Verification of Hybrid Systems An Assessment of Current Techniques Holly Bowen

Verification Methods Formal verification – determining whether given properties are true for a given model of a dynamic system Theorem proving – inferring/contradicting a specification using logical proof systems Not restricted to finite-state systems Model checking – using the state-transition relation in iterative computations to arrive at the set of states for which the specification is true Algorithmic technique

Model Checking Requires the construction of a finite-state approximation of the continuous dynamics Verification of properties for the finite-state approximation may be inconclusive! E.g. if a state is reachable in the finite-state approximation, that doesn’t imply it is reachable in the underlying hybrid system Tools can refine the approximation, but refinement will not necessarily terminate

Example: Batch Reactor System Exothermic reaction: 2A + B  D Discrete controller: v A, v B, v C, v O Variables: T R, V R, t R, c A

Operation Procedure Formal verification: Are the forbidden states (z 5, z 6 ) reachable?

Hybrid Model of System

Model Checking Tools UPPAAL HYTECH d/dt CheckMate VERDICT

UPPAAL Systems are represented as networks of timed automata (TA) Can analyze simple liveness properties and reachability properties Uses clock difference diagrams to represent TA in a compact format User must manually translate the process behavior into a set of concurrent TA

UPPAAL operation procedure reactor behavior Desired states: S2, S5, S9 Result: S10 is reachable!

HYTECH Specifications are given as temporal logic expressions Uses symbolic model checking in the continuous state space Can only model flows with form (linear hybrid automata)

HYTECH Three approaches to verify systems of higher complexity than LHA: Clock transition models – continuous state variables are replaced by clock variables (pure integrators with different rates) Constraints identify regions for which given rates are valid Rate translation – Retains original state variables, but approximates continuous behavior with piecewise- constant bounds on first derivatives Linear phase-portrait approximation – Derivatives of state variables can be constrained in linear combinations Gives a better approximation to original state equations

Rate Translation

d/dt Performs reachability analysis for hybrid systems with linear continuous dynamics Face-lifting – computing collections of orthogonal polyhedra to represent reachable sets Allows models with uncertainty in the input in the dynamics equations E.g. User must linearize system dynamics around the operating point of interest

Face Lifting Each face is moved by an amount that bounds all possible trajectories starting on the face

CheckMate MATLAB-based tool, handles systems with arbitrary nonlinear continuous dynamics Allows any Simulink/StateFlow blocks for simulation Verification: Logical operators (AND, OR, XOR, etc.) MUX/DEMUX Switched Continuous System Block (SCSB) Polyhedral Threshold Block (PTHB) Finite State Machine Block (FSMB)

CheckMate Computes finite-state approximation using general polyhedral over-approximation to sets of reachable states for continuous dynamics Can refine current approximation and attempt verification again if result is inconclusive Searches for states that led to failure, splits them, recomputes reachable states, evaluates logic expression again

CheckMate Controller (FSMB) PHTBs

CheckMate within 1 hour?

VERDICT Modular modelling/verification of timed/hybrid systems Structure of system is built in a modular manner Behavior of each module is described by a discrete, timed, or hybrid transition system Translates the model into the input languages of different model checkers for discrete/timed automata HYTECH, KRONOS, SMV, UPPAAL

VERDICT controller hybrid behavior

Comparisons Two key issues: Computation takes hours – only very small systems can be verified! Modularity: break systems down into smaller pieces Interpretation of results – cause of failure is not clear

Making Tools Useful for Industry Connecting with Existing Models Model-building process is time-consuming, could introduce errors Tools for Exploring Models and Results Useful results are obtained only when the user is directing the verification process Tools for Building Verification Specifications & Interpreting Results Difficult to translate requirement specifications into formal specifications to be verified