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Duality-Based Nested Controller Synthesis from STL Specifications for Stochastic Linear Systems

  • Susmit Jha
  • Sunny RajEmail author
  • Sumit Kumar Jha
  • Natarajan Shankar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11022)

Abstract

We propose an automatic synthesis technique to generate provably correct controllers of stochastic linear dynamical systems for Signal Temporal Logic (STL) specifications. While formal synthesis problems can be directly formulated as exists-forall constraints, the quantifier alternation restricts the scalability of such an approach. We use the duality between a system and its proof of correctness to partially alleviate this challenge. We decompose the controller synthesis into two subproblems, each addressing orthogonal concerns - stabilization with respect to the noise, and meeting the STL specification. The overall controller is a nested controller comprising of the feedback controller for noise cancellation and an open loop controller for STL satisfaction. The correct-by-construction compositional synthesis of this nested controller relies on using the guarantees of the feedback controller instead of the controller itself. We use a linear feedback controller as the stabilizing controller for linear systems with bounded additive noise and over-approximate its ellipsoid stability guarantee with a polytope. We then use this over-approximation to formulate a mixed-integer linear programming (MILP) problem to synthesize an open-loop controller that satisfies STL specifications.

Notes

Acknowledgements

The authors acknowledge support from the National Science Foundation (NSF) Cyber-Physical Systems #1740079 project, NSF Software & Hardware Foundation #1750009 and #1438989 projects, US ARL Cooperative Agreement W911NF-17-2-0196, and DARPA under contract FA8750-16-C-0043.

References

  1. 1.
    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)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Akametalu, A.K., Fisac, J.F., Gillula, J.H., Kaynama, S., Zeilinger, M.N., Tomlin, C.J.: Reachability-based safe learning with Gaussian processes. In: 53rd IEEE Conference on Decision and Control, pp. 1424–1431. IEEE (2014)Google Scholar
  3. 3.
    Bellman, R., Bellman, R.E., Bellman, R.E.: Introduction to the Mathematical Theory of Control Processes, vol. 2. IMA (1971)Google Scholar
  4. 4.
    Berkenkamp, F., Schoellig, A.P.: Safe and robust learning control with Gaussian processes. In: 2015 European Control Conference (ECC), pp. 2496–2501. IEEE (2015)Google Scholar
  5. 5.
    Blanchini, F.: Set invariance in control. Automatica 35(11), 1747–1767 (1999)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bogomolov, S., Schilling, C., Bartocci, E., Batt, G., Kong, H., Grosu, R.: Abstraction-based parameter synthesis for multiaffine systems. In: Piterman, N. (ed.) HVC 2015. LNCS, vol. 9434, pp. 19–35. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-26287-1_2CrossRefGoogle Scholar
  7. 7.
    Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory, vol. 15. SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  8. 8.
    Cassandras, C.G., Lygeros, J.: Stochastic Hybrid Systems, vol. 24. CRC Press, Boca Raton (2006)zbMATHGoogle Scholar
  9. 9.
    Dang, T., Dreossi, T., Piazza, C.: Parameter synthesis through temporal logic specifications. In: Bjørner, N., de Boer, F. (eds.) FM 2015. LNCS, vol. 9109, pp. 213–230. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-19249-9_14CrossRefGoogle Scholar
  10. 10.
    Donzé, A., Maler, O.: Robust satisfaction of temporal logic over real-valued signals. In: Chatterjee, K., Henzinger, T.A. (eds.) FORMATS 2010. LNCS, vol. 6246, pp. 92–106. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-15297-9_9CrossRefzbMATHGoogle Scholar
  11. 11.
    Eggers, A., Fränzle, M., Herde, C.: SAT modulo ODE: a direct SAT approach to hybrid systems. In: Cha, S.S., Choi, J.-Y., Kim, M., Lee, I., Viswanathan, M. (eds.) ATVA 2008. LNCS, vol. 5311, pp. 171–185. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-88387-6_14CrossRefGoogle Scholar
  12. 12.
    Fainekos, G.E., Girard, A., Kress-Gazit, H., Pappas, G.J.: Temporal logic motion planning for dynamic robots. Automatica 45(2), 343–352 (2009).  https://doi.org/10.1016/j.automatica.2008.08.008MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fainekos, G.E., Pappas, G.J.: Robustness of temporal logic specifications for continuous-time signals. Theor. Comput. Sci. 410(42), 4262–4291 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fan, C., Mathur, U., Mitra, S., Viswanathan, M.: Controller synthesis made real: reach-avoid specifications and linear dynamics. In: Chockler, H., Weissenbacher, G. (eds.) CAV 2018. LNCS, vol. 10981. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-96145-3_19CrossRefGoogle Scholar
  15. 15.
    Haddad, W.M., Chellaboina, V.: Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach. Princeton University Press, Princeton (2011)zbMATHGoogle Scholar
  16. 16.
    Huang, Z., Wang, Y., Mitra, S., Dullerud, G.E., Chaudhuri, S.: Controller synthesis with inductive proofs for piecewise linear systems: an SMT-based algorithm. In: 2015 54th IEEE Conference on Decision and Control (CDC), pp. 7434–7439, December 2015Google Scholar
  17. 17.
    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, Cham (2016).  https://doi.org/10.1007/978-3-319-40648-0_10CrossRefGoogle Scholar
  18. 18.
    Jha, S., Raman, V.: On optimal control of stochastic linear hybrid systems. In: Fränzle, M., Markey, N. (eds.) FORMATS 2016. LNCS, vol. 9884, pp. 69–84. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-44878-7_5CrossRefzbMATHGoogle Scholar
  19. 19.
    Jha, S., Raman, V., Sadigh, D., Seshia, S.A.: Safe autonomy under perception uncertainty using chance-constrained temporal logic. J. Autom. Reason. 60(1), 43–62 (2018).  https://doi.org/10.1007/s10817-017-9413-9MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kautsky, J., Nichols, N.K., Van Dooren, P.: Robust pole assignment in linear state feedback. Int. J. Control 41(5), 1129–1155 (1985)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kloetzer, M., Belta, C.: A fully automated framework for control of linear systems from temporal logic specifications. IEEE Trans. Autom. Control 53(1), 287–297 (2008).  https://doi.org/10.1109/TAC.2007.914952MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kong, S., Gao, S., Chen, W., Clarke, E.: dReach: \(\delta \)-reachability analysis for hybrid systems. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 200–205. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-46681-0_15CrossRefGoogle Scholar
  23. 23.
    Koutsoukos, X., 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).  https://doi.org/10.1007/11730637_29CrossRefzbMATHGoogle Scholar
  24. 24.
    Liu, J., Prabhakar, P.: Switching control of dynamical systems from metric temporal logic specifications. In: IEEE International Conference on Robotics and Automation (2014)Google Scholar
  25. 25.
    Maasoumy, M., Razmara, M., Shahbakhti, M., Vincentelli, A.S.: Handling model uncertainty in model predictive control for energy efficient buildings. Energy Build. 77, 377–392 (2014).  https://doi.org/10.1016/j.enbuild.2014.03.057. http://www.sciencedirect.com/science/article/pii/S0378778814002771CrossRefGoogle Scholar
  26. 26.
    Maasoumy, M., Sanandaji, B.M., Sangiovanni-Vincentelli, A., Poolla, K.: Model predictive control of regulation services from commercial buildings to the smart grid. In: 2014 American Control Conference (ACC), pp. 2226–2233. IEEE (2014)Google Scholar
  27. 27.
    Maler, O., Nickovic, D., Pnueli, A.: Real time temporal logic: past, present, future. In: Pettersson, P., Yi, W. (eds.) FORMATS 2005. LNCS, vol. 3829, pp. 2–16. Springer, Heidelberg (2005).  https://doi.org/10.1007/11603009_2CrossRefGoogle Scholar
  28. 28.
    Mitchell, I., Tomlin, C.J.: Level set methods for computation in hybrid systems. In: Lynch, N., Krogh, B.H. (eds.) HSCC 2000. LNCS, vol. 1790, pp. 310–323. Springer, Heidelberg (2000).  https://doi.org/10.1007/3-540-46430-1_27CrossRefzbMATHGoogle Scholar
  29. 29.
    Mitchell, I.M., Bayen, A.M., Tomlin, C.J.: A time-dependent hamilton-jacobi formulation of reachable sets for continuous dynamic games. IEEE Trans. Autom. Control 50(7), 947–957 (2005)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Ouaknine, J., Worrell, J.: Some recent results in metric temporal logic. In: Cassez, F., Jard, C. (eds.) FORMATS 2008. LNCS, vol. 5215, pp. 1–13. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-85778-5_1CrossRefGoogle Scholar
  31. 31.
    Pontryagin, L.: Optimal control processes. Usp. Mat. Nauk 14(3), 3–20 (1959)Google Scholar
  32. 32.
    Prabhakar, P., García Soto, M.: Formal synthesis of stabilizing controllers for switched systems. In: Proceedings of the 20th International Conference on Hybrid Systems: Computation and Control, HSCC 2017, pp. 111–120. ACM, New York (2017). http://doi.acm.org/10.1145/3049797.3049822
  33. 33.
    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
  34. 34.
    Prandini, M., Hu, J.: Stochastic reachability: theory and numerical approximation. Stochast. Hybrid Syst. Autom. Control Eng. Ser. 24, 107–138 (2006)CrossRefGoogle Scholar
  35. 35.
    Raman, V., Donz, A., Maasoumy, M., Murray, R.M., Sangiovanni-Vincentelli, A., Seshia, S.A.: Model predictive control with signal temporal logic specifications. In: 53rd IEEE Conference on Decision and Control, pp. 81–87, December 2014.  https://doi.org/10.1109/CDC.2014.7039363
  36. 36.
    Sadigh, D., Kapoor, A.: Safe control under uncertainty with probabilistic signal temporal logic. In: Robotics: Science and Systems XII (2016). http://www.roboticsproceedings.org/rss12/p17.html
  37. 37.
    Schrmann, B., Althoff, M.: Optimal control of sets of solutions to formally guarantee constraints of disturbed linear systems. In: 2017 American Control Conference (ACC), pp. 2522–2529, May 2017Google Scholar
  38. 38.
    Seto, D., Krogh, B.H., Sha, L., Chutinan, A.: Dynamic control system upgrade using the simplex architecture. IEEE Control Syst. 18(4), 72–80 (1998)CrossRefGoogle Scholar
  39. 39.
    Summers, S., Kamgarpour, M., Lygeros, J., Tomlin, C.: A stochastic reach-avoid problem with random obstacles. In: Proceedings of the 14th International Conference on Hybrid Systems: Computation and Control, pp. 251–260. ACM (2011)Google Scholar
  40. 40.
    Tabuada, P., Pappas, G.J.: Linear time logic control of discrete-time linear systems. IEEE Trans. Autom. Control 51(12), 1862–1877 (2006)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Wongpiromsarn, T., Topcu, U., Murray, R.M.: Receding horizon temporal logic planning. IEEE Trans. Autom. Control 57(11), 2817–2830 (2012).  https://doi.org/10.1109/TAC.2012.2195811MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Yordanov, B., Tumova, J., Cerna, I., Barnat, J., Belta, C.: Temporal logic control of discrete-time piecewise affine systems. IEEE Trans. Autom. Control 57(6), 1491–1504 (2012)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Susmit Jha
    • 1
  • Sunny Raj
    • 2
    Email author
  • Sumit Kumar Jha
    • 2
  • Natarajan Shankar
    • 1
  1. 1.Computer Science LaboratorySRI InternationalMenlo ParkUSA
  2. 2.Computer Science DepartmentUniversity of Central FloridaOrlandoUSA

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