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Modelling Hybrid systems

Modelling Hybrid systems. Hybrid Systems. Hybrid (combined) Modeling Framework Use more than one formalism Different formalisms to specify different levels of abstraction Top-down design step-wise refinement Ex: FSM + Difference equation Design and Implement the following waveform generator

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Modelling Hybrid systems

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  1. Modelling Hybrid systems

  2. Hybrid Systems • Hybrid (combined) Modeling Framework • Use more than one formalism • Different formalisms to specify different levels of abstraction • Top-down design • step-wise refinement • Ex: FSM + Difference equationDesign and Implement the following waveform generator • ObservationPeriodic, say-tooth waveformTc : Charging timeTd : Discharging time • Top-Level Spec in FSMM=<X, Y, S, , >S = { Ch, Dis}X = { charge, discharge}Y= { 0_to_V, V_to_0} V Tc Td

  3. Example: hierarchical control • Ex. Hierarchical control Operator Planning/scheduling Event-based control Discrete Event Controller Supervisory control Discrete state Command PID controller analog/digital Time-based control actuation Sensor Plant

  4. Motivation • Opto-electrical interfaces: transmitter and receiver • Conversion of electrical current into optical impulses, and viceversa.

  5. Motivation • Continuous systems analysis: • Different mathematical formalisms • Simulation: solutions to particular problems under certain experimental conditions of interest • Classical methods for continuous systems simulation • Based on numerical approximation • Require time discretization => of time • Inefficient in terms of execution times • Complex composition; difficulties in integration, multiresolution models • Benefits of DEVS for continuous system M&S • Discrete event models specification: continuous time base • Execution time reduction • Complex system definition using hierarchical modular models • Easier integration with discrete-event models

  6. GDEVS • Generalization of DEVS formalism • Polynomial of any degree to represent piecewise input-output trajectories • Introduction of a new event concept: coefficient-events • Advantages of GDEVS • Greater accuracy for continuous systems modeling • Unified approach to model hybrid systems • Piecewise linear trajectory specification w<t0;tn>  A a trajectory on a continuous time base finite set of instants t0, t1,…,tn associated with constant pairs (ai; bi) such that t  <ti; tj>, w(t) = ai t + bi, and w<t0; tn> = w<t0;t1>*w<t1;t2>*…*w<tn-1;tn>

  7. Discontinuities Event at t+d f1(x,u) f2(x,u) t-2h t-h t t+d t+h

  8. Overview of Hybrid OO modelling. • The occurrence of events should be notified to the simulation runtime. • Time events: calendar of events is known beforehand. f1(x,u) f2(x,u) t-2h t-h t t+d t+h • Stateevents: triggeredbystate of simulation

  9. Quantized DEVS • Continuous signal represented by crossing of an equal spaced set of boundaries, separated by a quantum size • Check for boundary crossing for every change in the model • Outputs generated only when a crossing occurs • Substantial reduction of the message updates frequency Signal Quantization

  10. DEVS Quantized models • Crossings of an equal spaced set of boundaries: quantum • Quantizer: checks for boundary crossings. • The sender computes a value, and generates outputs. • The number of messages involved is reduced. • The quantizer consumes CPU time. • The receiver will have some error, depending on q.

  11. Theoretical results on quantization #messages(D) = #messages(1)/D Error = c D 0 1 Quantum size, D Dtol = tolerance / c Tolerance to error

  12. Q-DEVS with hysteresis • strong stability, convergence and error bound properties. • If signal changes direction: use n*Q size (proof: n=2 provides best results) • If signal keeps current direction: use Q size

  13. QDEVS based models Uniform quantizer with hysteresis Uniform quantizer • Hysteresis assures legitimate DEVS models’ simulation • Avoids infinite iterations on finite time interval

  14. Multiple Model Controller

  15. Adaptive Control results q=0.02 q=0.2

  16. Higher-order Approximations

  17. Complete model • Multiple model controller allowed to operate as designed, and switch among plant identifying models • Controller was able to find it and use its parameters • Error existed only at the period coinciding w/each jump in plant parameters • Only at time 355 did a false model switch occur (due to two models having almost zero error )

  18. SF R Bond Graphs • Suitable for multiple domains: electrical, mechanical, hydraulic, etc. • Physical processes: vertices in a directed graph. Edges: represent ideal exchange of energy between components. • Interactions: 0-junction (connectors), 1-junction (interactions between serial components). • Causality: given a pair of elements connected through a bond, causality determine which component causes flow, which effort.

  19. Bond Graphs Formalism • Exchange of energy and information between elements of a system can be represented in a graphical form • Energy is the fundamental feature that is exchanged between elements of a system during interaction • Constrained interactions in Bond Graphs are represented by junctions • constraint equalizes the flow in the elements 1-junction • constraint equalizes the effort in the elements 0-junction

  20. A library for Bond Graph development on DEVS • Model library: modular approach to build systems; code reuse • Bond Graph library built to model and simulate continuous systems on different domains • Library designed using GDEVS formalism concepts • BG components developed as atomic GDEVS models of degree one • Multicomponent systems can be built as coupled DEVS components • Models implemented using the CD++ tool

  21. GDEVS Capacitor model • Equations: Flow arrives at component: dext. - Calculates effort: integrate input flow data (generate Capacitor’s load). - Value computed according to the elapsed time since last transition. - Output function transmits the previously computed value yout. - Internal transition: computes next state using a polynomial approximation. External transition . . . // time since last transition elapsedTime=msg.time().asMsecs()- time; // calculates load value c = c+a1/2*pow(elapsedTime,2)+a0*elapsedTime; . . . yout->updElementAtPos(1, c); yout->updElementAtPos(2, a1/2*dt + a0); holdIn( active, Time::Zero ); Internal transition // approximates load using order 1 polynomial. if ( a1 != 0 ) { // next state calculated using coefficients c = c + a1/2 * pow(dt,2) + a0*dt; a0 = a1*dt + a0; // coefficient values to send when dt elapsed yout->updElementAtPos(1,c); yout->updElementAtPos(2,a1/2*dt+a0); holdIn(active, Time(dt)); } else { passivate(); // slope is null }

  22. BG library class hierarchy

  23. Model execution examples • Electrical Circuit Simulation Bond Graph model construction of the electrical circuit Electrical circuit Bond Graph representation

  24. Electrical circuit simulation GDEVS Bond Graph model representation

  25. Bond-Graph model simulation in CD++ • Resistance (R1)=1 • Inductors: L1 = 48; L2 = 48. • Capacitance: C = 65. • Conductance: R2 = 0.001 • EffortSource: emits pulses; period = 2500 ms; duration = 2 ms. Pulse amplitude= 220 V Circuit current

  26. Model execution – Comparing Quantizers • Uniform quantizer vs. HQ (Quantizer with hysteresis) Test cases - Evaluation functions

  27. Modelica • object-oriented language for modeling physical systems • designed to support library development and model exchange • Modelica supports different formalisms such as: • ODEs • finite state automata • Petri nets • discrete events • bond graphs

  28. Creating Models in Modelica/CD++ Model circuit Modelica.Electrical.Analog.Sources.SineVoltage V(V=15,freqHz=60); Modelica.Electrical.Analog.Basic.Resistor R1(R=10); Modelica.Electrical.Analog.Basic.Ground Gnd; equation connect(V.p, R1.p); connect(R1.n, V.n); connect(R1.n, Gnd.p); end circuit1;

  29. M/CD++ Execution Example: Modelica model model circuit Modelica.Electrical.Analog.Sources. PulseVoltage V(V=10,width=50, period=2.5); Modelica.Electrical.Analog.Basic.Resistor R1(R=0.001); Modelica.Electrical.Analog.Basic.Inductor I1(L=500); Modelica.Electrical.Analog.Basic.Inductor I2(L=2000); Modelica.Electrical.Analog.Basic.Capacitor C(C=10); Modelica.Electrical.Analog.Basic.Resistor R2(R=1000); Modelica.Electrical.Analog.Basic.Ground Gnd; equation connect(V.p, R1.p); connect(R1.n, I1.p); connect(R1.n, I2.p); connect(I2.n, C.p); connect(I2.n, R2.p); connect(C.n, I1.n); connect(R2.n, C.n); connect(I1.n, V.n); connect(V.n, Gnd.p); end circuit;

  30. M/CD++ Execution Example: The results

  31. M/CD++ Execution Example: The results

  32. Modelica/DEVSLib Model • Transmitter model • The received electrical current (continuous) is translated into optical impulses (discrete). • The driver model can include any modulation and polarization circuit. No actions performed in this case, to simplify the model. • The laser translates the electrical signal into events using the crossUP model included in DEVSLib.

  33. Modelica/DEVSLib Model Detail of the crossUP model when (u > Value-Threshold) and x > 0 and time > 0 then e.Type := EType; e.Value := u; e.Port := 1; sendEvent(outport.queue, e); outport.event := pre(outport.event) + 1; end when;

  34. Cell-DEVS quantization

  35. Quantized Cell-DEVS • Every cell includes a quantizer. • The value produced by the local computing function is quantized. • The quantized value is compared with the quantum threshold. • If the threshold bound was reached, an output is provided. The output is delayed using transport or inertial delays. • If the threshold was not reached, the change is not sent to other models.

  36. Experimental results (a) Heat diffusion: two-dimensional model (10x10 cells). One "hot" cell. (b) Heat model with 87% of active cells. (c) Three-dimensional extension of the previous model. (d) Three-dimensional modification of the Life game (e) Four dimensional extension of the previous model. (f) Dynamic heat seeker: a three dimensional model consisting of two adjacent planes.

  37. Number of messages involved

  38. Error behavior

  39. Cell-DEVS w/Dynamic Quantization • Reduce error (improving precision) in each cell. • Rationale: active cell can appear as quiescent (Q size covering activity area). Q dynamically adjusted -> smaller error. • Dynamic increase Q in cells with steep update functions: execution improved at a low cost. • Strategy 1: reduce Q if cell's update does not cross threshold (increase precision). • Strategy 2: reduce Q if threshold is crossed (2nd chance in case of oscillations; higher precision).

  40. Test cases: a heart tissue model • Heart muscle excitable; responds to external stimuli by contracting muscular cells. • Equations defined by Hodgkin and Huxley

  41. A Watershed model

  42. Flow Injection Analysis Model

  43. Approximating the Corsica fire model

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