Skip to main content
SpringerLink
Log in

Concave Programming in Control Theory

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

We show in the present paper that many open and challenging problems in control theory belong the the class of concave minimization programs. More precisely, these problems can be recast as the minimization of a concave objective function over convex LMI (Linear Matrix Inequality) constraints. As concave programming is the best studied class of problems in global optimization, several concave programs such as simplicial and conical partitioning algorithms can be used for the resolution. Moreover, these global techniques can be combined with a local Frank and Wolfe feasible direction algorithm and improved by the use of specialized stopping criteria, hence reducing the overall computational overhead. In this respect, the proposed hybrid optimization scheme can be considered as a new line of attack for solving hard control problems.

Computational experiments indicate the viability of our algorithms, and that in the worst case they require the solution of a few LMI programs. Power and efficiency of the algorithms are demonstrated for a realistic inverted-pendulum control problem.

Overall, this dedication reflects the key role that concavity and LMIs play in difficult control problems.

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. Anderson, B. and Vongpanitlerd, S. (1973), Network analysis and synthesis: a modern systems theory, Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  2. Apkarian, P. and Gahinet, P. (1995), A Convex Characterization of Gain-Scheduled H1 Controllers, IEEE Trans. Aut. Control 40: 853-864. See also pp. 1681.

    Google Scholar 

  3. Apkarian, P., Gahinet, P. and Becker, G. (1995), Self-Scheduled H1 Control of Linear Parameter-Varying Systems: A Design Example, Automatica 31: 1251-1261.

    Google Scholar 

  4. Apkarian, P. and Tuan, H.D. (1998), Robust Control via Concave Minimization-Local and Global Algorithms, in Proc. IEEE Conf. on Decision and Control, Tampa, Florida.

  5. Bennett, K.P. and Mangasarian, O.L. (1993), Bilinear Separation of Two Sets in n-Space, Computational Optimization and Applications 2: 207-227.

    Google Scholar 

  6. Boyd, S., ElGhaoui, E., Feron, E. and Balakrishnan, V. (1994), Linear Matrix Inequalities in Systems and Control Theory, vol. 15 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia.

    Google Scholar 

  7. Coppel, W.A. (1974), Matrix quadratic equations, Bull. Austral. Math. Soc. 10: 377-401.

    Google Scholar 

  8. Doyle, J., Packard, A. and Zhou, K. (1991), Review of LFT's LMI's and µ, in Proc. IEEE Conf. on Decision and Control 2: 1227-1232.

    Google Scholar 

  9. Fan, M.K.H., Tits, A.L. and Doyle, J.C. (1991), Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics, IEEE Trans. Aut. Control AC-36: 25-38.

    Google Scholar 

  10. Gahinet, P. and Apkarian, P. (1994), A Linear Matrix Inequality Approach to H1 Control, Int. J. Robust and Nonlinear Control 4: pp. 421-448.

    Google Scholar 

  11. Gahinet, P., Nemirovski, A., Laub, A.J. and Chilali, M. (1995), LMI Control Toolbox, The MathWorks Inc.

  12. Geromel, J.C., deSouza, C.C. and Skelton, R.E. (1998), Static Output Feedback Controllers: Stability and Convexity, IEEE Trans. Aut. Control 43: 120-125.

    Google Scholar 

  13. Ghaoui, L.E., Oustry, F. and Aitrami, M. (1997), A Cone Complementary Linearization Algorithm for Static Output-Feedback and Related Problems, IEEE Trans. Aut. Control 42: 1171-1176.

    Google Scholar 

  14. Goh, K.C., Safonov, G. and Papavassilopoulos, G.P. (1994), A Global Optimization Approach to the BMI Problem, in Proc. IEEE Conf. on Decision and Control 45-49.

  15. Grigoriadis, K.M. and Skelton, R.E. (1996), Low-order Control Design for LMI Problems Using Alternating Projection Methods, Automatica 32: 1117-1125.

    Google Scholar 

  16. Helmersson, A. (1995), Methods for Robust Gain-Scheduling, Ph.D. Thesis, Linkoping University, Sweden.

    Google Scholar 

  17. Horst, R. and Pardalos, P. eds. (1995), Handbook of Global Optimization, Kluwer Academic Publishers.

  18. Horst, R. and Tuy, H. (1996), Global Optimization: deterministic approaches, Springer (3rd edition).

  19. Iwasaki, T. (1997), The dual iteration for fixed-order control, in Proc. American Control Conf., 1997, pp. 62-66.

  20. Iwasaki, T. and Hara, S. (1996), Well-Posedness Theorem: A classification of LMI/BMIreducible control problems, in Proc. Int. Symp. Intelligent Robotic Syst., Bangalore, pp. 145-157.

  21. Iwasaki, T. and Skelton, R.E. (1994), All Controllers for the General H1 Control Problem: LMI Existence Conditions and State Space Formulas, Automatica 30: 1307-1317.

    Google Scholar 

  22. Kajiwara, H., Apkarian, P. and Gahinet, P. (1999), LPV techniques for control of an inverted pendulum, Control System Magazie, 19: 44-54.

    Google Scholar 

  23. Kokotovic, P.V. (1975), A Riccati equation for block-diagonalisation of ill-conditioned systems, IEEE Trans. Aut. Control, 20, 812-814.

    Google Scholar 

  24. Konno, H., Thach, P. and Tuy, H. (1997), Optimization on Low Rank Nonconvex Structures, Kluwer Academic Publishers.

  25. Laub, A.J. (1979), A Schur method for solving algebraic Riccati equations, IEEE Trans. Aut. Control AC-24: 913-921.

    Google Scholar 

  26. Lyapunov, A.M. (1947), Problème général de la stabilité du mouvement, vol. 17 of Annals of Mathematics Studies, Princeton University Press, Princeton.

    Google Scholar 

  27. Mangasarian, O.L. and Pang, J.S. (1995), The Extended Linear Complementary Problem, SIAM J. on Matrix Analysis and Applications 16: 359-368.

    Google Scholar 

  28. Nesterov, Y.E. and Nemirovski, A.S. (1994), Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM Studies in applied Mathematics 13, SIAM, Philadelphia.

    Google Scholar 

  29. Packard, A. (1994), Gain Scheduling via Linear Fractional Transformations, Syst Control Letters, 22: 79-92.

    Google Scholar 

  30. Packard, A. and Becker, G. (1992), Quadratic Stabilization of Parametrically-Dependent Linear Systems using Parametrically-Dependent Linear, Dynamic Feedback, Advances in Robust and Nonlinear Control Systems, DSC-Vol. 43, pp. 29-36.

    Google Scholar 

  31. Packard, A., Zhou, K., Pandey, P. and Becker, G. (1991), A collection of robust control problems leading to LMI's in Proc. IEEE Conf. on Decision and Control 2: 1245-1250.

    Google Scholar 

  32. Scherer, G., Ganinet, P. and Chilali, M. (1997), Multi-Objective Output-Feedback Control via LMI Optimization, IEEE Trans. Aut. Control 42: 896-911.

    Google Scholar 

  33. Scorletti G. and Ghaoui, L.E. (1995), Improved Linear Matrix Inequality Conditions for Gain-Scheduling, in Proc. IEEE Conf. on Decision and Control, New Orleans, LA, 3626-3631.

  34. Trentelman, H.L. and Willems, J.C. (1991), The dissipation inequality and the algebraic riccati equation, in The Riccati Equation, S. Bittani, A.J. Laub, and J.C. Willems, eds., Communications and Control Engineering Series, Springer-Verlag, ch. 8, 197-242.

  35. Tuan, H., Apkarian, P., Hosoe, S. and Tuy, H., D.C. Optimization Approach to Robust Control: Feasibility Problems. To appear in Int. J. Control.

  36. Tuan, H.D., Apkarian, P., Nakashima, Y., A new Lagrangian dual global optimization algorithm for solving bilinear matrix inequalities, in Proc. of 1999 American Control Conference, San Diego, CA, 1851-1855; Also to appear in Inf. J. of Nonlinear Robust Control.

  37. Tuy, H. (1964), Concave Programming under linear constraints, Soviet Mathematics, 1437-1440.

  38. Tuy, H. (1998), Convex analysis and global optimization, Kluwer Academic Publishers.

  39. Yamada, Y. and Hara, S. (1996), The Matrix Eigenvalue Problem-Global Optimization for the Spectral Radius of a Matrix Product under Convex Constraints, in Proc. IEEE Conf. on Decision and Control, Kobe, 1326-1330.

Download references

Author information

Authors and Affiliations

  1. Control System Dept., ONERA-CERT, 2 av. Edouard Belin, 31055, Toulouse, France

    Pierre Apkarian

  2. Department of Control and Information, Toyota Technological Institute, Hisakata 2-12-1, Tenpaku, Nagoya, 468-8511, Japan

    Hoang Duong Tuan

Authors
  1. Pierre Apkarian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hoang Duong Tuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Apkarian, P., Tuan, H.D. Concave Programming in Control Theory. Journal of Global Optimization 15, 343–370 (1999). https://doi.org/10.1023/A:1008385006172

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008385006172

Search

Navigation