iosr-Jm   Volume-9 ~ Issue-5

Abstract: This paper investigates the application of the iterative Steepest Descent Method (SDM) to the solution of nonlinear two-point boundary-value problems for the optimal controls and trajectories of continuous-time linear-quadratic regulator problems. Numerical results show some improvement over the classical methods.

Keywords: Optimal Control, Continuous Linear Regulator, Steepest Descent Method..

[1] Athans, M. and Falb, P. L., (1966), Optimal Control: An Introduction to the Theory and Its Applications, McGraw-Hill, New York.
[2] Bryson, A. E. Jr., and W. F., Denham, (1964), "Optimal Programming Problems with Inequality Constraints II: Solution by Steepest Ascent," AIAA Journal, 25-34.
[3] Burghes, D. N. and Graham, A., (1980), Introduction To Control Theory, Including Optimal Control, John Wiley & Sons.
[4] George M. Siouris, (1996), An Engineering Approach To Optimal Control And Estimation Theory, John Wiley & Sons, Inc.
[5] Kelly, H. J., (1962), "Method of Gradient," Optimization Techniques with Applications To Aerospace Systems. G. Leitmann, Ed. New York: Academic Press Inc.

Abstract: This paper investigates and discusses the method of dynamic programming in solving Bolza's cost form of Linear Quadratic Regulator Problems (LQRP). It is the desire of the authors of this paper to experiment numerically the solution of this class of problem using dynamic programming to solve for the optimal controls and the trajectories compared with other numerical methods with a view to further improving the results. The method uses the principle of optimality to reduce mathematically the number of calculations required to determine the optimal control law as well as the corresponding optimal cost functional.

Keywords: Continuous-Time Linear Regulator Problem, Optimal Control, Discretization and Dynamic Programming.

[1] Boudarel, J., et al, (1971), Dynamic Programming and Its Application to Optimization Theory, Academic Press, Inc., 111 Fifth Avenue, New York, New York 10003.
[2] David, G. Hull, (2003), Optimal Control Theory for Applications, Mechanical Engineering Series, Springer-Verlag, New York, Inc., 175 Fifth Avenue, New York, NY 10010.
[3] Kirk, E. Donald, (2004), Optimal control theory: An introduction, Prentice-Hall, Inc., Englewood Cliff, New Jersey.
[4] George M. Siouris, (1996), An Engineering Approach To Optimal Control And Estimation Theory, John Wiley & Sons, Inc., 605 Third Avenue, New York, 10158- 00 12.

Abstract:The present paper deals with Latin squares, orthogonal Latin squares, mutually Orthogonal Latin squares, close connections between Latin squares and finite geometries. Moreover the great mathematician Leonhard Euler introduced Latin squares in 1783 as a "nouveau espece de carres magiques", a new kind of magic squares. He also defined what he meant by orthogonal Latin squares, which led to a famous conjecture of his that went unsolved for over 100 years.

[1] Mann, H. B., Analysis and Design of Experiments, Dover, New York, 1949.

[2] Denes, J. and A. D. Keedwell, Latin Squares and their Applications, Academic Press, New York, 1974.

[3] Bose, Ray Chandra, "On the Application of the Properties of Galois Fields to the Problem of Construction of Hyper-Graeco-Latin Squares," Sankhya (The Indian Journal of Statistics), 3, 323-338(1938).

[4] Ball, W. W. Rouse, Mathematical Recreations and Essays, rev. ed. Macmillan, London, 1939.

Abstract: In this paper, we constructed a control operator sequel to an earlier constructed control operator in one of our papers which enables an Extended Conjugate Gradient Method (ECGM) to be employed in solving discrete time linear quadratic regulator problems with delay parameter in the state variable. The construction of the control operator places scalar linear delay problems of the type within the class of problems that can be solved with the ECGM and it is aimed at reducing the rigours faced in using the classical methods in solving this class of problem.

[1] ADEBAYO,K. J. and ADERIBIGBE, F. M., (2014), On Construction of A Control Operator Applied In Conjugate Gradient Method Algorithm For Solving Continuous Time Linear Regulator Problems With Delay – I
[2] Aderibigbe, F. M., (1993), "An Extended Conjugate Gradient Method Algorithm For Control Systems with Delay-I, Advances in Modeling & Analysis, C, AMSE Press, Vol. 36, No. 3, pp 51-64.
[3] Athans, M. and Falb, P. L., (1966), Optimal Control: An Introduction to the Theory and Its Applications, McGraw-Hill, New York.
[4] David, G. Hull, (2003), Optimal Control Theory for Applications, Mechanical Engineering Series, Springer-Verlag, New York, Inc., 175 Fifth Avenue, New York, NY 10010.