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EECE 396-1 Hybrid and Embedded Systems: Computation T. John Koo, Ph.D. Institute for Software Integrated Systems Department of Electrical Engineering and Computer Science Vanderbilt University 300 Featheringill Hall March 16, 2004 john.koo@vanderbilt.edu http://www.vuse.vanderbilt.edu/~kootj

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2 Computational Tool: Reachability

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3 Unsafe Set Transition System Initial set

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4 Hybrid Automaton

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6 Reachable Sets

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10 Reachable Sets

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11 Reachable Sets

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12 Reachability Problem

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13 Reachability Problem

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14 Reachability Problem

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15 Reachability Problem

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16 Reachability Algorithm

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17 Reachability Algorithm

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18 Reachability Algorithm Unsafe!!

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19 Reachability Algorithm Safe!!

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20 Reachability Algorithm Keep iterating until when!?

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21 Reachability Algorithm

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22 Reachability Algorithm

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23 Reachability Algorithm Keep iterating until when!?

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24 Deciability Keep iterating until when!?

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25 Deciability Keep iterating until when!?

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26 Computational tools Basic computation includes Set-theoretic operations: Union, Intersection, Difference Reach set computations: Post d, Post c, Pre d, Pre c Verification Safety Property Forward algorithm Backward algorithm Liveness Property Properties specified by Temporal Logics Ref: Thomas A. Henzinger, The Symbolic Approach to Hybrid Systems, (CAV’02), UC Berkeley.

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27 Computational tools d/dt Library contributed by Thao Dang System Dynamics Linear systems Affine systems Linear systems with bounded inputs Set Representation Convex sets Basic (approximate) computation includes Set-theoretic operations: Union, Intersection, Difference Reach set computations: Post d, Post c, Pre d, Pre c Verification Specifications written as Temporal Logic Formula Algorithms

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28 Computational tools Projects Temporal Logic specifications Algorithms derivation d/dt based computational tool Verification Synthesis DC-DC Converters Controller verification Controller synthesis

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29 Computational tools Algorithm Reach Sets Specification Set Operations Dynamics Data Structure Temporal Logic Input Output

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30 Design Example: DC-DC Converters

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31 ENNA GmbH Power Electronics Power electronics found in: DC-DC converters Power supplies Electric machine drives Circuits can be defined as networks of: Voltage and current sources (DC or AC) Linear elements (R, L, C) Semiconductors used as switches (diodes, transistors)

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32 ENNA GmbH Power Electronics Discrete dynamics N switches, (up to) 2 N discrete states Only discrete inputs (switching): some discrete transitions under control, others not Continuous dynamics Linear or affine dynamics at each discrete state + + 2 3 =8 possible configurations

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33 Power Electronics : DC-DC Converters Have a DC supply (e.g. battery), but need a different DC voltage Different configurations depending on whether Vin Vout Control switching to maintain Vout with changes in load (R), and Vin V in L CR sw1 sw2 + - + - V out iLiL iLiL 212

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34 Two Output DC-DC Converter Want two DC output voltages Inductors are big and heavy, so only want to use one Similar to “two tank” problem V in L C2C2 R2R2 sw1 sw2 + - + - V outA iLiL + - V outB sw3 C3C3 R3R3 iLiL V outA V outB 123123

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35 Circuit Operation One and only one switch closed at any time Each switch state has a continuous dynamics sw1: i L , V outA , V outB sw2: i L , V outA , V outB sw3: i L , V outA , V outB

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36 Design Objective Objective: Regulate two output voltages and limit current by switching between three discrete states with continuous dynamics. i L , V outA , V outB i L , V outA , V outB i L , V outA , V outB

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37 Typical Circuit Analysis/Control Governing equations Time domain, steady state Energy balance System dynamics Discretization in time Switched quantity only sampled at discrete instants Assumes a fixed clock Averaging Switched quantity approximated by a moving average Assumes switching is much faster than system time constants Control Linearize with duty ( ) as input Use classical control techniques T T(1- )T i0i0 i1i1 i2i2 match! i L (t) i L [k]

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38 Problem Formulation

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39 Design Example: DC-DC Converters Controller Synthesis - Feasibility

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40 Problem Formulation Parallel Composition of Hybrid Automata Given a collection of Modes and Edges, design Guards

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41 Problem Formulation: Hybrid Automaton

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42 Formulation Capacitor Discharging Mode (q1)

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43 Formulation Capacitor Charging Mode (q2)

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44 Backward Reachable sets (qualitative) w = q2 – q1 q1q2

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45 d/dt Calculations result (quantitative) w = q2 – q1 NOT FEASIBLE

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46 Backward Reachable sets (qualitative) w = q1 – q2 q1q2

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47 d/dt Calculations result (quantitative) w = q1 – q2 FEASIBLE

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49 Design Example: DC-DC Converters Controller Synthesis – Switching Surfaces

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51 Switching Surface (Guard) – Go Forward! w = q1 – q2 q1 Switching Surface (Guard)

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52 Design Example: DC-DC Converters Controller Synthesis – Simulation

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53 Problem Formulation Parallel Composition of Hybrid Automata Given a collection of Modes and Edges, design Guards

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54 Semi-Analytic Calculation of Switching Time t sw =0.174 ms t sw =0.158 ms

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55 End

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