State Event Detecting Algorithm for Hybrid System Based on Interval Newton's Method

Abstract

Abstract: Accurately detecting of state events is critical for hybrid dynamic systems,especially for systems with singularities or event functions with multiple roots. A carefully constructed extrapolation polynomial among state value of past time points was applied to predict values of event functions. The polynomial was derived from a variable-step multi-step integration method. The extended interval Newton's method was employed to find all roots in a certain time interval. Due to no using of values of future time points, the polynomial can be employed to determine the integration step size by checking potential events in systems with model singularities. The procedure of root existence test and root finding was combined into one procedure by using of non-existence test of roots. Simulation results show the algorithm is effective for hybrid dynamic systems with several different critical situations. By this algorithm, model singularities are found on time and simulation failures are avoided.

Key words: hybrid system, state event detecting, vary step multi-step method, state event function predicting, interval Newton's method

CLC Number:

Wang Haiyan, Hu Yihuai. State Event Detecting Algorithm for Hybrid System Based on Interval Newton's Method[J]. Journal of System Simulation, 2015, 27(4): 738-746.

References

[1] Lygeros J, Tomlin C, Sastry S.Hybrid Systems: Modeling, Analysis and Control [R]. Berkeley, CA 94720, USA: EECS Department, University of California, 2008.
[2] 赖镇洲, 张淼, 王佳伟, 等. 基于混杂系统描述的卫星姿态控制系统研究[C]// 第32届中国控制会议论文集. 西安. USA: IEEE Computer Society, 2013: 4452-4457.
[3] 王雷, 周理, 蒋新华. 合金热挤压中基于混杂模型的突变预测与仿真[J]. 系统仿真学报, 2012, 24(12): 2587-2592.
[4] Taylor J H, Zhang J.Rigorous Hybrid Systems simulation with continuous-time discontinuities and discrete-time components[C]// Proceeding of 2007 Mediterranean Conference on Control and Automation. Athens, Greece: Inst. of Elec. and Elec. Eng. Computer Society, 2007.
[5] Paul C.A. H. The Treatment of Derivative Discontinuities in Differential Equations [R]. Manchester, UK: Manchester Centre for Computational Mathematics, 1999.
[6] 张博, 董领逊, 窦丽华. 一类综合控制系统的MLD建模与仿真研究[J]. 系统仿真学报, 2009, 21(2): 500-502.
[7] Esposito J M, Kumar V, Pappas G J.Accurate Event Detection for Simulating Hybrid Systems[J]. Lecture Notes in Computer Science (S0302-9743), 2001, 2034(2001): 204-217.
[8] Shampine L F, Gladwell I, Brankin R W.Reliable Solution of Special Event Location Problems for ODEs[J]. ACM transactions on Mathematical Software (S0098-3500), 1987, 17(1): 11-25.
[9] Park T, Barton P I.State Event Location in Differential-Algebraic Models[J]. ACM Trans. Model. Comput. Simul.(S1049-3301), 1996, 6(2): 137-165.
[10] Esposito J M, Kumar V.A state event detection algorithm for numerically simulating hybrid systems with model singularities[J]. ACM Transactions on Modeling and Computer Simulation (S1049-3301), 2007, 17(1): 1-26.
[11] Hong-Shan Zhao, Yan-bin Hu. Algorithm of Event Detection for Hybrid Power System Simulation[C]// Proceeding of DPRT 2008. Piscataway, USA: Inst. of Elec. and Elec. Eng. Computer Society, 2008: 990-994.
[12] 郅跃茹, 诸静, 左光华. 混合动态系统状态事件的精确定位方法[J]. 浙江大学学报(工学版), 2005, 39(9): 41-45.
[13] Griffiths D F, Higham D J.Numerical Methods for Ordinary Differential Equations (Initial Value Problems)[M]. 1st ed. London, UK: Springer, 2010.
[14] Navarro-López E M. Hybrid-automaton models for simulating systems with sliding motion: still a challenge[C]// Proceeding of the 3rd IFAC Conference on Analysis and Design of Hybrid System. Zaragoza, Spain: IFAC Secretariat, 2009: 322-327.
[15] M Berry M.A Variable-Step Double-Integration Multi-Step Integrator [D]. Blacksburg, VA, USA: Virginia Polytechnic Institute and State University, 2004.
[16] Gwaltney C R, Lin Y, Simoni L D, et al. Interval methods for nonlinear equation solving applications [K]. Pedrycz W, Skowron A, Kreinovich V. Handbook of Granular Computing. USA: John Wiley & Sons, 2008: 81-96.
[17] Moore R E, Kearfott R B, Cloud M J.Introduction to Interval Analysis[M]. 1st ed. Philadelphia, USA: Society for Industrial and Applied Mathematics, 2009.