Hello good day, the main difference is that when using u = kx are performing feedback control just like a proportional type which does not consider the error in steady state which will exist instead to use u = kx states + v variable v is the output of an integrator block is added to remove steady-state error to normal feedback control states which would like as servo system.

For more information on this subject check:

https://hellsingge.files.wordpress.com/2014/10/ingenieria-de-control-moderna-ogata-5ed.pdf

Provided that K is chosen such that A-BK has eigenvalues with strictly negative real parts, and that v is a constant, the difference is that u=Kx will stabilize the equilibrium point x=0, while u=Kx+v will stabilize the equilibrium point x=-(A-BK)^(-1)Bv.

I don't know if this helps, your question should be clarified (What is v? What do you want to achieve with the control?)

Thank you Professor Diaz. Have a good day. Your points are very helpful to me. The book suggested by you is in Spanish which I can not read. If you have some related material in English, please send it to me. Thanks again!

Thank you Professor Martin. Your answer really helps me a lot. Let me pose a more precise statement of the problem. ``If I consider my feedback control $u = Kx$ only, no noise (i.e., v); for some specific goal in mind. Is it OK from the control theoretic aspects or there is any problem?" Thank you once again!

Hello, Here's more information:

http://web.mit.edu/2.14/www/Handouts/StateSpace.pdf

http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction§ion=ControlStateSpace

https://www.researchgate.net/publication/228007299_Control_System_Design_Using_StateSpace_Methods

Chapter Control System Design Using State‐Space Methods

I am not sure I understand the question: yes, the feedback $u=Kx$ is ok from the control-theroretic point of view (I assume you know, or estimate, the state $x$

Thank you so much Professor Diaz. I went through these articles. They are very much beneficial to me.

Good morning Professor Martin and many thanks for such a nice answer. However, let me go in more details about the problem. Consider the following two cases:

1. If I apply $u=Kx$ on the system \dot{x} = Ax + Bu. The closed loop system is 

\dot{x} = (A+BK)x.

2. While if I apply $u=Kx+v$ on the system \dot{x} = Ax + Bu. The closed loop system is 

\dot{x} = (A+BK)x + Bv.

I am interested in the case (1). I want to derive some new properties for the closed loop system in case (1). Note that the properties may not retain in the case (2) as they solely depend upon the matrix A+BK. Will the results be interesting?

I still don't get it. You don't choose one feedback law or the other just for the sake of it, but with respect to a control objective. What is is you are trying to achieve with you feedback law?

Dear Professor Martin, many thanks! I got your points thoroughly. Well, suppose, I want my closed loop system matrix $A+BK$ has eigenvalues with negative real parts. Can I take my feedback as $u = Kx$ only in this case.

Placing the eigenvalues of A+BK obviously depends only on K, so yes using u=Kx wille achieve the desired eigenvalue placement, and the state x will converge to zero. But if you want x to converge to something else than zero, you will use u=Kx+v with a suitably chosen v.

In addition to the previous responses, I would like to add something. First control gives a regulator system. The second one is a tracking servo system where the output tracks v. 

Thank you, Professor Martin and Professor Movva. Your responses are really highly beneficial to me. Thanks again!

In addition to the previos responses, I'd like to add something more. The first equation can be regarded as a control under ideal conditions. The state vector is measured without noise. The second equation can be considered as forming a control vector in conditions of noise.

Thank you Professor Osadchiy for sharing such a nice physical interpretation for the two equations.