An introduction to hybrid systems Thanks to theory and applications School Organizers Maurice Heemels Bart De Schutter Alberto Bemporad George J. Pappas DISC Summer School on Departments of ESE and CIS Modeling and Control of Hybrid Systems Agnes van Regteren University of Pennsylvania Veldhoven, The Netherlands and DISC June 23-26, 2003 pappasg@ee.upenn.edu http://www.seas.upenn.edu/~pappasg http://lcewww.et.tudelft.nl/~disc ˙ hs/ Acknowledgments Goals for this mini-course Collaborators Postdocs Why hybrid systems ? Rajeev Alur, Datta Godbole, Tom Paulo Tabuada Henzinger, Ali Jadbabaie, John Koo, Emphasis on engineering and biological examples Herbert Tanner Vijay Kumar, Gerardo Lafferierre, Modeling of hybrid systems Insup Lee, John Lygeros, Shankar Ph.D Students Sastry, Omid Shakernia, Claire Emphasis on abstraction and refinement Ali Ahmazadeh Tomlin, Sergio Yovine Analysis of hybrid systems George Fainekos Support Hadas Kress Gazit Emphasis on algorithmic verification NSF Career Hakan Yazarel Synthesis of hybrid controllers NSF ITR Michael Zavlanos ARO MURI M.S. students Emphasis on temporal logic synthesis DARPA MoBIES Selcuk Bayraktar Honeywell Pranav Srivastava Warning : All questions and answers are biased and incomplete! Microsoft Some references Outline of this mini-course Bisimilar linear systems George J. Pappas Automatica. To appear in 2003. Model checking LTL over controllable linear systems is decidable Lecture 1 : Monday, June 23 Lecture 1 : Monday, June 23 Paulo Tabuada and George J. Pappas Hybrid Systems : Computation and Control, Lecture Notes in Computer Science, Prague, Czech Republic, April 2003 Examples of hybrid systems, modeling formalisms Modeling and analyzing biomolecular networks Rajeev Alur, Calin Belta, Vijay Kumar, Max Mintz, George J. Pappas, Harvey Rubin, and Jonathan Schug Lecture 2 : Monday, June 23 Computing in Science and Engineering, 4(1):20-31, January 2002. Symbolic reachability computations for families of linear vector fields Transitions systems, temporal logic, refinement notions G. Lafferriere, G. J. Pappas, and S. Yovine Journal of Symbolic Computation, 32(3):231-253, September 2001. Lecture 3 : Tuesday, June 24 Discrete abstractions of hybrid systems R.Alur, T. Henzinger, G. Lafferriere, G. Pappas Discrete abstractions of hybrid systems for verification Proceedings of the IEEE, 88(2):971-984, July 2000. Lecture 4 : Tuesday, June 24 Hierarchically consistent control systems George J. Pappas, Gerardo Lafferriere, and Shankar Sastry IEEE Transactions on Automatic Control, 45(6):1144-1160, June 2000. Discrete abstractions of continuous systems for control O-minimal hybrid systems Lecture 5 : Thursday, June 26 G. Lafferriere, G. J. Pappas, and S. Sastry Mathematics of Control, Signals, and Systems, 13(1):1-21, March 2000. Bisimilar control systems Decidable controller synthesis for classes of linear systems Omid Shakernia, George J. Pappas, and Shankar Sastry Hybrid Systems : Computation and Control, Lecture Notes in Computer Science, volume 1790, Springer, 2000 1
Enabling technologies Advances in sensor and actuator technology GPS, control of quantum systems Invasion of powerful microprocessors in physical devices Why hybrid ? Sophisticated software/hardware on board Networking everywhere Interconnects subsystems Emerging applications… Networked embedded systems… Network Controller Controller SW/HW SW/HW Actuator Sensor Actuator Sensor Physical Physical System System Latest BMW : 72 networked microprocessors Boeing 777 : 1280 networked microprocessors Networked embedded systems… Discrete and Continuous Network Control Theory Computer Science Continuous systems Transition systems Composition, abstraction Controller Controller Stability, control SW/HW SW/HW Feedback, robustness Concurrency models Actuator Sensor Actuator Sensor Hybrid Systems Physical Physical Software controlled systems System System Multi-modal systems Embedded real-time systems Multi-agent systems Physical system is continuous, software is discrete 2
Exporting Science Different views… Control Theory Computer Science Computer science perspective Transition systems Continuous systems View the physics from the eyes of the software Stability, control Composition, abstraction Modeling result : Hybrid automaton Feedback, robustness Concurrency models Control theory perspective View the software from the eyes of the physics Composition Robustness Abstraction Feedback Modeling result : Switched control systems Concurrency Stability Hybrid behavior arises in Hybrid dynamics Hybrid model is a simplification of a larger nonlinear model Quantized control of continuous systems Input and observation sets are finite Logic based switching Logic based switching Software is designed to supervise various dynamics/controllers Partial synchronization of many continuous systems Resource allocation for competing multi-agent systems Hybrid specifications of continuous systems Plant is continuous, but specification is discrete or hybrid... Nuclear reactor example Software model of nuclear reactor Without rods . T 0.1 T 50 = − With rod 1 . Rod1 NoRod Rod2 T = 0.1 T − 56 With rod 2 . T 0.1 T 60 = − Rod 1 and 2 cannot be used simultaneously Once a rod is removed, you cannot use it for 10 minutes Shutdown Specification : Keep temperature between 510 and 550 degrees. If T=550 then either a rod is available or we shutdown the plant. 3
Hybrid model of nuclear reactor Conflict Resolution in ATM* T = 510 ∧ y = 10 ∧ y = 10 1 2 Rod1 T 550 y 10 NoRod T 550 y 10 Rod2 = ∧ 1 ≥ = ∧ 2 ≥ . . . T = 0.1 T − 56 T = 0.1 T − 60 T 0.1 T 50 = − . . . . . . y 1 = 1 y 2 = 1 y 1 = 1 y 2 = 1 y 1 = 1 y 2 = 1 T 510 y 0 T 510 y 0 = → 1 = : = → 2 = : T ≥ 510 T ≤ 550 T ≥ 510 T 550 y 10 y 10 = ∧ < ∧ < 1 2 Analysis : Is shutdown reachable ? Analysis : Is shutdown reachable ? Shutdown . T = 0.1 T − 50 Algorithmic verification : NO Algorithmic verification : NO . . y 1 = 1 y 2 = 1 true Conflict Resolution Protocol Conflict Resolution Maneuver 1. Cruise until a miles away 1 2. Change heading by ∆Φ 3. Maintain heading until lateral distance d 4. Change to original heading 5. Change heading by - ∆Φ 6. Maintain heading until lateral distance - d 7. Change to original heading Is this protocol safe ? Computing Unsafe Sets Safe Sets 4
The train gate θ x raise lower exit approach Partial synchronization Controller (Concurrency) System = Train || Gate || Controller Safety specification : If train is within 10 meters of the crossing, then gate should completely closed. Liveness specification : Keep gate open as much as possible. Train model Gate model θ = 90 raising open raise raise x ≥ 2000 θ = 90 . . θ = 0 θ = 9 θ = 90 near θ ≤ 90 far past lower x = 1000 x = 0 . . . - 50 ≤ x ≤ − 40 approach - 50 ≤ x ≤ − 30 - 50 ≤ x ≤ − 30 raise lower raise x ≥ 1000 x ≥ 0 x ≥ -100 exit lowering closed x 10 0 x' [2000, ) = − → ∈ ∞ θ = 0 . . θ = − 9 θ = 0 θ ≥ 0 θ = 0 lower lower Controller model Synchronized transitions x ≥ 2000 near past y = : 0 far exit x = 1000 x = 0 . . . - 50 ≤ x ≤ − 40 - 50 ≤ x ≤ − 30 - 50 ≤ x ≤ − 30 approach true x ≥ 1000 x ≥ 0 x ≥ -100 exit Going tolower idle Going to raise x = − 10 0 → x' ∈ [2000, ∞ ) y = : 0 y = : 0 approach exit . . . y 1 y = 1 = y = 1 y = : 0 y ≤ d y ≤ d exit lower raise true true Going tolower idle Going to raise y = : 0 y = : 0 y = : 0 approach exit . . . y 1 y = 1 y 1 = = approach y ≤ d y ≤ d lower true raise y = : 0 approach 5
Verifying the controller θ x raise lower exit approach Hybrid dynamics Controller System = Train || Gate || Controller Safety specification : Can we avoid the set ? θ > 0 ∧ (-10 ≤ x ≤ 10) 49 YES if d Parametric HyTech verification : ≤ 5 Quorum sensing in V. fischeri Quorum sensing in V. fischeri 2102-01.jpg 2102-02.jpg 2102-03.jpg 2102-04.jpg - + CRP cAMP O L lux box O R luxR luxICDABEG CRP binding site + Ai LuxR 2102-05.jpg 2102-06.jpg 2102-07.jpg 2102-08.jpg - LuxA LuxI LuxR Ai 2102-09.jpg 2102-10.jpg 2102-11.jpg 2102-12.jpg LuxB Ai Substrate Ai 2102-13.jpg 2102-14.jpg 2102-14A.jpg 2102-15.jpg luciferase Ai Ai Ai Modeling of biological systems Modeling of biological systems d ( LuxR ) LuxR = − − T luxR l dt H d [ x ] d [ x ] = − ± sp ± = − ± ± synthesis decay transform transport synthesis decay transform transport dt dt − + r b Ai LuxR r d Co Φ CRP ( CRP ) luxR LuxR / Ai LuxR / Ai 1 CRP CRP 0.5 + + positive negative CRP 1 CRP 2 CRP sw sw START luxR gene STOP START luxR gene STOP regulation regulation transcription transcription negative positive - - translation translation protein protein d ( luxR ) luxR Ai Ai LuxR LuxR = T [ Φ CRP ( CRP ) Ψ Co ( Co ) + b ] − Co Co c luxR luxR dt H RNA chemical chemical reaction reaction Ai Ai 6
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