Name: routh table special cases
Uploaded: 2025-05-22T09:06:20.000Z
Duration: 1 min 19 s
Channel: JoVE

在劳斯-赫尔维茨判据中，可能出现两种不同的情况。
当劳斯表第一列中出现唯一的零时，会导致除零问题。
为避免此问题，通常使用一个很小的数（通常为 epsilon）来替代零。
分析首先假设 ε 的符号。若选择 ε 为正值，则符号变化表明系统不稳定，且在右半平面有两个极点。同样，若选择 ε 为负值，也会得出系统不稳定的相同结论。
第二种情况出现在整行均由零组成时。这表明原多项式含有偶次多项式因子。
利用零行上一行的系数构造一个辅助多项式。对该多项式求导，并用该导数的系数替换表中的零。
此外，采用常规方法构建剩余的劳斯表。
偶次多项式和奇次多项式分别列成表格，右半平面极点的总数等于偶次和奇次多项式表中所找到极点数的总和。&#160;

routh-hurwitz 准则 ii

在应用 Routh-Hurwitz 准则时，可能会出现两种使稳定性分析复杂化的特殊情况。

第一种情况是，在劳斯表的第一列中出现奇异零点。这种情况会产生除以零的问题。为了解决这个问题，用一个小的正数或负数（表示为 epsilon (ε)）代替零点。稳定性分析通过假设 ε 的符号展开。如果 ε 为正，劳斯表第一列中的任何符号变化都表明系统不稳定，两个极点位于右半 s 平面。相反，如果 ε 为负，则得出同样的不稳定结论。

第二种情况是，劳斯表中的整行仅由零组成。这种情况表明原始多项式有一个偶数多项式作为因数。为了解决这个问题，使用零行上方行的系数构造辅助多项式。然后对该辅助多项式进行微分，并用求导出来的系数替换劳斯表中的零。构建剩余劳斯表的标准流程从这一点开始。

当分别处理偶数多项式和奇数多项式时，右半s平面上的极点总数由偶数多项式和奇数多项式的劳斯表上的极点总数相加来确定。这种方法确保对系统稳定性的全面分析。

Routh-Hurwitz 准则中的这些特殊情况对于准确确定系统的稳定性至关重要。通过小心地处理第一列和第一行的零点，工程师可以避免对系统稳定性的误解。这种方法可以精确识别右半 S 平面极点，确保系统设计和分析稳健可靠。

Routh-Hurwitz 准则 II

在应用劳斯-赫尔维茨判据时，可能出现两种特定情况，使稳定性分析变得复杂。  
第一种情况发生在当一个……

Education

教育

JoVE Core

Electrical Engineering

电气工程

Unit 1

第一单元

Transient and Steady-state Response Analysis

瞬态与稳态响应分析

transient-and-steady-state-response-analysis

KEY TERMS AND CONCEPTS

关键术语与概念

Routh-Hurwitz Criterion II

routh table special cases

Routh table special cases can make stability analysis harder when using the Routh-Hurwitz criterion. Two situations need extra care. One is a zero in the first column. The other is an entire row of zeros.

When a single zero appears in the first column, division by zero would occur. The usual fix is to replace that zero with a very small number, epsilon (ε). The stability test then continues by assuming a sign for ε. If ε is positive, any sign change in the first column means the system is unstable, with two poles in the right-half s-plane. The same unstable result also follows if ε is negative.

A full row of zeros points to another case. It suggests that the original polynomial has an even polynomial factor. To handle it, an auxiliary polynomial is built from the coefficients in the row above the zero row. That auxiliary polynomial is then differentiated, and the derivative coefficients replace the zeros in the Routh table. After that, the standard Routh process continues.

When even and odd polynomials are handled separately, the total number of poles in the right-half s-plane comes from adding the poles found in both Routh tables. This step helps give a complete stability result. Careful treatment of first-column zeros and zero rows prevents wrong conclusions about system stability and improves the identification of right-half s-plane poles.

Routh Table Special Cases

transient and steady-state response

Control System Test Signals and Stability

Control system test signals help measure how a system reacts to change, stability, and accuracy. A ramp function is useful when a system changes gradually. A step function is used when a system faces a sudden disturbance. An impulse function is the best test signal for a shock input. These signals are used to study two main performance features: transient response and steady-state response. Transient response is the short-term change from the initial state to the final state. It shows how fast...

first order systems

RC System Responses to Step Inputs

RC circuits and other first-order systems show how a simple input can shape system response over time. These systems are often studied through their transfer function, which links the input and output under zero initial conditions. That relationship makes it easier to analyze how the system behaves when the input changes. A unit-step input gives a response that starts at zero and rises toward one. The Laplace transform of the step input is applied to the transfer function, then the result is...

second order systems i

Servo System Response in Second-Order Models

A servo system is a second-order model that links an input position to an output position. It uses a proportional controller and load elements to keep the output aligned with the input. This behavior can be written as a second-order differential equation. When the equation is transformed with the Laplace transform under zero initial conditions, it becomes a transfer function. The transfer function shows how the input is converted into the output. It can also be rewritten as a closed-loop...

second order systems ii

Step Response of Second-Order Systems

Second-order systems react to a unit-step input in different ways depending on the damping ratio. When the damping ratio, zeta, is between 0 and 1, the system is underdamped. In this case, inverse Laplace methods give an output that oscillates with a decreasing amplitude. The error signal, which is the difference between the input and output, also shows damped oscillations before it goes to zero at steady state. When zeta equals 0, the system is undamped. The response keeps oscillating forever...

stability

System Response and Stability

System response and stability describe how a linear time-invariant (LTI) system reacts to a change in input. The time response has two parts: transient response and steady-state response. The transient response is the first reaction and it fades to zero over time. The steady-state response remains after the transient effects have disappeared. An LTI system is stable when it gives a bounded output for a bounded input. For these systems, stability depends on the roots of the characteristic...

routh-hurwitz criterion i

Checking Stability with the Routh Table

Checking stability with the Routh table is a useful part of analyzing a closed-loop transfer function. It helps determine whether a system will remain stable when load conditions change or when a fault occurs. In a power grid, this kind of check is important because instability can lead to blackouts. The Routh-Hurwitz criterion uses a table built from the denominator polynomial of the transfer function. The rows of the Routh table are labeled by powers of the complex variable s, starting with...

routh-hurwitz criterion ii

Routh table special cases can make stability analysis harder when using the Routh-Hurwitz criterion. Two situations need extra care. One is a zero in the first column. The other is an entire row of zeros. When a single zero appears in the first column, division by zero would occur. The usual fix is to replace that zero with a very small number, epsilon (ε). The stability test then continues by assuming a sign for ε. If ε is positive, any sign change in the first column means the system is...

key terms and concepts

Learn how to handle zeros in the Routh table, use epsilon for first-column issues, and apply an auxiliary polynomial when a row of zeros appears.

Routh Table Special Cases | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Routh Table Special Cases

Description

Transcript

Tags