It is well known that given the continuous-time AutoRegressive representation |$A\left ( \rho \right ) \beta \left ( t\right ) =0,$| where |$\rho $| denotes the differential operator and |$A\left ( \rho \right ) $| a regular polynomial matrix, we can always construct the smooth behaviour of this system, by using the finite zero structure of |$A\left ( \rho \right ) $|⁠. The main theme of this work is to study the following inverse problem: given a specific smooth behaviour, find a family of regular polynomial matrices |$A\left ( \rho \right ) $|⁠, such that the system |$A\left ( \rho \right ) \beta \left ( t\right ) =0$| has exactly the prescribed behaviour. Following an idea coming from Antoulas & Willems (1993) and Willems (1986, 1991) we present an algorithm which solve this problem and can be easily implemented either in a computer programming language like C++ or in a computer algebra system like Mathematica.

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关于线性系统的精确建模

众所周知，给定连续时间AutoRegressive表示| $ A \ left（\ rho \ right）\ beta \ left（t \ right）= 0，$ | 其中| $ \ $ RHO | 表示微分运算符，| $ A \ left（\ rho \ right）$ | 常规多项式矩阵，我们总是可以使用| $ A \ left（\ rho \ right）$ |⁠的有限零结构来构造该系统的平稳行为。这项工作的主要主题是研究以下反问题：给定特定的平滑行为，找到一族正规多项式矩阵| $ A \ left（\ rho \ right）$ |⁠，这样系统| $ A \左（\ rho \ right）\ beta \左（t \ right）= 0 $ |完全符合规定的行为。跟随一个想法Antoulas＆Willems（1993）和 Willems（1986，1991）提出了一种解决此问题的算法，可以很容易地在计算机编程语言（如C ++）或计算机代数系统（如Mathematica）中实现。