全局模型式(15)，如果经过固定步数 ${T_0}$ ，存在一个不动点 ${X_e}$ ，使得对于全局模型的任意的初始状态 $X(0) =$ $\left( {x_1^0, x_2^0,\cdots ,x_n^0} \right)$ ，都有 $X(t) = X(e),t \geqslant {T_0}$ , 则称系统的全局模型是能稳定的。

3 非线性布尔网络系统动态性能分析

 \begin{aligned}{{{L}}_k} = & [{\rm{Bl}}{{\rm{k}}_1}({{{L}}_k}){\rm{Bl}}{{\rm{k}}_2}({{{L}}_k}) \cdots {\rm{Bl}}{{\rm{k}}_{{2^m}}}({{{L}}_k})] =\\& [{B_{k1}},{B_{k2}}, \cdots ,{B_{k{2^m}}}]\end{aligned} (16)

 ${{{M}}_k} = \sum\limits_{i = 1}^{{2^m}} {{\rm{Bl}}{{\rm{k}}_i}} ({{{L}}_k})$ (17)
3.1 能控性

1)设 $\alpha ,\beta ,{\alpha _i} \in {\cal{D}},i = 1,2, \cdots ,n$ ，则布尔加法定义为

 \left\{ \begin{aligned} & \alpha { + _{\cal{B}}}\beta = \alpha \vee \beta \\& \sum\limits_{i = {1_{\cal{B}}}}^n {{\alpha _i} = {\alpha _1} \vee {\alpha _2}\vee \cdots \vee {\alpha _n}} \end{aligned} \right. (18)

2)设 ${{A}} = ({a_{ij}}) \in {{\cal{B}}_{m \times n}},{{B}} = ({b_{ij}}) \in {{\cal{B}}_{n \times p}}$ ，则布尔乘法定义为

 ${{A}}{ \triangleright _{\cal{B}}}{{B}} = {{C}} \in {{\cal{B}}_{m \times p}}$ (19)

3)设 ${{A}} \triangleright {{A}}$ 有定义，则布尔幂定义为

 ${{{A}}^{(k)}} = \underbrace {{{A}}{ \triangleright _{\cal{B}}}{{A}}{ \triangleright _{\cal{B}}} \cdots { \triangleright _{\cal{B}}}{{A}}}_k$ (20)

 ${{{C}}_k} = \sum\limits_{j = 1}^{{2^{(m + n)}}} {_{\cal{B}}\sum\limits_{i = 1}^{{2^m}} {_{\cal{B}}} } {\rm{Bl}}{{\rm{i}}_i}({{L}}_{_k}^{(j)}) = \sum\limits_{j = 1}^{{2^{(m + n)}}} {_{\cal{B}}} M_{_k}^{(j)} \in {{\cal{B}}_{{2^m} \times {2^n}}}$ (21)

2)当且仅当矩阵 ${{C}} > 0$ 时，全局模型式(15)是能控的，即 ${{C}} = \displaystyle\sum\limits_{k = 1}^N {{\mu _k}(z){{{C}}_k}}$

$j = 1$ 时，由式(17)知，当 ${M_k} > 0$ 时，存在一个控制序列使状态 ${X_0}$ ${X_d}$ ,显然局部模型式(14)是能控的；假设当 $j = k'$ 时， ${{{C}}_k} = \displaystyle\sum\limits_{k' = 1}^{{2^{m + n}}} {M_k^{(k')}} > 0$ ，式(14)能控，则当 $j = k' + 1$ 时, ${{{C}}_k} = \displaystyle\sum\limits_{k' = 0}^{{2^{m + n}}} {M_k^{(k' + !)}} = \displaystyle\sum\limits_{k' = 1}^{{2^{m + n}}} {M_k^{(k')}}$ $+ {M_k}$ ,因为 $\displaystyle\sum\limits_{k' = 1}^{{2^{m + n}}} {M_k^{(k')}} > 0,$ ${M_k} > 0,$ 可知 ${{{C}}_k}>0$ ，且存在控制序列使局部模型式(14)能控。

3.2 能观性

 \left\{ \begin{aligned} & {{{\varOmega}} _{k0}} = \{ {{{H}}_k}\} \\ & {{{\varOmega}} _{k1}} = \{ {{{H}}_k}{{{B}}_i}\left| {i = 1,2, \cdots ,{2^m}} \right.\} \\ & \quad\quad\quad \vdots \\ & {{{\varOmega}} _{ks}} = \{ {{{H}}_k}{{{B}}_{{i_1}}}{{{B}}_{{i_2}}} \cdots {{{B}}_{{i_s}}}\left| {{i_1},{i_2}, \cdots ,{i_s} = 1,2, \cdots ,{2^m}} \right.\} \end{aligned} \right. (22)

 \left\{ {\begin{aligned}& {{{{\varGamma}} _{k0}} = {{{H}}_k}}\\& {{{{\varGamma}} _{k1}} = {{\left[ {{{{H}}_k}{{{B}}_1}\;\;{{{H}}_k}{{{B}}_2} \cdots {{{H}}_k}{{{B}}_{{2^m}}}} \right]}^{\rm{T}}}}\\& {{{{\varGamma}} _{k2}} = {{\left[ {{{{H}}_k}{{{B}}_1}\;\;{{{B}}_1}{{{H}}_k}{{{B}}_1}{{{B}}_2} \cdots {{{H}}_k}{{{B}}_{{2^m}}}{{{B}}_{{2^m}}}} \right]}^{\rm{T}}}}\end{aligned}} \right. (23)

 ${{{O}}_k} = \left[ {{{{\varGamma}} _{k0}}\;} \right.{{{\varGamma}} _{k1}}\; \cdots \;{\left. {{{{\varGamma}} _{ks*}}} \right]^{\rm T}}$ (24)

 ${\rm{Rank}}({{{O}}_k}) = {2^n}$ (25)

2)设系统全局模型式(14)是能控的，那么全局模型式(17)是能观的，当且仅当

 ${\rm{Rank}}({{O}}) = {2^n}$ (26)

3.3 稳定性

 $\left\{ \begin{gathered} {z_1} = {f_1}({x_1},{x_2}, \cdots ,{x_n}) \\ \;\;\;\;\;\;{\kern 1pt} {\kern 1pt} \vdots \\ {z_n} = {f_n}({x_1}, {x_2},\cdots ,{x_n}) \\ \end{gathered} \right.$ (27)

 {b_{ij}} = \left\{ {\begin{aligned}& {1,\quad {x_j}\left( {t + 1} \right)\text{依赖于}{x_i}\left( t \right)}\\& {0,\quad \text{其他}}\end{aligned}} \right. (28)

${{X}} = {[{x_1},{x_2}, \cdots ,{x_n}]^{\rm T}},{{F}}= {[{f_1},{f_2} \cdots ,{f_n}]^{\rm T}}$ ，则式(14)对应的布尔网络的逻辑映射可简记为

 \left\{\begin{aligned} &{{X}}(t + 1) = F({{X}}(t),{{U}}(t)) \\ &{{X}}(t) \in {{\cal{D}}^n},{{U}}(t) \in {{\cal{D}}^m}\end{aligned}\right. (29)

 ${{X}}(t + 1) = {F_k}({{X}}(t),{{U}}(t)){\kern 1pt}$ (30)

2)设 $\xi$ 是式(15)的一个不动点，则 $X(k)\bar \vee \xi \leqslant$ ${\cal I}{(F)^j} \times \left( {X(0)\bar \vee \xi } \right)$ ,如果存在 $j > 0$ ,使得 $[{\cal{I}}(F)]$ (j)=0，则称全部模型是能稳定的。其中 $F = \displaystyle\sum\limits_{k = 1}^N {{\mu _k}(z){F_k}}$

②充分性：假设存在 $j > 0$ ,使 ${[{\cal{I}}({F_k})]^{(j)}} = 0$ 成立，那么对于任意的 $X$ $F_k^j(X) = \xi$ $X(t) \in {{\cal{D}}^n}$ 。故对任意步数 $t \geqslant j, F_k^t(X) = F_k^j(F_k^{t - j}(X)) = \xi$ ，得证。

4 实验仿真

 \begin{aligned} & {R^1}:{\rm{if}}\;{z_1}\;{\rm{is}}\;{F_1}, {z_2}\;{\rm{is}}\;{F_2}\\ & {\rm{Then}}\left\{ \begin{aligned} & {x_1}(t + 1) = {x_2}(t) \wedge {u_1}(t) \\ & {x_2}(t + 1) = {x_3}(t) \vee {u_2}(t) \\ & {x_3}(t + 1) = {x_1}(t) \\ & {y_1}(t) = {x_1}(t) \\ & {y_2}(t) = \neg {x_2}(t) \\ \end{aligned} \right. \\ & {R^2}:{\rm{if}}\;{z_1}\;{\rm{is}}\;{F_2}, {z_2}\;{\rm{is}}\;{F_1} \\ &{\rm{Then}}\left\{ \begin{aligned} & {x_1}(t + 1) = {x_3}(t) \wedge {u_1}(t) \\ & {x_2}(t + 1) = \neg {u_2}(t) \\ & {x_3}(t + 1) = {x_1}(t) \vee {x_2}(t) \\ & {y_1}(t) = {x_1}(t) \\ & {y_2}(t) = {x_2}(t) \vee {x_3}(t) \\ \end{aligned} \right. \\ \end{aligned} (31)

 \begin{aligned} & {R^1}:{\rm{if}}\;{z_1}\;{\rm{is}}\;{F_1}, {z_2}\;{\rm{is}}\;{F_2}\\ & {\rm{Then}}\left\{ \begin{aligned} & x(t + 1) = {L_1}u(t)x(t) \\ & y(t) = {H_1}x(t) \\ \end{aligned} \right. \\ & {R^2}:{\rm{if}}\;{z_1}\;{\rm{is}}\;{F_2},{z_2}\;{\rm{is}}\;{F_1} \\& {\rm{Then}}\left\{ \begin{aligned} & x(t + 1) = {L_2}u(t)x(t) \\ & y(t) = {H_2}x(t) \\ \end{aligned} \right. \end{aligned} (32)

 \begin{aligned}&{{{L}}_1} = {\delta _8}[1\,1\,5\,5\,2\,2\,6\,6\,1\,3\,5\,7\,2\,4\,6\,8\,5\,5\,5\,5\,6\,6\,6\,6\,5\,7\,5\,7\,6\,8\,6\,8]\end{aligned}
 \begin{aligned}& {{{L}}_2} = {\delta _8}[3\,1\,7\,5\,3\,1\,7\,5\,7\,5\,7\,5\,7\,5\,7\,5\,3\,1\,7\,5\,3\,1\,7\,5\,8\,6\,8\,6\,8\,6\,8\,6]\end{aligned}
 ${{{H}}_1} = {\delta _4}[2{\kern 1pt}\;\; 1{\kern 1pt}\;\; 2{\kern 1pt}\;\; 1{\kern 1pt}\;\; 4{\kern 1pt}\;\; 3{\kern 1pt}\;\; 4{\kern 1pt}\;\; 3],\quad{{{H}}_2} = {\delta _4}[1{\kern 1pt}\;\; 1{\kern 1pt}\;\; 1{\kern 1pt}\;\; 2{\kern 1pt}\;\; 3{\kern 1pt}\;\; 3{\kern 1pt}\;\; 3{\kern 1pt}\;\; 4]$
 ${{L}} = {\mu _1}(z){{{L}}_1} + {\mu _2}(z){{{L}}_2},\quad{{H}} = {\mu _1}(z){{{H}}_1} + {\mu _2}(z){{{H}}_2},$
 ${\mu _1} = {\mu _{{F_{11}}}}({z_1}) \wedge {\mu _{{F_{21}}}}({z_2}),\quad{\mu _2} = {\mu _{{F_{12}}}}({z_1}) \wedge {\mu _{{F_{22}}}}({z_2}){\text{。}}$

 \left\{ \begin{aligned} & {x_1}(0) = 0,\quad {x_2}(0) = 1,\quad{x_3}(0) = 0 \\ & {u_1}(0) = 1,\quad{u_2}(0) = 0 \\ & {y_1}(0) = 0,\quad{y_2}(0) = 0 \\ \end{aligned} \right.
4.1 能控性

 \begin{aligned}{{{L}}_1} & = [{\rm{Bl}}{{\rm{k}}_1}({{{L}}_1})\;\;{\rm{Bl}}{{\rm{k}}_2}({{{L}}_1})\;\;{\rm{Bl}}{{\rm{k}}_3}({{{L}}_1})\;\;{\rm{Bl}}{{\rm{k}}_4}({{{L}}_1})]=\\& [{{{B}}_{11}}\;{{{B}}_{12}}\;{{{B}}_{13}}\;{{{B}}_{14}}]\end{aligned}
 ${{{B}}_{11}} = {\delta _8}[1\;1\;5\;5\;2\;2\;6\;6],\quad {{{B}}_{12}} = {\delta _8}[1\;3\;5\;7\;2\;4\;6\;8],$
 ${{{B}}_{13}} = {\delta _8}[5\;5\;5\;5\;6\;6\;6\;6],\quad {{{B}}_{14}} = {\delta _8}[5\;7\;5\;7\;6\;8\;6\;8]$

 $\begin{array}{c} {{{M}}_1} = \sum\limits_{i = 1}^{{2^m}} {_{\cal{B}}{\rm{Bl}}{{\rm{k}}_i}({{{L}}_1})} = \sum\limits_{i = 1}^4 {_{\cal{B}}{\rm{Bl}}{{\rm{k}}_i}({{{L}}_1})} = \\ \left[ {\begin{array}{*{20}{l}}1&1&0&0&0&0&0&0\\0&0&0&0&1&1&0&0\\0&1&0&0&0&0&0&0\\0&0&0&0&0&1&0&0\\1&1&1&1&0&0&0&0\\0&0&0&0&1&1&1&1\\0&1&0&1&0&0&0&0\\0&0&0&0&0&1&0&1\end{array}} \right]\end{array}$

 ${{{C}}_1} = \sum\limits_{j = 1}^{{2^{m + n}}} {_{\cal{B}}\sum\limits_{i = 1}^m {_{\cal{B}}{B_{1i}}^{(j)}} } = \sum\limits_{j = 1}^{{2^5}} {_{\cal{B}}{M_1}^{(j)}} = {\rm{10^{12}}} \times \\\left[ \begin{gathered} {\rm{1}}{\rm{.311\;5\;\;2}}{\rm{.122\;0\;\;0}}{\rm{.810\;5\;\;1}}{\rm{.311\;5\;\;2}}{\rm{.122\;0\;\;3}}{\rm{.433\;5\;\;1}}{\rm{.311\;5\;\;2}}{\rm{.122\;0}} \\ {\rm{2}}{\rm{.122\;0\;\;3}}{\rm{.433\;5\;\;1}}{\rm{.311\;5\;\;2}}{\rm{.122\;0\;\;3}}{\rm{.433\;5\;\;5}}{\rm{.555\;6\;\;2}}{\rm{.122\;0\;\;3}}{\rm{.433\;5}} \\ {\rm{0}}{\rm{.810\;5\;\;1}}{\rm{.311\;5\;\;0}}{\rm{.500\;9\;\;0}}{\rm{.810\;5\;\;1}}{\rm{.311\;5\;\;2}}{\rm{.122\;0\;\;0}}{\rm{.810\;5\;\;1}}{\rm{.311\;5}} \\ {\rm{1}}{\rm{.311\;5\;\;2}}{\rm{.122\;0\;\;0}}{\rm{.810\;5\;\;1}}{\rm{.311\;5\;\;2}}{\rm{.122\;0\;\;3}}{\rm{.433\;5\;\;1}}{\rm{.311\;5\;\;2}}{\rm{.122\;0}} \\ {\rm{2}}{\rm{.122\;0\;\;3}}{\rm{.433\;5\;\;1}}{\rm{.311\;5\;\;2}}{\rm{.122\;0\;\;3}}{\rm{.433\;5\;\;5}}{\rm{.555\;6\;\;2}}{\rm{.122\;0\;\;3}}{\rm{.433\;5}} \\ {\rm{3}}{\rm{.433\;5\;\;5}}{\rm{.555\;6\;\;2}}{\rm{.122\;0\;\;3}}{\rm{.433\;5\;\;5}}{\rm{.555\;6\;\;8}}{\rm{.989\;1\;\;3}}{\rm{.433\;5\;\;5}}{\rm{.555\;6}} \\ {\rm{1}}{\rm{.311\;5\;\;2}}{\rm{.122\;0\;\;0}}{\rm{.810\;5\;\;1}}{\rm{.311\;5\;\;2}}{\rm{.122\;0\;\;3}}{\rm{.433\;5\;\;1}}{\rm{.311\;5\;\;2}}{\rm{.122\;0}} \\ {\rm{2}}{\rm{.122\;0\;\;3}}{\rm{.433\;5\;\;1}}{\rm{.311\;5\;\;2}}{\rm{.122\;0\;\;3}}{\rm{.433\;5\;\;5}}{\rm{.555\;6\;\;2}}{\rm{.122\;0\;\;3}}{\rm{.433\;5}} \end{gathered} \right] > 0$

 Download: 图 1 控制变量和状态变量的关系 Fig. 1 Schematic diagram of relationship between the control variables and state variables

 ${{{B}}_{21}} = {\delta _8}[3\;1\;7\;5\;3\;1\;7\;5],\quad {{{B}}_{22}} = {\delta _8}[7\;5\;7\;5\;7\;5\;7\;5],$
 ${{{B}}_{23}} = {\delta _8}[3\;1\;7\;5\;3\;1\;7\;5],\quad{{{B}}_{24}} = {\delta _8}[8\;6\;8\;6\;8\;6\;8\;6]$
 \begin{aligned} {{{M}}_2} & = \displaystyle\sum\limits_{i = 1}^{{2^m}} {_{\cal{B}}{\rm{Bl}}{{\rm{k}}_i}({{{L}}_2})} = \displaystyle\sum\limits_{i = 1}^4 {_{\cal{B}}{\rm{Bl}}{{\rm{k}}_i}({{{L}}_2})}=\\[-4pt]&\left[ {\begin{array}{*{20}{l}}1&1&0&0&0&0&0&0\\[-4pt]0&0&0&0&1&1&0&0\\[-4pt]0&1&0&0&0&0&0&0\\[-4pt]0&0&0&0&0&1&0&0\\[-4pt]1&1&1&1&0&0&0&0\\[-4pt]0&0&0&0&1&1&1&1\\[-4pt]0&1&0&1&0&0&0&0\\[-4pt]0&0&0&0&0&1&0&1\end{array}} \right]\end{aligned}

 ${{{C}}_2} = \sum\limits_{j = 1}^{{2^{m + n}}} {_{\cal{B}}\sum\limits_{i = 1}^m {_{\cal{B}}{B_{2i}}^{(j)}} } = \sum\limits_{j = 1}^{{2^5}} {_{\cal{B}}{M_2}^{(j)}} = {\rm{10^{11}}} \times \\\left[ \begin{gathered} {\rm{1}}{\rm{.298\;6\;\;1}}{\rm{.836\;5\;\;0}}{\rm{.918\;2\;\;1}}{\rm{.298\;6\;\;1}}{\rm{.298\;6\;\;1}}{\rm{.836\;5\;\;0}}{\rm{.918\;2\;\;1}}{\rm{.298\;6}} \\ {\rm{0 \quad\quad\quad 0 \quad\quad\quad 0 \quad\quad\quad 0 \quad\quad\quad 0 \quad\quad\quad 0 \quad\quad\quad 0 \quad\quad\quad 0}} \\ {\rm{1}}{\rm{.836\;5\;\;2}}{\rm{.597\;2\;\;1}}{\rm{.298\;6\;\;1}}{\rm{.836\;5\;\;1}}{\rm{.836\;5\;\;2}}{\rm{.597\;2\;\;1}}{\rm{.298\;6\;\;1}}{\rm{.836\;5}} \\ {\rm{0 \quad\quad\quad 0 \quad\quad\quad 0 \quad\quad\quad 0 \quad\quad\quad 0 \quad\quad\quad 0 \quad\quad\quad 0 \quad\quad\quad 0}} \\ {\rm{3}}{\rm{.135\;1\;\;4}}{\rm{.433\;7\;\;2}}{\rm{.216\;8\;\;3}}{\rm{.135\;1\;\;3}}{\rm{.135\;1\;\;4}}{\rm{.433\;7\;\;2}}{\rm{.216\;8\;\;3}}{\rm{.135\;1}} \\ {\rm{3}}{\rm{.135\;1\;\;4}}{\rm{.433\;7\;\;2}}{\rm{.216\;8\;\;3}}{\rm{.135\;1\;\;3}}{\rm{.135\;1\;\;4}}{\rm{.433\;7\;\;2}}{\rm{.216\;8\;\;3}}{\rm{.135\;1}} \\ {\rm{4}}{\rm{.433\;7\;\;6}}{\rm{.270\;1\;\;3}}{\rm{.135\;1\;\;4}}{\rm{.433\;7\;\;4}}{\rm{.433\;7\;\;6}}{\rm{.270\;1\;\;3}}{\rm{.135\;1\;\;4}}{\rm{.433\;7}} \\ {\rm{4}}{\rm{.433\;7\;\;6}}{\rm{.270\;1\;\;3}}{\rm{.135\;1\;\;4}}{\rm{.433\;7\;\;4}}{\rm{.433\;7\;\;6}}{\rm{.270\;1\;\;3}}{\rm{.135\;1\;\;4}}{\rm{.433\;7}} \end{gathered} \right] = 0$

4.2 能观性

${{{\varOmega}} _{k\left( {s + 1} \right)}} \subset \bigcup\limits_{i = 1}^s {{{{\varOmega}} _i}} ,$ $s = 3$ ，则式(25)中 ${{{\varGamma}} _{10}}\text{、}{{{\varGamma}} _{11}}$ ${{{\varGamma}} _{12}}$ ${{{\varGamma}} _{13}}$ 分别为 ${{{H}}_1} = {\delta _4}[2\;1\;2\;1\;4\;3\;4\;3]$

 ${{{\varGamma}} _{10}} = {{{H}}_1} = \left[ {\begin{array}{*{20}{l}}0&1&0&1&0&0&0&0\\1&0&1&0&0&0&0&0\\0&1&0&0&0&1&0&1\\0&0&0&0&1&0&1&0\end{array}} \right]$
 {{{\varGamma}} _{11}}=\left[ {\begin{aligned}{{{H}}{}_1{{{B}}_{11}}}\\{{{H}}{}_1{{{B}}_{12}}}\\{{{H}}{}_1{{{B}}_{13}}}\\{{{H}}{}_1{{{B}}_{14}}}\end{aligned}} \right]=\\\left[ {\begin{array}{*{20}{l}}0&1&0&0&0&1&0&0&0&0&0&1&0&0&0&1\\0&1&0&0&0&1&0&0&0&0&0&1&0&0&0&1\\0&0&0&1&0&0&0&1&0&0&0&1&0&0&0&1\\0&1&0&0&0&1&0&0&0&0&0&1&0&0&0&1\\1&0&0&0&1&0&0&0&0&0&1&0&0&0&1&0\\1&0&0&0&1&0&0&0&0&0&1&0&0&0&1&0\\0&0&1&0&0&0&1&0&0&0&1&0&0&0&1&0\\0&0&1&0&0&0&1&0&0&0&1&0&0&0&1&0\end{array}} \right]
 ${{{\varGamma}} _{12}} = \left[ \begin{array}{c}H{}_1{B_{11}}{B_{11}}\\ \vdots \\{H_1}{B_{12}}{B_{11}}\\ \vdots \\{H_1}{B_{13}}{B_{11}}\\ \vdots \\{H_1}{B_{14}}{B_{14}}\end{array} \right] = \left[\begin{array}{*{20}{c}}0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad\cdots\quad0\quad0\quad1\quad0\\0\quad1\quad0\quad0\quad0\quad0\quad0\quad1\quad1\quad0\quad0\quad0\quad0\quad0\quad1\quad0\quad\cdots\quad0\quad0\quad1\quad0\\1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad\cdots\quad0\quad0\quad1\quad0\\1\quad0\quad0\quad0\quad0\quad0\quad1\quad0\quad1\quad0\quad0\quad0\quad0\quad0\quad1\quad0\quad\cdots\quad0\quad0\quad1\quad0\\0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad\cdots\quad0\quad0\quad1\quad0\\0\quad1\quad0\quad0\quad0\quad0\quad0\quad1\quad1\quad0\quad0\quad0\quad0\quad0\quad1\quad0\quad\cdots\quad0\quad0\quad1\quad0\\1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad\cdots\quad0\quad0\quad1\quad0\\1\quad0\quad0\quad0\quad0\quad0\quad1\quad0\quad1\quad0\quad0\quad0\quad0\quad0\quad1\quad0\quad\cdots\quad0\quad0\quad1\quad0\end{array} \right]_{8 \times 64}^{\rm T}$

Γ12为64×8维的布尔矩阵，由于篇幅的限制，上式中只列出了一部分。

 ${{{\varGamma}} _{13}} = \left[ \begin{array}{c}{H_1}{B_{11}}{B_{11}}{B_{11}}\\{H_1}{B_{11}}{B_{11}}{B_{12}}\\ \vdots \\{H_1}{B_{14}}{B_{14}}{B_{13}}\\{H_1}{B_{14}}{B_{14}}{B_{14}}\end{array} \right] = \left[\begin{array}{*{20}{c}}0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad\cdots\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\\0\quad1\quad0\quad0\quad0\quad0\quad1\quad0\quad0\quad1\quad0\quad0\quad\cdots\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\\0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad\cdots\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\\0\quad1\quad0\quad0\quad0\quad0\quad1\quad0\quad0\quad1\quad0\quad0\quad\cdots\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\\0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad\cdots\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\\0\quad1\quad0\quad0\quad0\quad0\quad1\quad0\quad0\quad1\quad0\quad0\quad\cdots\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\\0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad\cdots\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\\0\quad1\quad0\quad0\quad0\quad0\quad1\quad0\quad0\quad1\quad0\quad0\quad\cdots\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\end{array} \right]_{8 \times 256}^{\rm T}$

Γ13为256×8维的布尔矩阵，由于篇幅的限制，上式中只列出了一部分。

 ${{{O}}_{1}} = \left[ \begin{array}{l}{{{\varGamma}}_{10}}\\{{{\varGamma}}_{11}}\\{{{\varGamma}}_{12}}\\{{{\varGamma}}_{13}}\end{array} \right] = \left[\begin{array}{*{20}{c}}0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad\cdots\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\\1\quad0\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad\cdots\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\\0\quad1\quad0\quad0\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad\cdots\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\\1\quad0\quad0\quad0\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad\cdots\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\\0\quad0\quad0\quad1\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad\cdots\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\\0\quad0\quad1\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad\cdots\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\\0\quad0\quad0\quad1\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad\cdots\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\\0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\quad\cdots\quad0\quad0\quad1\quad0\quad0\quad0\quad1\quad0\end{array} \right]_{8 \times 340}^{\rm T}$

O1为340×8维的布尔矩阵，由于篇幅的限制，文中只列出了一部分。

 Download: 图 2 输出变量和状态变量的关系 Fig. 2 Schematic diagram of relationship between the output variables and state variables under rule 1
4.3 稳定性

${{\cal{I}}}({F_1}) = \left[ {\begin{array}{*{20}{c}} 0&1&0 \\ 0&0&1 \\ 1&0&0 \end{array}} \right]$

${[{\cal{I}}({F_1})]^{(5)}} = {\cal{I}}({F_1}) \ne 0$ ，即 ${[{\cal{I}}({F_1})]^{(k)}} \ne 0$ ,则规则1下的局部模型是不稳定的。

${\cal{I}}({F_2}) = \left[ {\begin{array}{*{20}{c}} 0&0&1 \\ 0&0&0 \\ 1&1&0 \end{array}} \right]$

${[{\cal{I}}({F_2})]^{(3)}} = 0$ ，则规则2下的局部模型是稳定的。

5 结束语

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