1. 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 18, 2004

2. Project: DC-DC Converter2

3. Computational tools Temporal Logic Specification Input Data Structure
Set
Operations
Algorithm
Dynamics
Reach Sets
Output 3

4. 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: Postd, Postc, Pred, Prec
Verification
Specifications written as Temporal Logic Formula
Algorithms 4

5. Computational tools Projects Temporal Logic specifications
Algorithms derivation
d/dt based computational tool
Verification
Synthesis
DC-DC Converters
Controller verification
Controller synthesis 5

6. Project: Algorithm 1 Hybrid plant Initial sets Temporal Specification
Final sets
Temporal
Logic
Specification
Data
Structure
Set
Operations
Algorithm 1
Dynamics
Reach Sets
Feasible
sequences
ddt
GME 6

7. Project: Algorithm 2 Temporal Specification Logic Feasible sequence
Data
Structure
Set
Operations
Algorithm 2
Dynamics
Reach Sets
Switching
surfaces
ddt
GME 7

8. Project: Controller Implementation
Feasible
sequence and
switching
surfaces
Algorithm 3
Embedded Computing Systems Laboratory
QNX Computation Clusters
FPGA testbed
Hybrid
controller
Hybrid plant
Initial set (point)
Final set
Simulink
OPAL-RT 8

9. Project: Verification of Actual Design of DC-DC Converters
Hybrid Automaton
Initial sets
Final sets
Temporal
Logic
Specification
Data
Structure
Set
Operations
Algorithm 4
Dynamics
Reach Sets
Reachable:Yes/No
ddt
GME
Alternative: Use ddt program instead of ddt library to verify a DC-DC converter design.
No need to develop GME program 9

10. Design Example: DC-DC Converters10

11. 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)
ENNA GmbH 11

12. Power Electronics Discrete dynamics Continuous dynamics + +
N switches, (up to) 2N discrete states
Only discrete inputs (switching): some discrete transitions under control, others not
Continuous dynamics
Linear or affine dynamics at each discrete state
ENNA GmbH + +
23=8 possible configurations 12

13. Power Electronics : DC-DC Converters
Vin L C R
sw1
sw2 + -
Vout iL iL
Vout 2 1
Have a DC supply (e.g. battery), but need a different DC voltage
Different configurations depending on whether Vin<Vout or Vin>Vout
Control switching to maintain Vout with changes in load (R), and Vin 13

14. Two Output DC-DC Converter
Vin L C2 R2
sw1
sw2 + -
VoutA iL
VoutB
sw3 C3 R3 iL
VoutA
VoutB 1 2 3
Want two DC output voltages
Inductors are big and heavy, so only want to use one
Similar to “two tank” problem
US Patent since it can reduce the number of inductors used by having extra switches!! 14

15. Circuit Operation sw1: iL, VoutA, VoutB sw2: iL , VoutA , VoutB
One and only one switch closed at any time
Each switch state has a continuous dynamics
sw1: iL, VoutA, VoutB
sw2: iL , VoutA , VoutB
sw3: iL , VoutA, VoutB  15

16. Design Objective iL , VoutA, VoutB  iL, VoutA, VoutB
Objective: Regulate two output voltages and limit current by switching between three discrete states with continuous dynamics. 16

17. Typical Circuit Analysis/Control
match!
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
iL(t)
iL(t)
iL[k] 17

18. Problem Formulation
Connectivity between each pair of modes defines an edge 18

19. Design Example: DC-DC Converters Controller Synthesis - Feasibility19

20. Problem Formulation Parallel Composition of Hybrid Automata
Given a collection of Modes and Edges, design Guards
Given a design of DC-DC converter, one can model it as a hybrid automaton. State-feedback is used. But the control is a discrete symbol which corresponds to a specific switch configuration.
Notice that the design is event-triggered and exact. While using duty cycle, the controller is based on time domain analysis on a discrete-time averaged system model. 20

21. Problem Formulation: Hybrid Automaton21

22. Formulation Capacitor Discharging Mode (q1)
Keep the flow pattern and direction in mind 22

23. Formulation Capacitor Charging Mode (q2)
Keep the flow pattern and direction in mind 23

24. Backward Reachable sets (qualitative) w = q2 – q124

25. d/dt Calculations result (quantitative) w = q2 – q1
NOT
FEASIBLE 25

26. Backward Reachable sets (qualitative) w = q1 – q226

27. d/dt Calculations result (quantitative) w = q1 – q2
FEASIBLE 27

28. bwdsearchresult.eps 28

29. Design Example: DC-DC Converters Controller Synthesis – Switching Surfaces29

30. fwdsurfacealgorithm.eps 30

31. Switching Surface (Guard) – Go Forward! w = q1 – q231

32. Design Example: DC-DC Converters Controller Synthesis – Simulation32

33. Problem Formulation Parallel Composition of Hybrid Automata
Given a collection of Modes and Edges, design Guards
Given a design of DC-DC converter, one can model it as a hybrid automaton. State-feedback is used. But the control is a discrete symbol which corresponds to a specific switch configuration.
Notice that the design is event-triggered and exact. While using duty cycle, the controller is based on time domain analysis on a discrete-time averaged system model. 33

34. Semi-Analytic Calculation of Switching Time
tsw=0.174 ms
tsw=0.158 ms 34

35. End 35