P11.1 A first-order system is represented by the time-do

main differential equation

x t ( ) = + x t( ) u t( ).

A feedback controller is to be designed such that

u t( ) = −2 , kx( )t 

and the desired equilibrium condition is x t( ) = 0 as

t → ∞. The performance integral is defined as

and the initial value of the state variable is x( ) 0 3 = .

Obtain the value of k in order to make J a minimum.

Is this k physically realizable? Select a practical value for the gain k, and evaluate the performance index with that gain. Is the system stable without the feed

back due to u t( )?

x dt, 0 xS = 1 J