Analyzing Reachability of Linear Dynamic Systems with Parametric Uncertainties

  • Matthias AlthoffEmail author
  • Bruce H. Krogh
  • Olaf Stursberg
Part of the Mathematical Engineering book series (MATHENGIN, volume 3)


As an important approach to analyzing safety of a dynamic system, this paper considers the task of computing overapproximations of reachable sets, i.e. the set of states which is reachable from a given initial set of states. The class of systems under investigation are linear, time-invariant systems with parametric uncertainties and uncertain but bounded input. The possible set of system matrices due to uncertain parameters is represented by matrix zonotopes and interval matrices – computational techniques for both representations are presented. The reachable set is represented by zonotopes, which makes it possible to apply the approach to systems of 100 continuous state variables with computation times of a few minutes. This is demonstrated for randomized examples as well as a transmission line example.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Glover, J.D., Schweppe, F.C.: Control of linear dynamic systems with set constrained disturbances. IEEE Transactions on Automatic Control 16(5), 411–423 (1971)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Clarke, E., Fehnker, A., Han, Z., Krogh, B.H., Ouaknine, J., Stursberg, O., Theobald, M.: Abstraction and counterexample-guided refinement in model checking of hybrid systems. International Journal of Foundations of Computer Science 14(4), 583–604 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Schlaepfer, F.M., Schweppe, F.C.: Continuous-time state estimation under disturbances bounded by convex sets. IEEE Transactions on Automatic Control 17(2), 197–205 (1972)zbMATHCrossRefGoogle Scholar
  4. 4.
    Althoff, M., Stursberg, O., Buss, M.: Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization. In: Proc. of the 47th IEEE Conference on Decision and Control, pp. 4042–4048 (2008)Google Scholar
  5. 5.
    Althoff, M., Stursberg, O., Buss, M.: Computing reachable sets of hybrid systems using a combination of zonotopes and polytopes. Nonlinear Analysis: Hybrid Systems 4(2), 233–249 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Henzinger, T.: Verification of Digital and Hybrid Systems, NATO ASI Series F: Computer and Systems Sciences, vol. 170, chap. The theory of hybrid automata, pp. 265–292. Springer (2000)Google Scholar
  7. 7.
    Henzinger, T.A., Ho, P.H., Wong-Toi, H.: Algorithmic analysis of nonlinear hybrid systems. IEEE Transactions on Automatic Control 43(4), 540–554 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Frehse, G.: PHAVer: Algorithmic verification of hybrid systems past HyTech. In: Hybrid Systems: Computation and Control, LNCS 3413, pp. 258–273. Springer (2005)Google Scholar
  9. 9.
    Lafferriere, G., Pappas, G.J., Yovine, S.: Symbolic reachability computation for families of linear vector fields. Symbolic Computation 32, 231–253 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Chutinan, A., Krogh, B.H.: Computational techniques for hybrid system verification. IEEE Transactions on Automatic Control 48(1), 64–75 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Kurzhanskiy, A.B., Varaiya, P.: Ellipsoidal techniques for reachability analysis of discretetime linear systems. IEEE Transactions on Automatic Control 52(1), 26–38 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Stursberg, O., Krogh, B.H.: Efficient representation and computation of reachable sets for hybrid systems. In: Hybrid Systems: Computation and Control, LNCS 2623, pp. 482–497. Springer (2003)Google Scholar
  13. 13.
    Kuhn, W.: Rigorously computed orbits of dynamical systems without the wrapping effect. Computing 61, 47–67 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Girard, A.: Reachability of uncertain linear systems using zonotopes. In: Hybrid Systems: Computation and Control, LNCS 3414, pp. 291–305. Springer (2005)Google Scholar
  15. 15.
    Tomlin, C., Mitchell, I., Bayen, A., Oishi, M.: Computational techniques for the verification and control of hybrid systems. Proceedings of the IEEE 91(7), 986–1001 (2003)CrossRefGoogle Scholar
  16. 16.
    Girard, A., Guernic, C.L.: Efficient reachability analysis for linear systems using support functions. In: Proc. of the 17th IFAC World Congress, pp. 8966–8971 (2008)Google Scholar
  17. 17.
    Henzinger, T.A.,Horowitz, B.,Majumdar, R.,Wong-Toi, H.: Beyond HyTech: Hybrid systems analysis using interval numerical methods. In: Hybrid Systems: Computation and Control, LNCS 1790, pp. 130–144. Springer (2000)Google Scholar
  18. 18.
    Ramdani, N., Meslem, N., Candau, Y.: Reachability analysis of uncertain nonlinear systems using guaranteed set integration. In: Proc. of the 17th IFAC World Congress, pp. 8972–8977 (2008)Google Scholar
  19. 19.
    Ramdani, N., Meslem, N., Candau, Y.: Reachability of uncertain nonlinear systems using a nonlinear hybridization. In: Hybrid Systems: Computation and Control, LNCS 4981, pp. 415–428. Springer (2008)Google Scholar
  20. 20.
    Nedialkov, N.S., Jackson, K.R.: Perspectives on Enclosure Methods, chap. A New Perspective on the Wrapping Effect in Interval Methods for Initial Value Problems for Ordinary Differential Equations, pp. 219–264. Springer-Verlag (2001)Google Scholar
  21. 21.
    Krasnochtanova, I., Rauh, A., Kletting, M., Aschemann, H., Hofer, E.P., Schoop, K.M.: Interval methods as a simulation tool for the dynamics of biological wastewater treatment processes with parameter uncertainties. Applied Mathematical Modeling 34(3), 744–762 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Rauh, A., Auer, E., Hofer, E.P.: Valencia-ivp: A comparison with other initial value problem solvers. In: CD-Proc. of the 12th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics. IEEE Computer Society (2007)Google Scholar
  23. 23.
    Asarin, E., Dang, T., Frehse, G., Girard, A., Le Guernic, C.,Maler, O.: Recent progress in continuous and hybrid reachability analysis. In: Proc. of the 2006 IEEE Conference on Computer Aided Control Systems Design, pp. 1582–1587 (2006)Google Scholar
  24. 24.
    Prajna, S.: Barrier certificates for nonlinear model validation. Automatica 42(1), 117–126 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Girard, A., Pappas, G.J.: Verification using simulation. In: Hybrid Systems: Computation and Control, LNCS 3927, pp. 272–286. Springer (2006)Google Scholar
  26. 26.
    Kapinski, J., Donz´e, A., Lerda, F.,Maka, H.,Wagner, S., Krogh, B.H.: Control software model checking using bisimulation functions for nonlinear systems. In: Proc. of the 47th IEEE Conference on Decision and Control, pp. 4024–4029 (2008)Google Scholar
  27. 27.
    Althoff, M.: Reachability analysis and its application to the safety assessment of autonomous cars. Dissertation, TU M¨unchen (2010). URL: Scholar
  28. 28.
    Althoff, M., Le Guernic, C., Krogh, B.H.: Reachable set computation for uncertain time varying linear systems. In: Hybrid Systems: Computation and Control (2011)Google Scholar
  29. 29.
    Lohner, R.: Perspectives on Enclosure Methods, chap. On the Ubiquity of theWrapping Effect in the Computation of the Error Bounds, pp. 201–217. Springer (2001)Google Scholar
  30. 30.
    Dang, T.: V´erification et synth`ese des syst`emes hybrides. Ph.D. thesis, Institut National Polytechnique de Grenoble (2000)Google Scholar
  31. 31.
    Rump, S.M.: Fast and parallel interval arithmetic. BIT Numerical Mathematics 39(3), 534–554 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Moler, C., Loan, C.V.: Nineteen dubious ways to compute the exponential of a matrix, twenty five years later. SIAM Review 45(1), 3–49 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Coxson, G.E.: Computing exact bounds on elements of an inverse interval matrix is NP-hard. Reliable Computing 5(2), 137–142 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Kolev, L.V.: Outer interval solution of the eigenvalue problem under general form parametric dependencies. Reliable Computing 12(2), 121–140 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Rugh, W.J.: Linear System Theory. Prentice Hall (1996)Google Scholar
  36. 36.
    Kosheleva, O., Kreinovich, V., Mayer, G., Nguyen, H.T.: Computing the cube of an interval matrix is NP-hard. In: Proc. of the ACM symposium on Applied computing, pp. 1449–1453 (2005)Google Scholar
  37. 37.
    Ahn, H.S., Chen, Y.Q.,Moore, K.L.:Maximum singular value and power of an interval matrix. In: Proc. of the 2006 IEEE International Conference on Mechatronics and Automation, pp. 678–683 (2006)Google Scholar
  38. 38.
    Dyer, M., Gritzmann, P., Hufnagel, A.: On the complexity of computing mixed volumes. SIAM Journal on Computing 27(2), 356–400 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Girard, A., Guernic, C.L., Maler, O.: Efficient computation of reachable sets of linear time in variant systems with inputs. In: Hybrid Systems: Computation and Control, LNCS 3927, pp. 257–271. Springer (2006)Google Scholar
  40. 40.
    Pillage, L.T., Rohrer, R.A.: Asymptotic waveform evaluation for timing analysis. IEEE Transactions on Computer-Aided Design 9(4), 352–366 (1990)CrossRefGoogle Scholar
  41. 41.
    Han, Z.: Reachability analysis of continuous dynamic systems using dimension reduction and decomposition. Ph.D. thesis, Carnegie Mellon University, Electrical and Computer Engineering Department (2005)Google Scholar
  42. 42.
    Rump, S.M.: Developments in Reliable Computing, chap. INTLAB - IN Terval L A Boratory, pp. 77–104. Kluwer Academic Publishers (1999)Google Scholar
  43. 43.
    Dietzenbacher, E.: A limiting property for the powers of a non-negative, reducible matrix. Structural Change and Economic Dynamics 4, 353–366 (1993)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Matthias Althoff
    • 1
    Email author
  • Bruce H. Krogh
    • 1
  • Olaf Stursberg
    • 2
  1. 1.Carnegie Mellon UniversityPittsburghUSA
  2. 2.Control and System Theory (FB16)University of KasselKasselGermany

Personalised recommendations