Discrete-Event Dynamic Systems Research

1. Discrete-Event Dynamic Systems Research • Systems being studied • Time discretisation vs state discretisation • Industrial interest • Supervisory control – one reason to do research • Chosen philosophy • Mapping continuous plants into automata • A brief example: Pendulum • Dimensional explosion • Status

2. DMUX DMUX Systems being studied

3. state variable time Time vs State Discretisation Sampled signal Discrete-event signal state variable Definition State Event : A state variable just crossed a state boundary time

4. Industrial interest • Supervisory control • Recipe execution and control • Start-up • Shut-down (normal and emergency) • Fault analysis • detection • isolation

5. continuous discrete ControlledPlant Discrete-event-dynamic plant Supervision of Controlled Plant Supervisor Environment EventDetection

6. Supervisory control – a justification for research Example PLC project : industrial dairy product plant Task man months % Specs 12.0 26 Coding data 5.6 12 Coding logic 16.8 37 Bench testing 5.6 12 Commissioning 6.0 13 after 18 months code out of date

7. Philosophy of approach • think in terms of state space – • discretise state, x, and input, u, (output, y, follows trivially). • if dim(x) = dim(y) no problem – take either x or y, after appropriate transformation, as state. • if state observable – usually no problem as this type of control operates on the long time scale at which one can assume the observer to be instantaneous. • Result: Non-deterministic automaton • Note: Several transitions may be possible being in a discrete state and applying a discrete input. level model does not know what the current trajectory is temperature

8. Modelling approach • Definition of event : Trajectory crosses in one of the state dimensions a defined boundary • note: only one boundary at the time can be crossed. Simultaneous crossings are not possible.Argument: use of serial line for the communications of the events or use of a serially operating computing device.

9. Modelling 2 • Transition can be computed by analysing boundary only.Proved also for non-analytical functions f(x,u) as long as continuity is given. Consequence: no integration necessary • Component equilibrium hypersurfaces • Assumptions • linear hypersurfaces • nonlinear hypersurfaces • monotone hypersurfaces • non-monotone hypersurfaces • Characteristic • Component-equilibrium hypersurface often passes through corner of hypercube

10. Pendulum in the Box position in one co- ordinate is only detected when a light beam is interrupted

11. 1 1,3 2,3 3,3 0.8 0.6 2,2 0.4 0.2 3,2 1,2 x2 0 -0.2 -0.4 -0.6 1,1 2,1 3,1 -0.8 -1 -1 -0.5 0 0.5 1 x1 Pendulum phase behaviour d/dt x1 = 0 d/dt x2 = 0

12. Pendulum automaton 1 0.5 1,3 2,3 3,3 x2 -0.5 2,2 -1 3,2 1,2 -1 -0.5 0 0.5 1 x1 1,1 2,1 3,1

13. Dimensionality explosion • One of the commonly used arguments • Claim: is not a problem because • analysis and thus mapping is done component by component • only those states and inputs must be analysed that are present in the right-hand side of the differential equation for each component only • the latter coupling is usually week, as equipment further away is not directly affecting the state being analysed. It is only the equipment-internal state and the coupled pieces’ state that may affect the state being analysed.

14. Status • Theory, algorithms and software for mapping continuous models that are observed with event detectors. • process may be nonlinear • monotone component-equilibrium surfaces are an advantage • linear is extremely simple • state-explosion does essentially not exist • Conditions and algorithm to reconstruct continuous trajectory given a set of events. • Some results on control, based on a reachability analyses using control invariant sets and forced transitions between such sets. • Min and max time for the transitions can be computed for those parts of the flow field that is monotone, thus for monotone systems completely. • Results for the construction of automata, that is how to select the boundaries as to be able to detect not directly observed faults.