Decomposition of stability proofs for hybrid systems

Hybride Systeme dienen der Beschreibung der Interaktion von zeitkontinuierlichem und zeitdiskretem Verhalten, wie sie beispielsweise in eingebetteten Systemen stattfindet. Ein Beschreibungsmittel fur solche Systeme sind hybride Automatenmodelle, endliche Automaten angereichert mit Differentialgleichungen. Im Rahmen dieser Arbeit wird eine wichtige Eigenschaft solcher Systeme untersucht: Stabilitat. Stabile Systeme sind in der Lage, temporare Storungen auszuregeln, indem sie wieder zu einem Arbeitspunkt zuruckkehren. Der Hauptbeitrag der Arbeit ist eine Methodik zum automatischen, dekompositionellen Nachweise dieser Eigenschaft. Den Kern bildet hier die graphentheoretische Zerlegung des Automaten mit Hilfe von Ljapunow-Funktionen, welche die Komponierbarkeit sicherstellen. Hiermit ist es moglich, die Komplexitat der zu losenden Stabilitatsbeweise signifikant zu reduzieren, sowie Aussagen uber das Ergebnis einer Komposition zweier Automaten zu treffen.

[1]  W. P. M. H. Heemels,et al.  Input-to-State Stability of Discontinuous Dynamical Systems with an Observer-Based Control Application , 2007, HSCC.

[2]  Alberto Bemporad,et al.  On the Optimal Control Law for Linear Discrete Time Hybrid Systems , 2002, HSCC.

[3]  Chaohong Cai,et al.  Smooth Lyapunov Functions for Hybrid Systems Part II: (Pre)Asymptotically Stable Compact Sets , 2008, IEEE Transactions on Automatic Control.

[4]  Lijun Zhang,et al.  Safety Verification for Probabilistic Hybrid Systems , 2010, Eur. J. Control.

[5]  Alexandre M. Bayen,et al.  Network Congestion Alleviation Using Adjoint Hybrid Control: Application to Highways , 2004, HSCC.

[6]  André Platzer,et al.  KeYmaera: A Hybrid Theorem Prover for Hybrid Systems (System Description) , 2008, IJCAR.

[7]  D. Liberzon,et al.  A small-gain approach to stability analysis of hybrid systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[8]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[9]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[10]  Masakazu Kojima,et al.  Implementation and evaluation of SDPA 6.0 (Semidefinite Programming Algorithm 6.0) , 2003, Optim. Methods Softw..

[11]  Debasish Chatterjee,et al.  Stability analysis of deterministic and stochastic switched systems via a comparison principle and multiple Lyapunov functions , 2006, SIAM J. Control. Optim..

[12]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..

[13]  S. Sastry,et al.  HYBRID LIMIT CYCLES AND HYBRID POINCARÉ-BENDIXSON , 2002 .

[14]  Carsten Thomassen.  On the Complexity of Finding a Minimum Cycle Cover of a Graph , 1997, SIAM J. Comput..

[15]  W. P. M. H. Heemels,et al.  Input-to-state stability and interconnections of discontinuous dynamical systems , 2008, Autom..

[16]  Jeremy Sproston.  Decidable Model Checking of Probabilistic Hybrid Automata , 2000, FTRTFT.

[17]  Thomas A. Henzinger,et al.  Reachability Verification for Hybrid Automata , 1998, HSCC.

[18]  M. Branicky.  Stability of switched and hybrid systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[19]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[20]  A. Papachristodoulou.  Analysis of nonlinear time-delay systems using the sum of squares decomposition , 2004, Proceedings of the 2004 American Control Conference.

[21]  Olga Taussky-Todd.  SOME CONCRETE ASPECTS OF HILBERT'S 17TH PROBLEM , 1996 .

[22]  Ian M. Mitchell,et al.  Level Set Methods for Computing Reachable Sets of Hybrid Systems with Differential Algebraic Equation Dynamics , 2008, HSCC.

[23]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[24]  Antoine Girard,et al.  Computation and Stability Analysis of Limit Cycles in Piecewise Linear Hybrid Systems , 2003, ADHS.

[25]  Stefan Pettersson,et al.  Analysis and Design of Hybrid Systems , 1999 .

[26]  Anders Rantzer,et al.  Computation of piecewise quadratic Lyapunov functions for hybrid systems , 1997, 1997 European Control Conference (ECC).

[27]  S. Shankar Sastry,et al.  Probabilistic reachability for stochastic hybrid systems: theory, computations, and applications , 2007 .

[28]  Dragan Nesic,et al.  Stability Analysis of Hybrid Systems Via Small-Gain Theorems , 2006, HSCC.

[29]  Jorge M. Gonçalves,et al.  Regions of stability for limit cycle oscillations in piecewise linear systems , 2005, IEEE Transactions on Automatic Control.

[30]  João Pedro Hespanha,et al.  Lyapunov conditions for input-to-state stability of impulsive systems , 2008, Autom..

[31]  Ian A. Hiskens,et al.  Stability of limit cycles in hybrid systems , 2001, Proceedings of the 34th Annual Hawaii International Conference on System Sciences.

[32]  Jamal Daafouz,et al.  Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach , 2002, IEEE Trans. Autom. Control..

[33]  Eduardo D. Sontag,et al.  Mathematical control theory: deterministic finite dimensional systems (2nd ed.) , 1998 .

[34]  Gang Feng,et al.  Stability analysis of piecewise discrete-time linear systems , 2002, IEEE Trans. Autom. Control..

[35]  G. Frehse,et al.  Assume-guarantee reasoning for hybrid I/O-automata by over-approximation of continuous interaction , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[36]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[37]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

[38]  Jan H. van Schuppen,et al.  Reachability and control synthesis for piecewise-affine hybrid systems on simplices , 2006, IEEE Transactions on Automatic Control.

[39]  Amir Ali Ahmadi,et al.  Non-monotonic Lyapunov functions for stability of discrete time nonlinear and switched systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[40]  James F. Manwell,et al.  LEAD-ACID-BATTERY STORAGE MODEL FOR HYBRID ENERGY-SYSTEMS , 1993 .

[41]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[42]  Antonis Papachristodoulou,et al.  Positive Forms and Stability of Linear Time-Delay Systems , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[43]  Ali Jadbabaie,et al.  Safety Verification of Hybrid Systems Using Barrier Certificates , 2004, HSCC.

[44]  Dragan Nesic,et al.  Discrete-time Lyapunov-based small-gain theorem for parameterized interconnected ISS systems , 2003, IEEE Trans. Autom. Control..

[45]  H. Kushner.  Stochastic Stability and Control , 2012 .

[46]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[47]  K. Loparo.  Stability of Stochastic Systems , 2010 .

[48]  Pablo A. Parrilo,et al.  Semidefinite Programming Relaxations and Algebraic Optimization in Control , 2003, Eur. J. Control.

[49]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[50]  G. Zhai,et al.  Quadratic stabilizability of switched linear systems with polytopic uncertainties , 2003 .

[51]  Mircea Lazar,et al.  On infinity norms as Lyapunov functions for piecewise affine systems , 2010, HSCC '10.

[52]  Bengt Lennartson,et al.  Stability of limit cycles in hybrid systems using discrete-time Lyapunov techniques , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[53]  Xiong Zhang,et al.  Solving Large-Scale Sparse Semidefinite Programs for Combinatorial Optimization , 1999, SIAM J. Optim..

[54]  S. Pettersson,et al.  Stability and robustness for hybrid systems , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[55]  Goran Frehse,et al.  PHAVer: algorithmic verification of hybrid systems past HyTech , 2005, International Journal on Software Tools for Technology Transfer.

[56]  D. G. Korenevskii.  Stability with probability 1 of solutions of systems of linear ito stochastic differential-difference equations , 1987 .

[57]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[58]  Andreas Podelski,et al.  Model Checking of Hybrid Systems: From Reachability Towards Stability , 2006, HSCC.

[59]  A. Michel,et al.  Stability theory for hybrid dynamical systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[60]  Stefan Ratschan,et al.  Providing a Basin of Attraction to a Target Region of Polynomial Systems by Computation of Lyapunov-Like Functions , 2010, SIAM J. Control. Optim..

[61]  T. Zolezzi,et al.  Differential Inclusions and Sliding Mode Control , 2002 .

[62]  Amir Pnueli,et al.  Towards Component Based Design of Hybrid Systems: Safety and Stability , 2010, Essays in Memory of Amir Pnueli.

[63]  F. Kozin,et al.  Stability of the linear stochastic system , 1972 .

[64]  C. Jansson.  VSDP : A MATLAB software package for Verified Semidefinite Programming , 2006 .

[65]  李幼升,et al.  Ph , 1989 .

[66]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[67]  Oliver E. Theel,et al.  Fully Automated Stability Verification for Piecewise Affine Systems , 2007, HSCC.

[68]  P. Olver.  Nonlinear Systems , 2013 .

[69]  Manfred Morari,et al.  Analysis of discrete-time piecewise affine and hybrid systems , 2002, Autom..

[70]  A. Teel,et al.  Results on input-to-state stability for hybrid systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[71]  Chaohong Cai,et al.  Smooth Lyapunov Functions for Hybrid Systems—Part I: Existence Is Equivalent to Robustness , 2007, IEEE Transactions on Automatic Control.

[72]  Edmund M. Clarke,et al.  Computing differential invariants of hybrid systems as fixedpoints , 2008, Formal Methods Syst. Des..

[73]  A. Liapounoff,et al.  Problème général de la stabilité du mouvement , 1907 .

[74]  Debasish Chatterjee,et al.  On Stability of Randomly Switched Nonlinear Systems , 2007, IEEE Transactions on Automatic Control.

[75]  U. Rieder,et al.  Markov Decision Processes , 2010 .

[76]  D. Dimarogonas,et al.  Lyapunov-like stability of switched stochastic systems , 2004, Proceedings of the 2004 American Control Conference.

[77]  R. Kalman,et al.  Control system analysis and design via the second method of lyapunov: (I) continuous-time systems (II) discrete time systems , 1959 .

[78]  Thomas A. Henzinger,et al.  Hybrid Automata: An Algorithmic Approach to the Specification and Verification of Hybrid Systems , 1992, Hybrid Systems.

[79]  Oliver E. Theel,et al.  A Decompositional Proof Scheme for Automated Convergence Proofs of Stochastic Hybrid Systems , 2009, ATVA.

[80]  Martin Fränzle,et al.  Efficient Solving of Large Non-linear Arithmetic Constraint Systems with Complex Boolean Structure , 2007, J. Satisf. Boolean Model. Comput..

[81]  John N. Tsitsiklis,et al.  The Stability of Saturated Linear Dynamical Systems Is Undecidable , 2000, J. Comput. Syst. Sci..

[82]  Jan Willem Polderman,et al.  Stability Analysis for Hybrid Automata Using Conservative Gains , 2003, ADHS.

[83]  Carla Piazza,et al.  Decidable Compositions of O-Minimal Automata , 2008, ATVA.

[84]  John N. Tsitsiklis,et al.  Complexity of stability and controllability of elementary hybrid systems , 1999, Autom..

[85]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[86]  Frank L. Lewis,et al.  Control engineering series , 1998 .

[87]  Sergey Dashkovskiy,et al.  Stability of Interconnections of ISS Systems , 2008 .

[88]  Oliver E. Theel,et al.  Decompositional Construction of Lyapunov Functions for Hybrid Systems , 2009, HSCC.

[89]  T. A. Burton,et al.  ON THE CONSTRUCTION OF LYAPUNOV FUNCTIONS , 1969 .

[90]  Daniel Liberzon,et al.  Common Lyapunov functions for families of commuting nonlinear systems , 2005, Syst. Control. Lett..

[91]  J. Tsitsiklis,et al.  The boundedness of all products of a pair of matrices is undecidable , 2000 .

[92]  Bengt Lennartson,et al.  A Converse Theorem for Exponential Stability using Piecewise Quadratic Lyapunov Functions , 2007 .

[93]  Radu Grosu,et al.  Modular and Visual Specification of Hybrid Systems: An Introduction to HyCharts , 2002, Formal Methods Syst. Des..

[94]  Martin L. Puterman,et al.  Markov Decision Processes: Discrete Stochastic Dynamic Programming , 1994 .

[95]  Nathan van de Wouw,et al.  Stability Analysis of Networked Control Systems Using a Switched Linear Systems Approach , 2009, IEEE Transactions on Automatic Control.

[96]  B. Borchers.  CSDP, A C library for semidefinite programming , 1999 .

[97]  A. Papachristodoulou,et al.  On the construction of Lyapunov functions using the sum of squares decomposition , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[98]  Orest Iftime,et al.  Proceedings of the 16th IFAC World congress , 2006 .

[99]  J. Cortés.  Discontinuous dynamical systems , 2008, IEEE Control Systems.

[100]  John N. Tsitsiklis,et al.  Deciding stability and mortality of piecewise affine dynamical systems , 2001, Theor. Comput. Sci..

[101]  J. Polderman,et al.  Tools for Stability of Switching Linear Systems: Gain Automata and Delay Compensation , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[102]  Antonis Papachristodoulou,et al.  Analysis of Switched and Hybrid Systems , 2004 .

[103]  Anders Rantzer,et al.  On the necessity of barrier certificates , 2005 .

[104]  Ernst-Rüdiger Olderog,et al.  Automating Verification of Cooperation, Control, and Design in Traffic Applications , 2007, Formal Methods and Hybrid Real-Time Systems.

[105]  A. Papachristodoulou,et al.  Analysis of switched and hybrid systems - beyond piecewise quadratic methods , 2003, Proceedings of the 2003 American Control Conference, 2003..

[106]  Jianghai Hu,et al.  Stochastic Hybrid Systems , 2013 .

[107]  Sumit Gulwani,et al.  Constraint-Based Approach for Analysis of Hybrid Systems , 2008, CAV.

[108]  Sigurdur Hafstein,et al.  A CONSTRUCTIVE CONVERSE LYAPUNOV THEOREM ON EXPONENTIAL STABILITY , 2004 .

[109]  M. Branicky.  Multiple Lyapunov functions and other analysis tools for switched and hybrid systems , 1998, IEEE Trans. Autom. Control..

[110]  G. Chesi.  On the estimation of the domain of attraction for uncertain polynomial systems via LMIs , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).