Skip to main content
SpringerLink
Log in

Modified quasilinearization algorithm for optimal control problems with bounded state

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper considers the numerical solution of optimal control problems involving a functionalI subject to differential constraints, a state inequality constraint, and terminal constraints. The problem is to find the statex(t), the controlu(t), and the parameter π so that the functional is minimized, while the constraints are satisfied to a predetermined accuracy.

A modified quasilinearization algorithm is developed. Its main property is the descent property in the performance indexR, the cumulative error in the constraints and the optimality conditions. Modified quasilinearization differs from ordinary quasilinearization because of the inclusion of the scaling factor (or stepsize) α in the system of variations. The stepsize is determined by a one-dimensional search on the performance indexR. Since the first variation δR is negative, the decrease inR is guaranteed if α is sufficiently small. Convergence to the solution is achieved whenR becomes smaller than some preselected value.

Here, the state inequality constraint is handled in a direct manner. A predetermined number and sequence of subarcs is assumed and, for the time interval for which the trajectory of the system lies on the state boundary, the control is determined so that the state boundary is satisfied. The state boundary and the entrance conditions are assumed to be linear inx and π, and the modified quasilinearization algorithm is constructed in such a way that the state inequality constraint is satisfied at each iteration and along all of the subarcs composing the trajectory.

At first glance, the assumed linearity of the state boundary and the entrance conditions appears to be a limitation to the theory. Actually, this is not the case. The reason is that every constrained minimization problem can be brought to the present form through the introduction of additional state variables.

In order to start the algorithm, some nominal functionsx(t),u(t), π and nominal multipliers λ(t), ρ(t), σ, μ must be chosen. In a real problem, the selection of the nominal functions can be made on the basis of physical considerations. Concerning the nominal multipliers, no useful guidelines have been available thus far. In this paper, an auxiliary minimization algorithm for selecting the multipliers optimally is presented: the performance indexR is minimized with respect to λ(t), ρ(t), σ, μ. Since the functionalR is quadratically dependent on the multipliers, the resulting variational problem is governed by optimality conditions which are linear and, therefore, can be solved without difficulty.

The numerical examples illustrating the theory demonstrate the feasibility as well as the rapidity of convergence of the technique developed in this paper.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Miele, A., Well, K. H., andTietze, J. L.,Modified Quasilinearization Algorithm for Optimal Control Problems with Bounded State Variables, Part 1, Theory, Rice University, Aero-Astronautics Report No. 103, 1972.

  2. Miele, A., Well, K. H., andTietze, J. L.,Modified Quasilinearization Algorithm for Optimal Control Problems with Bounded State Variables, Part 2, Examples, Rice University, Aero-Astronautics Report No. 105, 1972.

  3. Kelley, H. J.,Method of Gradients, Optimization Techniques, Edited by G. Leitmann, Academic Press, New York, New York, 1962.

    Google Scholar 

  4. Lasdon, L. S., Waren, A. D., andRice, R. K.,An Interior Penalty Function Method for Inequality Constrained Optimal Control Problems, IEEE Transactions on Automatic Control, Vol. AC-12, No. 4, 1967.

  5. McGill, R.,Optimal Control, Inequality State Constraints, and the Generalized Newton-Raphson Algorithm, SIAM Journal on Control, Vol. 3, No. 2, 1965.

  6. Jacobson, D. H., andLele, M. M.,A Transformation Technique for Optimal Control Problems with a State Variable Inequality Constraint, IEEE Transactions on Automatic Control, Vol. AC-14, No. 5, 1969.

  7. Dreyfus, S. E.,Variational Problems with State Variable Inequality Constraints, The RAND Corporation, Report No. P-2605-1, 1962.

  8. Bryson, A. E., Jr., Denham, W. F., andDreyfus, S. E.,Optimal Programming Problems with Inequality Constraints, I, Necessary Conditions for Extremal Solutions, AIAA Journal, Vol. 1, No. 11, 1963.

  9. Denham, W. F., andBryson, A. E., Jr.,Optimal Programming Problems with Inequality Constraints, II, Solution by Steepest Ascent, AIAA Journal, Vol. 2, No. 1, 1964.

  10. Speyer, J. L., Mehra, R. K., andBryson, A. E., Jr.,The Separate Computation of Arcs for Optimal Flight Paths with State Variable Inequality Constraints, Harvard University, Division of Engineering and Applied Physics, TR No. 526, 1967.

  11. Hamilton, W. E., andHaas, V. B.,On the Solution of Optimal Control Problems with State Variable Inequality Constraints, Purdue University, TR No. EE-70-8, 1970.

  12. Fong, T. S.,Method of Conjugate Gradients for Optimal Control Problems with State Variable Constraints, University of California at Los Angeles, School of Engineering and Applied Science, TR No. 70-30, 1970.

  13. Van Schieveen, H. M., andKwakernaak, H.,Solution of State-Constrained Optimal Control Problems Through Quasilinearization, Journal of Engineering Mathematics, Vol. 4, No. 1, 1970.

  14. Williamson, W. E., andTapley, B. D.,A Modified Perturbation Method for Solving Optimal Control Problems with State Variable Inequality Constraints, AIAA Journal, Vol. 9, No. 11, 1971.

  15. Miele, A., Iyer, R. R., andWell, K. H.,Modified Quasilinearization and Optimal Initial Choice of the Multipliers, Part 2, Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 6, No. 5, 1970.

  16. Bliss, G. A.,Lectures on the Calculus of Variations, The University of Chicago Press, Chicago, Illinois, 1946.

    MATH  Google Scholar 

  17. Hestenes, M. R.,Calculus of Variations and Optimal Control Theory, John Wiley and Sons, New York, New York, 1966.

    MATH  Google Scholar 

  18. Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., andMishchenko, E. F.,The Mathematical Theory of Optimal Processes, John Wiley and Sons (Interscience Publishers), New York, New York, 1962.

    MATH  Google Scholar 

  19. Bryson, A. E., Jr., andHo, Y. C.,Applied Optimal Control, Blaisdell Publishing Company, Waltham, Massachusetts, 1969.

    Google Scholar 

  20. Miele, A., Editor,Theory of Optimum Aerodynamic Shapes, Academic Press, New York, New York, 1965.

    MATH  Google Scholar 

  21. Jacobson, D. H., Lele, M. M., andSpeyer, J. L.,New Necessary Conditions of Optimality for Control Problems with State-Variable Inequality Constraints, Harvard University, Division of Engineering and Applied Physics, TR No. 597, 1969.

  22. Huang, H. Y., andEsterle, A.,Some Properties of the Sequential Gradient-Restoration Algorithm and the Modified Quasilinearization Algorithm for Optimal Control Problems with Bounded State Variables, Rice University, Aero-Astronautics Report No. 106, 1972.

  23. Miele, A.,Method of Particular Solutions for Linear, Two-Point Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 2, No. 4, 1968.

  24. Heideman, J. C.,Use of the Method of Particular Solutions in Nonlinear, Two-Point Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 2, No. 6, 1968.

  25. Miele, A., andIyer, R. R.,General Technique for Solving Nonlinear, Two-Point Boundary-Value Problems via the Method of Particular Solutions, Journal of Optimization Theory and Applications, Vol. 5, No. 5, 1970.

  26. Miele, A., andIyer, R. R.,Modified Quasilinearization Method for Solving Nonlinear, Two-Point Boundary-Value Problems, Journal of Mathematical Analysis and Applications, Vol. 36, No. 3, 1971.

  27. Ralston, A.,Numerical Integration Methods for the Solution of Ordinary Differential Equations, Mathematical Methods for Digital Computers, Vol. 1, Edited by A. Ralston and H. S. Wilf, John Wiley and Sons, New York, New York, 1960.

    Google Scholar 

  28. Leitmann, G.,An Introduction to Optimal Control, McGraw-Hill Book Company, New York, New York, 1966.

    MATH  Google Scholar 

  29. Hennig, G. R., andMiele, A.,Sequential Gradient-Restoration Algorithm for Optimal Control Problems with Bounded State Variables, Part 1, Theory, Rice University, Aero-Astronautics Report No. 101, 1972.

  30. Hennig, G. R., andMiele, A.,Sequential Gradient-Restoration Algorithm for Optimal Control Problems with Bounded State Variables, Part 2, Examples, Rice University, Aero-Astronautics Report No. 102, 1972.

  31. Hennig, G. R., andMiele, A.,Sequential Gradient-Restoration Algorithm for Optimal Control Problems with Bounded State, Journal of Optimization Theory and Applications, Vol. 12, No. 1, 1973.

Download references

Author information

Authors and Affiliations

  1. Rice University, Houston, Texas

    A. Miele (Professor of Astronautics and Mathematical Sciences)

  2. Department of Mechanical and Aerospace Engineering and Materials Science, Rice University, Houston, Texas

    K. H. Well (Graduate Student)

  3. Department of Electrical Engineering, Rice University, Houston, Texas

    J. L. Tietze (Graduate Student)

Authors
  1. A. Miele
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. H. Well
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. L. Tietze
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-72-2185. The authors are indebted to Dr. R. R. Iyer and Mr. A. K. Aggarwal for helpful discussions as well as analytical and numerical assistance. This paper is a condensation of the investigations described in Refs. 1–2.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Miele, A., Well, K.H. & Tietze, J.L. Modified quasilinearization algorithm for optimal control problems with bounded state. J Optim Theory Appl 12, 285–319 (1973). https://doi.org/10.1007/BF00935110

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00935110

Keywords

Search

Navigation