Towards ComputingPhase Portraits ofPolygonal Di�erential Inclusions

E. Asarin G. Schneider S. YovineVERIMAG, France

Overview of the presentation� Motivation� Polygonal Di�erential Inclusion System (SPDI)� One cycle analysis{ Viability and Controllability Kernels{ Properties

� Global analysis� Conclusions

Overview of the presentation� Motivation� Polygonal Di�erential Inclusion System (SPDI)� One cycle analysis{ Viability and Controllability Kernels{ Properties

� Global analysis� Conclusions

Motivation� For Hybrid Systems{ Veri�cation (reachability, ...):{ Qualitative behavior (Phase Portrait, ...)

� For a class of Non-deterministic systems(SPDI){ Veri�cation (HSCC'01){ Phase Portrait (this work)

Di�culties� In most cases: Undecidable{ We consider planar systems

� Phase Portrait of Non-deterministic systems{ De�nition (?){ Objects (?)� What is a Limit Cycle?

Overview of the presentation� Motivation� Polygonal Di�erential Inclusion System (SPDI)� One cycle analysis{ Viability and Controllability Kernels{ Properties

� Global analysis� Conclusions

SPDI: Polygonal Di�erential InclusionSystem� SPDI:{ A partition of the plane into convexpolygonal regions{ A constant di�erential inclusion for eachregion

_x 2 \biai if x 2 Ri

x0Ri

xb a

b\ba

a

SPDI: Polygonal Di�erential InclusionSystem

Example: An SPDI and a trajectory segment

R6 R8

R3

R7

R2R4

R5 R1e5e4

e3 e2e1e8

e7e6

y x

� A signature � is a sequence of traversededges (� = e2; e3; � � � ; e8; e1; e2; e3)

Computation of Successors (by �)

� One step (� = e2e3)

������������������������������������������������������������������������������������������

������������������������������������������������������������������������������������������e4

e5 e11e12e9e10I 0 xe2e3

e1 e13e8e7e6

I 0 = Succe2e3(x)� Several steps (� = e1e2e3 � � � e8e1)

e5e4

e8e10 e9 e11e12 e13Succ�(x)Succ�(x)e3 e2 e1

e7e6

x

Overview of the presentation� Motivation� Polygonal Di�erential Inclusion System (SPDI)� One cycle analysis{ Viability and Controllability Kernels{ Properties

� Global analysis� Conclusions

One cycle analysis

� Fix � = e1 � � � e8e1

R6 R8

R3

R7

R2R4

R5 R1e5e4

e3 e2e1e8

e7e6

y x

� Explore trajectories with signature �

One cycle analysisDe�ning the set

� Given � = e1 � � � e8e1, take

K� = k[i=1(int(Pi) [ ei)

e6 e7

e2e3e4e5R6 R7 R8

R1

R2R3R4

R5 e8e1

� Cycling in � =) remaining in K�

Overview of the presentation� Motivation� Polygonal Di�erential Inclusion System (SPDI)� One cycle analysis{ Viability and Controllability Kernels{ Properties

� Global analysis� Conclusions

Viability Kernel

� M is a viability domain if 8x 2M , 9 at leastone trajectory �, starting in x and remain-ing in M� Viab(K): Viability kernel of K is the largestviability domain M contained in K

KB

A wx z

Viab(K) = A [B

One CycleViability Kernel (Computation)

� We can easily compute the Viability Kernelfor one cycle, which is a polygon� Theorem: Viab(K�) = Pre�(Dom(Succ�))

R6 R8

R4 R3

R7

R1R5

R2

e5e4

e3 e2e1

e8e7e6

Overview of the presentation� Motivation� Polygonal Di�erential Inclusion System (SPDI)� One cycle analysis{ Viability and Controllability Kernels{ Properties

� Global analysis� Conclusions

Controllability Kernel

� M is controllable if 8x;y 2 M , 9 a trajec-tory segment � starting in x that reaches anarbitrarily small neighborhood of y withoutleaving M� Controllability kernel of K, denoted Cntr(K),is the largest controllable subset of K

A

Kxw

z

Cntr(K) = A

One CycleControllability Kernel (Computation)

� Theorem: Cntr(K�) = (Succ�\Pre�)(CD(�))(We know how to compute the special intervalCD(�) = [l; u])

R6 R8

R3

R7

R4 R2

R1R5 l ue2e3

e1

e7e6

e4

e8e5

Overview of the presentation� Motivation� Polygonal Di�erential Inclusion System (SPDI)� One cycle analysis{ Viability and Controllability Kernels{ Properties

� Global analysis� Conclusions

Properties

� Theorem: Any viable trajectory in K�converges to Cntr(K�)

R6 R8

R4 R2R3

R7

R5 R1

e6 e7e8

e1e2e3

e4e5

� Controllability Kernel: \Weak" analog oflimit cycle� Viability Kernel: Its \local" attraction basin

Overview of the presentation� Motivation� Polygonal Di�erential Inclusion System (SPDI)� One cycle analysis{ Viability and Controllability Kernels{ Properties

� Global analysis� Conclusions

Global AnalysisPhase Portrait - Algorithm

� To compute Limit Sets and Attraction Basins:for each simple cycle � computeCntr(K�) and Viab(K�)

R6 R8

R4 R3

R7

R5R2 R11 R12

R13R14R15R1 e14e15e8e5

e4e3 e2

e7e6e13e12

e11e10e1

Poincar�e-Bendixson's like Theorem

� Controllability Kernel yields an analog ofPoincar�e-Bendixson theorem:Every trajectory with in�nite signaturewithout self-crossings converges to thecontrollability kernel of some simple edge-cycle

R5R4 R3 R2 R11 R12

e12R13R14R15R1R8R7R6

e5e4

e3 e2e1e8e7e6 e15 e14

e13

e11e10

Conclusions (Achievements)

� Algorithmic analysis of qualitative behaviorof non-deterministic planar hybrid systems� Our algorithm enumerates all the \limit cy-cles" and their attraction basins� Properties of controllability and convergenceto the set of limit cycles

Conclusions (Future Work)

� Identify and analyse other structures{ Stable and unstable manifolds{ Orbits{ Bifurcation points

� Limit behavior of self-intersecting trajecto-ries� Extension of the tool SPeeDI

Poincar�e-Bendixson Theorem

A non-empty compact limit set of C1 planardynamical system that contains no equilibriumpoints is a close orbit (limit cycle)

One CycleControllability Kernel (Computation)

Pre�(CD(�)) Succ�(CD(�))

R6 R8

R3

R7

R5 R1

R4 R2l u

R6 R8

R4 R2

R7

R1R5

R3l u

e6 e7e5e4

e3 e2e1

e8 e8e1

e2e3e4e5

e6 e7