This research note establishes a specific framework for transforming nonlinear multi-input and multi-output diffeomorphism systems into extended observable normal forms without using differential geometry techniques. For this purpose, the nonlinear MIMO systems whose nonlinear terms do not need to be Lipschitz, are proposed. First, a change of coordinates is designed to eliminate the square items and coupled items for each nonlinear dynamical subsystem. Second, coupled auxiliary dynamics are constructed to transform the nonlinear multi-input and multi-output diffeomorphism systems into extended observable normal forms such that the finite-time and robust step-by-step sliding mode observer can be applied. Then, the state variables for the considered nonlinear dynamical systems are estimated by using the inverse of the transformations. Finally, the validity of the proposed design methods is verified by two numerical examples.

中文翻译：

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具有输出注入和输出微分同态的非线性 MIMO 可观察范式


本研究说明建立了一个特定的框架，用于在不使用微分几何技术的情况下将非线性多输入和多输出微分同态系统转换为扩展的可观察范式。为此，提出了非线性 MIMO 系统，其非线性项不需要 Lipschitz。首先，设计坐标更改以消除每个非线性动力学子系统的平方项和耦合项。其次，构建耦合辅助动力学，将非线性多输入和多输出微分同构系统转换为扩展的可观察范式，以便可以应用有限时间和稳健的逐步滑模观测器。然后，通过使用变换的逆函数来估计所考虑的非线性动力系统的状态变量。最后，通过两个数值算例验证了所提设计方法的有效性。