Explanation Given: The below block diagram representation of a system: - Concept Used: Since we know that closed loop transfer function is as below: T ( s ) = G ( s ) 1 + G ( s ) H ( s ) use this to obtain characteristic equation i.e. 1 + G ( s ) H ( s ) , which will be the denominator of closed loop transfer function. For this 1 + G ( s ) H ( s ) make Routh table. Since we know that number of sign changes in the first column of the Routh table will be equal to number of closed loop right half plane poles. So, we will find the range of K so that system will be stable. Calculation: Since we know that closed loop transfer function is as below: T ( s ) = G ( s ) 1 + G ( s ) H ( s ) Thus, on solving the given block diagram we will get the closed loop transfer function as below G ( s ) = K × 1 s + 0.5 × 0.7 s 2 + 1.7 s + 0.3 , H ( s ) = 0.1 s + 0.1 T ( s ) = G ( s ) 1 + G ( s ) H ( s ) = 0.7 K ( s + 0.5 ) ( s 2 + 1.7 s + 0.3 ) 1 + 0.7 K ( s + 0.5 ) ( s 2 + 1.7 s + 0.3 ) × 0.1 s + 0.1 = 0.7 K ( s + 0.1 ) ( s + 0.5 ) ( s 2 + 1.7 s + 0.3 ) ( s + 0.5 ) ( s 2 + 1.7 s + 0.3 ) ( s + 0.1 ) + 0.07 K ( s + 0.5 ) ( s 2 + 1.7 s + 0.3 ) T ( s ) = 0.7 K ( s + 0.1 ) ( s 3 + 1.7 s 2 + 0.3 s + 0.5 s 2 + 0.85 s + 0.15 ) ( s + 0.1 ) + 0.07 K T ( s ) = 0.7 K ( s + 0.1 ) ( s 4 + 0.1 s 3 + 1.7 s 3 + 0.17 s 2 + 0.3 s 2 + 0.03 s + 0.5 s 3 + 0.05 s 2 + 0.85 s 2 + 0.085 s + 0.15 s + 0.015 + 0.07 K T ( s ) = 0.7 K ( s + 0.1 ) ( s 4 + 2.2 s 3 + 1.14 s 2 + 0.193 s + 0.07 K + 0.015 ) As we know the denominator part of closed loop transfer function gives us the characteristic equation 1 + G ( s ) H ( s ) . Thus, 1 + G ( s ) H ( s ) = ( s 4 + 2.2 s 3 + 1.14 s 2 + 0.193 s + 0.07 K + 0.015 ) Use this characteristic equation to make Routh table. s 4 1 1

7th Edition

ISBN: 9781119185666

Author: NISE

Publisher: WILEY

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Question

Chapter 6, Problem 57P

To determine

Range of gain K for the given system to be stable.

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