hurwitz's criterion

简明释义

赫尔维兹稳定性判据

英英释义

例句

1.When analyzing polynomial roots, Hurwitz's criterion 赫尔维茨准则 provides a necessary condition for stability.

在分析多项式根时，赫尔维茨准则提供了稳定性的必要条件。

2.In control theory, we often use Hurwitz's criterion 赫尔维茨准则 to determine the stability of a system.

在控制理论中，我们常常使用赫尔维茨准则来确定系统的稳定性。

3.In our project, we demonstrated how Hurwitz's criterion 赫尔维茨准则 helps in designing stable controllers.

在我们的项目中，我们展示了赫尔维茨准则如何帮助设计稳定的控制器。

4.The engineer applied Hurwitz's criterion 赫尔维茨准则 to verify that the feedback loop was stable.

工程师应用赫尔维茨准则来验证反馈回路是稳定的。

5.Using Hurwitz's criterion 赫尔维茨准则, we can assess whether the characteristic equation has roots with negative real parts.

通过使用赫尔维茨准则，我们可以评估特征方程是否具有负实部的根。

作文

In the field of control theory and systems engineering, stability is a fundamental concept that engineers must grasp to ensure the reliable operation of systems. One important tool used to assess the stability of a polynomial is Hurwitz's criterion, which provides a systematic method for determining whether all roots of a polynomial have negative real parts. This criterion is particularly useful in the analysis of linear time-invariant (LTI) systems, where the characteristic polynomial plays a critical role in defining system behavior.To understand Hurwitz's criterion, we first need to consider the general form of a polynomial of degree n: P(s) = a_n s^n + a_{n-1} s^{n-1} + ... + a_1 s + a_0. The roots of this polynomial correspond to the system's poles, and their locations in the complex plane directly influence the stability of the system. If all poles are located in the left half of the complex plane, the system is considered stable; conversely, if any poles lie in the right half or on the imaginary axis, the system is unstable.Hurwitz's criterion provides a way to determine the stability of the polynomial without explicitly calculating its roots. It involves constructing the Hurwitz matrix, which is a square matrix formed from the coefficients of the polynomial. The first step is to arrange the coefficients of the polynomial in descending order of power. If any coefficients are missing, they should be filled with zeros. The Hurwitz matrix is then constructed by taking the coefficients and arranging them in a specific pattern.The next step is to compute the determinants of the leading principal minors of the Hurwitz matrix. For a polynomial to be stable according to Hurwitz's criterion, all leading principal minors must be positive. If any minor is non-positive, the system is deemed unstable. This method allows engineers to quickly assess the stability of a system without the need for complex calculations or root-finding algorithms.One of the key advantages of using Hurwitz's criterion is its applicability to higher-order polynomials. As systems become more complex, the degree of the characteristic polynomial increases, making traditional root-finding methods less practical. Hurwitz's criterion simplifies this process and provides a clear, systematic approach to stability analysis.In practice, engineers often use Hurwitz's criterion in conjunction with other stability analysis techniques, such as the Routh-Hurwitz stability criterion or Nyquist stability criterion. By combining these methods, they can gain a comprehensive understanding of system stability and make informed design decisions. This is particularly important in industries such as aerospace, automotive, and robotics, where system failures can have catastrophic consequences.In conclusion, Hurwitz's criterion serves as a vital tool in the toolbox of engineers working with dynamic systems. Its ability to assess stability through a straightforward matrix approach makes it invaluable in both theoretical studies and practical applications. As systems continue to grow in complexity, the importance of tools like Hurwitz's criterion will only increase, ensuring that engineers can maintain control over the systems they design and implement.

在控制理论和系统工程领域，稳定性是工程师必须掌握的一个基本概念，以确保系统的可靠运行。用于评估多项式稳定性的一个重要工具是Hurwitz准则，它提供了一种系统的方法来确定多项式的所有根是否具有负实部。该准则在分析线性时不变（LTI）系统时特别有用，因为特征多项式在定义系统行为中发挥着关键作用。要理解Hurwitz准则，我们首先需要考虑一般形式的n次多项式：P(s) = a_n s^n + a_{n-1} s^{n-1} + ... + a_1 s + a_0。该多项式的根对应于系统的极点，其在复平面中的位置直接影响系统的稳定性。如果所有极点位于复平面的左半部分，则系统被认为是稳定的；相反，如果任何极点位于右半部分或在虚轴上，则系统是不稳定的。Hurwitz准则提供了一种方法来判断多项式的稳定性，而无需明确计算其根。它涉及构建Hurwitz矩阵，这是一个由多项式系数形成的方阵。第一步是将多项式的系数按降幂排列。如果缺少任何系数，则应用零填充。然后，通过按照特定模式排列系数来构建Hurwitz矩阵。下一步是计算Hurwitz矩阵的主子行列式的行列式。根据Hurwitz准则，为了使多项式稳定，所有主子行列式必须为正。如果任何一个子行列式为非正，则系统被认为是不稳定的。这种方法使工程师能够快速评估系统的稳定性，而无需复杂的计算或求根算法。使用Hurwitz准则的一个主要优点是它适用于高阶多项式。随着系统复杂性的增加，特征多项式的次数也随之增加，使得传统的求根方法不太实际。Hurwitz准则简化了这一过程，并提供了一种清晰、系统的稳定性分析方法。在实践中，工程师通常将Hurwitz准则与其他稳定性分析技术结合使用，如Routh-Hurwitz稳定性准则或Nyquist稳定性准则。通过结合这些方法，他们可以全面了解系统的稳定性，并做出明智的设计决策。这在航空航天、汽车和机器人等行业尤为重要，因为系统故障可能会导致灾难性后果。总之，Hurwitz准则作为工程师在动态系统工作中的重要工具。它通过直接的矩阵方法评估稳定性的能力，使其在理论研究和实际应用中都具有不可或缺的价值。随着系统复杂性的不断增加，像Hurwitz准则这样的工具的重要性只会增加，确保工程师能够控制他们设计和实施的系统。