How to design a control law for the system 1/(s-1) using one of the frequency response techniques?

I tried to work it out with bode plot design using lag and lead compensators but couldn't get a good result for this !, whenever I'm trying one of the techniques, the system is still unstable.


any suggestions?How to design a control law for the system 1/(s-1) using one of the frequency response techniques?
1/ (s-1) is a first order open loop unstable process. To write this out a bit better it is...





Gm(s) = Kp / ( 蟿p s - 1 )





where:


Kp = 1


蟿p = 1





The system has an open loop pole in the RHP at s = +1/蟿p. The unit step response is an exponential that goes off to infinity as time increases.





If you close the loop with a proportioning controller you can make it conditionally closed loop stable depending on the gain of the controller.





Applying a proportioning controller B(s) = Kc then closing the loop yields:





1 + Gm(s)B(s) = 1 + [ ( Kp Kc ) / ( 蟿p s - 1 ) ] = 0





then, there is a closed loop root at s = ( 1 - Kc Kp ) / 蟿p





The loop is closed loop unstable for small values of controller gain. Once the controller gain equals 1/Kp, the closed loop root is at the origin of a root locus plot. When the controller gain is larger than 1/Kp, the root is in the LHP so the system is closed loop stable.





Since this explanation applies a closed loop solution applying root locus stability analysis, I'll leave dealing with it in the frequency domain using Bode and Nyquist to you...