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Routh Hurwitz criterion is used to determine
- a)peak response of the system
- b)time response of the system
- c)absolute stability of the system
- d)roots of the characteristic equation graphically
Correct answer is option 'C'. Can you explain this answer?
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Routh Hurwitz criterion is used to determinea)peak response of the sys...
C) Absolute Stability of the System
The Routh-Hurwitz criterion is a mathematical method used to determine the absolute stability of a linear time-invariant (LTI) system. It provides a way to analyze the stability of a system without solving the characteristic equation directly or finding the roots of the equation.
The Routh-Hurwitz criterion is based on the coefficients of the characteristic equation of the system. The characteristic equation is obtained by setting the denominator of the transfer function equal to zero. For a system with a transfer function G(s), the characteristic equation can be written as:
1 + G(s) = 0
To apply the Routh-Hurwitz criterion, the coefficients of the characteristic equation are arranged in a special tabular form called the Routh array. The Routh array is created by organizing the coefficients in descending powers of 's' and then filling in the array using a set of recursive equations. The first two rows of the array are obtained directly from the coefficients of the characteristic equation.
The Routh array has a specific pattern, and the sign changes in the first column of the array are used to determine the stability of the system. If all the sign changes in the first column are positive (no sign changes), then the system is stable. If there are any sign changes, the number of sign changes determines the number of poles in the right-half plane (RHP). If there are no poles in the RHP, the system is stable. If there are any poles in the RHP, the system is unstable.
The Routh-Hurwitz criterion allows engineers to quickly determine the stability of a system without the need to find the roots of the characteristic equation. This is advantageous because finding the roots of higher-order polynomials can be complex and time-consuming. By using the Routh-Hurwitz criterion, engineers can easily determine the stability of a system and make design decisions accordingly.
In summary, the Routh-Hurwitz criterion is used to determine the absolute stability of a system by analyzing the sign changes in the first column of the Routh array. It is a powerful tool in control system analysis and design, allowing engineers to assess the stability of a system without solving the characteristic equation or finding the roots of the equation.
The Routh-Hurwitz criterion is a mathematical method used to determine the absolute stability of a linear time-invariant (LTI) system. It provides a way to analyze the stability of a system without solving the characteristic equation directly or finding the roots of the equation.
The Routh-Hurwitz criterion is based on the coefficients of the characteristic equation of the system. The characteristic equation is obtained by setting the denominator of the transfer function equal to zero. For a system with a transfer function G(s), the characteristic equation can be written as:
1 + G(s) = 0
To apply the Routh-Hurwitz criterion, the coefficients of the characteristic equation are arranged in a special tabular form called the Routh array. The Routh array is created by organizing the coefficients in descending powers of 's' and then filling in the array using a set of recursive equations. The first two rows of the array are obtained directly from the coefficients of the characteristic equation.
The Routh array has a specific pattern, and the sign changes in the first column of the array are used to determine the stability of the system. If all the sign changes in the first column are positive (no sign changes), then the system is stable. If there are any sign changes, the number of sign changes determines the number of poles in the right-half plane (RHP). If there are no poles in the RHP, the system is stable. If there are any poles in the RHP, the system is unstable.
The Routh-Hurwitz criterion allows engineers to quickly determine the stability of a system without the need to find the roots of the characteristic equation. This is advantageous because finding the roots of higher-order polynomials can be complex and time-consuming. By using the Routh-Hurwitz criterion, engineers can easily determine the stability of a system and make design decisions accordingly.
In summary, the Routh-Hurwitz criterion is used to determine the absolute stability of a system by analyzing the sign changes in the first column of the Routh array. It is a powerful tool in control system analysis and design, allowing engineers to assess the stability of a system without solving the characteristic equation or finding the roots of the equation.
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Routh Hurwitz criterion is used to determinea)peak response of the sys...
Routh-Hurwitz criterion:
- Using the Routh-Hurwitz method, the stability information can be obtained without the need to solve the closed-loop system poles. This can be achieved by determining the number of poles that are in the left-half or right-half plane and on the imaginary axis.
- This involves checking the roots of the characteristic polynomial of a linear system to determine its stability.
- It is used to determine the absolute stability of a system.
Important points
Other methods of determining stability include:
Root locus:
- This method gives the position of the roots of the characteristic equation as the gain K is varied.
- With Root locus (unlike the case with Routh-Hurwitz criterion), we can do both analysis (i.e., for each gain value we know where the closed-loop poles are) and design (i.e., on the curve we can search for a gain value that results in the desired closed-loop poles).
Nyquist plot:
- This method is mainly used for assessing the stability of a system with feedback.
- While Nyquist is a graphical technique, it only provides a limited amount of intuition for why a system is stable or unstable, or how to modify an unstable system to be stable.
Techniques like Bode plots, while less general, are sometimes a more useful design tool.
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