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In the formation of Routh-Hurwitz array for a polynomial, all the elements of a row have zero values. This premature termination of the array indicates the presence of
- a pair of real roots with opposite sign
- complex conjugate roots on the imaginary axis
- a pair of complex conjugate roots with opposite real parts
Which of the above statements are correct?
- a)Only 2
- b)2 and 3
- c)Only 3
- d)1, 2 and 3
Correct answer is option 'D'. Can you explain this answer?
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In the formation of Routh-Hurwitz array for a polynomial, all the elem...
Introduction:
The Routh-Hurwitz array is a mathematical tool used in control systems engineering to determine the stability of a system described by a polynomial equation. By examining the array, we can determine the number of roots of the polynomial that have positive real parts, as well as the presence of any purely imaginary roots.
Explanation:
The Routh-Hurwitz array is formed by taking the coefficients of the polynomial and arranging them in a specific pattern. Each row of the array depends on the previous two rows. The elements of the first row are the coefficients of the even powers of the polynomial, while the elements of the second row are the coefficients of the odd powers. Subsequent rows are calculated using a set of formulas.
Premature Termination:
If a row of the Routh-Hurwitz array has all zero elements, it indicates that there is a pair of complex conjugate roots with opposite real parts. This is because the elements of each row of the array are derived from the coefficients of the polynomial, and if all the elements in a row are zero, it means that all the coefficients that contribute to those elements are also zero. In the context of the Routh-Hurwitz array, this corresponds to having a pair of complex conjugate roots on the imaginary axis.
Opposite Sign:
In addition to complex conjugate roots, the premature termination of a row with all zero elements also indicates that the complex roots have opposite signs. This means that one of the roots is positive and the other is negative. This can be inferred from the fact that the coefficients of the polynomial alternate in sign, and if all the elements in a row are zero, it means that the signs of the coefficients in that row alternate as well.
Conclusion:
In conclusion, the premature termination of a row in the Routh-Hurwitz array with all zero elements indicates the presence of a pair of complex conjugate roots with opposite real parts and opposite signs. Therefore, both statements 2 and 3 are correct. Additionally, since statement 1 is a general case that includes the specific cases mentioned in statements 2 and 3, it is also correct. Hence, the correct answer is option D, which includes all three statements.
The Routh-Hurwitz array is a mathematical tool used in control systems engineering to determine the stability of a system described by a polynomial equation. By examining the array, we can determine the number of roots of the polynomial that have positive real parts, as well as the presence of any purely imaginary roots.
Explanation:
The Routh-Hurwitz array is formed by taking the coefficients of the polynomial and arranging them in a specific pattern. Each row of the array depends on the previous two rows. The elements of the first row are the coefficients of the even powers of the polynomial, while the elements of the second row are the coefficients of the odd powers. Subsequent rows are calculated using a set of formulas.
Premature Termination:
If a row of the Routh-Hurwitz array has all zero elements, it indicates that there is a pair of complex conjugate roots with opposite real parts. This is because the elements of each row of the array are derived from the coefficients of the polynomial, and if all the elements in a row are zero, it means that all the coefficients that contribute to those elements are also zero. In the context of the Routh-Hurwitz array, this corresponds to having a pair of complex conjugate roots on the imaginary axis.
Opposite Sign:
In addition to complex conjugate roots, the premature termination of a row with all zero elements also indicates that the complex roots have opposite signs. This means that one of the roots is positive and the other is negative. This can be inferred from the fact that the coefficients of the polynomial alternate in sign, and if all the elements in a row are zero, it means that the signs of the coefficients in that row alternate as well.
Conclusion:
In conclusion, the premature termination of a row in the Routh-Hurwitz array with all zero elements indicates the presence of a pair of complex conjugate roots with opposite real parts and opposite signs. Therefore, both statements 2 and 3 are correct. Additionally, since statement 1 is a general case that includes the specific cases mentioned in statements 2 and 3, it is also correct. Hence, the correct answer is option D, which includes all three statements.
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In the formation of Routh-Hurwitz array for a polynomial, all the elem...
Routh-Hurwitz Stability Criterion: It is used to test the stability of an LTI system.
The characteristic equation for a given open-loop transfer function G(s) is 1 + G(s) H(s) = 0
According to the Routh tabulation method,
The system is said to be stable if there are no sign changes in the first column of the Routh array
The number of poles lies on the right half of s plane = number of sign changes
A row of zeros in a Routh table:
This situation occurs when the characteristic equation has
The characteristic equation for a given open-loop transfer function G(s) is 1 + G(s) H(s) = 0
According to the Routh tabulation method,
The system is said to be stable if there are no sign changes in the first column of the Routh array
The number of poles lies on the right half of s plane = number of sign changes
A row of zeros in a Routh table:
This situation occurs when the characteristic equation has
- a pair of real roots with opposite sign (±a)
- complex conjugate roots on the imaginary axis (± jω)
- a pair of complex conjugate roots with opposite real parts (-a ± jb, a ± jb)
The procedure to overcome this as follows:
- Form the auxiliary equation from the preceding row to the row of zeros
- Complete Routh array by replacing the zero row with the coefficients obtained by differentiating the auxiliary equation.
- The roots of the auxiliary equation are also the roots of the characteristic equation.
- The roots of the auxiliary equation occur in pairs and are of the opposite sign of each other.
- The auxiliary equation is always even in order.
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