Semi-Infinite Programming in Control

• Ekkehard W. Sachs
Chapter
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 25)

Abstract

Optimal control problems represent a special class of optimization problems which describe dynamical processes. There is a wide range of applications for optimal control problems in engineering and economics. A few of these problems are described and it is shown how they are related and lead to semi-infinite programming problems.

Among the vast literature in the area of semi-infinite programming we want to point out two review articles which include also the aspect of optimal control problems: Polak [14] reviews engineering applications of semi-inifinite programming and Hettich and Kortanek [10] give an overview of applications, algorithms and theory in this area.

The dynamical system can be given by a system of difference equations, ordinary differential equations or partial differential equations. With a proper objective function this often leads to optimization problems in function spaces.

A general class of optimal control problems for ordinary differential equations is presented. The controls and the states have to satisfy pointwise bounds. It is shown that a discretization of the control space leads to a semi-infinite programming problem.

A heat conduction process in food industry coupled with the decay of microorganisms is discussed as an important application of optimal control problems. This process, however, also leads to reduction in the vitamines. The optimal control problem consists in finding numerically a control which optimizes the loss of vitamines and satisfies the requirement of achieved sterility for the product. The mathematical model is described by a system of nonlinear partial differential equations. After discretization one obtains a semi-infinite programming problem with nonlinear constraints.

Another application of semi-infinite programming occurs in the control of flutter of an aircraft wing. The avoidance of flutter has to be guaranteed over a certain velocity range at which the aircraft operates. A proper formulation of this problem leads to a semi-infinite programming problem. The dependence of the functions on the indices describing the constraints, however, is not necessarily differentiable and therefore computationally difficult to handle. Therefore, a new approach is presented using the characterization of stability with the Lyapunov equation. This leads to an interesting and novel combination of positive-definite and semi-infinite programming. It is shown how this problem can be solved by a barrier function technique.

Keywords

Optimal Control Problem Optimal Design Problem Lyapunov Equation Aircraft Wing Fixed Design
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.

References

1. [1]
S. Baxnett. Polynomials and Linear Control Systems. New York, 1983.Google Scholar
2. [2]
R. L. Bisplinghoff, H. Ashley, and R. L. Halfman. Aeroelasticity. Reading, 1955.
3. [3]
S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM Studies in Applied Mathematics, 1994.
4. [4]
P. Dierolf. Aeroelastische Phänomene, Flattern. Vorlesung, Universität München, 1980.Google Scholar
5. [5]
M. Fahl. Zur Strukturoptimierung unter Flatterrestriktionen. Diploma Thesis, Universität Trier, 1996.Google Scholar
6. [6]
M. Fahl and E. W. Sachs. Modern optimization methods for structural design under flutter constraints. Technical report, Universität Trier, Germany, 1997.Google Scholar
7. [7]
H. W. Försching. Grundlagen der Aeroelastik. Springer, Berlin-Heidelbeg-New York, 1974.
8. [8]
E. Haaren-Retagne. Semi-Infinite Programming Algorithm for Robot Trajectory Planning. PhD thesis, Universität Trier, 1992.Google Scholar
9. [9]
R. T. Haftka. Parametric constraints with application to optimization using a continuous flutter constraint. AIAA Journal, 13:471–475, 1975.
10. [10]
R. Hettich and K. O. Kortanek. Semi-infinite programming: theory, methods, and applications. SIAM Review, 35:380–429, 1993.
11. [11]
D. Kleis. Augmented Lagrange SQP Methods and Application to the Sterilization of Prepackaged Food. PhD thesis, Universität Trier, 1997.Google Scholar
12. [12]
D. Kleis and E. W. Sachs. Optimal control of the sterilization of prepackaged food. Technical report, Universität Trier, Germany, 1997.Google Scholar
13. [13]
H.-D. Kothe. Störungstheorie des Eigenwert problems. Diploma Thesis, Universität Trier, 1988.Google Scholar
14. [14]
E. Polak. On the mathematical foundations of nondifferentiable optimization in engineering design. SIAM Review, 29:21–89, 1987.
15. [15]
U. T. Ringertz. Eigenvalues in optimum structural design. Technical Report 96–8, KTH Stockholm, Department of Aeronautics, 1996.Google Scholar
16. [16]
C. L. M. Silva, F. A. R. Oliveira, and M. Hendrickx. Modelling optimum processing conditions for the sterilization of prepackaged foods. Food Control, 4(2):67–78, 1993.