Skip to main content

Part of the book series: SpringerBriefs in Electrical and Computer Engineering ((BRIEFSCONTROL))

  • 807 Accesses

Abstract

This book focuses on a class of control problems that can be translated to an optimization-based decision over a feasible region which is neither convex nor compact.

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

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • 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

Institutional subscriptions

Notes

  1. 1.

    NP-hard (Non-deterministic Polynomial-time hard), in computational complexity theory, is a class of problems that are, informally, “at least as hard as the hardest problems in NP”. More precisely, a decision problem H is NP-hard when for any problem L in NP, there is a polynomial-time reduction from L to H [18].

References

  1. Marafioti, G., Stoican, F., Bitmead, R.R., Hovd, M.: Persistently exciting model predictive control for siso systems. In: 4th IFAC Nonlinear Model Predictive Control Conference International Federation of Automatic Control, pp. 448–453. Noordwijkerhout, Netherlands (2012)

    Google Scholar 

  2. Stoican, F., Olaru, S., Seron, M., De Doná, J.: Reference governor design for tracking problems with fault detection guarantees. J. Process Control 22(5), 829–836 (2012)

    Google Scholar 

  3. Grundel, D., Murphey, R., Pardalos, P.: Cooperative Systems, Control and Optimization, vol. 588. Springer (2007)

    Google Scholar 

  4. Prodan, I., Olaru, S., Stoica, C., Niculescu, S.I.: On the tight formation for multi-agent dynamical systems. In: KES—Agents and Multi-agent Systems—Technologies and Applications, pp. 554–565. Springer (2012)

    Google Scholar 

  5. Prodan, I., Bitsoris, G., Olaru, S., Stoica, C., Niculescu, S.: On the limit behavior for multi-agent dynamical systems. In: The IFAC Workshop on Navigation, Guidance and Control of Underwater Vehicles, pp. 106–111. Porto, Portugal (2012)

    Google Scholar 

  6. Prodan, I., Olaru, S., Stoica, S., Niculescu, S.I.: Predictive control for trajectory tracking and decentralized navigation of multi-agent formations. Int. J. Appl. Math. Comput. Sci. 23(1), 91–102 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Blanchini, F., Miani, S.: Set-Theoretic Methods in Control, 2nd edn. Springer (2014)

    Google Scholar 

  8. Aubin, J.: Viability Theory. Birkhauser, Boston (1991)

    Google Scholar 

  9. Jünger, M., Junger, M., Liebling, T., Naddef, D., Nemhauser, G., Pulleyblank, W.: 50 Years of Integer Programming 1958-2008: From the Early Years to the State-of-the-Art. Springer (2009)

    Google Scholar 

  10. Fourier, J.: Solution d’une question particulière du calcul des inégalités. Nouveau Bulletin des Sciences par la Société Philomatique de Paris, pp. 317–319 (1826)

    Google Scholar 

  11. Farkas, G.: On the applications of the mechanical principle of fourier. Mathematikai és Természettudományi Értesoto 12, 457–472 (1894)

    Google Scholar 

  12. Minkowski, H.: Geometrie der Zahlen (Erste Lieferung). Teubner, Leipzig (1896)

    Google Scholar 

  13. Dantzig, G.: Maximization of a linear function of variables subject to linear inequalities. In: Koopmans, T.C. (ed.) Activity Analysis of Production and Allocation, pp. 339–347. Wiley, New York (1951)

    Google Scholar 

  14. Dantzig, G., Fulkerson, R., Johnson, S.: Solution of a large-scale traveling-salesman problem. J. Oper. Res. Soc. Am. 2(4), 393–410 (1954)

    MathSciNet  Google Scholar 

  15. Kuhn, H.W.: The hungarian method for the assignment problem. Naval Res. Logist. Q. 2(1–2), 83–97 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hoffman, A.J., Kruskal, J.B.: Integral boundary points of convex polyhedra. In: 50 Years of Integer Programming 1958–2008, pp. 49–76. Springer (2010)

    Google Scholar 

  17. Gomory, R.E., et al.: Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc. 64(5), 275–278 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  18. Van Leeuwen, J., Leeuwen, J.: Handbook of Theoretical Computer Science: Algorithms and Complexity, vol. 1. Elsevier (1990)

    Google Scholar 

  19. Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  20. Edmonds, J.: Systems of distinct representatives and linear algebra. J. Res. Nat. Bur. Stan. B 71, 241–245 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  21. Garey, M., Johnson, D.: Computers and intractability. A guide to the theory of NP-completeness. A Series of Books in the Mathematical Sciences. WH Freeman and Company, San Francisco, Calif (1979)

    Google Scholar 

  22. Khachiyan, L.: A polynomial algorithm in linear programming. Sov. Math. Dokl. 20, 191–194 (1979)

    MATH  Google Scholar 

  23. Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pritsker, A.A.B., Waiters, L.J., Wolfe, P.M.: Multiproject scheduling with limited resources: a zero-one programming approach. Manag. Sci. 16(1), 93–108 (1969)

    Article  Google Scholar 

  25. Sahinidis, N., Grossmann, I.E.: Reformulation of multiperiod milp models for planning and scheduling of chemical processes. Comput. Chem. Eng. 15(4), 255–272 (1991)

    Article  Google Scholar 

  26. Grundel, D., Pardalos, P.: Theory and Algorithms for Cooperative Systems, vol. 4. World Scientific Publishing Co Inc. (2004)

    Google Scholar 

  27. Earl, M., D’Andrea, R.: Modeling and control of a multi-agent system using mixed integer linear programming. In: Proceedings of the 40th IEEE Conference on Decision and Control, vol. 1, pp. 107–111. Orlando, Florida, USA (2001)

    Google Scholar 

  28. Propoi, A.: Use of linear programming methods for synthesizing sampled-data automatic systems. Autom. Remote Control 24(7), 837–844 (1963)

    Google Scholar 

  29. Richalet, J., Rault, A., Testud, J., Papon, J.: Model predictive heuristic control: applications to industrial processes. Automatica 14(5), 413–428 (1978)

    Article  Google Scholar 

  30. Mayne, D.Q., Schroeder, W.: Robust time-optimal control of constrained linear systems. Automatica 33, 2103–2118 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Schouwenaars, T., De Moor, B., Feron, E., How, J.: Mixed integer programming for multi-vehicle path planning. In: Proceedings of the 2nd IEEE European Control Conference, pp. 2603–2608. Citeseer, Porto, Portugal (2001)

    Google Scholar 

  32. Richards, A., How, J.: Model predictive control of vehicle maneuvers with guaranteed completion time and robust feasibility. In: Proceedings of the 24th American Control Conference, vol. 5, pp. 4034–4040. Portland, Oregon, USA (2005)

    Google Scholar 

  33. Ousingsawat, J., Campbell, M.: Establishing trajectories for multi-vehicle reconnaissance. In: Proceedings of the 22nd AIAA Guidance, Navigation, and Control Conference, pp. 2188–2199. Providence, Rhode Island, USA (2004)

    Google Scholar 

  34. Beard, R., McLain, T., Goodrich, M., Anderson, E.: Coordinated target assignment and intercept for unmanned air vehicles. Proc. IEEE Trans. Robot. Autom. 18(6), 911–922 (2002)

    Article  Google Scholar 

  35. Richards, A., Bellingham, J., Tillerson, M., How, J.: Coordination and control of multiple UAVs. In: AIAA Guidance, Navigation, and Control Conference, Monterey, CA (2002)

    Google Scholar 

  36. Schumacher, C.: UAV task assignment with timing constraints. Tech. rep., Air Force research lab Wright-Patterson AFB of air vehicles directorate (2003)

    Google Scholar 

  37. Bemporad, A., Morari, M.: Control of systems integrating logic, dynamics, and constraints. Automatica 35, 407–428 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  38. Bemporad, A., Borrelli, F., Morari, M.: Optimal controllers for hybrid systems: stability and piecewise linear explicit form. In: Proceedings of the 39th IEEE Conference on Decision and Control, vol. 2, pp. 1810–1815. IEEE (2000)

    Google Scholar 

  39. Bellingham, J., Richards, A., How, J.: Receding horizon control of autonomous aerial vehicles. In: IEEE (IEEE (ed.): Proceedings of the 21th American Control Conference, pp. 138–143. Anchorage, Alaska, USA (2002)

    Google Scholar 

  40. Prodan, I., Olaru, S., Stoica Maniu, C., Niculescu, S.I.: Predictive control for tight group formation of multi-agent systems. In: Proceedings of the 18th IFAC World Congress. Milano, Italy (2011), pp. 138–143

    Google Scholar 

  41. Prodan, I., Stoican, F., Olaru, S., Stoica, C., Niculescu, S.I.: Mixed-integer programming techniques in distributed MPC problems. In: Distributed MPC Made Easy, vol. 69, pp. 273–288. Springer (2013)

    Google Scholar 

  42. Stoican, F., Olaru, S., Seron, M., De Doná, J.: Reference governor for tracking with fault detection capabilities. In: Proceedings of the 2010 Conference on Control and Fault Tolerant Systems, pp. 546–551. Nice, France (2010)

    Google Scholar 

  43. Hovd, M.: Multi-level Programming for Designing Penalty Functions for MPC Controllers. In: Proceedings of the 18th IFAC World Congress, pp. 6098–6103. Milano, Italy (2011)

    Google Scholar 

  44. Hovd, M., Stoican, F.: On the design of exact penalty functions for MPC using mixed integer programming. Comput. Chem. Eng. 70, 104–113 (2013)

    Google Scholar 

  45. Earl, M.G., D’Andrea, R.: Iterative MILP methods for vehicle-control problems. IEEE Trans. Robot. 21(6), 1158–1167 (2005)

    Google Scholar 

  46. Vitus, M.P., Pradeep, V., Hoffmann, G., Waslander, S.L., Tomlin, C.J.: Tunnel-milp: path planning with sequential convex polytopes. In: AIAA Guidance, Navigation, and Control Conference, pp. 1–13. Honolulu, Hawaii, USA (2008)

    Google Scholar 

  47. Vielma, J., Nemhauser, G.: Modeling disjunctive constraints with a logarithmic number of binary variables and constraints. Math. Prog. 128(1), 49–72 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  48. Stoican, F., Prodan, I., Olaru, S.: On the hyperplanes arrangements in mixed-integer techniques. In: Proceedings of the 30th American Control Conference, pp. 1898–1903. San Francisco, California, USA (2011)

    Google Scholar 

  49. Stoican, F., Prodan, I., Olaru, S.: Enhancements on the hyperplane arrangements in mixed integer techniques. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference, pp. 3986–3991. Orlando, Florida, USA (2011)

    Google Scholar 

  50. Zaslavsky, T.: Facing up to arrangements: face-count formulas for partitions of space by hyperplanes. American Mathematical Society (1975)

    Google Scholar 

  51. Edelsbrunner, H., Seidel, R., Sharir, M.: On the zone theorem for hyperplane arrangements. In: New Results and New Trends in Computer Science, pp. 108–123 (1991)

    Google Scholar 

  52. Orlik, P., Terao, H.: Arrangements of Hyperplanes, vol. 300. Springer (1992)

    Google Scholar 

  53. Geyer, T., Torrisi, F., Morari, M.: Optimal complexity reduction of piecewise affine models based on hyperplane arrangements. In: Proceedings of the 23th American Control Conference, vol. 2, pp. 1190–1195. Boston, Massachusetts, USA (2004)

    Google Scholar 

  54. Orlik, P.: Hyperplane arrangements. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization, pp. 1545–1547. Springer, US (2009)

    Chapter  Google Scholar 

  55. Cairns, S.S.: Polyhedral approximations to regular loci. Ann. Math. 409–415 (1936)

    Google Scholar 

  56. Padberg, M., Rinaldi, G.: A branch and cut algorithm for the resolution of large-scale symmetric traveling salesmen problems. SIAM Rev. 33, 60–100 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  57. Land, A., Doig, A.: An automatic method of solving discrete programming problems. Econometrica 28, 497–520 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  58. Gomory, R.: Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc. 64, 275–278 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  59. Lodi, A.: Mixed integer programming computation. In: 50 Years of Integer Programming 1958–2008, pp. 619–645. Springer (2010)

    Google Scholar 

  60. Bixby, R.: Solving real-world linear programs: a decade and more of progress. Oper. Res. 50, 315 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  61. Bixby, R.: Mixed-integer programming: a progress report, the sharpest cut: the impact of manfred padberg and his work. In: Grotschel, M. (ed.) MPS-SIAM Series on Optimization pp. 309–325 (2004)

    Google Scholar 

  62. Optimization, G.: Gurobi Optimizer Reference Manual. http://www.gurobi.com (2012)

  63. Achterberg, T.: SCIP-a framework to integrate constraint and mixed integer programming. Konrad-Zuse-Zentrum für Informationstechnik (2004)

    Google Scholar 

  64. Achterberg, T.: Scip: solving constraint integer programs. Math. Prog. Comput. 1(1), 1–41 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  65. D’Ambrosio, C., Lodi, A.: Mixed integer nonlinear programming tools: a practical overview. 4OR 9(4), 329–349 (2011)

    Google Scholar 

  66. Borghetti, A., D’Ambrosio, C., Lodi, A., Martello, S.: An milp approach for short-term hydro scheduling and unit commitment with head-dependent reservoir. IEEE Trans. Power Syst. 23(3), 1115–1124 (2008)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ionela Prodan .

Rights and permissions

Reprints and permissions

Copyright information

© 2016 The Author(s)

About this chapter

Cite this chapter

Prodan, I., Stoican, F., Olaru, S., Niculescu, SI. (2016). Introduction. In: Mixed-Integer Representations in Control Design. SpringerBriefs in Electrical and Computer Engineering(). Springer, Cham. https://doi.org/10.1007/978-3-319-26995-5_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-26995-5_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-26993-1

  • Online ISBN: 978-3-319-26995-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics