(1) Mathematical design of Linear-quadratic-Gaussian (LQG) control on paper without MATLAB can follow the guidance of the following LQG Wiki.

http://en.wikipedia.org/wiki/Linear-quadratic-Gaussian_control

(2) Mathematical design of Linear-quadratic regulator (LQR) on paper without MATLAB can be found in the following paper.

Y. Hong and O.W.W. Yang, "Self-Tuning Optimal PI Rate Controller for End-to-End Congestion With LQR Approach," Proceedings of 20th International Teletraffic Congress (ITC-20), Ottawa, Canada, June 2007, pp.829-840. Available in the following RG link.

http://www.researchgate.net/publication/221584579_Self-tuning_Optimal_PI_Rate_Controller_for_End-to-End_Congestion_With_LQR_Approach

Springer subscriber can download the paper from link.springer.com.

http://link.springer.com/chapter/10.1007%2F978-3-540-72990-7_72#

(3) Quote Reza Amoui's follow-up question "We have weighting function for this controller.Which weighting functions are optimal for solution with quantity facet?how can i determine weighting function for optimal solution?"

In most control designs based on linear quadratic optimal control, the weighting matrices Q and R are design parameters. We can always choose Q to guarantee detectability of (Q,A). For example, we can choose Q > 0. We will then have (Q,A) detectable (apply Popov-Belevitch-Hautus (PBH) test). In such design problems, the sufficient condition for the solution of the optimal control problem is just stabilizability of (A,B), the same as the condition for solving state regulation problems using pole assignment. We can interpret the use of LQR design as a better way of generating a stabilizing control law compared to pole placement. It results in a unique control law, has reasonably good interpretation for the choice of the design parameters Q and R, avoid the need to specify desired pole locations (often chosen without good design reasons: for example, is −2,−2 a better choice than −2 ± i?), has good transient response, and has better numerical properties than pole placement design. For these reasons, LQR design is often preferred over pole placement in control design practice. However, the implementation cost of LQR design is more expensive than that of pole placement design.

(4) Recent discussions on control system design approaches can be found in the following RG link..

"What are trends in control theory and its applications in physical systems (from a research point of view)?"

https://www.researchgate.net/post/What_are_trends_in_control_theory_and_its_applications_in_physical_systems_from_a_research_point_of_view2

Conference Paper Self-tuning Optimal PI Rate Controller for End-to-End Conges...

Thanks a lot Yang Hong.

LQ optimal control developed after the invention of programmable computers partly because it is very hard to obtain the solutions by hand for most systems of order higher than 2.

Thanks alot J.F.Whidborne .We have weighting function for this controller.Which weighting functions are optimal for solution with quantity facet?how can i determine weighting function for optimal solution?

I agree with James above. You can calculate by pencil-and-paper for systems of order 1 and 2, sometimes with special cases of order 3. But very rarely people will calculate solution manually, rather use an interactive way designing with Matlab, Scilab or similar software tools.

You seem to ask questions in an incremental way, i.e. lately on weighting functions. Note that LQR-type controllers are designed by tuning weighting matrices for control effort and states. The tuning is also done "per-application". Have a look in the well known control books in the literature, i.e. By Skogestad and Postlethwaite: MFC, or by Franklin-et-al for further discussions and details. Note that LQG can be setup as a H2 controller design problem.

Argyrios Zolotas : Thanks a lot .