EECE 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,

2 Computational Tool: Reachability

3 Unsafe Set Transition System Initial set

4 Hybrid Automaton

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6 Reachable Sets

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10 Reachable Sets

11 Reachable Sets

12 Reachability Problem

13 Reachability Problem

14 Reachability Problem

15 Reachability Problem

16 Reachability Algorithm

17 Reachability Algorithm

18 Reachability Algorithm Unsafe!!

19 Reachability Algorithm Safe!!

20 Reachability Algorithm Keep iterating until when!?

21 Reachability Algorithm

22 Reachability Algorithm

23 Reachability Algorithm Keep iterating until when!?

24 Deciability Keep iterating until when!?

25 Deciability Keep iterating until when!?

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.

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

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

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

30 Design Example: DC-DC Converters

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)

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 =8 possible configurations

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 sw V out iLiL iLiL 212

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 sw V outA iLiL + - V outB sw3 C3C3 R3R3 iLiL V outA V outB

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 

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 

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]

38 Problem Formulation

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

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

41 Problem Formulation: Hybrid Automaton

42 Formulation Capacitor Discharging Mode (q1)

43 Formulation Capacitor Charging Mode (q2)

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

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

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

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)

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

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

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

55 End