Lectures 25 & 26: Consensus and vehicular formation problems

Transcription

1 EE 8235: Lectures 25 & 26 Lectures 25 & 26: Consensus and vehicular formation problems Consensus Make subsystems (agents, nodes) reach agreement Distributed decision making Vehicular formations How does performance scale with size? Are there any fundamental limitations? Is it enough to only look at neighbors? Should information be broadcast to all?

2 EE 8235: Lectures 25 & 26 2 Collective behavior in nature COLLECTIVE MOTION IN 3D SNOW GEESE STRING FORMATION WILDEBEEST HERD MIGRATION

3 FORMATION FLIGHT FOR AERODYNAMIC ADVANTAGE e.g. additional lift in V-formations MICRO-SATELLITE FORMATIONS e.g. for synthetic aperture precise control needed EE 8235: Lectures 25 & 26 3 Coordinated control of formations MAKE VEHICLES SMALLER AND CHEAPER USE MANY cooperative control becomes a major issue

4 AUTOMATED CONTROL OF EACH VEHICLE tight spacing at highway speeds KEY ISSUES (also in: control of swarms, flocks, formation flight) EE 8235: Lectures 25 & 26 4 Vehicular strings Is it enough to only look at neighbors? How does performance scale with size? Are there any fundamental limitations? FUNDAMENTALLY DIFFICULT PROBLEM (scales poorly) Jovanović & Bamieh, IEEE TAC 05 Bamieh, Jovanović, Mitra, Patterson, IEEE TAC (to appear)

5 EE 8235: Lectures 25 & 26 5 String instability PROBLEM: STRING INSTABILITY FLIGHT FORMATION EXAMPLE (Allen et al., 2002) ONE APPROACH: design a follower cruise control chain into a formation Figure 6. Cascaded formation flight system. Figure 5. Formation reference frame.

6 CHAINING OF A FOLLOWER CONTROLLER STRING INSTABILITY Allen et al., 2002 STRING INSTABILITY: BETTER DESIGN: EE 8235: Lectures 25 & 26 6

7 EE 8235: Lectures 25 & 26 7 Control of vehicular platoons ACTIVE RESEARCH AREA FOR 40 YEARS (Levine & Athans, Melzer & Kuo, Chu, Ioannou, Varaiya, Hedrick, Swaroop,...) SPATIO-TEMPORAL SYSTEMS signals depend on time & discrete spatial variable n Estimation Error sensor networks m variance estimate arrays of micro-cantilevers arrays of micro-mirrors UAV formations satellite constellations INTERACTIONS CAUSE COMPLEX BEHAVIOR ce to the reference string instability in vehicular platoons SPECIAL STRUCTURE every unit has sensors and actuators

8 EE 8235: Lectures 25 & 26 8 Controller architectures: platoons CENTRALIZED: G 0 G G 2 K FULLY DECENTRALIZED: G 0 G G 2 best performance excessive communication not safe! K 0 K K 2 LOCALIZED: G 0 G G 2 many possible architectures K0 K K2

9 spatially invariant theory CENTRALIZED: G 0 G G 2 K performance vs. size EE 8235: Lectures 25 & 26 9 FUNDAMENTAL LIMITATIONS LOCALIZED: G 0 G G 2 is it enough to look only at nearest neighbors? K 0 K K 2

10 EE 8235: Lectures 25 & 26 0 Optimal control of vehicular platoons FINITE PLATOONS Levine & Athans, IEEE TAC 66 Melzer & Kuo, IEEE TAC 7 INFINITE PLATOONS Melzer & Kuo, Automatica 7

11 EE 8235: Lectures 25 & 26 Control objective DYNAMICS OF n-th VEHICLE: ẍ n = u n CONTROL OBJECTIVE: desired cruising velocity v d := const. inter-vehicular distance L := const. COUPLING ONLY THROUGH FEEDBACK CONTROLS ABSOLUTE DESIRED TRAJECTORY x nd (t) := v d t nl

12 EE 8235: Lectures 25 & 26 2 Optimal control of finite platoons [ ṗn absolute position error: p n (t) := x n (t) v d t + nl absolute velocity error: v n (t) := ẋ n (t) v d v n = [ [ pn v n + [ 0 u n, n {,..., M} J := ( M+ 0 n= (p n (t) p n (t)) 2 + ) M (vn(t) 2 + u 2 n(t)) dt n=

13 [ ṗn v n = J := [ ( M+ 0 [ pn v n + [ 0 (p n (t) p n (t)) 2 + n= 0 max Re (λ{a cl }): u n, n {,..., M} M (vn(t) 2 + u 2 n(t)) n= ) dt EE 8235: Lectures 25 & max Re ( λ { A cl } ) M

14 EE 8235: Lectures 25 & 26 4 Optimal control of infinite platoons [ ṗn v n A θ = = J := [ MAIN IDEA: EXPLOIT SPATIAL INVARIANCE [ n Z [ pn, Q θ = v n + [ 0 u n, n Z ( (pn (t) p n (t)) 2 + v 2 n(t) + u 2 n(t) ) dt SPATIAL Z θ-transform [ 2( cos θ) 0 0, 0 θ < 2π pair (Q θ, A θ ) not detectable at θ = 0 POSSIBLE FIX: PENALIZE ABSOLUTE POSITION ERRORS IN J

15 [ ṗn v n = J := [ n Z [ pn CLOSED-LOOP SPECTRUM: v n + [ 0 u n, n Z ( q p 2 n (t) + (p n (t) p n (t)) 2 + v 2 n(t) + u 2 n(t) ) dt q = 0: q = : EE 8235: Lectures 25 & 26 5 Im ( λ { A cl }) Im (λ { A cl }) Re (λ { A cl }) Re (λ { A cl })

16 ψ n (t) t M = 20, q = : M = 50, q = : ψ n (t) t EE 8235: Lectures 25 & M = 20, q = 0: M = 50, q = 0: 2 ψ n (t) t ψ n (t) t

17 EE 8235: Lectures 25 & 26 7 Problematic initial conditions INFINITE PLATOONS: non-zero mean initial conditions cannot be driven to zero p n (0) 0 lim n Z t n Z p n (t) 0 many modes have very slow rates of convergence

18 EE 8235: Lectures 25 & 26 8 [ ṗ v = J = [ [ [ 0 I p 0 + u 0 0 v I ( ) p T (t) Q p p(t) + q v v T (t) v(t) + r u T (t) u(t) dt 0 Q p = Q T p = V Λ V > 0, q v 0, r > 0 spectrum of large-but-finite platoon dense in the spectrum of infinite platoon Key: entries into ARE jointly unitarily diagonalizable by V Jovanović & Bamieh, IEEE TAC 05

19 EE 8235: Lectures 25 & 26 9 Consensus by distributed computation Relative information exchange with neighbors Simple distributed averaging algorithm ẋ i (t) = ( ) x i (t) x j (t) Questions j N i Will the network asymptotically equilibrate? lim t x n(t)? = x(t) := N N n = x n (t) Quantify performance (e.g., rate of convergence, response to disturbances)

20 EE 8235: Lectures 25 & Convergence to deviation from average Write dynamics as ẋ (t).. ẋ N (t) Let A be such that = A ẋ(t) = A x(t) + d(t) x (t). x N (t) + d (t). d N (t) All rows and columns sum to zero A = 0 T A = 0 T := [ T is an equilibrium point, A = 0 All other eigenvalues of A have negative real parts

21 Deviation from average scalar form: vector form: x (t).. x N (t) x n (t) = x n (t) x(t) = x(t) = x (t).. x N (t).. (I N ) T x(t) N [ x (t).. x N (t) } {{ } x(t) EE 8235: Lectures 25 & 26 2 x(t) := N (x (t) + + x N (t)) = N T x(t) x(t) = x(t) }{{} + x(t)

22 Write x(t) as x(t) = ψ (t) u + + ψ N (t) u N = [ u u N }{{} U Coordinate transformation x(t) = x(t) + x(t) = [ U [ ψ(t) x(t) = [ U N T [ ψ(t) x(t) x(t) ψ (t). } ψ N (t) {{ } ψ(t) EE 8235: Lectures 25 & {u,..., u N } orthonormal basis of

23 In new coordinates [ U [ ψ(t) x(t) [ ψ(t) x(t) = A [ U [ ψ(t) x(t) = = [ U N T A [ U [ U A U U A N T A U N T A + d(t) [ ψ(t) x(t) [ ψ(t) x(t) + + [ U N T [ U N T d(t) d(t) EE 8235: Lectures 25 & ẋ(t) = A x(t) + d(t) Use structure of A to obtain ψ(t) = U A U ψ(t) + U d(t) x(t) = 0 x(t) + N T d(t)

24 EE 8235: Lectures 25 & Spatially invariant systems over circle Circulant A-matrix Use DFT to obtain ẋ(t) = A x(t) + d(t) z(t) = (I N ) T x(t) ˆx k (t) = â k ˆx k (t) + ˆd k (t) ẑ k (t) = ( δ k ) ˆx k (t) Variance of the network (i.e., the H 2 norm from d to z) solve Lyapunov equation and sum over spatial frequencies H 2 2 = N k = (â k + â k )

25 EE 8235: Lectures 25 & An example Nearest neighbor information exchange ẋ n (t) = (x n (t) x n (t)) (x n (t) x n+ (t)) + d n (t), n Z N Use DFT to obtain Variance per node ˆx k (t) = 2 ( cos ( )) 2 π k ˆxk (t) + ˆd k (t) ẑ k (t) = ( δ k ) ˆx k (t) N N H 2 2 = N N k = 4 ( cos ( )) = 2 π k N N N k = 8 sin 2 ( π k N ) = N 2 24 N Will the scaling trends change if we { use information from more neighbors? work in 2D or 3D?

26 EE 8235: Lectures 25 & Problem setup: double-integrator vehicles ẍ n = u k + d n control disturbance Desired trajectory: Deviations: { xn := v d t + n constant velocity p n := x n x n, v n := ẋ n v d Controls: u = K p p K v v Closed loop: [ ṗ(t) v(t) = [ K p K v v(t) 0 I [ p(t) + [ 0 I d(t) K p, K v : feedback gains

27 EE 8235: Lectures 25 & Structured feedback design Example: design K p and K v to use nearest neighbor feedback e.g. use a simple rule like: u n = K + p (x n+ x n ) K p (x n x n ) K + v (v n+ v n ) K v (v n v n ) select K p and K v to guarantee global stability

28 EE 8235: Lectures 25 & 26 LOCAL FEEDBACK : GLOBAL STABILITY UMN raf t VI C, Incoherence phenomenon 28 ll vehicles subject to stochastic disturbances (N = long vs. short range deviations N = 00 VEHICLES Figure 2: Absolute positions of all vehicles.

29 high frequency disturbance quickly regulated low frequency disturbance penetrates further into formation ly rstfirst vehicle vehicle subject subject to stochastic disturbance (N (N = 0 = random disturbance acting on lead vehicle x n (t) x n (t) N = 00 VEHICLES x n (t) x n (t) EE 8235: Lectures 25 & poor macroscopic performance: not string instability! Bamieh, Jovanović, Mitra, Patterson, IEEE TAC (to appear)

30 EE 8235: Lectures 25 & Role of dimensionality M = N d vehicles arranged in d-dimensional torus Z d N STRUCTURAL FEATURES: spatial invariance locality mirror symmetry ẍ (n,...,n d ) = u (n,...,n d ) + w (n,...,n d ), n i Z n desired trajectory: x k := vt + k RELATIVE vs. ABSOLUTE MEASUREMENTS u n = K + p (x n+ x n ) K p (x n x n ) K + v (v n+ v n ) K v (v n v n ) K 0 p (x n (v d t + n )) K 0 v (v n v d )

31 EE 8235: Lectures 25 & 26 3 Performance measures Microscopic: local position deviation (x n+ x n ) Macroscopic: deviation from average or long range deviation How does variance per vehicle scale with system size?

32 MICROSCOPIC ERROR: bounded for any dimension d ASYMPTOTIC SCALING OF MACROSCOPIC ERROR: d = d = 2 d 3 M log M bounded! EE 8235: Lectures 25 & relative position & absolute velocity feedback: Same scaling obtained in standard consensus problem

33 ASYMPTOTIC SCALING OF MICROSCOPIC ERROR: d = d = 2 d = 3 M log M bounded ASYMPTOTIC SCALING OF MACROSCOPIC ERROR: d = M 3 d = 2 M d = 3 M /3 EE 8235: Lectures 25 & relative position & relative velocity feedback: Only local feedback: large tight formations in D not possible!

34 esistive EE 8235: LecturesNetwork 25 & 26 Analogy 34 Resistive network analogy D : Net resistance = R M 2D : 3D : Net resistance = O (log(m)) Net resistance is bounded! nd, Feb 0 slide 3/39