Objective function design for robust optimality of linear control under state-constraints and uncertainty

Fabio Bagagiolo; Dario Bauso

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 1, page 155-177
  • ISSN: 1292-8119

Abstract

top
We consider a model for the control of a linear network flow system with unknown but bounded demand and polytopic bounds on controlled flows. We are interested in the problem of finding a suitable objective function that makes robust optimal the policy represented by the so-called linear saturated feedback control. We regard the problem as a suitable differential game with switching cost and study it in the framework of the viscosity solutions theory for Bellman and Isaacs equations.

How to cite

top

Bagagiolo, Fabio, and Bauso, Dario. "Objective function design for robust optimality of linear control under state-constraints and uncertainty." ESAIM: Control, Optimisation and Calculus of Variations 17.1 (2011): 155-177. <http://eudml.org/doc/276336>.

@article{Bagagiolo2011,
abstract = { We consider a model for the control of a linear network flow system with unknown but bounded demand and polytopic bounds on controlled flows. We are interested in the problem of finding a suitable objective function that makes robust optimal the policy represented by the so-called linear saturated feedback control. We regard the problem as a suitable differential game with switching cost and study it in the framework of the viscosity solutions theory for Bellman and Isaacs equations. },
author = {Bagagiolo, Fabio, Bauso, Dario},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Optimal control; viscosity solutions; differential games; switching; flow control; networks; optimal control},
language = {eng},
month = {2},
number = {1},
pages = {155-177},
publisher = {EDP Sciences},
title = {Objective function design for robust optimality of linear control under state-constraints and uncertainty},
url = {http://eudml.org/doc/276336},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Bagagiolo, Fabio
AU - Bauso, Dario
TI - Objective function design for robust optimality of linear control under state-constraints and uncertainty
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2011/2//
PB - EDP Sciences
VL - 17
IS - 1
SP - 155
EP - 177
AB - We consider a model for the control of a linear network flow system with unknown but bounded demand and polytopic bounds on controlled flows. We are interested in the problem of finding a suitable objective function that makes robust optimal the policy represented by the so-called linear saturated feedback control. We regard the problem as a suitable differential game with switching cost and study it in the framework of the viscosity solutions theory for Bellman and Isaacs equations.
LA - eng
KW - Optimal control; viscosity solutions; differential games; switching; flow control; networks; optimal control
UR - http://eudml.org/doc/276336
ER -

References

top
  1. F. Bagagiolo, Minimum time for a hybrid system with thermostatic switchings, in Hybrid Systems: Computation and Control, A. Bemporad, A. Bicchi and G. Buttazzo Eds., Lect. Notes Comput. Sci.4416, Springer-Verlag, Berlin, Germany (2007) 32–45.  Zbl1221.49053
  2. F. Bagagiolo and M. Bardi, Singular perturbation of a finite horizon problem with state-space constraints. SIAM J. Contr. Opt.36 (1998) 2040–2060.  Zbl0953.49031
  3. F. Bagagiolo and D. Bauso, Robust optimality of linear saturated control in uncertain linear network flows, in Decision and Control, 2008, CDC 2008, 47th IEEE Conference (2008) 3676–3681.  
  4. M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston, USA (1997).  Zbl0890.49011
  5. M. Bardi, S. Koike and P. Soravia, Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximation. Discrete Contin. Dyn. Syst.6 (2000) 361–380.  Zbl1158.91323
  6. D. Bauso, F. Blanchini and R. Pesenti, Robust control policies for multi-inventory systems with average flow constraints. Automatica42 (2006) 1255–1266.  Zbl1097.90002
  7. A. Bemporad, M. Morari, V. Dua and E.N. Pistikopoulos, The explicit linear quadratic regulator for constrained systems. Automatica38 (2002) 320.  Zbl0999.93018
  8. A. Ben Tal and A. Nemirovsky, Robust solutions of uncertain linear programs. Oper. Res.25 (1998) 1–13.  
  9. D.P. Bertsekas and I. Rhodes, Recursive state estimation for a set-membership description of uncertainty. IEEE Trans. Automatic Control16 (1971) 117–128.  
  10. D. Bertsimas and A. Thiele, A robust optimization approach to inventory theory. Oper. Res.54 (2006) 150–168.  Zbl1167.90314
  11. P. Cardialaguet, M. Quincampoix and P. Saint-Pierre, Pursuit differential games with state constraints. SIAM J. Contr. Opt.39 (2001) 1615–1632.  
  12. J. Casti, On the general inverse problem of optimal control theory. J. Optim. Theory Appl.32 (1980) 491–497.  Zbl0421.49029
  13. X. Chen, M. Sim, P. Sun and J. Zhang, A linear-decision based approximation approach to stochastic programming. Oper. Res.56 (2008) 344–357.  Zbl1167.90609
  14. M.G. Crandall, L.C. Evans and P.L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.282 (1984) 487–502.  Zbl0543.35011
  15. S. Dharmatti and M. Ramaswamy, Zero-sum differential games involving hybrid controls. J. Optim. Theory Appl.128 (2006) 75–102.  Zbl1099.91022
  16. R.J. Elliot and N.J. Kalton, The existence of value in differential games, Mem. Amer. Math. Soc.126. AMS, Providence, USA (1972).  
  17. L.C. Evans and H. Ishii, Differential games and nonlinear first order PDE on bounded domains. Manuscripta Math.49 (1984) 109–139.  Zbl0559.35013
  18. M. Garavello and P. Soravia, Representation formulas for solutions of HJI equations with discontinuous coefficients and existence of value in differential games. J. Optim. Theory Appl.130 (2006) 209–229.  Zbl1123.49033
  19. S. Koike, On the state constraint problem for differential games. Indiana Univ. Math. J.44 (1995) 467–487.  Zbl0840.49016
  20. O. Kostyukova and E. Kostina, Robust optimal feedback for terminal linear-quadratic control problems under disturbances. Math. Program.107 (2006) 131–153.  Zbl1089.49035
  21. V.B. Larin, About the inverse problem of optimal control. Appl. Comput. Math2 (2003) 90–97.  Zbl1209.49045
  22. T.T. Lee and G.T. Liaw, The inverse problem of linear optimal control for constant disturbance. Int. J. Control43 (1986) 233–246.  Zbl0584.93027
  23. P. Soravia, Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J.51 (2002) 451–477.  Zbl1032.35055
  24. H.M. Soner, Optimal control problems with state-space constraints I. SIAM J. Contr. Opt.31 (1986) 132–146.  
  25. A. Visintin, Differential Models of Hysteresis. Springer-Verlag, Berlin, Germany (1996).  Zbl0906.93006

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.