The observability of a dynamical system is affected by the presence of external inputs, either known (such as control actions) or unknown (disturbances). Inputs of unknown magnitude are especially detrimental for observability, and they also complicate its analysis. Hence the availability of computational tools capable of analysing the observability of nonlinear systems with unknown inputs has been limited until lately. Two symbolic algorithms based on differential geometry, ORC-DF and FISPO, have been recently proposed for this task, but their critical analysis and comparison is still lacking. Here we perform an analytical comparison of both algorithms and evaluate their performance on a set of problems, discussing their strengths and limitations. Additionally, we use these analyses to provide insights about certain aspects of the relationship between inputs and observability. We find that, while ORC-DF and FISPO follow a similar approach, they differ in key aspects that can have a substantial influence on their applicability and computational cost. The FISPO algorithm is more generally applicable, since it can analyse any nonlinear ODE model. The ORC-DF algorithm analyses models that are affine in the inputs, and if those models have known inputs it is sometimes more efficient. Thus, the optimal choice of a method depends on the characteristics of the problem under consideration. To facilitate the use of both algorithms we implement the ORC-DF algorithm in a new version of STRIKE-GOLDD, a MATLAB toolbox for structural identifiability and observability analysis. Since this software tool already had an implementation of the FISPO algorithm, the new release allows modellers and model users the convenience of choosing between different algorithms in a single tool, without changing the coding of their model.

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具有已知和未知输入的非线性可观测性算法：分析和实现

动态系统的可观察性受外部输入的影响，无论是已知的（例如控制动作）还是未知的（干扰）。未知量级的输入对可观测性尤其不利，并且它们也使分析复杂化。因此，能够分析具有未知输入的非线性系统的可观测性的计算工具的可用性直到最近才受到限制。最近为该任务提出了两种基于微分几何的符号算法 ORC-DF 和 FIspo，但仍缺乏对它们的批判性分析和比较。在这里，我们对两种算法进行分析比较并评估它们在一组问题上的性能，讨论它们的优点和局限性。此外，我们使用这些分析来提供有关输入和可观察性之间关系的某些方面的见解。我们发现，虽然 ORC-DF 和 Fispo 遵循类似的方法，但它们在关键方面有所不同，这可能对其适用性和计算成本产生重大影响。FIPO 算法更普遍适用，因为它可以分析任何非线性 ODE 模型。ORC-DF 算法分析输入中的仿射模型，如果这些模型具有已知输入，则有时效率更高。因此，方法的最佳选择取决于所考虑问题的特征。为了便于使用这两种算法，我们在新版本的 STRIKE-GOLDD 中实现了 ORC-DF 算法，STRIKE-GOLDD 是一个用于结构可识别性和可观察性分析的 MATLAB 工具箱。