Name: checking stability with the routh table
Uploaded: 2024-11-21T14:54:10.000Z
Duration: 1 min 15 s
Channel: JoVE

考虑一个电力系统，其稳定性对于防止停电至关重要。利用劳斯-赫尔维茨判据，可以评估该系统在不同负载条件或故障情况下的稳定性。
考虑一个闭环传递函数。为了构建劳斯表，各行应从最高次幂开始，用 s 的各次幂进行标记。
第一行从最高次幂开始，水平填入分母的每隔一个的系数。第二行则从次高次幂开始，列出被跳过的系数。
后续项通过前一行的负行列式计算，再除以上方第一列的对应项得到。
对于一个系统，需计算劳斯表的各行。各行可独立地乘以正的常数因子。
劳斯-赫尔维茨判据指出，多项式在右半平面的根的个数等于劳斯表第一列中符号变化的次数，表明系统不稳定。
如果所有极点均位于 s 平面的左半部分，则系统稳定，这相当于劳斯表第一列中无符号变化。

劳思-赫尔维茨稳定判据 i

以电网为例，稳定性对于防止停电至关重要。劳思-赫尔维茨稳定判据是评估系统在不同负载条件或故障下的稳定性的有力工具。通过分析闭环传递函数，劳思-赫尔维茨稳定判据有助于确定系统是否保持稳定。

为了应用劳思-赫尔维茨稳定判据，需要构建一个劳思表。表中的行标有复频率变量 s 的幂，从最高幂开始。变量 s 表示复数，通常采用以下形式:

<img alt="Equation1" src="/files/ftp_upload/16116/16116_PT%20Equation_1.svg" style="height: 14px;">

其中 σ 是实部，jω 是虚部。第一行每隔一个系数水平填充分母多项式的系数，从最高幂开始。第二行则填充第一行中跳过的系数。

劳思表中的后续条目使用前几行的行列式进行计算。具体来说，每个条目都是通过取其上方两个条目的负行列式并将其除以其正上方列中的第一个条目来确定的。此过程持续进行，直到填满整个表格。

为了说明这一点，请考虑一个计算劳思表行的系统。如果需要，每行都通过正常数单独缩放以简化计算。根据劳思-赫尔维茨稳定判据，右半 s 平面中多项式的根数对应于劳思表第一列中符号改变的次数。这些符号变化表示不稳定性。

相反，如果所有极点都在左半 s 平面，即劳思表第一列的符号没有变化，则系统稳定。确保没有发生符号变化可确认系统在各种操作条件下都保持稳定。

理解和应用劳思-赫尔维茨稳定判据对于维持电网等复杂系统的稳定性至关重要。通过验证所有极点是否位于左半 s 平面，工程师可以确保系统可靠运行，防止停电等问题并确保持续供电。该方法提供了一种系统的稳定性分析方法，可识别和缓解系统中的潜在不稳定性。

劳思-赫尔维茨稳定判据 I

考虑一个电力系统，其中稳定性对于防止停电至关重要。利用劳斯-赫尔维茨判据，分析系统在……下的稳定性。

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 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 the highest power. Here, s is a complex number with a real part, σ, and an imaginary part, jω.

The first row is filled with every other coefficient from the denominator polynomial. It starts with the coefficient from the highest power term. The second row then uses the coefficients that were skipped in the first row.

The remaining entries are found from the rows above them. Each new value is calculated from the negative determinant of the two entries above it, divided by the first entry in the column directly above. If needed, each row can be scaled by a positive constant to make the calculations simpler.

Once the table is complete, the first column gives the stability result. The number of sign changes in that column matches the number of roots in the right-half s-plane. Sign changes show instability. If there are no sign changes, then all poles are in the left-half s-plane and the system is stable. This makes the Routh-Hurwitz criterion a systematic way to test whether a system can operate reliably under different conditions.

Checking Stability with the Routh Table

transient and steady-state response

Control System Test Signals and Stability

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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 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

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 the Routh-Hurwitz criterion uses a Routh table to test system stability by tracking sign changes and pole locations in the s-plane.

Checking Stability with the Routh Table | 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

Checking Stability with the Routh Table

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