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Aditya Zutshi Sriram Sankaranarayanan Ashish Tiwari TIMED RELATIONAL ABSTRACTIONS FOR SAMPLED DATA CONTROL SYSTEMS

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SAMPLED DATA CONTROL SYSTEMS

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CONTROL SYSTEMS

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SAMPLED CONTROL SYSTEMS Discrete Controller compute wait Ts actuatesense Plant: hybrid system Hybrid Plant M2 ODE2 M1 ODE1 M3 ODE3 Controller: software

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System speed time SAMPLED CONTROL SYSTEMS Discrete Controller Physical Plant SA Desired Speed

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System speed time SAMPLED CONTROL SYSTEMS Discrete Controller Physical Plant SA uphill!

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PLANT – HYBRID AUTOMATON Down shift Up shift

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RELATIONALIZATION Discrete System [Discrete Transition System] Physical System [Hybrid Automaton] Actuate (Ts) Sense (Ts) Abstract the plant dynamics using relations

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RELATIONALIZATION Discrete System [Discrete Transition System] Physical System [Hybrid Automaton] Actuate (Ts) Sense (Ts)

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RELATIONALIZATION Discrete System [Discrete Transition System] Actuate (Ts) Sense (Ts) ODE 1 ODE 2

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RELATIONALIZATION Discrete System [Discrete Transition System] Actuate (Ts) Sense (Ts) R1R1 R2R2

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RELATIONALIZATION Discrete System [Discrete Transition System] Physical System [Discrete Transition System] Actuate (Ts) Sense (Ts) Use existing tools to verify safety properties

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TIMED RELATIONAL ABSTRACTIONS Plant state time Plant Dynamics

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TIMED RELATIONAL ABSTRACTIONS Plant state time System Dynamics

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TIMED RELATIONAL ABSTRACTIONS Plant state time Relational Abstraction R R R

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TIMED RELATIONAL ABSTRACTIONS Relation R Captures states reachable in one sampling period Resulting abstraction is equivalent: when only controlled transitions are present sound: when autonomous transitions are present Plant state time R R R

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CONTROLLED TRANSITIONS Relationalize

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AUTONOMOUS TRANSITIONS

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time m1 ODE1 Controlled Transitions m1 ODE1 m2 ODE2 Autonomous Transitions M1 ODE1 M2 ODE2 M5 ODE5 M3 ODE3 M4 ODE4 Dwell Time Restriction

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AUTONOMOUS TRANSITIONS time m1 ODE1 Controlled Transitions m1 ODE1 m2 ODE2 Autonomous Transitions

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The resulting abstraction is a quantified formula over exponentials. AUTONOMOUS TRANSITIONS Relationalize

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Solution Using interval arithmetic rewrite the formula as a Interval linear inequalities Reformulate as a Linear Complementarity Problem Linearize the dynamics around the midpoint and iteratively find the bounds AUTONOMOUS TRANSITIONS

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IMPLEMENTATION

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Experiments: NAV and Heat benchmark set [Ivancic + Fehnker] Benchmarks formulated in the paper Results: Promising for systems with many controlled transitions + few autonomous transitions Precision loss as number of autonomous transition increases Our Approach: Is sound Provides proofs when the property is inductive Is exact for controlled transitions EXPERIMENTAL RESULTS

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QUESTIONS?

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