This paper pilots Schulz generalised matrix inverse algorithm as a paradigm in demonstrating how computer aided reachability analysis and theoretical numerical analysis can be combined effectively in developing verification methodologies and tools for matrix iterative solvers. It is illustrated how algorithmic convergence to computed solutions with required accuracy is mathematically quantified and used within computer aided reachability analysis tools to formally verify convergence over predefined sets of multiple problem data. In addition, some numerical analysis results are used to form computational reliability monitors to escort the algorithm on-line and monitor the numerical performance, accuracy and stability of the entire computational process. For making the paper self-contained, formal verification preliminaries and background on tools and approaches are reported together with the detailed numerical analysis in basic mathematical language. For demonstration purposes, a custom made reachability analysis program based on affine arithmetic is applied to numerical examples.
Citation: |
[1] | AAFLIB - An Affine Arithmetic C++, 2019. Available from: http://aaflib.sourceforge.net. |
[2] | R. Alur, C. Courcoubetis, N. Halbwachs, T. Henzinger, P. H. Ho, X. Nicollin, A. Olivero, J. Sifakis and S. Yovine, The algorithmic analysis of hybrid systems, Theoretical Computer Science, 138 (1995), 3-34. doi: 10.1016/0304-3975(94)00202-T. |
[3] | R. Alur, T. A. Henzinger, G. Lafferriere and G. Pappas, Discrete abstractions of hybrid systems, Proceedings of the IEEE 88, 7 (2000), 971-984. |
[4] | R. Alur, T. Dang and F. Ivancic, Counter example-guided predicate abstraction of hybrid systems, Theoretical Computer Science, 354 (2006), 250-271. doi: 10.1016/j.tcs.2005.11.026. |
[5] | Y. Annapureddy, C. Liu, G. Fainekos and S. Sankaranarayanan, S-TaLiRo: A tool for temporal logic falsification for hybrid systems, Proc. of Tools and Algorithms for the Construction and Analysis of Systems, (2011), 254–257. |
[6] | E. Asarin, T. Dang and O. Maler, The d/dt tool for verification of hybrid systems, Int. Conf. on Computer Aided Verification, LNCS, Springer-Verlag, (2002), 365–350. |
[7] | A. Ben-Israel and D. Cohen, On iterative computation of generalized inverses and associated projections, SIAM J. Numer. Anal., 3 (1966), 410-419. doi: 10.1137/0703035. |
[8] | A. Ben-Israel and T. N. E. Greville, Generalized Inverses Theory and Applications, Springer, 2003, ISBN 978-0-387-00293-4. |
[9] | A. Bhatia and E. Frazzoli, Incremental search methods for reachability analysis of continuous and hybrid systems, Hybrid Systems: Computation and Control, LNCS, Springer-Verlag, 2993 (2004), 142–156. |
[10] | O. Botchkarev and S. Tripakis, Verification of hybrid systems with linear differential inclusions using ellipsoidal approximations, Hybrid Systems: Computation and Control, LNCS, Springer-Verlag, 1790 (2000), 73–78. |
[11] | M. S. Branicky, M. M. Curtiss, J. Levine and S. Morgan, Sampling-based planning, control, and verification of hybrid systems, Control Theory and Applications, 153 (2006), 575-590. |
[12] | C. Bu, X. Zhang, J. Zhou, W. Wang and Y. Wei, The inverse, rank and product of tensors, Linear Algebra and Its Applications, 446 (2014), 269-280. doi: 10.1016/j.laa.2013.12.015. |
[13] | X. Chen, E. Abraham and S. Sankaranarayanan, Flow* An analyzer for non-linear hybrid systems, Proc. of CAV13, LNCS, Springer, 8044 (2013), 258–263. doi: 10.1007/978-3-642-39799-8_18. |
[14] | X. Chen, Reachability Analysis of Non-Linear Hybrid Systems Using Taylor Models, PhD Thesis in RWTH Aachen University, Germany, 2015. |
[15] | C. Chutinan and B. H. Krogh, Computational techniques for hybrid system verification, IEEE Transactions on Automatic Control, 48 (2003), 64-75. doi: 10.1109/TAC.2002.806655. |
[16] | E. M Clarke, W. Klieber, M. Novcek and P. Zuliani, Model checking and the state explosion problem, Tools for Practical Software Verification, Springer, (2012), 1–30. |
[17] | E. M. Clarke, Th. A. Henzinger, H. Veith and R. Bloem, Handbook of Model Checking, Springer International Publishing, 2018. doi: 10.1007/978-3-319-10575-8. |
[18] | J-J. Climent, N. Thome and Y. Wei, A geometrical approach on generalized inverses by Neumann-type series, Linear Algebra and Its Applications, 1 (2001), 533-540. doi: 10.1016/S0024-3795(01)00309-3. |
[19] | B. Datta, Numerical Methods for Linear Control Systems, Elsevier Academic Press, 2004. |
[20] | M. Daumas, D. Lester and C. Muoz, Verified real number calculations, A library for interval arithmetic, IEEE Transactions on Computers, 58 (2009), 226-237. doi: 10.1109/TC.2008.213. |
[21] | A. Donze, Breach, a toolbox for verification and parameter synthesis of hybrid systems, Proc. of Computer Aided Verification, (2010), 167–170. |
[22] | P. Duggirala, S. Mitra, M. Viswanathan and M. Potok, C2E2 A verification tool for Stateflow models, Proc. of TACAS15, LNCS, Springer, 9035 (2015), 68–82. |
[23] | J. M. Esposito, J. Kim and V. Kumar, Adaptive RRTs for validating hybrid robotic control systems, Workshop on Algorithmic Foundations of Robotics, Zeist, Netherlands, (2004), 107–132. |
[24] | J. Kim, J. M. Esposito and V. Kumar, An RRT-based algorithm for testing and validating multi-robot controllers, Robotics: Science and Systems, Boston, MA, (2005), 249–256. |
[25] | L. H. de Figueiredo and J. Stolfi, Affine arithmetic: concepts and applications, Numerical Algorithms, 37 (2004), 147-158. doi: 10.1023/B:NUMA.0000049462.70970.b6. |
[26] | G. Frehse, C. L. Guernic, A. Donz, R. Ray, O. Lebeltel, R. Ripado, A. Girard, T. Dang and O. Maler, SpaceEx Scalable verification of hybrid systems, Proc. of CAV11, LNCS, Springer, 6806 (2011), 379–395. doi: 10.1007/978-3-642-22110-1_30. |
[27] | G. Frehse, R. Kateja and C. Le Guernic, Flowpipe approximation and clustering in space-time, Proc. of HSCC13, 9035 (2013), 203-212. doi: 10.1145/2461328.2461361. |
[28] | M. Gameiro and P. Manolios, Formally verifying an algorithm based on interval arithmetic for checking transversality, Workshop on ACL2 Prover and Applications, 2004. |
[29] | N. Giorgetti, G.J. Pappas and A. Bemporad, Bounded model checking for hybrid dynamical systems, IEEE Conference on Decision and Control, Seville, Spain, (2005), 672–677. |
[30] | C. Le Guernic, Reachability Analysis of Hybrid Systems with Linear Continuous Dynamics, PhD Thesis in Universit Joseph-Fourier-Grenoble I, France, 2009. |
[31] | T. Henzinger, P. Kopke, A. Puri and P. Varaiya, What's decidable about hybrid automata?, ACM Symposium on Theory of Computing, (1995), 273–282. doi: 10.1016/0895-7177(96)00072-6. |
[32] | T. Henzinger, The theory of hybrid automata, Symposium on Logic in Computer Science, (1996), 278–292. doi: 10.1109/LICS.1996.561342. |
[33] | F. Immler, Tool presentation Isabelle/hol for reachability analysis of continuous systems, in ARCH14-15, 1st and 2nd International Workshop on Applied veRification for Continuous and Hybrid Systems (eds. M. Frehse and M. Althoff), Academic Press, (1971), 33–75. EPiC Series in Computer Science, 34 (2015), 180–187. |
[34] | A. A. A. Julius, G. E. Fainekos, M. Anand, I. Lee and G. J. Pappas, IRobust test generation and coverage for hybrid systems, Hybrid Systems: Computation and Control, LNCS, Springer-Verlag, 4416 (2007), 329–342. |
[35] | S. Kong, S. Gao and W. Chen, Reachability analysis for hybrid systems, Proc. of TACAS15, LNCS, Springer, 9035 (2015), 200–205. |
[36] | M. Konstantinov, D. Gu, V. Mehrmann and P. Petkov, Perturbation Theory for Matrix Equations, 2$^{nd}$ edition, Elsevier, Amsterdam, 2003, ISBN-9780444513151. |
[37] | G. A. Kumar, T. V. Subbareddy, B. M. Redd, N. Raju and V. Elamaran, An approach to design a matrix inversion hardware module using FPGA, Int. Conf. on Control, Instrumentation, Comm. and Comput. Technologies, 230 (2014), 87-90. |
[38] | G. Lafferriere, G. Pappas and S. Yovine, A new class of decidable hybrid systems, Hybrid Systems: Computation and Control, LNCS, 1569 (1999), 137-151. |
[39] | LAPACK-Linear Algebra PACKage, 2019. Available from: http://www.netlib.org/lapack/. |
[40] | W. Levine, The Control Handbook, IEEE Press, 1996. |
[41] | C. Livadas and N. Lynch, A new class of decidable hybrid systems, Hybrid Systems: Computation and Control, LNCS, 1386 (1998), 253-272. |
[42] | Matlab-Mathworks, 2019. Available from: https://www.mathworks.com |
[43] | F. Messine and A. Touhami, A general reliable quadratic form: an extension of affine arithmetic, Reliable Computing, 12 (2006), 171-192. doi: 10.1007/s11155-006-7217-4. |
[44] | D. Monniaux, Toward verifiably correct control implementations, The impact of control technology Part 2: Challenges for control research, Report of IEEE Control Systems Society, 2nd Ed, 2011. Available from: http://ieeecss.org/general/IoCT2-report. |
[45] | D. Monniaux and A. Mine, Verification of control system software, the impact of control technology, Part 1: Success stories for control, Report of IEEE Control Systems Society, 2nd Ed, 2011. Available from: http://ieeecss.org/general/IoCT2-report. |
[46] | R. E. Moore, Methods and Applications of Interval Analysis, SIAM Studies in Applied and Numerical Mathematics, Philadelphia, 1987, ISBN-10: 0898711614. |
[47] | R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, SIAM, 2009. doi: 10.1137/1.9780898717716. |
[48] | C. Munoz and D. Lester, Real number calculations and theorem proving, 18th Int. Conf. on Theorem Proving in Higher Order Logics, England, (2005), 239–254. doi: 10.1007/11541868_13. |
[49] | T. Nahhal and T. Dang, Test coverage for continuous and hybrid systems, Int. Conf. on Computer Aided Verification, LNCS, 4590 (2007), 449-462. |
[50] | I. Pasca, Formal Verifcation for Numerical Methods, PhD Thesis in Universit Nice Sophia Antipolis, France, 2010. |
[51] | P. Petkov, M. Konstantinov and N. Christov, Computational Methods for Linear Control Systems, 2$^{nd}$ edition, Prentice- Hall, Hemel Hempstead, 1991, ISBN-9780444513151. |
[52] | M. D. Petkovic and P. S. Stanimirovic, Generalized matrix inversion is not harder than matrix multiplication, J. Comput. and Applied Math., 230 (2009), 270-282. doi: 10.1016/j.cam.2008.11.012. |
[53] | M. D. Petkovic and P. S. Stanimirovic, Iterative method for computing Moore-Penrose inverse based on Penrose equations, J. Comput. and Applied Math., 235 (2011), 1604-1613. doi: 10.1016/j.cam.2010.08.042. |
[54] | M. S. Petkovic, Iterative methods for bounding the inverse of a matrix (a survey), FILOMAT Nis, Algebra Logic & Discrete Mathematics, 9 (1995), 543-577. |
[55] | E. Plaku, L. Kavraki and M. Vardi, Hybrid systems: from verification to falsification by combining motion planning and discrete search, Formal Methods in System Design, 34 (2009), 157-182. |
[56] | A. Platzer and J-D Quesel, KeYmaera a hybrid theorem prover for hybrid systems (system description), Automated Reasoning, IJCAR 2008, Lecture Notes in Computer Science (eds. M. Frehse and M. AlthoffArmando A., Baumgartner P., Dowek G.), Springer Berlin Heidelberg, 5195 (2008), 163–183. doi: 10.1007/978-3-540-71070-7_15. |
[57] | A. Puri, Theory of Hybrid Systems and Discrete Event Systems, PhD Thesis in University of California, Berkeley, 1995. |
[58] | PyInterval - Interval arithmetic in Python, 2019. Available from: https://github.com/taschini/pyinterval |
[59] | S. Qiao, X. Wang and Y. Wei, Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse, Linear Algebra and Its Applications, 542 (2018), 101-117. doi: 10.1016/j.laa.2017.03.014. |
[60] | Y. Rahman, A. Xie and S. Bernstein, Retrospective cost adaptive control: pole placement, frequency response, and connections with LQG control, IEEE Control System Magazine, (2017), 28–69. doi: 10.1109/MCS.2017.2718825. |
[61] | S. Ratschan and Z. She, Safety verification of hybrid systems by constraint propagation based abstraction refinement, ACM Transactions on Embedded Computing Systems, 6 (2007), Article No. 8. doi: 10.1145/1210268.1210276. |
[62] | RTCA Formal Methods Supplement to DO-178C and DO-278A, RTCA DO-333, December 13, 2011. |
[63] | RTCA, Software Considerations in Airborne Systems and Equipment Certification, RTCA DO-178C, December 13, 2011. |
[64] | S. Schupp, E. Abraham, X. Chen, I. B. Makhlouf, G. Frehse, S. Sankaranarayanan and S. Kowalewski, Current challenges in the verification of hybrid systems, Workshop on Design, Modeling, and Evaluation of Cyber Physical Systems, (2015), 8–24. |
[65] | S. Schupp and E. Abraham, Efficient dynamic error reduction for hybrid systems reachability analysis, Proc. of the 24th International Conference on Tools and Algorithms for the Construction and Analysis of Systems TACAS18, LNCS, Springer, 2018. |
[66] | B. I. Silva and B. H. Krogh, Formal verification of hybrid systems using CheckMate: A case study, American Control Conference, (2000), 1679–1683. |
[67] | G. Stewart and J. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990. |
[68] | O. Stursberg and B. H. Krogh, Efficient representation and computation of reachable sets for hybrid systems, Hybrid Systems: Computation and Control, LNCS, 2623 (2003), 482-497. |
[69] | C. J. Tomlin, I. Mitchell, A. Bayen and M. Oishi, Computational techniques for the verification and control of hybrid systems, Proceedings of the IEEE 91, Springer-Verlag, 7 (2003), 986–1001. |
[70] | V. A. Tsachouridis, Numerical analysis of $H_{\infty} $ filter for system parameter identification, Int. J. Modelling, Identification and Control, 30 (2018), 163-183. |
[71] | University of Florida Sparse Matrix Collection, 2019. Available from: https://www.cise.ufl.edu/research/sparse/matrices/list_by_id.html |
[72] | G. Wang, Y. Wei and S. Qiao, Generalized Inverses: Theory and Computations, Springer, Singapore: Science Press Beijing, Beijing, 2018. doi: 10.1007/978-981-13-0146-9. |
[73] | Y. Wei, P. Stanimirovic and M. Petkovic, Numerical and Symbolic Computations of Generalized Inverses, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018. |
[74] | J. Wilkinson, Rounding Errors in Algebraic Processes, Prentice Hall, Englewood Cliff, NJ, 1963. |
[75] | H. Wolkovicz and G. P. H. Stayan, Bounds for eigenvalues using traces, Linear Algebra and Its Applications, 29 (1980), 471-506. doi: 10.1016/0024-3795(80)90258-X. |
[76] | P. Xie, H. Xiang and Y. Wei, Randomized algorithms for total least squares problems, Numerical Linear Algebra with Applications, 26 (2019), e2219, Available from: https://doi.org/10.1002/nla.2219 doi: 10.1002/nla.2219. |
Hybrid automaton for a
Hybrid automaton for a
Hybrid automaton for a
Affine arithmetic approach for bound convergence expression (34) for a
Affine arithmetic approach for bound convergence expression (34) for a
Variation of condition number of
Computational reliability monitor for