Hybrid Systems Verification and Robotics Andr e Platzer - PowerPoint PPT Presentation

Hybrid Systems Verification and Robotics Andr´ e Platzer aplatzer@cs.cmu.edu Computer Science Department Carnegie Mellon University, Pittsburgh, PA http://symbolaris.com/ 0.5 0.4 0.3 0.2 1.0 0.1 0.8 0.6 0.4 0.2 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 1 / 25

Outline Hybrid Systems Applications 1 Logic for Hybrid Systems 2 Model Checking 3 Successive Image Computation Image Computation in Hybrid Systems Approximation Refinement Model Checking Summary Proofs for Hybrid Systems 4 Proof Rules Soundness and Completeness Survey 5 Summary 6 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 1 / 25

Can you trust a computer to control physics? Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 2 / 25

Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 0.2 a v p 2.5 p x 0.8 0.1 2.0 10 t 0.6 2 4 6 8 1.5 0.4 � 0.1 1.0 p y � 0.2 0.2 0.5 � 0.3 10 t 10 t 2 4 6 8 2 4 6 8 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 3 / 25

Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 0.2 a d Ω 1.0 d x 0.00008 0.1 0.8 0.00006 10 t 0.6 2 4 6 8 0.00004 � 0.1 0.4 0.00002 � 0.2 0.2 10 t 2 4 6 8 � 0.3 d y 10 t 2 4 6 8 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 3 / 25

Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 a 1.0 v p 0.2 8 0.8 10 t 2 4 6 8 6 0.6 � 0.2 p x 4 0.4 � 0.4 � 0.6 0.2 2 p y � 0.8 10 t 10 t 2 4 6 8 2 4 6 8 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 4 / 25

Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 a d Ω 1.0 d x 0.2 0.5 10 t 2 4 6 8 0.5 10 t 2 4 6 8 � 0.2 � 0.5 10 t � 0.4 2 4 6 8 d y � 0.6 � 1.0 � 0.5 � 0.8 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 4 / 25

Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 a 1.0 v p 10 t p x 2 4 6 8 4 0.8 � 1 3 0.6 � 2 2 0.4 � 3 1 0.2 p y 10 t � 4 10 t 2 4 6 8 2 4 6 8 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 5 / 25

Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 a d Ω 10 t 1.0 2 4 6 8 d x 0.5 � 1 0.5 10 t 2 4 6 8 � 2 10 t � 0.5 2 4 6 8 � 3 d y � 1.0 � 0.5 � 4 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 5 / 25

Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 a 1.2 v 7 p 0.4 6 1.0 0.2 5 0.8 10 t 4 2 4 6 8 0.6 p x � 0.2 3 0.4 � 0.4 2 0.2 1 � 0.6 p y 10 t 10 t 2 4 6 8 2 4 6 8 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 6 / 25

Hybrid Systems Analysis Challenge (Hybrid Systems) Fixed rule describing state 3.5 evolution with both 3.0 2.5 Discrete dynamics 2.0 (control decisions) 1.5 1.0 Continuous dynamics 0.5 (differential equations) 0.0 0 1 2 3 4 5 6 a d Ω 1.0 d x 0.4 0.5 0.2 0.5 10 t 10 t d y 2 4 6 8 2 4 6 8 � 0.2 � 0.5 10 t 2 4 6 8 � 0.4 � 1.0 � 0.6 � 0.5 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 6 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = DL + HP 1.0 0.5 0.0 0 1 2 3 4 6 5 Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R 1.0 0.5 0.0 0 1 2 3 4 6 5 v v 2 ≤ 2 b ( M − z ) z M Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R 1.0 0.5 0.0 0 1 2 3 4 6 5 v v ≤ 1 z M Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R 1.0 0.5 0.0 0 1 2 3 4 6 5 v v ≤ 1 ∧ v 2 ≤ 2 b ( M − z ) z M Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R 1.0 0.5 0.0 0 1 2 3 4 6 5 v v ≤ 1 ∨ v 2 ≤ 2 b ( M − z ) z M Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R 1.0 0.5 0.0 0 1 2 3 4 6 5 v v ≤ 1 ∨ v 2 ≤ 2 b ( M − z ) ∀ M ∃ SB . . . ∀ t ≥ 0 . . . z M Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + ML 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b � v 2 ≤ 2 b v 2 ≤ 2 b Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + DL 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b ] v 2 ≤ 2 b [ v 2 ≤ 2 b Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + DL + HP 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b [ z ′′ = a ] v 2 ≤ 2 b v 2 ≤ 2 b Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + DL + HP 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b [ if ( z > SB ) a := − b ; z ′′ = a ] v 2 ≤ 2 b v 2 ≤ 2 b Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + DL + HP 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b [ if ( z > SB ) a := − b ; z ′′ = a ] v 2 ≤ 2 b � �� v 2 ≤ 2 b hybrid program Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + DL + HP 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b C → [ if ( z > SB ) a := − b ; z ′′ = a ] v 2 ≤ 2 b � �� v 2 ≤ 2 b hybrid program Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Logic for Hybrid Systems 3.5 3.0 2.5 differential dynamic logic 2.0 1.5 d L = FOL R + DL + HP 1.0 0.5 0.0 0 1 2 3 4 6 5 v 2 ≤ 2 b v 2 ≤ 2 b C → [ if ( z > SB ) a := − b ; z ′′ = a ] v 2 ≤ 2 b � �� v 2 ≤ 2 b hybrid program Initial condition Andr´ e Platzer (CMU) Hybrid Systems Verification and Robotics RSS-FMRA 7 / 25

Hybrid Systems Verification and Robotics Andr e Platzer - PowerPoint PPT Presentation

Hybrid Systems Verification and Robotics Andr e Platzer aplatzer@cs.cmu.edu Computer Science Department Carnegie Mellon University, Pittsburgh, PA http://symbolaris.com/ 0.5 0.4 0.3 0.2 1.0 0.1 0.8 0.6 0.4 0.2 Andr e Platzer

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Assume-Guarantee verification of Hybrid Systems in A RIADNE Davide Bresolin and Tiziano Villa

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Hybrid Systems Verification and Robotics Andr e Platzer - PowerPoint PPT Presentation

Hybrid Systems Verification and Robotics Andr e Platzer aplatzer@cs.cmu.edu Computer Science Department Carnegie Mellon University, Pittsburgh, PA http://symbolaris.com/ 0.5 0.4 0.3 0.2 1.0 0.1 0.8 0.6 0.4 0.2 Andr e Platzer

Verification of hybrid dynamical systems Jri Vain Tallinn Technical University/Institute of

Verification of Robotics and Autonomous Deep Learning Verification Systems Safety Definition

Verification of Hybrid Systems George Pappas University of Pennsylvania, USA

An Algorithmic Approach to Global Asymptotic Stability Verification of Hybrid Systems Miriam

An Algorithmic Approach to Stability Verification of Hybrid Systems: A Summary Miriam Garca Soto

Algorithmic Verification of Stability of Hybrid Systems Pavithra Prabhakar Kansas State

Deductive Verification Of Hybrid Systems Lectures on Formal Methods for Cyber-Physical Systems

Verification of Hybrid Systems Using Linear Hybrid Automata Bruce H. Krogh Department of

Verification of Hybrid Controlled Processing Systems based on Decomposition and Deduction Goran

Verification and Qualitative analysis of hybrid systems: dimension 2 Joint work with Gerardo

A CEGAR Approach for Stability Verification of Linear Hybrid Systems Miriam Garca Soto

Verification for Robotics and Autonomous Systems Clare Dixon Department of Computer Science

AVERIST Algorithmic Verifier for Stability of Linear Hybrid Systems Miriam Garca Soto and

Invariant-Based Verification and Synthesis for Hybrid Systems Naijun Zhan Institute of Software,

Modeling and Verification of Real-time/Hybrid/Cyber-Physical Systems via Concurrent Co-inductive

From Robotic Systems to Robotics-grounded AI Systems Maxim Likhachev Robotics Institute &amp;

Whole Systems Energy Transparency Kerstin Eder Design Automation and Verification,

Assume-Guarantee verification of Hybrid Systems in A RIADNE Davide Bresolin and Tiziano Villa

Algorithmic Verification of Stability of Hybrid Systems Pavithra Prabhakar Kansas State

Formal Verification of V2I aided Autonomous Driving Ishan Pardesi Dhruv Mahajan A hybrid

Weak Singular Hybrid Automata Formal Modeling and Verification for Cyber-Physical Systems Krishna

APPROACH TO HYBRID SYSTEMS SAFETY VERIFICATION Andreas Mller andreas.mueller@jku.at Stefan

Hybrid Systems Parasara Sridhar Duggirala, Chuchu Fan, Matthew Potok, Bolun Qi, Sayan Mitra,

Theory of Reals for Verification and Synthesis of Hybrid Dynamical Systems Ashish Tiwari

From Robotic Systems to Robotics-grounded AI Systems Maxim Likhachev Robotics Institute &