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Chapter5 - CHAPTER 5 5.1 CONCEPTS OF STABILITY AND THE...

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CHAPTER 5 CONCEPTS OF STABILITY AND THE ROUTH STABILITY CRITERION 5.1 (a) All the elements in the first column of the Routh array are +ve. Therefore, all the roots are in the left-half plane. (b) Two sign changes are found in the first column of the Routh array. Therefore, two roots are in the right-half plane and the rest in the left- half plane. (c) s 3 -row of the Routh array has zero pivot element, but the entire row is not all zeros. We replace the pivot element by e and then proceed with the construction of the Routh array. As e fi 0, two sign changes are found in the first column of the Routh array. Therefore, two roots are in right-half plane and the rest in the left-half plane. (d) s 1 -row of the Routh array is an all-zero row. Auxiliary polynomial formed using the elements of s 2 -row is given by A ( s ) = s 2 + 1 We replace the elements of s 1 -row with the coefficients of dA s ds ( ) = 2 s + 0 and proceed with the construction of the Routh array. There are no sign changes in the resulting Routh array; the characteristic polynomial does not have any root in the right-half plane. The roots of the 2nd-order auxiliary polynomial are therefore purely imaginary. The given characteristic equation has two roots on the imaginary axis and the rest in the left-half plane. (e) Since all the coefficients of the given characteristic polynomial are not of the same sign, the system is unstable. The Routh array formation is required only if the number of roots in the right-half plane are to be determined.
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