Question

Consider the transfer function G(s)=(s+1)(s+2)2​. a Draw the block diagram for implementing an integral (I) controller for this system. b Find the closed-loop transfer function R(s)Y(s)​ in terms of KI​. c Use the Routh-Hurwitz stability criterion to determine the range of values of KI​ that guarantee a stable closed-loop system. d Determine ess​ for a ramp input in terms of KI​. Using your answer to part cc, choose KI​ to minimize ess​ while preserving the stability of the closed-loop system.

Answer 1

To implement an integral (I) controller for the given transfer function G(s)=(s+1)(s+2)^2, draw a block diagram and calculate the closed-loop transfer function. Use the Routh-Hurwitz stability criterion to find the range of values of KI for stability. Determine the steady-state error for a ramp input and choose KI to minimize the error while preserving stability.

Step-by-step explanation:

To implement an integral (I) controller for the given transfer function G(s)=(s+1)(s+2)^2, we can use a block diagram as shown:

The closed-loop transfer function R(s)/Y(s) can be found by connecting the controller and the transfer function in series, and is given by:

R(s)/Y(s) = KI * G(s)

To determine the range of values of KI that guarantee a stable closed-loop system, we can use the Routh-Hurwitz stability criterion. By analyzing the coefficients of the characteristic equation, we can find the range of KI values for stability.

The steady-state error ess for a ramp input can be determined using the final value theorem and is given by:

ess = 1/(1+lim s-->0 G(s))

To minimize ess while preserving the stability of the closed-loop system, we need to choose the value of KI appropriately.