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

No critério de Routh-Hurwitz, duas situações distintas podem ocorrer.
Quando um zero singular aparece exclusivamente na primeira coluna da tabela de Routh, ele aumenta a divisão por zero.
Para contornar esse problema, um pequeno número, geralmente épsilon, substitui o zero.
A interpretação começa assumindo um sinal para épsilon. Se o épsilon for escolhido como positivo, a mudança de sinal indica que o sistema é instável com dois pólos no meio-plano direito. Da mesma forma, escolher um negativo leva à mesma conclusão de instabilidade.
O segundo cenário ocorre quando uma linha inteira compreende apenas zeros. Isso sugere que um polinômio par é um fator do polinômio original.
Um polinômio auxiliar é formado usando os coeficientes da linha acima da linha zero. Isso é diferenciado e os coeficientes dessa derivada substituem os zeros na tabela.
Além disso, o método usual para construir a tabela de Routh restante é implementado.
Polinômios pares e ímpares são tabulados separadamente, e o número total de pólos do meio-plano direito é a soma daqueles encontrados nas tabelas de polinômios pares e ímpares.

critério routh-hurwitz ii

Na aplicação do critério Routh-Hurwitz, dois cenários específicos podem surgir que complicam a análise de estabilidade.

O primeiro cenário ocorre quando um zero singular aparece na primeira coluna da tabela Routh. Essa situação cria problemas de divisão por zero. Para resolver isso, um pequeno número positivo ou negativo, denotado como épsilon (ε), é substituído pelo zero. A análise de estabilidade prossegue assumindo um sinal para ε. Se ε for positivo, qualquer mudança de sinal na primeira coluna da tabela Routh indica que o sistema é instável, com dois polos localizados no plano s da metade direita. Por outro lado, se ε for negativo, a mesma conclusão de instabilidade é alcançada.

O segundo cenário surge quando uma linha inteira na tabela Routh consiste apenas de zeros. Essa ocorrência sugere que o polinômio original tem um polinômio par como fator. Para resolver isso, um polinômio auxiliar é construído usando os coeficientes da linha acima da linha zero. Este polinômio auxiliar é então diferenciado, e os coeficientes da derivada substituem os zeros na tabela de Routh. O procedimento padrão para construir a tabela de Routh restante continua deste ponto.

Ao lidar com polinômios pares e ímpares separadamente, o número total de polos no plano s da metade direita é determinado pela soma dos polos encontrados nas tabelas de Routh dos polinômios pares e ímpares. Este método garante uma análise abrangente da estabilidade do sistema.

Esses casos especiais no critério de Routh-Hurwitz são cruciais para determinar com precisão a estabilidade de um sistema. Ao gerenciar cuidadosamente os zeros na primeira coluna e linhas de zeros, os engenheiros podem evitar interpretações errôneas da estabilidade do sistema. Esta abordagem permite a identificação precisa dos polos do plano s da metade direita, garantindo um projeto e análise de sistema robustos e confiáveis.

Critério Routh-Hurwitz II

No critério de Routh-Hurwitz, duas situações distintas podem ocorrer.
Quando um zero singular aparece exclusivamente na primeira coluna da tabela de Routh, ele ...

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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.

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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...

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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.

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Routh Table Special Cases

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