1. Discrete Controller Synthesis
for Incrementally Stable Switched Systems
Javier Cámara, Antoine Girard, Gregor Goessler
VEDECY Project Meeting – June 1st, 2010 1

2. Symbolic Approach to Control Synthesis
A three step approach to controller synthesis:
Computation of a symbolic abstraction of the physical system.
Discrete controller synthesis for the symbolic abstraction.
Hybrid controller synthesis via discrete controller refinement.

3. Outline Introduction Discrete Controller Synthesis for ISSS
Switched Systems
Discrete Uniform Abstractions
Example: DC-DC Converter
Discrete Controller Synthesis for ISSS
Binary Decision Diagrams (BDDs)
Discrete Controller Synthesis for ISSS using BDDs
Multi-Scale Controller Synthesis
Discrete Multi-Scale Abstractions
Application to Time-Optimal Control Problems

4. Switched Systems A switched system is a tuple where :
is the state space;
is the finite set of modes;
is the collection of vector fields.
For a switching signal , initial state denotes the trajectory of given by:
For denotes the trajectory of associated to the constant switching signal

5. Switched Systems as Transition Systems
Consider a switched system and a time sampling parameter
Let be the transition system where:
The set of states is ;
The set of inputs is ;
The transition relation is given by
The set of outputs is ;
The output map H is the identity map over
The transition system is uncountable, but we can compute a symbolic abstraction.

6. Discrete Uniform Abstraction
We start by approximating the set of states by:
Where is a state sampling parameter:
Approximation of the transition relation:

7. Discrete Uniform Abstraction
We define the transition system where:
The set of states is ;
The set of inputs is ;
The transition relation is given by
The set of outputs is ;
The output map is given by
The transition system is discrete and deterministic.
and are approximately bisimilar if switched system is incrementally stable.

8. Example: DC-DC Converter
Power converter with switching control:
State variable:
System dynamics:

9. Example: DC-DC Converter
Symbolic abstraction

10. Outline Introduction Discrete Controller Synthesis for ISSS
Switched Systems
Discrete Uniform Abstractions
Example: DC-DC Converter
Discrete Controller Synthesis for ISSS
Binary Decision Diagrams (BDDs)
Discrete Controller Synthesis for ISSS using BDDs
Multi-Scale Controller Synthesis
Discrete Multi-Scale Abstractions
Application to Time-Optimal Control Problems

11. Binary Decision Diagrams (BDDs)
Acceptable resolutions result in large enumerative models.
Excessive computation time and memory consumption
BDDs provide an alternative to enumerative models:
Canonical representation of boolean formulas (for a fixed order of variables)
Compact
Efficient manipulation
Useful for concise representation of relations over finite domains
Extensively used in Symbolic Model Checking
Enable the verification of systems with large state spaces

12. Representing the Symbolic Model using BDDs
Let be an n-ary relation over can be represented by the BDD associated to its characteristic function:
Boolean formulas can be used to represent set of states and transition relation.

13. Binary Decision Tree Example (Two-bit Comparator)

14. Binary Decision Diagram Example (Two-bit Comparator)

15. Representing the Symbolic Model using BDDs
To encode the set of states in , we define a bijection
Encoding of partial transition relations for separate bits.

16. Encoding the Symbolic Model into BDDs

17. Controller for Reachability Specifications
Goal: guide the state of the system to a desired state, keeping the system safe along the way.
Let and be a controller for .
Let be a set of outputs associated with safe states and be a set of outputs associated with target states.
The entry time of from for specification is the smallest such that for all trajectories of of length starting from there exists
The entry time is noted by
If such does not exist, then we define

18. Controller for Reachability Specifications6 1 5 5
The controlled-predecessor of is: 2 3 4

19. Symbolic Computation of Controlled-Predecessor

20. Time-Optimal Controller
Control Objective: minimize entry time.
A controller is time-optimal for specification if, for all controllers :
Let be the controller for defined, for all by:
Then, is a time-optimal controller for the specification

21. Outline Introduction Discrete Controller Synthesis for ISSS
Switched Systems
Discrete Uniform Abstractions
Example: DC-DC Converter
Discrete Controller Synthesis for ISSS
Binary Decision Diagrams (BDDs)
Discrete Controller Synthesis for ISSS using BDDs
Multi-Scale Controller Synthesis
Discrete Multi-Scale Abstractions
Application to Time-Optimal Control Problems

22. Multi-Scale Controller Synthesis

23. Multi-Scale Abstractions
Sub-transition system of with transitions of duration in the set
Let be the transition system where:
The set of states is ;
The set of inputs is ;
The transition relation is given by
The set of outputs is ;
The output map is the identity map over
The set of Initial states is

24. Multi-Scale Abstractions
Approximation of state set as a sequence of embedded lattices:
We may now define
The set of states is
The set of inputs is
The transition relation is given by:
The set of outputs is
The observation map is the natural inclusion map of to
The set of initial states is

25. Multi-Scale Controller Synthesis
=0.5s
Entry time path A: 2.5s
3/2
/2
Entry time path B: 2s 3 2  
5/2 4 3 2 4 5

26. Time-Optimal Reachability: DC-DC Example

27. Time-Optimal Reachability: DC-DC Example

28. Time-Optimal Reachability: Automatic Refinement
Need of an automatic way to detect areas of the model that should be refined.
Heuristic: alternate switching pattern detection.

29. Time-Optimal Reachability: Automatic Refinement
Detection of refinement areas too extensive.
Statistical Data Analyisis: Clustering algorithms

30. Summary
Approach to control of incrementally stable switched systems using discrete controller synthesis:
Binary Decision Diagrams for symbolic fixpoint computation
Multi-Scale controller construction
Prototype tool under implementation:
Symbolic/Enumerative state space exploration
Colorado University Decision Diagram Package (CUDD) for BDD Manipulation
Next Steps:
Heuristics for automatic model refinement in time-optimal control problems, and other kinds of properties
Multi-Scale controller synthesis for safety specifications