An introduction to hybrid systems theory and applications. Thanks to. Goals for this mini-course. Acknowledgments. Some references

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1 An introduction to hybrid systems theory and applications Thanks to School Organizers Maurice Heemels Bart De Schutter George J Pappas Departments of ESE and CIS University of Pennsylvania pappasg@eeupennedu DISC Summer School on Modeling and Control of Hybrid Systems Veldhoven, The Netherlands June 23-26, 23 hs/ Alberto Bemporad Agnes van Regteren and DISC Acknowledgments Goals for this mini-course Postdocs Paulo Tabuada Herbert Tanner PhD Students Ali Ahmazadeh George Fainekos Hadas Kress Gazit Hakan Yazarel Michael Zavlanos MS students Selcuk Bayraktar Pranav Srivastava Collaborators Rajeev Alur, Datta Godbole, Tom Henzinger, Ali Jadbabaie, John Koo, Vijay Kumar, Gerardo Lafferierre, Insup Lee, John Lygeros, Shankar Sastry, Omid Shakernia, Claire Tomlin, Sergio Yovine Support NSF Career NSF ITR ARO MURI DARPA MoBIES Honeywell Microsoft Why hybrid systems? Emphasis on engineering and biological examples Modeling of hybrid systems Emphasis on abstraction and refinement Analysis of hybrid systems Emphasis on algorithmic verification Synthesis of hybrid controllers Emphasis on temporal logic synthesis Warning : All questions and answers are biased and incomplete! Bisimilar linear systems George J Pappas Automatica To appear in 23 Some references Outline of this mini-course Model checking LTL over controllable linear systems is decidable Paulo Tabuada and George J Pappas Hybrid Systems : Computation and Control, Lecture Notes in Computer Science, Prague, Czech Republic, April 23 Modeling and analyzing biomolecular networks Rajeev Alur, Calin Belta, Vijay Kumar, Max Mintz, George J Pappas, Harvey Rubin, and Jonathan Schug Computing in Science and Engineering, 4(1):2-31, January 22 Symbolic reachability computations for families of linear vector fields G Lafferriere, G J Pappas, and S Yovine Journal of Symbolic Computation, 32(3): , September 21 Discrete abstractions of hybrid systems RAlur, T Henzinger, G Lafferriere, G Pappas Proceedings of the IEEE, 88(2): , July 2 Hierarchically consistent control systems George J Pappas, Gerardo Lafferriere, and Shankar Sastry IEEE Transactions on Automatic Control, 45(6): , June 2 O-minimal hybrid systems G Lafferriere, G J Pappas, and S Sastry Mathematics of Control, Signals, and Systems, 13(1):1-21, March 2 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 179, Springer, 2 Lecture 1 : Monday, June 23 Examples of hybrid systems, modeling formalisms Lecture 2 : Monday, June 23 Transitions systems, temporal logic, refinement notions Lecture 3 : Tuesday, June 24 Discrete abstractions of hybrid systems for verification Lecture 4 : Tuesday, June 24 Discrete abstractions of continuous systems for control Lecture 5 : Thursday, June 26 Bisimilar control systems

2 Enabling technologies Advances in sensor and actuator technology GPS, control of quantum systems Why hybrid? Invasion of powerful microprocessors in physical devices Sophisticated software/hardware on board Networking everywhere Interconnects subsystems Emerging applications Networked embedded systems Network SW/HW SW/HW Actuator Sensor Actuator Sensor Physical System Physical System Latest BMW : 72 networked microprocessors Boeing 777 : 128 networked microprocessors Networked embedded systems Discrete and Continuous Network Control Theory Computer Science SW/HW SW/HW Continuous systems Stability, control Feedback, robustness Transition systems Composition, abstraction Concurrency models Actuator Sensor Actuator Sensor Physical System Physical System Hybrid Systems Software controlled systems Multi-modal systems Embedded real-time systems Multi-agent systems Physical system is continuous, software is discrete

3 Exporting Science Different views Control Theory Continuous systems Stability, control Feedback, robustness Computer Science Transition systems Composition, abstraction Concurrency models Computer science perspective View the physics from the eyes of the software Modeling result : Hybrid automaton Composition Abstraction Concurrency Robustness Feedback Stability Control theory perspective View the software from the eyes of the physics Modeling result : Switched control systems 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 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 Logic based switching Nuclear reactor example Software model of nuclear reactor Without rods With rod 1 With rod 2 T = 1 T 5 T = 1 T 56 T = 1 T 6 Rod1 NoRod Rod2 Rod 1 and 2 cannot be used simultaneously Once a rod is removed, you cannot use it for 1 minutes Specification : Keep temperature between 51 and 55 degrees If T=55 then either a rod is available or we shutdown the plant Shutdown

4 Hybrid model of nuclear reactor Conflict Resolution in ATM* T = 51 y1 = 1 y2 = 1 Rod1 T = 55 y1 1 NoRod T = 55 y2 1 Rod2 T = 1 T 56 T = 1 T 5 T = 1 T 6 y 1 = 1 y 2 = 1 y 1 = 1 y 2 = 1 y 1 = 1 y 2 = 1 T = 51 y1 : = T = 51 y2 : = T 51 T 55 T 51 T = 55 y1 < 1 y2 < 1 Analysis : Is shutdown reachable? Algorithmic verification : NO Shutdown T = 1 T 5 y 1 = 1 y 2 = 1 Conflict Resolution Protocol Conflict Resolution Maneuver 1 Cruise until a miles away 2 Change heading by Φ 1 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

5 The train gate x θ Partial synchronization (Concurrency) System = Train Gate Safety specification : If train is within 1 meters of the crossing, then gate should completely closed Liveness specification : Keep gate open as much as possible Train model Gate model θ = 9 x 2 raising θ = 9 θ = 9 open θ = far - 5 x 4 x = 1 near past x = - 5 x 3-5 x 3 θ 9 θ = 9 x 1 x x -1 x = 1 x' [2, ) ing closed θ = 9 θ = θ = θ θ = model x 2 Synchronized transitions far near past Goingto idle Goingto - 5 x 4 x = 1-5 x = x 3-5 x 3 x 1 x x -1 x = 1 x' [2, ) y d y d Going to idle Going to y d y d

6 Verifying the controller x θ Hybrid dynamics System = Train Gate Safety specification : Can we avoid the set θ > (-1 x 1)? Parametric HyTech verification : YES if 49 d 5 Quorum sensing in V fischeri Quorum sensing in V fischeri 212-1jpg 212-2jpg 212-3jpg 212-4jpg O L + camp - lux box O R luxr luxicdabeg 212-5jpg 212-6jpg 212-7jpg 212-8jpg binding site - + LuxI LuxA 212-9jpg 212-1jpg jpg jpg LuxB Substrate jpg jpg Ajpg jpg luciferase Modeling of biological systems d ( ) = Tl luxr d [ x ] H sp = synthesis decay ± transform ± transport b d r + r Co / / Modeling of biological systems d [ x ] = synthesis decay ± transform ± transport Φ luxr () 1 5 positive + START luxr gene STOP negative + START luxr gene STOP 1 sw 2 sw regulation negative - transcription translation regulation positive - transcription translation Co protein Co protein d( luxr) Co luxr = Tc[ ΦluxR ( ) ΨluxR ( Co) + b] H RNA chemical reaction chemical reaction

7 V fischeri mathematical model BioCharon = BioSketchPad + Charon d ( luxr ) luxr = Tc ( plij + b) H RNA d ( luxicdabeg ) luxicdabeg = Tc ( prkl + b) H RNA d ( ) = TlluxR k1 _ i + k 1Co H sp d ( LuxI ) LuxI = TlluxICDABEG H sp Co % act Co<Co_sw_l <_sw_l Co<Co_sw_l >_sw_l Co>Co_sw_l <_sw_l Co>Co_sw_l >_sw_l pl pl 1 pl 1 pl 11 luxr BioSketchPad Biologist-friendly environment for representing, storing, simulating, and analyzing biomolecular networks d ( _ i) _ i = ( k2 LuxI ) S k1 _ i + k 1Co + n( _ e _ i) H d ( Co ) Co = + k1 _ i k 1Co H sp d ( _ e) _ e = n( _ e _ i) H d ( ) = S2 Co % act Co<Co_sw_r <_sw_r pr Co<Co_sw_r >_sw_r pr 1 Co>Co_sw_r <_sw_r pr 1 Co>Co_sw_r >_sw_r pr 11 luxicdabeg Charon A programming language for modeling, simulating, analyzing, and designing hybrid systems GENE TRANSCRIPTION REGULATION mrna TRANSLATION DEGRADATION PROTEIN + Co TRANSPORT SOURCE SMALL MOLECULE Concentration in nm seconds

8 Modeling Issues Research Issues Well posedness, robustness, zenoness Analysis Stability issues, qualitative theory, parametric analysis Verification Algorithmic methods that verify system performance Synthesis Algorithmic methods that design hybrid controllers Simulation Mixed signal simulation, event detection, modularity Code generation From hybrid models to embedded code Comply Compositionality and hierarchies Tools : HyTech, Checkmate, d/, HYSDEL, Stateflow, Charon