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

SAMPLED DATA CONTROL SYSTEMS

CONTROL SYSTEMS

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

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

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

PLANT – HYBRID AUTOMATON Down shift Up shift

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

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

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

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

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

TIMED RELATIONAL ABSTRACTIONS Plant state time Plant Dynamics

TIMED RELATIONAL ABSTRACTIONS Plant state time System Dynamics

TIMED RELATIONAL ABSTRACTIONS Plant state time Relational Abstraction R R R

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

CONTROLLED TRANSITIONS Relationalize

AUTONOMOUS TRANSITIONS

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

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

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

 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

IMPLEMENTATION

 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

QUESTIONS?