Distributed Computation of Minimum Time Consensus for Multi-Agent Systems

Transcription

1 Distributed Computation of Minimum Time Consensus for Multi-Agent Systems Ameer Mulla, Deepak Patil, Debraj Chakraborty IIT Bombay, India Conference on Decision and Control 2014 Los Angeles, California, USA December 9, / 21

2 Minimum time consensus There are N double integrator agents with bounded inputs. Goal: Achieve consensus in minimum time. Strategy: Computation (in advance) of the consensus state. Key result: Computation can be distributed (By Helly s Theorem). Tool: Attainability set. 2 / 21

3 Problem Definition Outline 1 Minimum time consensus Problem Definition 2 Single integrators 3 / 21

4 Problem Definition Problem Statement Assume N identical agents with double integrator dynamics [ ] [ ] ẋ i (t) = x 0 0 i (t) + u 1 i (t) }{{}}{{} A b i = 1,...,N [ ri (t) with x i (t) = v i (t) Problem ] [ ri0, x i (0) = x i0 = v i0 Find x and min t such that, for all i, j x i (t) x j (t) 0 as t t x i ( t) = x j ( t) = x and x i (t) = x j (t) for all t t ] and u i (t) 1. 4 / 21

5 Problem Definition Definitions Attainable Set from point p at time t The set of all points that an agent can reach from initial condition p at time t using admissible input i.e. u i (t) 1 { t } A p (t) = x : x = e At p + e A(t τ) bu(τ)dτ, u(t) : u(t) 1 0 [ 1 t For double integrator e At = 0 1 Boundary of attainability set ts t x(t) = e At p ± 0 e A(t τ) Bdτ e A(t τ) Bdτ ; 0 t s t < t s Quadratic in t and t s ] 5 / 21

6 Problem Definition Definitions Figure: A p (t) 6 / 21

7 Outline 1 Minimum time consensus Problem Definition 2 Single integrators 7 / 21

8 Way to Consensus Possible if attainable sets of all agents at time t intersect Let t be minimum t such that A i(t) φ (A i (t):=a xi0 (t)) 1 i N Requires solution of coupled polynomial equations and inequalities Computation cannot be distributed 8 / 21

9 Main tool:helly s Theorem Helly s Theorem F a finite family of convex sets in R n Every n + 1 members of F have a point in common all the members of F have a point in common A i (t) R 2 is convex for all i and t Attainability set of every triplet intersect attainability set of all intersect Allows to distribute computation for all ( N 3) triplets evenly among N agents 9 / 21

10 Way to Consensus Lemma Let t ijk : Minimum time to consensus for agents {a i, a j, a k } Then, t max t ijk 1 i,j,k N If A i (t ) A j (t ) A k (t ) φ for some t > 0, then A i (t) A j (t) A k (t) φ for all t t Thus, t = max t ijk 1 i,j,k N Computing closed form expressions for three agents case suffices 10 / 21

11 Consensus: Three agents Theorem Let {a p, a q, a r } be the triple of agents and t pqr = The minimum time to consensus t = t pqr The corresponding consensus point x = x pqr Theorem For two different {a p, a q, a r } and {a p, a q, a r } If t pqr = t p q r = max t ijk then x pqr = x p q r 1 i,j,k N max 1 i,j,k N t ijk. 11 / 21

12 Consensus: Three agents Two ways towards consensus of three agents For a triple {a i, a j, a k }, t ijk max{ t ij, t jk, t ik } WLOG let t ij = max{ t ij, t jk, t ik }. Two cases: Case 1: t ijk = t ij = max{ t ij, t jk, t ik } i.e x ij A k ( t ij ) Case 2: t ijk > max{ t ij, t jk, t ik } i.e. x ij / A k ( t ij ) 12 / 21

13 Consensus: Three agents Case 1: t ijk = t ij = max{ t ij, t jk, t ik } i.e x ij A k ( t ij ) Figure: Case 1 13 / 21

14 Consensus: Three agents Case 2: t ijk > max{ t ij, t jk, t ik } i.e. x ij / A k ( t ij ) Figure: Case 2: x ij / A k ( t ij ) 14 / 21

15 Consensus: Three agents Case 2: t ijk > max{ t ij, t jk, t ik } i.e. x ij / A k ( t ij ) Figure: Case 2: Consensus of three agents 15 / 21

16 Role of Agents All N agents do not play a role in computation of t and x Triplet{a i, a j, a k } for which t = t ijk determine x and t For these agents, x A i ( t) Must use minimum-time control law to reach x at t In other words, x i0, x j0, x k0 R x ( t) For all other agents (say a q ), x inta q ( t), Must use a control law that allow them to reach x at t May use scaled down version of the input bounds 16 / 21

17 Six Agents Consensus Figure: Attainable Sets of Agents at t 17 / 21

18 Six Agents Consensus Definition The set of all initial conditions from which an agent can reach p at time t using admissible input i.e. u i (t) 1 Figure: Reachable Set to x at t 18 / 21

19 Single integrators Minimum time consensus: Single integrator N agents with single integrator dynamics ẋ i = u i, x i R and u i 1 Attainable set of agents are intervals in R given by x i (t) = x(0) + t 0 u i(t)dt Initial condition p then at time t attainable set is [p t,p + t] If every pair of the intervals intersect, then all the intervals intersect The minimum time is t ab = max ij t ij = max ij x i0 x j0 2 for some agents a and b (endpoints) That is t = max ij x i0 x j0 2 and x = max ij (x i0 +x j0 ) 2. For all other agents, any input that drives x i (t) to x in time t t ab For example, u i (t) = sign( x x i ) 19 / 21

20 Single integrators More can be said Single integrators interacting over a connected undirected graph G = (V,E) Prior computation of the t and corresponding x is not required Control law u i (t) = sign( (i,j) E (x j x i )) ensures proper input Agreement dynamics: ṗ(t) = sign(lp(t)), where L is Laplacian matrix Consensus occurs at x = max ij (x i0 +x j0 ) 2 20 / 21

21 Single integrators Conclusions An algorithm for computing minimum time consensus Computation is to be done for each possible triplet. Can be distributed evenly among each agent Final values of x and t are broad-casted to all Agreement becomes easier for single integrator agents 21 / 21