Commonly, variance-based global sensitivity analysis methods are popular and applicable to quantify the impact of a set of input variables on output response. However, for many engineering practical problems, the output response is not single but multiple, which makes some traditional sensitivity analysis methods difficult or unsuitable. Therefore, a novel global sensitivity analysis method is presented to evaluate the importance of multi-input variables to multi-output responses. First, assume that a multi-input multi-output system (MIMOS) includes $n$ variables and $m$ responses. A set of summatory functions $G(x)$ and $H(x)$ are constructed by the addition and subtraction of any two response functions. Naturally, each response function is represented using a set of summatory function. Subsequently, the summatory functions $G(x)$ and $H(x)$ are further decomposed based on the high dimensional model representation (HDMR), respectively. Due to the orthogonality of all the decomposed function sub-terms, the variance and covariance of each response function can be represented using the partial variances of all the decomposed function sub-terms on the corresponding summatory functions, respectively. The total fluctuation of MIMOS is calculated by the sum of the variances and covariances on all the response functions. Further, the fluctuation is represented as the sum of the total partial variances for all the $s$-order function sub-terms, and the total partial variance is the sum of $n$ partial variances for the corresponding $s$-order function sub-terms. Then, the function sensitivity index (FSI) ${\mathrm{\text{FSI}}}_s$ for s-order function sub-terms is defined by the ratio of the total partial variance and total fluctuation, which includes first-order, second-order, and high-order FSI. The variable sensitivity index ${\mathrm{\text{VSI}}}_i$ of variable $x_i$ is calculated by the sum of all the FSIs including the contribution of variable $x_i$. Finally, numerical example and engineering application are employed to demonstrate the accuracy and practicality of the presented global sensitivity analysis method for MIMOS.

通常，基于方差的全局敏感性分析方法很流行，适用于量化一组输入变量对输出响应的影响。然而，对于许多工程实际问题，输出响应不是单一的而是多重的，这使得一些传统的灵敏度分析方法变得困难或不适用。因此，提出了一种新的全局敏感性分析方法来评估多输入变量对多输出响应的重要性。首先，假设多输入多输出系统 (MIMOS) 包括$n$变量和$米$回应。一组求和函数$G(X)$和$H(X)$由任意两个响应函数的加减法构成。自然地，每个响应函数都使用一组求和函数来表示。随后，求和函数$G(X)$和$H(X)$分别基于高维模型表示（HDMR）进一步分解。由于所有分解函数子项的正交性，每个响应函数的方差和协方差可以分别用所有分解函数子项在相应求和函数上的偏方差来表示。MIMOS 的总波动由所有响应函数的方差和协方差之和计算得出。此外，波动表示为所有部分的总偏方差之和$s$-阶函数子项，总偏方差是$n$对应的部分方差$s$-阶函数子项。函数灵敏度指数（FSI）${\mathrm{\text{FSI}}}_s$对于s阶函数子项定义为总偏方差与总波动的比值，包括一阶、二阶和高阶 FSI。可变灵敏度指数${\mathrm{\text{VSI}}}_{\mathrm{一世}}$变量的$X_{\mathrm{一世}}$由所有 FSI 的总和计算，包括变量的贡献$X_{\mathrm{一世}}$. 最后，通过数值算例和工程应用，证明了所提出的 MIMOS 全局灵敏度分析方法的准确性和实用性。