切换布尔网络的分析、控制及应用
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摘要
随着系统生物学和医学的快速发展,布尔网络已经成为当前控制领域的研究热点.然而,在实际的布尔网络中,由于存在试图重构给定网络的外部干扰或控制,以及网络进化异步行为的影响,多模态切换现象广泛存在.这些多模态切换行为使得布尔网络的性能分析与控制设计变得异常复杂.本文利用矩阵的半张量积方法研究切换布尔网络的若干分析与控制问题,并将所得结果应用于系统生物学、组合电路的故障诊断和有限域网络的趋同分析.主要研究内容如下：
1.建立了布尔网络Lyapunov函数的概念和构造方法.利用函数摄动对状态转移矩阵的影响,得到了布尔网络在函数摄动影响下拓扑结构变化的判别准则.利用输出矩阵筛选状态反馈增益矩阵的方法,给出了布尔控制网络输出反馈镇定控制器的设计算法.通过构造一族合适的能达集合,建立了布尔控制网络输出跟踪控制器的设计方法.利用矩阵的列展开技术,给出了一组布尔控制网络可同时镇定控制器的设计方法.
2.建立了适用于切换布尔网络的共同Lyapunov函数方法,并给出了共同Lyapunov函数的构造方法.为了避免构造共同Lyapunov函数所带来的繁琐计算,通过定义切换布尔网络切换点能达的概念,给出了一个更易验证的判断切换布尔网络任意切换下稳定的充要条件.
3.研究了切换布尔网络的稳定切换信号设计问题.基于切换点能达建立了切换布尔网络逐点切换可稳的充要条件.给出了仅依赖于时间的稳定切换信号和状态反馈稳定切换信号的设计方法.
4.为切换布尔控制网络定义了切换-输入-状态关联矩阵并给出该矩阵的结构和意义.基于切换-输入-状态关联矩阵,建立了切换布尔控制网络能控和能达的充要条件,并给出了一个有效的算法来实现切换布尔控制网络在最短时间内能达.
5.利用冗余变量分离技术研究了切换布尔控制网络的干扰解耦控制器设计问题.通过将切换布尔控制网络转化为合适的输出友好坐标形式,分别为系统关于切换信号的解耦、关于外部干扰的解耦以及关于切换信号和外部干扰的解耦设计了所有状态反馈控制器和输出反馈控制器.
6.研究了状态和输入受限的切换布尔网络的分析与控制问题.基于受限关联矩阵建立了状态和输入受限的切换布尔网络能控的充分必要条件和Mayer型最优控制的求解方法.通过将受限切换布尔控制网络转化为等价的非受限系统,分别给出了状态和输入受限的切换布尔网络在开环控制和闭环控制作用下可镇定的充分必要条件.
7.将所得的理论结果分别应用于系统生物学、组合电路的故障诊断和有限域网络的趋同等实际问题.为大肠杆菌乳糖操纵子网络设计了输出反馈镇定控制器和输出跟踪控制器,并利用逻辑矩阵分解技术分析了信号转导子网络的拓扑结构.给出了高阶布尔导数基于矩阵半张量积的计算公式,并据此建立了组合电路多个故障检测向量集合的求解方法.基于切换点能达建立了切换拓扑下有限域网络在任意切换下趋同的充要条件.
With the rapid development of systems biology and medical science, Boolean networks have attracted much attention in the current control field. However, due to the existence of external interventions or control inputs that attempt to re-engineer a given Boolean network, and the effect of asynchronous dynamics in Boolean net-works, the multi-mode switching phenomenon is often encountered in a real Boolean network. These multi-mode switching behaviors make the analysis and control de-sign of Boolean networks more complex. This paper investigates the analysis and control of switched Boolean networks by using the semi-tensor product method, and applies the obtained results to the systems biology, the fault detection of combina-tional circuits and the consensus of finite-field networks. The main contents of this paper are listed as follows:
1. The concept of Lyapunov function for Boolean networks is defined and a construction method is proposed. Using the effect of function perturbations on the state transition matrix, several necessary and sufficient conditions are obtained for the topological structure analysis of Boolean networks with function perturbations. Based on the selection technique of state feedback gain matrices from the output equation, a general procedure is given for the design of output feedback stabilizers of Boolean networks. By constructing a series of reachable sets, a control design method is established for the output tracking control of Boolean networks. Using the column stacking form of matrices, an effective procedure is proposed to design simultaneous stabilizers for a set of Boolean networks.
2. The common Lyapunov function method is established for switched Boolean networks, and a constructive procedure is given for constructing common Lyapunov functions. To avoid the tedious computation of constructing common Lyapunov functions, an easily verifiable necessary and sufficient condition is presented for the stability of switched Boolean networks under arbitrary switching signal by defining the concept of switching point reachability.
3. The switching signal design for the stabilizability of switched Boolean net-works is studied. Based on the switching point reachability, a necessary and suf-ficient condition is given for the point-wise switching stabilizability of switched Boolean networks. Moreover, several methods are proposed to design both time-dependent and state feedback switching signals for the consistent switching stabiliz-ability of switched Boolean networks.
4. A kind of switching-input-state incidence matrix is proposed for switched Boolean control networks, based on which, a necessary and sufficient condition is presented for the controllability and reachability. Moreover, an effective algorithm is established to realize the reachability in the shortest time.
5. The disturbance decoupling problem (DDP) of switched Boolean control networks is considered by using the redundant variable separation technique. By converting switched Boolean control networks into a proper output-friendly coordi-nate frame, all the state feedback and output feedback controllers are designed for DDPs with respect to the switching signal, the exogenous disturbances, and both the switching signal and the exogenous disturbances, respectively.
6. The analysis and control of switched Boolean networks with state and input constraints are investigated. Based on a kind of constrained incidence matrix, a nec-essary and sufficient condition is obtained for the controllability of the constrained system, and a novel design method is proposed for the Mayer-type optimal control. By converting the constrained system into an equivalent unconstrained one, some necessary and sufficient conditions are presented for the design of both open-loop and closed-loop stabilizers.
7. The obtained results are applied to the systems biology, the fault detec-tion of combinational circuits and the consensus of finite-field networks. Both the output feedback stabilizers and the output tracking controllers are designed for the lac operon in the bacterium Escherichia coli. Using the logical matrix factorization technique, the topological structure of subnetwork of signal transduction networks is analyzed. A general formula is given for the calculation of higher-order Boolean derivative, based on which a new method is proposed to solve the test vector set of multi-fault detection of combinational circuits. Based on the switching point reachability, a necessary and sufficient condition is presented for the consensus of finite-field networks under arbitrary switching signal.
1.建立了布尔网络Lyapunov函数的概念和构造方法.利用函数摄动对状态转移矩阵的影响,得到了布尔网络在函数摄动影响下拓扑结构变化的判别准则.利用输出矩阵筛选状态反馈增益矩阵的方法,给出了布尔控制网络输出反馈镇定控制器的设计算法.通过构造一族合适的能达集合,建立了布尔控制网络输出跟踪控制器的设计方法.利用矩阵的列展开技术,给出了一组布尔控制网络可同时镇定控制器的设计方法.
2.建立了适用于切换布尔网络的共同Lyapunov函数方法,并给出了共同Lyapunov函数的构造方法.为了避免构造共同Lyapunov函数所带来的繁琐计算,通过定义切换布尔网络切换点能达的概念,给出了一个更易验证的判断切换布尔网络任意切换下稳定的充要条件.
3.研究了切换布尔网络的稳定切换信号设计问题.基于切换点能达建立了切换布尔网络逐点切换可稳的充要条件.给出了仅依赖于时间的稳定切换信号和状态反馈稳定切换信号的设计方法.
4.为切换布尔控制网络定义了切换-输入-状态关联矩阵并给出该矩阵的结构和意义.基于切换-输入-状态关联矩阵,建立了切换布尔控制网络能控和能达的充要条件,并给出了一个有效的算法来实现切换布尔控制网络在最短时间内能达.
5.利用冗余变量分离技术研究了切换布尔控制网络的干扰解耦控制器设计问题.通过将切换布尔控制网络转化为合适的输出友好坐标形式,分别为系统关于切换信号的解耦、关于外部干扰的解耦以及关于切换信号和外部干扰的解耦设计了所有状态反馈控制器和输出反馈控制器.
6.研究了状态和输入受限的切换布尔网络的分析与控制问题.基于受限关联矩阵建立了状态和输入受限的切换布尔网络能控的充分必要条件和Mayer型最优控制的求解方法.通过将受限切换布尔控制网络转化为等价的非受限系统,分别给出了状态和输入受限的切换布尔网络在开环控制和闭环控制作用下可镇定的充分必要条件.
7.将所得的理论结果分别应用于系统生物学、组合电路的故障诊断和有限域网络的趋同等实际问题.为大肠杆菌乳糖操纵子网络设计了输出反馈镇定控制器和输出跟踪控制器,并利用逻辑矩阵分解技术分析了信号转导子网络的拓扑结构.给出了高阶布尔导数基于矩阵半张量积的计算公式,并据此建立了组合电路多个故障检测向量集合的求解方法.基于切换点能达建立了切换拓扑下有限域网络在任意切换下趋同的充要条件.
With the rapid development of systems biology and medical science, Boolean networks have attracted much attention in the current control field. However, due to the existence of external interventions or control inputs that attempt to re-engineer a given Boolean network, and the effect of asynchronous dynamics in Boolean net-works, the multi-mode switching phenomenon is often encountered in a real Boolean network. These multi-mode switching behaviors make the analysis and control de-sign of Boolean networks more complex. This paper investigates the analysis and control of switched Boolean networks by using the semi-tensor product method, and applies the obtained results to the systems biology, the fault detection of combina-tional circuits and the consensus of finite-field networks. The main contents of this paper are listed as follows:
1. The concept of Lyapunov function for Boolean networks is defined and a construction method is proposed. Using the effect of function perturbations on the state transition matrix, several necessary and sufficient conditions are obtained for the topological structure analysis of Boolean networks with function perturbations. Based on the selection technique of state feedback gain matrices from the output equation, a general procedure is given for the design of output feedback stabilizers of Boolean networks. By constructing a series of reachable sets, a control design method is established for the output tracking control of Boolean networks. Using the column stacking form of matrices, an effective procedure is proposed to design simultaneous stabilizers for a set of Boolean networks.
2. The common Lyapunov function method is established for switched Boolean networks, and a constructive procedure is given for constructing common Lyapunov functions. To avoid the tedious computation of constructing common Lyapunov functions, an easily verifiable necessary and sufficient condition is presented for the stability of switched Boolean networks under arbitrary switching signal by defining the concept of switching point reachability.
3. The switching signal design for the stabilizability of switched Boolean net-works is studied. Based on the switching point reachability, a necessary and suf-ficient condition is given for the point-wise switching stabilizability of switched Boolean networks. Moreover, several methods are proposed to design both time-dependent and state feedback switching signals for the consistent switching stabiliz-ability of switched Boolean networks.
4. A kind of switching-input-state incidence matrix is proposed for switched Boolean control networks, based on which, a necessary and sufficient condition is presented for the controllability and reachability. Moreover, an effective algorithm is established to realize the reachability in the shortest time.
5. The disturbance decoupling problem (DDP) of switched Boolean control networks is considered by using the redundant variable separation technique. By converting switched Boolean control networks into a proper output-friendly coordi-nate frame, all the state feedback and output feedback controllers are designed for DDPs with respect to the switching signal, the exogenous disturbances, and both the switching signal and the exogenous disturbances, respectively.
6. The analysis and control of switched Boolean networks with state and input constraints are investigated. Based on a kind of constrained incidence matrix, a nec-essary and sufficient condition is obtained for the controllability of the constrained system, and a novel design method is proposed for the Mayer-type optimal control. By converting the constrained system into an equivalent unconstrained one, some necessary and sufficient conditions are presented for the design of both open-loop and closed-loop stabilizers.
7. The obtained results are applied to the systems biology, the fault detec-tion of combinational circuits and the consensus of finite-field networks. Both the output feedback stabilizers and the output tracking controllers are designed for the lac operon in the bacterium Escherichia coli. Using the logical matrix factorization technique, the topological structure of subnetwork of signal transduction networks is analyzed. A general formula is given for the calculation of higher-order Boolean derivative, based on which a new method is proposed to solve the test vector set of multi-fault detection of combinational circuits. Based on the switching point reachability, a necessary and sufficient condition is presented for the consensus of finite-field networks under arbitrary switching signal.
引文
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