On Optimal Control of Stochastic Linear Hybrid Systems

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9884)


Cyber-physical systems are often hybrid consisting of both discrete and continuous subsystems. The continuous dynamics in cyber-physical systems could be noisy and the environment in which these stochastic hybrid systems operate can also be uncertain. We focus on multimodal hybrid systems in which the switching from one mode to another is determined by a schedule and the optimal finite horizon control problem is to discover the switching schedule as well as the control inputs to be applied in each mode such that some cost metric is minimized over the given horizon. We consider discrete-time control in this paper. We present a two step approach to solve this problem with respect to convex cost objectives and probabilistic safety properties. Our approach uses a combination of sample average approximation and convex programming. We demonstrate the effectiveness of our approach on case studies from temperature-control in buildings and motion planning.


  1. 1.
    Abate, A., Amin, S., Prandini, M., Lygeros, J., Sastry, S.S.: Computational approaches to reachability analysis of stochastic hybrid systems. In: Bemporad, A., Bicchi, A., Buttazzo, G. (eds.) HSCC 2007. LNCS, vol. 4416, pp. 4–17. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Abate, A., Prandini, M., Lygeros, J., Sastry, S.: Probabilistic reachability and safety for controlled discrete time stochastic hybrid systems. Automatica 44(11), 2724–2734 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Acikmese, B., Ploen, S.R.: Convex programming approach to powered descent guidance for Mars landing. J. Guidance Control Dyn. 30(5), 1353–1366 (2007)CrossRefGoogle Scholar
  4. 4.
    Alur, R.: Formal verification of hybrid systems. In: EMSOFT, pp. 273–278. IEEE (2011)Google Scholar
  5. 5.
    Asarin, E., Bournez, O., Dang, T., Maler, O., Pnueli, A.: Effective synthesis of switching controllers for linear systems. Proc. IEEE 88(7), 1011–1025 (2000)CrossRefGoogle Scholar
  6. 6.
    Barr, N.M., Gangsaas, D., Schaeffer, D.R.: Wind models for flight simulator certification of landing and approach guidance and control systems. Technical report, DTIC Document (1974)Google Scholar
  7. 7.
    Bellman, R.E.: Introduction to the Mathematical Theory of Control Processes, vol. 2. IMA (1971)Google Scholar
  8. 8.
    Blackmore, L., Ono, M., Bektassov, A., Williams, B.C.: A probabilistic particle-control approximation of chance-constrained stochastic predictive control. IEEE Trans. Robot. 26(3), 502–517 (2010)CrossRefGoogle Scholar
  9. 9.
    Campi, M.C., Garatti, S., Prandini, M.: The scenario approach for systems and control design. Ann. Rev. Control 33(2), 149–157 (2009)CrossRefGoogle Scholar
  10. 10.
    Cassandras, C.G., Lygeros, J.: Stochastic Hybrid Systems, vol. 24. CRC Press, Boca Raton (2006)MATHGoogle Scholar
  11. 11.
    Charnes, A., Cooper, W.W., Symonds, G.H.: Cost horizons and certainty equivalents: an approach to stochastic programming of heating oil. Manage. Sci. 4(3), 235–263 (1958)CrossRefGoogle Scholar
  12. 12.
    Deori, L., Garatti, S., Prandini, M.: A model predictive control approach to aircraft motion control. In: American Control Conference, ACC 2015, 1–3 July 2015, Chicago, IL, USA, pp. 2299–2304 (2015)Google Scholar
  13. 13.
    Fang, C., Williams, B.C.: General probabilistic bounds for trajectories using only mean and variance. In: ICRA, pp. 2501–2506 (2014)Google Scholar
  14. 14.
    Frank, P.M.: Advances in Control: Highlights of ECC. Springer Science & Business Media, New York (2012)Google Scholar
  15. 15.
    Gonzalez, H., Vasudevan, R., Kamgarpour, M., Sastry, S., Bajcsy, R., Tomlin, C.: A numerical method for the optimal control of switched systems. In: CDC 2010, pp. 7519–7526 (2010)Google Scholar
  16. 16.
    Gonzalez, H., Vasudevan, R., Kamgarpour, M., Sastry, S.S., Bajcsy, R., Tomlin, C.J.: A descent algorithm for the optimal control of constrained nonlinear switched dynamical systems (2010)Google Scholar
  17. 17.
    Jha, S., Gulwani, S., Seshia, S.A., Tiwari, A.: Synthesizing switching logic for safety and dwell-time requirements. In: ICCPS, pp. 22–31 (2010)Google Scholar
  18. 18.
    Jha, S., Raman, V.: Automated synthesis of safe autonomous vehicle control under perception uncertainty. In: Rayadurgam, S., Tkachuk, O. (eds.) NFM 2016. LNCS, vol. 9690, pp. 117–132. Springer, Heidelberg (2016). doi: 10.1007/978-3-319-40648-0_10 CrossRefGoogle Scholar
  19. 19.
    Jha, S., Seshia, S.A., Tiwari, A.: Synthesis of optimal switching logic for hybrid systems. In: EMSOFT, pp. 107–116 (2011)Google Scholar
  20. 20.
    Kall, P., Wallace, S.: Stochastic Programming. Wiley-Interscience Series in Systems and Optimization. Wiley, New York (1994)MATHGoogle Scholar
  21. 21.
    Kamgarpour, M., Soler, M., Tomlin, C.J., Olivares, A., Lygeros, J.: Hybrid optimal control for aircraft trajectory design with a variable sequence of modes. In: 18th IFAC World Congress, Italy (2011)Google Scholar
  22. 22.
    Kariotoglou, N., Summers, S., Summers, T., Kamgarpour, M., Lygeros, J.: Approximate dynamic programming for stochastic reachability. In: ECC, pp. 584–589. IEEE (2013)Google Scholar
  23. 23.
    Kleywegt, A.J., Shapiro, A., Homem-de Mello, T.: The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12(2), 479–502 (2002)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Koutsoukos, X.D., Riley, D.: Computational methods for reachability analysis of stochastic hybrid systems. In: Hespanha, J.P., Tiwari, A. (eds.) HSCC 2006. LNCS, vol. 3927, pp. 377–391. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  25. 25.
    Li, P., Arellano-Garcia, H., Wozny, G.: Chance constrained programming approach to process optimization under uncertainty. Comput. Chem. Eng. 32(1–2), 25–45 (2008)CrossRefGoogle Scholar
  26. 26.
    Li, P., Wendt, M., Wozny, G.: A probabilistically constrained model predictive controller. Automatica 38(7), 1171–1176 (2002)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Liberzon, D.: Switching in Systems and Control. Springer Science & Business Media, New York (2012)MATHGoogle Scholar
  28. 28.
    Ma, Y.: Model predictive control for energy efficient buildings. Ph.D. Thesis, Department of Mechanical Engineering, UC Berkeley (2012)Google Scholar
  29. 29.
    Margellos, K., Prandini, M., Lygeros, J.: A compression learning perspective to scenario based optimization. In: CDC 2014, pp. 5997–6002 (2014)Google Scholar
  30. 30.
    Miller, B.L., Wagner, H.M.: Chance constrained programming with joint constraints. Oper. Res. 13(6), 930–945 (1965)CrossRefMATHGoogle Scholar
  31. 31.
    Nemirovski, A., Shapiro, A.: Convex approximations of chance constrained programs. SIAM J. Optim. 17(4), 969–996 (2006)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Ono, M., Blackmore, L., Williams, B.C.: Chance constrained finite horizon optimal control with nonconvex constraints. In: ACC, pp. 1145–1152. IEEE (2010)Google Scholar
  33. 33.
    Pontryagin, L.: Optimal control processes. Usp. Mat. Nauk 14(3), 3–20 (1959)Google Scholar
  34. 34.
    Prajna, S., Jadbabaie, A., Pappas, G.J.: A framework for worst-case and stochastic safety verification using barrier certificates. IEEE Trans. Autom. Control 52(8), 1415–1428 (2007)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Prandini, M., Garatti, S., Lygeros, J.: A randomized approach to stochastic model predictive control. In: CDC 2012, pp. 7315–7320 (2012)Google Scholar
  36. 36.
    Prandini, M., Hu, J.: Stochastic reachability: theory and numerical approximation. Stochast. Hybrid Syst. Autom. Control Eng. Ser. 24, 107–138 (2006)CrossRefMATHGoogle Scholar
  37. 37.
    Prékopa, A.: Stochastic Programming, vol. 324. Springer Science & Business Media, New York (2013)MATHGoogle Scholar
  38. 38.
    Sastry, S.S.: Nonlinear Systems: Analysis, Stability, and Control. Interdisciplinary Applied Mathematics. Springer, New York (1999). Numrotation dans la coll. principaleCrossRefMATHGoogle Scholar
  39. 39.
    Van Hessem, D., Scherer, C., Bosgra, O.: LMI-based closed-loop economic optimization of stochastic process operation under state and input constraints. In: 2001 Proceedings of the 40th IEEE Conference on Decision and Control, vol. 5, pp. 4228–4233. IEEE (2001)Google Scholar
  40. 40.
    Vichik, S., Borrelli, F.: Identification of thermal model of DOE library. Technical report, ME Department, Univ. California at Berkeley (2012)Google Scholar
  41. 41.
    Vitus, M.P., Tomlin, C.J.: Closed-loop belief space planning for linear, Gaussian systems. In: ICRA, pp. 2152–2159. IEEE (2011)Google Scholar
  42. 42.
    Xue, D., Chen, Y., Atherton, D.P.: Linear feedback control: analysis and design with MATLAB, vol. 14. SIAM (2007)Google Scholar
  43. 43.
    Zhang, Y., Sankaranarayanan, S., Somenzi, F.: Statistically sound verification and optimization for complex systems. In: Cassez, F., Raskin, J.-F. (eds.) ATVA 2014. LNCS, vol. 8837, pp. 411–427. Springer, Heidelberg (2014)Google Scholar
  44. 44.
    Zhu, F., Antsaklis, P.J.: Optimal control of switched hybrid systems: a brief survey. Discrete Event Dyn. Syst. 23(3), 345–364 (2011). ISISMathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.United Technology Research CenterBerkeleyUSA

Personalised recommendations