You can find what you are looking for in the following paper:

R.El-Khazali, ”Fractional-Order Controller Design ,”Comp. & Math. With

Applications, Vol. 66, No. 5, pp. 539-646, September 2013,

First find the parameter (steady state error, peak overshoot, rise time , settling time, phase margin , gain margin) value of your plant without using any type of controller such as P,PI and PID in closed loop. After that according to your requirement tune the PID controller parameter Kp, Ki and Kd.

probably no need to use controller and compensator with each other and then using PID, because your new plant properties would be change with PID or Lead-Lag controller.

Let's say what I have here is  - I got the closed control system (already tuned PID and plant ) in which the performance is not so good. And I would like to keep the tuned PID controlled value remain the same. Then as shown in case 2 in pic, I will like to determine the time constant value from compensator value from step response of G(s). Then I will use that time constant value in compensator equation , then( multiply compensator and plant function) which will be G1(s). Then I will just use with the same PID controller value which I got earlier.Meaning I just tune the PID first , but I may not get the desired value and would like to improve performance by compensating the plant.  May I know where can find the procedure in anywhere ?Thank you very much indeed for your kind suggestions.

1. You may look at Guillemin-Truxal method that shows how to obtain desired system with desired transfer function by using compensator. In essence it is based on cancelling undesired poles and zeros, and adding desired one. 

In general you should not cancel unstable poles and zeros. 

Ideal case would be to cancel whole transfer function of the plant with its inverse 

(1/G(s))*G(s)=1, 

but 1/G(s) is in most of the cases physically unrealizable. 

http://www.atp.ruhr-uni-bochum.de/rt1/syscontrol/node73.html

2. Two-Degrees-of-Freedom PID, Quantitative Feedback Theory, or other advanced PID structures may help, that use prefilters and/or feedforward part: 

Tutorial on 2-DOF PID:

http://ijcas.com/admin/paper/files/401-411.pdf

Book on QFT: 

Houpis, Rasmussen, Garcia-Sanz, "Quantitative Feedback Theory: Fundamentals and Applications"

Book on Advanced PID:

Astrom, Hagglund, "Advanced PID Control"

You can refer to Ogata's book, modern control engineering, for compensator design.

Your PID tuning may not be good. You may try the following MATALB tuning method.

PID Controller Tuning using Simulink for Multi Area Power Systems

International Journal of Engineering Research & Technology (IJERT) 09/2012;

At the risk of indulging in shameless self-promotion, you could take a look at my book "Practical Control Engineering" where compensation and PI controllers are considered. However, as the book suggests, your best bet is to first thoroughly characterize the process via open loop step changes. After the characterization you will likely have a great amount of insight and will be able to design a simple controller to achieve your goals.

You can also refer "Morden Control Theory" by Dorf and Bishop. In this book you find examples of compensation along with optimal control and observer design. You also find compensation in z-domain.

I suggest to have a look to Ogata's book:"M0dern Control Engineering"