Information Flow and Cooperative Control of Vehicle Formations

1. Information Flow and Cooperative Control of Vehicle Formations 김상훈 CDSL 2008-1-9

2. Previous SeminarRelative Position Control in Vehicle Formations • Goal • Cooperative Formation Control / Decentralized Control • Relative Formation / Offset (Inter-vehicle spacing) • Graph Theory / Laplacian Matrix / Perron-Frobenius Theorem  λ(L) properties • Theorem 3 • Individual (3)∼(7) system  identical N systems  (8) dynamics • Stabilizing (8) (BIG) ⇔ Stabilizing (11) (SMALL)  Check a single vehicle (with the same dynamics, only modified by a scalar λi)  Easy, Local Controller • 사실Laplacian L뿐만 아니라 일반적인 Matrix의 λ에도 동일하게 사용가능 • Theorem 4 • 가정 • Output y(Absolute Formation)는 관심 없다  Pc1을 0로  Relative Formation dynamics • SISO for individual vehicle • Nyquist Criterion 을 변형 (-1  -λ-1) 하여 Robust한 Controller Design • Evaluating Formations Via λ(L) • L의 Topology에 따른 -λ-1 Table 1  Stability를 짐작할 수 있다. (Stability Margin)

3. Previous SeminarFormation Stability when PC1=0; relative formations N vehicles together Theorem 3 Theorem 4 When SISO K N individual vehicles # of encirclements of -1

4. Information flow in Vehicle FormationA. Properties of the information flow loop • 주제 A • Problem • Graph의 Characteristic에 Stability가 큰 영향… • Controller로 들어가는 정보를 처리하고 싶은데… • What to do • Robustness to uncertainty in the Graph • To achieve desired response • How to • Block diagram에서 L(Graph)만 떼어서 생각 • L을 stabilizing 하는 R과 그것을 connection 방법을 제안 Information Flow Law • Stability Analysis by Theorem 4  Theorem 9 • Convergent Value는? 그것의 의미? Consensus란?  Theorem 10 • Result • Theorem9  이것에 맞추어 R을 잘 결정(또는 F결정) by Nyquist-like method  To achieve Robustness to L and Desired response • Theorem10  Consensus가 이루어지기 위한 필요조건

5. Information flow in Vehicle FormationB. Information Flow in the Loop • 주제B • Problem • Information Flow를 K, P와 함께 연결하면 기대한대로 작동?  Information Flow를 design할 때 생각했던 stability margin 은 L에 관한 것이다.  여전히 P의 uncertainty에 취약하다. • What to do • P의 uncertainty 대항력을 키우자. • 독립적으로 design K to stabilize P 과 design F to stabilize L 을 하고 싶다. • How to do • Vehicle의 motion 정보가 Information Flow Loop에 필요 Predictor 도입 (H) (Feedforward Information Compensator) • H를 잘 설정하면 K와 F를 독립적으로 설계해도 좋아 Theorem 11 • Result • Theorem 11  Separation Principle To design K, F by H

6. Information flow in Vehicle FormationAssumption & Preliminary • Assumption • Discrete-time Dynamical System • SISO for individual vehicle • Relative Sensing / Relative Formation Dynamics • Preliminary • Decentralized Control • Information  What to Receive from others, Compute and Transmit to others • Information flow law  Information 흐름에 관한 scheme • 각 Vehicle가 가진 Information은 Decentralized Control의 기초 • 여러 Vehicle의 Information은 공통 부분이 생기지 않을까?  Steady state조사  Consensus 확인 예) 다음 번 inform는 자신의 과거 inform, 이웃의 현재 inform, 추가되는 정보 (ex. sensing relative positions)에 의해 결정

7. Information flow in Vehicle FormationOverall Block Diagram ∵ for sensing & transmission delay (Ass. Strictly Proper) (Averaging information)

8. Properties of the information flow loopSimple Case : R(z) = (1/z) Information flow loop p y Let One step delay 1. Stability of Information flow loop? 2. Steady State Value of Information? ∵ delay  What Consensus is? • Stability of Information flow loop  neutrally stable • ∵ Perron root(spectral radius) of G <=1 • Steady State Value of Information •  Theorem 6

9. Theorem 6

13. Properties of the information flow loopConsensus as the formation Center 본 paper에서 even하게 부여하였지만 uneven 하게도 가능. 하지만 그경우global knowledge가 필요할 수 있음 주의 무엇일까? Graph에 의해 weighting된 Formation Center 모든 Vehicle의 inform.에서 공통으로 수렴 되는 값 “Formation reach consensus as its Center “ 라고 한다. (다른 논문에서 그림 가져 왔음)

14. Properties of the information flow loopGeneral Case : R(z) is general strictly proper • Stability (Theorem 9) • Steady State Value (Theorem 10)

15. Theorem 9

18. Theorem 10

20. Theorem 10 : About consensus

21. Properties of the information flow loopExample of F : stability and response F로 원하는 Response를 설계할 수 있다 (Shaping) F가 Information을 Stabilization 시킨다 Aperiodic graph의 경우 모든 가 -0.5 좌측에 위치(Fig.3)하므로  F1는 AperiodicGraph를 Stabilization한다  Theorem 6확인 모든 가 -0.5 좌측 또는 그 위에 위치(Fig.3)하므로  F2는 항상 Graph를 Stabilization한다

22. Information Flow in the loopStabilities in L and P isolating L • K와 P에 Information Flow Loop를 함께 연결하여 L의 양단에서 Nyquist Criterion 적용 • (L에 대항하는 Robustness)  여전히 P의 uncertainty에 취약하다. • P의 uncertainty 대항력을 키우자. • 독립적으로 design K to stabilize P 과 design F to stabilize L 을 하고 싶다. • Vehicle의 motion 정보가 Information Flow Loop에 필요 Predictor 도입 (H) (Feedforward Information Compensator) • H를 잘 설정하면 K와 F를 독립적으로 설계해도 좋아  Theorem 11

23. Information Flow in the loopSeparation Principle • Theorem 11 Block1 – by assumption, neutrally stable Block2 – equivalent to information flow law stabilization Block3 – by stabilization p through k Block4 – by assumption of theorem 9

24. Examples

25. Discussion • Results • Design of Dynamical System  Vehicles to achieve consensus on the formation center • Feedforward Compensation  to render sensed and transmitted information timely • Limitation • Need for an exact model P • No sensitivity analysis in modeling errors • Sensitivity to mismatches in initial conditions of vehicles Predictor 보다 Observer는 어때? • Linear System with fixed time delays • Nonlinear vehicle 또는 system with variable time delays 으로 확장해보는 것은 어때? • Nonlinear System의 Center Manifold  on which, information flow is restricted  Can extend information flow principle to such nonlinear systems?? • Feedforward term  stability separation  Can be extended to systems with variable time delays?? • Constraint c=1 in information flow  Information flow law : Neutral stable  Never decays out Sensitive to initial conditions 주기적으로 information을 reset하는 상위 레벨의 protocol을 생각해볼 수 있다.

26. Relative sensing • Identical multi-agent • Decentralized control • Formation Control을 했어 • 간단한 scheme에서 Stability를 보였어 • Scheme을 조금 복잡하게하여 Stability도 생각해봤어 • Consensus의 조건도 간략하게 보였고 • 어떤 응용을 해볼까??? • 무엇에 쓸까…..…. >_<

27. AppendixNyquist Criterion for Discrete System