Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Analytical Solution for Optimal Control by the Second Kind Chebyshev Polynomials Expansion

  • Research Paper
  • Published:
Iranian Journal of Science and Technology, Transactions A: Science Aims and scope Submit manuscript

Abstract

Second kind Chebyshev polynomials are modified set of defined Chebyshev polynomials by a slightly different generating function. This paper presents new and efficient algorithm for achieving an analytical approximate solution to optimal control problems. The proposed solution is based on state parameterization, such that the state variable is approximated by the second kind Chebyshev polynomials with unknown coefficients. At first, the equation of motion, boundary conditions and performance index are changed into some algebraic equations. This task converts the optimal control problem into an optimization problem, which can then be solved easily. The presented technique approximates the control and state variables as a function of time. After optimizing, the system is converted into a feedback mode for having the closed loop profits. The results proved the algorithm convergence. Finally by analyzing two numerical examples, the reliability and effectiveness of the proposed method by comparing two different methods is demonstrated.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  • Ahmed NU (1988) Elements of finite-dimensional systems and control theory. Longman Scientific and Technical, England

    MATH  Google Scholar 

  • Alipanah A, Razzaghi M, Dehgan M (2007) Nonclassical pseudospectral method for the solution of brachistochrone problem. Chaos Solitons Fractals 34:1622–1628

    Article  MathSciNet  MATH  Google Scholar 

  • Ashoori A, Moshiri B, Ramezani A, Bakhtiari MR, Khaki-Sedigh A (2009) Optimal control of a nonlinear fed-batch fermentation process using model predictive approach. J Process Control 19:1162–1173

    Article  MATH  Google Scholar 

  • Bellman R (1957) Dynamic programming. University Press, Princeton

    MATH  Google Scholar 

  • Bryson AE (1996) Optimal control—1950 to 1985. IEEE Control Syst Mag 16:26–33

    Article  Google Scholar 

  • Bryson A, Ho YC (1975) Applied optimal control. Hemisphere Publishing Corporation, Washington DC

    Google Scholar 

  • Chang R, Yang S (1986) Solutions of two-point-boundary-value problems by generalized orthogonal polynomials and applications to control of lumped and distributed parameter systems. Int J Control 43:1785–1802

    Article  MathSciNet  MATH  Google Scholar 

  • Craven BD (1995) Control and optimization. Chapman & Hall, London

    Book  MATH  Google Scholar 

  • Fleming WH, Rishel RW (1975) Deterministic and stochastic optimal control. Springer, New York

    Book  MATH  Google Scholar 

  • Kafash B, Delavarkhalafi A, Karbassi SM (2012) Application of Chebyshev polynomials to derive efficient algorithms for the solution of optimal control problems. Sci Iran 19(3):795–805

    Article  MATH  Google Scholar 

  • Kirk D (1970) Optimal control theory: an introduction. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • Mason JC, Handscomb DC (2003) Chebyshev polynomials. A CRC Press Company, Boca Raton

    MATH  Google Scholar 

  • Mehne HH, Hashemi Borzabadi A (2006) A numerical method for solving optimal control problems using state parametrization. Numer Algorithms 42:165–169

    Article  MathSciNet  MATH  Google Scholar 

  • Miele A (1980) Control and dynamic systems: advances in theory and applications. In: Leondes CT (ed) Gradient algorithms for the optimization of dynamic systems, vol 16, pp 1–52

  • Pantelev AB, Bortakovski AC, Letova TA (1996) Some issues and examples in optimal control. MAI Press, Moscow (in Russian)

    Google Scholar 

  • Pinch ER (1993) Optimal control and the calculus of variations. Oxford University Press, London

    MATH  Google Scholar 

  • Pontryagin LS (1962) The mathematical theory of optimal processes. In: Interscience. CRC press, USA

  • Razzaghi M, Arabshahi A (1989) Optimal control of linear distributed parameter systems via polynomial series. Int J Syst Sci 20:1141–1148

    Article  MATH  Google Scholar 

  • Razzaghi M, Elnagar G (1994) Linear quadratic optimal control problems via shifted Legendre state parameterization. Int J Syst Sci 25:393–399

    Article  MATH  Google Scholar 

  • Rubio JE (1986) Control and optimization: the linear treatment of non-linear problems. Manchester University Press, Manchester

    MATH  Google Scholar 

  • Sadek I, Bokhari M (1998) Optimal control of a parabolic distributed parameter system via orthogonal polynomials. Optim Control Appl Methods 19:205–213

    Article  MathSciNet  Google Scholar 

  • Sakawa Y, Shindo Y (1980) On the global convergence of an algorithm for optimal control. IEEE Trans Autom Control 25:1149–1153

    Article  MathSciNet  MATH  Google Scholar 

  • Stoer J, Bulirsch R (1993) Introduction to numerical analysis, 2nd edn. Springer, New York (Translated by R. Bartels, W. Gautschi, C. Witzgall)

    Book  MATH  Google Scholar 

  • Taraba P, Kozak S (2003) MPC control using AR-Volterra models. In: Proceedings of the 11th mediterranean conference on control and automation MED’03, Rhodos

  • Teo KL, Goh CJ, Wong KH (1991) A unified computational approach to optimal control problems. Longman Scientific and Technical, England

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, University of Tafresh, Tafresh, Iran

    Navid Razmjooy & Mehdi Ramezani

Authors
  1. Navid Razmjooy
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mehdi Ramezani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Navid Razmjooy.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Razmjooy, N., Ramezani, M. Analytical Solution for Optimal Control by the Second Kind Chebyshev Polynomials Expansion. Iran J Sci Technol Trans Sci 41, 1017–1026 (2017). https://doi.org/10.1007/s40995-017-0336-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40995-017-0336-4

Keywords

Search

Navigation