摘要
混合动态系统是由连续(或离散)时间变量和离散事件(或逻辑)变量相互作用的复杂系统。在描述和研究许多复杂的物理现象和实际应用的过程中,混合动态系统能有效地提供一个数学框架(模型)。它在混合自动机控制、数据采样控制、飞机控制、自动列车控制以及网络控制等实际问题中都有着广泛的应用。因此,对于混合动态系统的研究具有重要的理论意义和实用价值。
本文主要研究了几类混合动态系统的稳定性分析以及相关的控制器设计问题。主要工作概括如下:
1.研究一类混合动态系统的Lyapunov稳定性问题。通过结合多Lyapunov函数和向量Lyapunov函数,提出一种新的分析工具——多向量Lyapunov函数。利用该工具,弱化对系统能量函数的假设,给出系统渐近稳定的充分条件。用实例验证该方法的有效性。
2.研究一类混合动态系统的实用稳定化问题。所研究的系统具有时变子系统和时变跳转函数。首先,通过状态跳转函数,确定系统一条严格递增的切换时间序列。然后在该序列中的每一段时间区间内,明确地构造出相应的线性状态反馈控制律,由此实现闭环系统的实用稳定。
3.研究一类混合动态系统的有限时间稳定性及其稳定化问题。首先,通过放松对Lyapunov函数的限制,给出一个新的非线性系统有限时间稳定的充分必要条件。然后,给出混合动态系统有限时间稳定的概念,同时利用前述条件以及前人的结果(Bhat & Bernstein(2000), Moulay & Perruquetti(2003))进一步研究混合动态系统的有限时间稳定性,给出若干个充分条件。最后,根据系统连续和重置部分的状态划分,构造出一个混合反馈控制器使闭环系统有限时间稳定。
4.研究基于有限状态自动机的混合动态系统稳定性及其稳定化问题。首先,通过系统两部分的有机结合,对整个混合动态系统进行分析,提出稳定性概念,给出新的判别定理。然后,基于混合观测器进一步研究混合动态系统的混合反馈控制问题。通过设计混合反馈控制律使闭环系统达到渐近稳定。最后,将前述结果推广到不确定混合动态系统,其中不确定因素包括:自动机跳转的不确定性,连续子系统参数的不确定性以及控制器参数不确定性。基于最优路径的思想,通过求解线性矩阵不等式LMI,给出一种非脆弱的控制方法使闭环系统达到渐近稳定。
5.研究基于微分Petri网的混合动态系统稳定性问题。首先,针对混合动态系统的特点,在其微分Petri网(DPN)模型的基础上,给出它的稳定性概念和稳定性引理。然后通过引入两类辅助函数G和,利用关联矩阵的信息,构造出混合Lyapunov函数得到DPN的稳定性定理。另一方面,通过放松触发条件依赖弧权的限制以及扩展弧权的定义,提出了一种新的Petri网扩展形式——扩展微分Petri网。该Petri网结合了广义微分Petri网和混合自动机的优点。利用这种新的扩展形式对混合动态系统进行建模,通过结合系统两部分(离散和连续)的稳定性,给出系统渐近稳定的充分条件。此外,利用关联函数矩阵的信息,构造出一个新的混合Lyapunov函数,得到线性混合动态系统的稳定性结果。H
Hybrid dynamical systems are the complicated systems that consist of continuous (or discrete) time dynamics, discrete event (or logical) dynamics, and the interaction between them. Hybrid dynamical systems can provide an effective framework for mathematical modeling and analysis of many complex physical phenomena and practical applications. They have a variety of applications such as hybrid automata, sampled-data control systems, aircraft control system, automotive control systems and network control systems etc. Thus, the study of hybrid dynamical systems has important significance both in theory and applications.
In this dissertation, stability analysis and control problem of several classes hybrid dynamical systems are studied. The main contributions and original ideas included in the dissertation are summarized as follows.
1.The stability of a class of hybrid dynamical systems is studied. By combining the advantages of multiple Lyapunov functions and vector Lyapunov function, multiple vector Lyapunov functions (MVLF) is proposed. By using the MVLF, we weaken the hypothesis on the Lyapunov function of systems, and give a sufficient condition of asymptotic stability for the systems. A simulation is given to illustrate the effectiveness of the proposed method.
2.The practical stabilization of a class of hybrid dynamical systems is studied. These hybrid dynamical systems have time-varying subsystems and time-varying state jump. Firstly, by using the state jump function, we determine a strict increased switch- ing sequence for the system. Then, for each time interval of the switching sequence, we explicitly construct a corresponding linear state feedback control laws, which practically stabilize the closed-loop systems.
3.The finite time stability and stabilization of a class of hybrid dynamical systems are studied. At first, by relaxing the restrictions on Lyapunov function of the system, a new necessary and sufficient condition of finite time stability is given for nonlinear system. Then, we present the concept of finite stability for hybrid dynamical systems. By using the condition mentioned above and the results of previous works (Bhat & Bernstein, 2000; Moulay & Perruquetti, 2003), we further study the finite time stability of hybrid dynamical system, several sufficient conditions are derived. Finally, based on the state partition of continuous and resetting parts of system, a hybrid feedback controller is constructed, which stabilizes the closed-loop systems in finite time.
4.Based on the finite state machine (FSM), the stability and stabilization of hybrid dynamical systems are studied. Firstly, by concerning the stability of two parts, a concept of asymptotical stability for the whole hybrid dynamical systems is proposed. Moreover, a new stability criterion is given. Then, based on the hybrid observer, we further construct a hybrid feedback controller, which asymptotically stabilize the closed-loop system. Finally, we extend the results obtained above to the uncertain hybrid dynamical systems, where uncertainties appear in the place jump of FSM, the parameter of continuous subsystems and the parameter of controller. Based on optimal road idea, we construct a non-fragile controller by solving an LMI, which asympto- tically stabilize the closed-loop system.
5. Based on the (extended) differential Petri net (DPN), the stability of hybrid dynamical systems is studied. At first, according to the characteristics of hybrid dynamical systems, we propose its differential Petri net model, and give the concept and lemma of stability for DPN. Then, by using two auxiliary functions G , and the information of index matrix, we construct a new hybrid Lyapunov function and present the stability theorem for DPN. On the other hand, by relaxing the restriction on the enabling condition depends on the weight of arc, and extending the definition of weight for arc, extended differential Petri net (EDPN), a new model tool, is proposed. EDPN adopts both the advantages of generalized differential Petri nets and hybrid automata. Using this model, a sufficient condition of asymptotic stability for hybrid dynamical systems is obtained by combining the stability of two parts. In addition, by using the information of index matrix and a new hybrid Lyapunov function, the stability theorem of linear HDS is obtained. H
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