通过将单射浸入变换为满射自同态来表示优选坐标中的观察者
Expressing an observer in preferred coordinates by transforming an injective immersion into a surjective diffeomorphism
摘要：
在设计非线性系统的观测器时，给定系统的动态和所设计的观测器的动态通常不在相同的坐标中表示，甚至具有在不同空间中演化的状态。通常，给出一种状态与另一种状态的函数（表示为$τ$（或其逆函数，表示为$τ^*$））不是明确已知的，这产生了实现问题。我们建议通过在与给定系统相同的坐标中表示观测器动态来解决这个问题。但这可能会增加额外的坐标，我们称之为增强的问题。这也可能迫使修改域或范围的扩大”$τ$或$τ^*$，问题，我们称之为扩展。我们表明，扩增问题可以部分地解决一个自由的向量族的连续完成和扩展问题可以解决的功能扩展，使图像的扩展功能的整个空间。我们还展示了如何在不修改观测器动态特性的情况下进行扩充和扩展，从而保持收敛性。
英文摘要：
When designing observers for nonlinear systems, the dynamics of the given system and of the designed observer are usually not expressed in the same coordinates or even have states evolving in different spaces. In general, the function, denoted $τ$ (or its inverse, denoted $τ^*$) giving one state in terms of the other is not explicitly known and this creates implementation issues. We propose to round this problem by expressing the observer dynamics in the the same coordinates as the given system. But this may impose to add extra coordinates, problem that we call augmentation. This may also impose to modify the domain or the range of the augmented" $τ$ or $τ^*$, problem that we call extension. We show that the augmentation problem can be solved partly by a continuous completion of a free family of vectors and that the extension problem can be solved by a function extension making the image of the extended function the whole space. We also show how augmentation and extension can be done without modifying the observer dynamics and therefore with maintaining convergence.Several examples illustrate our results.
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