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

Routh-Hurwitz 기준에서는 두 가지 뚜렷한 상황이 발생할 수 있습니다.
Routh 테이블의 첫 번째 열에 단수 0이 독점적으로 나타나면 0으로 나누기 문제가 발생합니다.
이 문제를 피하기 위해 일반적으로 엡실론과 같은 작은 숫자가 0을 대체합니다.
해석은 엡실론에 대한 기호를 가정하는 것으로 시작됩니다. 엡실론이 양수로 선택되면 부호 변경은 시스템이 오른쪽 반평면에 두 개의 극이 있어 불안정하다는 것을 나타냅니다. 마찬가지로, 음수를 선택하면 동일한 불안정성 결론에 도달하게 됩니다.
두 번째 시나리오는 전체 행이 0으로만 구성되는 경우에 발생합니다. 이것은 짝수 다항식이 원래 다항식의 인수임을 시사합니다.
보조 다항식은 0 행 위에 있는 행의 계수를 사용하여 형성됩니다. 이것은 미분되며, 이 도함수의 계수가 표의 0을 대체합니다.
또한 나머지 Routh 테이블을 구성하는 일반적인 방법이 구현됩니다.
짝수 다항식과 홀수 다항식은 별도로 표로 작성되며, 오른쪽 절반 평면 극점의 총 수는 짝수 및 홀수 다항식 표에서 발견된 극점의 합입니다.

루스-허위츠 기준 ii

Routh-Hurwitz 기준을 적용할 때 안정성 분석을 복잡하게 만드는 두 가지 특정 시나리오가 발생할 수 있습니다.

첫 번째 시나리오는 Routh 표의 첫 번째 열에 단일 0이 나타날 때 발생합니다. 이 상황은 0으로 나누는 문제를 일으킵니다. 이를 해결하기 위해 엡실론(ε)으로 표시되는 작은 양수 또는 음수가 0으로 대체됩니다. 안정성 분석은 ε에 대한 부호를 가정하여 진행됩니다. ε가 양수이면 Routh 표의 첫 번째 열에서 부호가 변경되면 시스템이 불안정하고 두 극이 오른쪽 반 s 평면에 위치한다는 것을 나타냅니다. 반대로 ε가 음수이면 불안정하다는 동일한 결론에 도달합니다.

두 번째 시나리오는 Routh 표의 전체 행이 0으로만 구성된 경우 발생합니다. 이 현상은 원래 다항식에 인수로 짝수 다항식이 있음을 시사합니다. 이를 해결하기 위해 0 행 위의 행의 계수를 사용하여 보조 다항식을 구성합니다. 그런 다음 이 보조 다항식을 미분하고, 미분의 계수를 Routh 표의 0으로 대체합니다. 나머지 Routh 표를 구성하는 표준 절차는 이 지점에서 계속됩니다.

짝수와 홀수 다항식을 따로 다룰 때, 오른쪽 반 s-평면의 극의 총 수는 짝수와 홀수 다항식의 Routh 표에서 발견된 극을 합산하여 결정됩니다. 이 방법은 시스템 안정성에 대한 포괄적인 분석을 보장합니다.

Routh-Hurwitz 기준 내의 이러한 특수한 경우는 시스템의 안정성을 정확하게 결정하는 데 중요합니다. 엔지니어는 첫 번째 열과 행의 0을 신중하게 관리함으로써 시스템 안정성을 잘못 해석하는 것을 피할 수 있습니다. 이 접근 방식을 사용하면 오른쪽 반 s 평면 극을 정확하게 식별하여 견고하고 신뢰할 수 있는 시스템 설계 및 분석을 보장할 수 있습니다.

루스-허위츠 기준 II

Routh-Hurwitz 기준에서는 두 가지 뚜렷한 상황이 발생할 수 있습니다.
단수 0이 Routh 테이블의 첫 번째 열에 독점적으로 나타나면 ...

Education

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JoVE Core

Electrical Engineering

전기공학

Unit 1

제1단원

Transient and Steady-state Response Analysis

과도 및 정상 상태 응답 분석

transient-and-steady-state-response-analysis

KEY TERMS AND CONCEPTS

주요 용어 및 개념

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

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