Skip to main content

Optimization of multistage processes described by differential-algebraic equations

Part of the Lecture Notes in Mathematics book series (LNM,volume 1230)

Abstract

The paper describes an algorithm for the computation of optimal design and control variables for a multistage process, each stage of which is described by a system of nonlinear differential-algebraic equations of the form:

$$f(t,\dot x(t),x(t),u(t),v) = 0$$

where t is the time, x(t) the state vector, \(\dot x(t)\) its time derivative, u(t) the control vector, and v a vector of design parameters. The system may also be subject to end-point or interior-point constraints, and the switching times may be explicitly or implicitly defined. Methods of dealing with path constraints are also discussed.

Keywords

  • Adjoint System
  • Matrix Pencil
  • Path Constraint
  • Ordinary Differential Equation System
  • Kronecker Canonical Form

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   46.00
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Biegler, L.T., 1983, "Solution of Dynamic Optimization Problems by Successive Quadratic Programming and Orthogonal Collocation", Comp. and Chem. Eng., 8 (3/4), 243–248.

    Google Scholar 

  2. Burrage, K., 1982, "Efficiently Implementable Algebraically Stable Runge-Kutta Methods", SIAM J. Numer. Anal. 19 (2), 245–258, (April, 1982).

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. Cameron, I.T., 1983, "Solution of Differential-Algebraic Systems Using Diagonally Implicit Runge-Kutta Methods", IMA Journal of Numerical Analysis, 3 (3), 273–290, (July, 1983).

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Cobb, D., 1983, "Descriptor Variable Systems and Optimal State Regulation", IEEE Trans. Auto. Control, AC-28, 601–611.

    Google Scholar 

  5. Gantmacher, F.R., 1959, "Applications of the Theory of Matrices", Interscience (New York, 1959).

    MATH  Google Scholar 

  6. Gear, C.W., 1971, "Simultaneous Numerical Solution of Differential-Algebraic Equations", IEEE Trans. Circuit Theory, CT-18, 89–95.

    CrossRef  Google Scholar 

  7. Gear, C.W., and L.R. Petzold, 1984, "ODE Methods for the Solution of Differential-Algebraic Systems", SIAM J. Numer. Anal. 21, 716.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Jacobson, D.H., and M.M. Lele, 1969, "A Transformation Technique for Optimal Control Problems with a State Variable Inequality Constraint", IEEE Trans. Auto. Control., AC-14, 457–464.

    CrossRef  Google Scholar 

  9. Pantelides, C.C., 1985, "The Consistent Initialization of Differential-Algebraic Systems", submitted for publication.

    Google Scholar 

  10. Petzold, L.R., 1982, "Differential-Algebraic Equations are not ODEs", SIAM J. Sci. Stat. Comput., 3 (3), 367–384, (September 1982).

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. Sargent, R.W.H. and G.R. Sullivan, 1977, "The Development of an Efficient Optimal Control Package", in J. Stoer (ed) "Optimization Techniques-Proceedings of the 8th IFIP Conference on Optimization Techniques, Wurzburg, 1977", Part 2, pp 158–167, Springer-Verlag, (Berlin, 1978).

    Google Scholar 

  12. Sargent, R.W.H., 1981, "Recursive Quadratic Programming Algorithms and their Convergence Properties", in J.P. Hennart (ed.), "Numerical Analysis-Proceedings, Cocoyoc, Mexico, 1981", Lecture Notes in Mathematics, pp 208–225, Springer Verlag, (Berlin, 1982).

    Google Scholar 

  13. Sargent, R.W.H., 1985, "The Existence and Regularity of Solutions of Differential-Algebraic Systems", submitted for publication.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1986 Springer-Verlag

About this paper

Cite this paper

Morison, K.R., Sargent, R.W.H. (1986). Optimization of multistage processes described by differential-algebraic equations. In: Hennart, JP. (eds) Numerical Analysis. Lecture Notes in Mathematics, vol 1230. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072673

Download citation

  • DOI: https://doi.org/10.1007/BFb0072673

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17200-0

  • Online ISBN: 978-3-540-47379-4

  • eBook Packages: Springer Book Archive