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针对一类具有更复杂动力学行为的忆阻混沌系统, 本文基于新型幂次趋近律设计两种滑模控制协议分别实现了系统的有限时间、固定时间同步. 首先对于有限时间同步问题, 基于Lyapunov稳定性理论和有限时间稳定性理论, 推导了实现全局有限时间同步的充分条件, 得到了与系统初始条件有关的稳定时间上限, 并证明了系统的稳定性. 对于固定时间同步问题, 利用固定时间稳定性理论, 推导得到不随系统初始值变化的收敛时间上确界. 最后, 通过设置两组对照实验, 比较了两种滑模控制律对系统同步状态的影响, 其仿真结果与数值分析相符, 从而验证了本文的有效性和可行性.Two innovative sliding mode control laws based on the convergence principle of reaching law are presented in this work. These control laws are used to achieve both finite-time and fixed-time synchronization for a specific class of memristive chaotic system, which are known for their intricate and complex dynamical behaviors. By utilizing these control strategies, we can effectively manage the synchronization process and ensure rapid convergence. Firstly, for the finite-time synchronization issue, a novel power reaching law is derived. Compared with the conventional reaching law, the reaching law presented in this work has a prominent advantage that the chattering of the sliding mode control is reduced to a lesser extent and the speed of reaching the sliding surface is quicker. An upper bound of the stabilization time, which is dependent on the initial conditions of the system, is obtained and the system is proved stable. For the fixed time synchronization problem, a new double power reaching law is put forward to minimize the chattering and accelerate the convergence. Then, by utilizing the fixed time stability theory, the upper bound of the convergence time that remains invariant with the initial value of the system is derived. Finally, in order to verify the effectiveness and feasibility of the theoretical derivation in this paper, two sets of control experiments are set up and the influences of the two control laws on the system synchronization state are compared. The experimental phenomenon strongly proves the accuracy of the proposed theorem.
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Keywords:
- finite-time synchronization /
- fixed-time synchronization /
- new power reaching law /
- memristor chaotic systems
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图 2 忆阻混沌系统(1)吸引子相图 (a) x-y平面; (b) x-w平面; (c) x-z平面; (d) y-w平面
Fig. 2. Attractor phase diagrams of the memristor chaotic system (1): (a) x-y plane; (b) x-w plane; (c) x-z plane; (d) y-w plane.
图 3 初始状态为$ {x_1}(0) = ( - 1, {\text{ }}2, {\text{ }}3, {\text{ }} - 4) $时有限时间同步误差
Fig. 3. Finite time synchronization error when the initial state is $ {x_1}(0) = ( - 1, {\text{ }}2, {\text{ }}3, {\text{ }} - 4) $.
图 4 初始状态为$ {x_2}(0) = ( - 10, {\text{ }}20, {\text{ }}30, {\text{ }} - 40) $时有限时间同步误差
Fig. 4. Finite time synchronization error when the initial state is $ {x_2}(0) = ( - 10, {\text{ }}20, {\text{ }}30, {\text{ }} - 40) $.
图 5 初始状态为$ {x_1}(0) = ( - 1, {\text{ }}2, {\text{ }}3, {\text{ }} - 4) $时固定时间同步误差
Fig. 5. Fixed time synchronization error when the initial state is $ {x_1}(0) = ( - 1, {\text{ }}2, {\text{ }}3, {\text{ }} - 4) $.
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