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Want to impress your boy friend / girl friend? SAT Modulo Ordinary Differential Equations An Analysis Method for Hybrid Systems Martin Fränzle 1 with slides, L A T EX souce, etc., by Andreas Eggers 1 · Christian Herde 1 Nacim Ramdani 2 · Nedialko S. Nedialkov 3 SFB/TR 14 AVACS [YouTube video] 1 Carl von Ossietzky Universität · Oldenburg, Germany 2 Université d’Orléans PRISME Bourges, France · · 3 McMaster University Hamilton, Ontario, Canada · Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 2 / 85 Want to impress your boy friend / girl friend? What is a hybrid system? [www.popsci.com] That’s why we build hybrid systems! Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 3 / 85 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 4 / 85

What is a hybrid system? What is a hybrid system? Hybrid (from Greece) means arrogant, Hybrid (from Greece) means arrogant, presumptuous. presumptuous. After H. Menge: Griechisch/Deutsch, After H. Menge: Griechisch/Deutsch, Langenscheidt 1984 Langenscheidt 1984 Hybrid stems from Latin hybrida ’off- spring of a tame sow and wild boar, child of a freeman and slave, etc.’ From the Compact Oxford English Dictionary, 2008 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 4 / 85 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 4 / 85 Hybrid Systems Hybrid Systems Plant Plant disturbances ("noise") disturbances ("noise") environmental observable environmental observable state state influence influence control control Plant Plant Control Control Analog Analog switch switch Continuous Continuous controllers controllers Loads of continuous selection selection computations A/D A/D interleaved D/A D/A with discrete setpoints setpoints part of part of decisions observable observable state state Discrete Discrete setpoints setpoints supervisor supervisor task selection task selection active control law active control law Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 5 / 85 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 5 / 85

Hybrid systems are ensembles of interacting discrete and continuous subsystems: � Technical systems: � physical plant + multi-modal control � physical plant + embedded digital system � mixed-signal circuits � multi-objective scheduling problems (computers / distrib. energy management / traffic management / ...) � Biological systems: Hybrid Systems � Delta-Notch signaling in cell differentiation � Blood clotting The Formal Model � ... � Economy: � cash/good flows + decisions � ... � Medicine/health/epidemiology: � infectious diseases + vaccination strategies � ... Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 6 / 85 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 7 / 85 A Formal Model: Hybrid Automata A Formal Model: Hybrid Automata x = 20 . 0 ∧ y = 0 . 0 x = 20 . 0 ∧ y = 0 . 0 20 20 y y • • x = y x = y • • y = − 9 . 81 y = − 9 . 81 10 10 x ≥ 0 x ≥ 0 0 0 x = 0 . 0 ∧ y ≤ 0 . 0 / x = 0 . 0 ∧ y ≤ 0 . 0 / y ′ = − 0 . 8 · y y ′ = − 0 . 8 · y −10 −10 x : x : vertical position of the ball vertical position of the ball y : velocity y : velocity x x y > 0 y > 0 ball is moving up ball is moving up −20 −20 y < 0 y < 0 ball is moving down ball is moving down 0 5 10 15 20 0 5 10 15 20 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 8 / 85 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 8 / 85

A Formal Model: Hybrid Automata A Formal Model: Hybrid Automata x = 20 . 0 ∧ y = 0 . 0 x = 20 . 0 ∧ y = 0 . 0 20 20 y y • • x = y x = y • • y = − 9 . 81 y = − 9 . 81 10 10 x ≥ 0 x ≥ 0 0 0 x = 0 . 0 ∧ y ≤ 0 . 0 / x = 0 . 0 ∧ y ≤ 0 . 0 / y ′ = − 0 . 8 · y y ′ = − 0 . 8 · y −10 −10 x : x : vertical position of the ball vertical position of the ball y : y : velocity velocity x x y > 0 y > 0 ball is moving up ball is moving up −20 −20 y < 0 y < 0 ball is moving down ball is moving down 0 5 10 15 20 0 5 10 15 20 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 8 / 85 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 8 / 85 A Formal Model: Hybrid Automata A Formal Model: Hybrid Automata x = 20 . 0 ∧ y = 0 . 0 x = 20 . 0 ∧ y = 0 . 0 20 20 y y • • x = y x = y • • y = − 9 . 81 y = − 9 . 81 10 10 x ≥ 0 x ≥ 0 0 0 x = 0 . 0 ∧ y ≤ 0 . 0 / x = 0 . 0 ∧ y ≤ 0 . 0 / y ′ = − 0 . 8 · y y ′ = − 0 . 8 · y −10 −10 x : x : vertical position of the ball vertical position of the ball y : velocity y : velocity x x y > 0 y > 0 ball is moving up ball is moving up −20 −20 y < 0 y < 0 ball is moving down ball is moving down 0 5 10 15 20 0 5 10 15 20 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 8 / 85 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 8 / 85

A Formal Model: Hybrid Automata A Formal Model: Hybrid Automata x = 20 . 0 ∧ y = 0 . 0 x = 20 . 0 ∧ y = 0 . 0 20 20 y y • • x = y x = y • • y = − 9 . 81 y = − 9 . 81 10 10 x ≥ 0 x ≥ 0 0 0 x = 0 . 0 ∧ y ≤ 0 . 0 / x = 0 . 0 ∧ y ≤ 0 . 0 / y ′ = − 0 . 8 · y y ′ = − 0 . 8 · y −10 −10 x : x : vertical position of the ball vertical position of the ball y : y : velocity velocity x x y > 0 y > 0 ball is moving up ball is moving up −20 −20 y < 0 y < 0 ball is moving down ball is moving down 0 5 10 15 20 0 5 10 15 20 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 8 / 85 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 8 / 85 A Formal Model: Hybrid Automata State and Dimension Explosion Number of continuous variables linear in num- x = 20 . 0 ∧ y = 0 . 0 20 ber of cars y � Positions, speeds, accelerations, � torque, slip, ... • x = y Number of discrete states exponential in num- • y = − 9 . 81 10 ber of cars x ≥ 0 � Operational modes, control modes, � state of communication subsystem, ... Size-dependent dynamics 0 � Latency in ctrl. loop depends on number x = 0 . 0 ∧ y ≤ 0 . 0 / of cars due to communication subsystem. y ′ = − 0 . 8 · y � Coupled dynamics yields long hidden −10 channels chaining signal transducers. x : vertical position of the ball y : velocity x y > 0 ball is moving up −20 y < 0 ball is moving down 0 5 10 15 20 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 8 / 85 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 9 / 85

State and Dimension Explosion Hybrid Control — A Case Study Number of continuous variables linear in num- ber of cars � Positions, speeds, accelerations, � torque, slip, ... ! target position Number of discrete states exponential in num- ber of cars ! � Operational modes, control modes, � state of communication subsystem, ... Size-dependent dynamics � Latency in ctrl. loop depends on number of cars due to communication subsystem. � Coupled dynamics yields long hidden channels chaining signal transducers. ! • ω = α ! satellite position ⇒ Need a scalable approach v ⇒ Let’s try to achieve this through SAT/SMT-based methods. α !! ( x, y ) Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 9 / 85 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 10 / 85 Hybrid Control — A Case Study Hybrid Control — A Case Study 2 4 1 trigger y T − y α ′ T = sin − 1 ( √ ( x T − x ) 2 +( y T − y ) 2 ) v_set v ( x T , y T ) e c n a 1.5 t s i d 1 2 alpha asin(x) 1 3 1 alpha omega 1 s vx x α ′ P_v 0.5 omega x T alpha_target 0 ( x, y ) 1 −0.5 1 s v 2 • v v = a 2 dist_to_target −1 vy s y v y −1.5 controller −1 −0.5 0 0.5 1 • cos ω 1 1 s omega s alpha sin P_omega 3 6 u2 1.6 alpha y_target 4 u asin distance ypos α ′ u2 T Sqrt 5 >= 0 3 5 x_target 3 alpha_target xpos omega π − α ′ T 1 pi alpha_target omega_set time−triggered controller continuous−time robot motion proportional control Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 11 / 85 Martin Fränzle · SAT/SMT School 2012 · SAT Modulo Ordinary Differential Equations · 12 / 85