This paper focuses on a problem of sliding mode output tracking for non-minimum-phase nonlinear systems whose zero dynamics is not stable. Sliding mode consists of continuous power functions instead of switch functions, and the power of the function has the term of fraction whose denominator and numerator are all positive odd numbers, so the chattering phenomenon can be avoided. The method of the design is based on the centre manifold theory, and a sliding manifold can be constructed by the combination of the polynomial of tracing error and internal dynamics variable.

This paper reports a new four-dimensional continuous autonomous chaotic system, in which each equation in the system contains a 3-term cross product. Basic properties of the system are analyzed by means of Lyapunov exponents and bifurcation diagrams.

通过构造快速滑模控制器，实现了混沌系统的有限时同步，指出许多混沌系统经过状态变换后，都可以通过快速收敛的有限时控制器在有限时间内实现驱动系统和响应系统的完全同步，并以陈氏混沌系统和四阶细胞神经网络超混沌系统为例进行了系统仿真。结果显示，该有限时控制器不仅具有良好的性能，而且具备较强的鲁棒性。

Abstract-A linear model of recurrent neural networks, called the Elman networks, is combined with the simple nonlinear modulo (mod) operation on its linear activated function so as to generate chaos purposely. Conditions on the weight matrix are obtained, under which the generated chaos satisfies the mathematical definition of chaos in the sense of Li and Yorke. Some simple and representative weight matrices are constructed for designing such Elman networks that can generate Li-Yorke chaos. Several numerical simulations are shown to verify and visualize the design.

倒立摆是一种复杂的非线性控制系统。通过对其进行控制能够检验控制器的鲁棒性。基于一个能量形式的Lyapunov 函数设计了倒立摆稳定控制器使得摆趋于上平衡位置,并且使得小车位移和角度都收敛于零。该控制策略基于系统的总能量，利用其耗散特性设计了Lyapunov函数，并证明了控制系统的稳定性。理论分析及仿真试验表明该控制器对于倒立摆控制具有很强的鲁棒性。