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?