AASCIT - Journal - All Issues

ISSN: 2375-3927

International Journal of Mathematical Analysis and Applications

The Journal

Manuscript Information

On the Generalization of the Discretized Continuous Algorithm for Optimal Proportional Control Problem

International Journal of Mathematical Analysis and Applications
Vol.1 , No. 3, Publication Date: Aug. 15, 2014, Page: 49-58

1585 Views Since August 15, 2014, 713 Downloads Since Apr. 14, 2015

Paper in PDF (552K)

Follow on

Authors

[1]	Dawodu Kazeem Adebowale, Department of Mathematical Sciences, Federal University of Technology, Akure, P.M.B 704, Akure, Ondo-state, Nigeria.
[2]	Olotu Olusegun, Department of Mathematical Sciences, Federal University of Technology, Akure, P.M.B 704, Akure, Ondo-state, Nigeria.

Abstract

This paper seeks to find solution to the generalization of a class of continuous-time optimal control model with special preference to those whose control efforts are proportional to the state of the dynamical system with and without delay in the state variables. The adoption of the Augmented Lagrangian method in the transformation of the constrained problem into an unconstrained sequential nonlinear quadratic problem allows for the use of the well-posed Broydon-Fletcher-Goldberg-Shanno (BFGS) embedded Quasi-Newton algorithm. The symmetric and positive definite properties of the constructed control operator guarantees the invertibility of the BFGS embedded in the algorithm and as well induces faster convergence. Numerical results were considered; result tested and responded favourably to the analytical solution with linear convergence.

Keywords

Generalized Model, Analytical Solution, Discretization, Quasi-Newton Method, Augmented Lagrangian Method, BFGS Updates Formula and Quadratic Programming

Reference

[01]	Bertsekas, D.P. (1973), Convergence rate of Penalty and Multiplier Methods. Proc. 1973 IEEE Confer. Decision Control, San Diego, Calif, pp. 260 – 264.
[02]	Bertsekas, D.P. (1996), Constrained Optimization and Lagrange Multiplier Methods. Athena scientific, Belmont, Massachusetts.
[03]	Betts J.T. (2001), Practical Methods for Optimal Control Problem Using Nonlinear Programming. SIAM, Philadelphia
[04]	Burden, R. L. and Faires, J. D. (1993), Numerical Analysis. PWS Publishers, Boston.
[05]	Colonius, F. and Hinrichsen, D. (1978), Optimal Control of Functional Differential Systems. SIAM Journal on Control and Optimization 1978; 16(6), Pp. 861–879.
[06]	Fiacco, A. V. and McCormick, G. P. (1968), Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York.
[07]	Gollman, L., Kern, D. and Maurer, H. (2008), Optimal Control Problem with Delays in State and Control Variables Subject to Mixed Control –State Constraints. Optimal Control Applications and Methods, John Wiley & Sons, USA.
[08]	Hestenes M.R. (1969), Multiplier and Gradient Methods. J.O.T.A. 4. 303 – 320.
[09]	Ibiejugba M.A and Onumanyi p. (1984), A Control Operator and Some of its Applications. Journal of Mathematics Analysis and Application Vol. 103, 31-47.
[10]	Nocedal, J. and Wright, S.J. (2006), Numerical Optimization (2nd Ed.), Springer Publisher, NY, USA.
[11]	Olotu, O. and Adekunle, A.I. (2010), Analytic and Numeric Solutions of Discretized Constrained Optimal Control Problem with Vector and Matrix Coefficients. The Journal of Advanced Modeling and Optimization, Vol. 12, Number 1, pp. 119-131.
[12]	Olotu, O. and Adekunle, A.I. (2012). An Algorithm for Optimal Control of Delay Differential Equation. The Pacific Journal of Science and Technology, Vol. 13, Number 1, Pp. 228-237.
[13]	Olotu, O. and Akeremale , O.C. (2012), Augmented Lagrangian Method for Discretized Optimal Control Problems, Journal of the Nigerian Association of mathematical physics, vol.2 Pp.185-192
[14]	Olotu, O. and Dawodu, K. A. (2013), Quasi Newton Embedded Augmented Lagrangian Algorithm for Discretized Optimal Proportional Control Problem, Journal of Mathematical Theory and Modelling by IISTE, USA. Vol.3, No.3, Pp. 67-79.
[15]	Olotu, O. and Dawodu, K. A. (2013), On the Discretized Algorithm for Optimal Proportional Control Constrained by Delay Differential Equation, Journal of Mathematical Theory and Modelling by IISTE, USA. Vol.3, No.8, Pp. 157-169.
[16]	Olotu, O. and Olorunsola S.A. (2009), Convergence profile of a discretized scheme for constrained problem via the penalty –multiplier method. Journal of the Nigerian Association of Mathematical Physics, Vol.14 (1), 341-348.
[17]	Polak, E. and Tits, A.L. (1979), A Globally Convergent, Implementable Multiplier Method with Automatic Penalty Limitation. Res, Lab., Univ. of California, Berkeley. Applied Maths Optim, Vol. 6, Pp. 335-360.
[18]	Poljak, B.T., (1971), The Convergence Rate of the Penalty Function Method. Z.V.ychist. Mat, Fiz,3-11.
[19]	Powell, M.J.D. (1969), A Method for Nonlinear Constraints in Minimization Problems in Optimization, pp. 283 – 298. Academic Press, New York.
[20]	Soliman, M.A and Ray, W.H. (1972), On the Optimal Control of Systems having Pure Time Delays and Singular arcs. International Journal of Control; 16(5), Pp. 963–976.