Distributed estimation based on multi-hop subspace decomposition

Transcription

1 Distributed estimation based on multi-hop subspace decomposition Luca Zaccarian 2,3 Joint work with A.R. del Nozal 1, P. Millán 1, L. Orihuela 1, A. Seuret 2 1 Departamento de Ingeniería, Universidad Loyola Andalucía, Sevilla, Spain 2 LAAS-CNRS, Université de Toulouse, CNRS, Toulouse, France 3 Department of Industrial Engineering, University of Trento, Italy IEEE RTSI 2018 Conference Palermo (Italy), September 10, 2018 Research partially supported by AEI/FEDER (TEC P) 1/19

2 Overview 1 Motivation and Preliminaries 2 Multi-hop Decomposition 3 Observer design 4 Distributed setup and operation 5 Simulation results 6 Conclusions and future work 2/19

3 A large plant with scattered sensors Motivation Increasing number of embedded sensors dispersedly integrated at complex plants is fostering the application of distributed estimation and control schemes. Distributed strategies oer interesting advantages such as scalability, exibility, fault tolerance and robustness. 3/19

4 Abstraction involves multi-agent structure Plant denition G = (V, E), x + = Ax, y i = C i x i V, ( ) Goals Reconstruct the whole state at every node. Design the observer in a distributed way. Fix an arbitrary convergence rate for the estimation error. Exploit linearity to provide intuitive structure. Reduce the exchange of information. 4/19

5 Multi-hop output matrix generalization The ρ-hop output matrix of agent i The ρ-hop output matrix of agent i, C i,ρ, is composed by its output matrix C i and the output matrices of all agents j with a direct path to i involving ρ or less edges. That is: [ ] C C i,ρ i,ρ 1, ρ > 0, col(c j,ρ 1 ) j Ni where C i,0 := C i. 5/19

6 Multi-hop Unobservable Subspaces The ρ-hop unobservable subspace from agent i, Ō i,ρ The ρ-hop unobservable subspace from agent i, Ō i,ρ := Im( V i,ρ ), is the unobservable subspace related to pair (C i,ρ, A). The ρ-hop observable subspace from agent i, O i,ρ The ρ-hop observable subspace from agent i, O i,ρ := Im(V i,ρ ) is the orthogonal complement of Ō i,ρ. Property: [ Vi,ρ V i,ρ ] is nonsingular, O i,ρ 1 O i,ρ, i V, ρ {1,..., l i } Innovation Matrix W i,ρ Innovation matrix W i,ρ that generates O i,ρ (O i,ρ 1 ) satises W i,ρ W i,ρ = 0, ρ, ρ {1,..., l i } such that ρ ρ. Im(W j,ρ 1 ) Im(V i,ρ ), j N i. 6/19

7 Multi-hop Transformation Matrix l i is selected later as a suitable positive integer Multi-hop transformation matrix T i It is orthogonal Ti = Ti 1 by construction: T i := [ ] V i,li W i,li W i,ρ+1 W i,ρ W i,0 }{{}}{{} V i,ρ V i,ρ Property: [ Vi,ρ V i,ρ ] is nonsingular, O i,ρ 1 O i,ρ, i V, ρ {1,..., l i } Innovation Matrix W i,ρ Innovation matrix W i,ρ that generates O i,ρ (O i,ρ 1 ) satises W i,ρ W i,ρ = 0, ρ, ρ {1,..., l i } such that ρ ρ. Im(W j,ρ 1 ) Im(V i,ρ ), j N i. 7/19

8 Multi-hop observable decomposition enjoys nice structure T i := [ ] V i,li W i,li W i,ρ+1 W i,ρ W i,0 }{{}}{{} V i,ρ V i,ρ Proposition Orthogonal transformation T i provides Multi-hop Observable Decomposition: V i,li AW i,li... V i,li AW i,1 V i,li AW i,0 0 Wi,l i AW i,li... Wi,l i AW i,1 Wi,l i AW i,0 V i,l i A V i,li Ti AT i = Wi,1 AW i,1 Wi,1 AW i, Wi,0 AW i,0 C i,0 T i = [ ] C i,li T i = [ ] 0 8/19

9 Example of multi-hop observable decomposition A = , C 1 = [ ] C 2 = [ ] C 3 = [ ] W i,0 = 0 0, Wi,1 = , Wi,2 = T 1 AT 1 = C 1,0T 1 = C 1T 1 = [ ] C 1,1T 1 = [ ] C 1,2T 1 = /19

10 Proposed observer structure ˆx + i (A) (B) (C) l i = Aˆx }{{} i + W i,0 L i (y i ŷ i ) + W i,ρ N i,j,ρ W }{{} j,ρ 1(ˆx j ˆx i ) ( ) ρ=1 j N (A) i (B) }{{} (C) Model-based open-loop term. Luenberger-term for the observable modes: The dierence between the locally measured y i and predicted ŷ i = C iˆx i outputs is multiplied by gain L i whose action is limited to the (local) subspace W i,0. Recall that y i ŷ i = C i x C iˆx i = C i ε i Consensus-term for the unobservable modes: The estimation dierence with the neighbors ˆx i ˆx j is multiplied by the gain N i,j,ρ, whose action is limited to the (non-local) subspaces W i,ρ. Recall that ˆx j ˆx i = ˆx j x + x ˆx i = ε i ε j 10/19

11 Design goal and basic (necessary) assumption Problem α Given α (0, 1), plant ( ), and the interconnection graph G = (V, E), design the gains L i and N i,j,ρ in ( ) such that all estimates ˆx i converge to x exponentially fast with exponential rate α. Collective α-detectability Plant ( ) is collectively α-detectable if for each agent i V, there exist a nite number of hops l i Z > 0 such that pair (C i,li, A) is α-detectable. Collective α-detectability is necessary Collective α-detectability is a necessary and sucient condition for solving Problem α. 11/19

12 Assumption α Given α (0, 1), plant ( ) is collectively α-detectable. Figure 1: Assume that pair ( C, A) with C = [C 1, C 2, C 3 ] is α-detectable. Although strong connectivity does not hold, Assumption is met. Collective α-detectability is necessary Collective α-detectability is a necessary and sucient condition for solving Problem α. 12/19

13 Main Result: design guidelines Theorem α (Design of the distributed observer) Consider plant ( ) and observer structure ( ). If matrices Wi,0AW i,0 L i C i W i,0, Wi,ρAW i,ρ N i,j,ρ Wj,ρ 1W i,ρ, }{{} j N i =D i,(0,0) }{{} =D i,(ρ,ρ) for all ρ {1,..., l i }, have spectral radius smaller than α, then Problem α is solved: ˆx 1 (0) x(0) ˆx i (k) x(k) Mα k., i V. ˆx N (0) x(0) Coarse bound from the Theorem Interesting additional bounds emerge from multi-hop decomposition (next slide). 13/19

14 Sketch of the proof Transformed estimation error dynamics of agent i at hop ρ (ε i,ρ = W i,ρ (x ˆx i)): ε + i,0 = [W i,0aw i,0 L i C i W i,0 ]ε i,0, ε 0 := col(ε i,0 ) ρ ε + i,ρ = D i,(ρ,r) ε i,r + N i,j,ρ ε j,ρ 1, r=0 j N i ε ρ := col(ε i,ρ ) i V:li +1 ρ Stacking the estimation error at each hop for every agent ε l. ε 1 ε 0 + l = ε l. ε 1 ε 0 with ρ = D 1,(ρ,ρ) D p,(ρ,ρ) T i := [ ] V i,li W i,li W i,ρ+1 W i,ρ W i,0 }{{}}{{} V i,ρ V i,ρ 14/19

15 Design feasibility It is always possible, under Assumption α, to nd a set of matrices L i and N i,j,ρ that satisfy the conditions of Theorem α Sketch of the proof Existence of L i from observability of pair (C i W i,0, W i,0 AW i,0 ). For N i,j,ρ rewrite expression as D i,(ρ,ρ) = W i,ρaw i,ρ j N i N i,j,ρ W j,ρ 1W i,ρ, W i,ρaw i,ρ N i,ρ Λ i,ρ with N i,ρ = col(n i,j,ρ ) j N i, Λ i,ρ = col(w j,ρ 1 ) j N i W i,ρ, The the pair (Λ i,ρ, W i,ρ AW i,ρ) is observable from the Popov- Belevitch-Hautus test. 15/19

16 Distributed observer setup For every agent/node i do: a. Compute O i,0 and construct matrix W i,0. Set ρ = 0. b. Perform the two steps: Exchange W i,ρ with the neighbors. Construct O i,ρ+1 and construct matrix W i,ρ+1. c. If the ρ-hop unobservable modes have speed α, then stop. Otherwise increment ρ and go to (b). Gain selection phase Each agent designs gains (L i, N i,j,ρ ). Running phase Each agent will exchange with its neighbors a portion of the state dened by W j,ρ 1ˆx i, for all ρ = 1,..., l i. 16/19

17 Simulation example with 7 nodes + x x 1 x 2 x 3 = x x 3 x x 4 y 1 = x 1, y 2 = x 2, y 3 = x 4, y 4 = x 1, y 5 = x 3, y 6 = x 1, y 7 = x 4 Estimates of Agent 2 Estimates of Agent 5 17/19

18 Results directly extend to continuous-time Continuous-time Simulation Example ẋ = x, x(1) x(2) x(3) x(4) ˆx1(1) ˆx1(2) ˆx1(3) ˆx1(4) Estimates for Agent Convergence Speed α 18/19

19 Conclusions Distributed observer with distributed design Arbitrary convergence rate α of the estimation error Intuitive method based on observable decomposition Future work Continuous- and Discrete-time results Necessary and sucient conditions Optimal design of gains L i and N i,j,ρ (LMI-based) Time-varying graphs, delays or packet losses Designs reducing online information exchange References P. Millan Gata, A. Rodriguez del Nozal, L. Zaccarian, L. Orihuela Espina, and A. Seuret. Distributed implementation and design for state estimation. In Proc. of the IFAC World Congress, pages , Toulouse, France, July A. Rodriguez del Nozal, P. Millan Gata, L. Orihuela Espina, and A. Seuret, L. Zaccarian. Distributed Estimation based on multi-hop subspace decomposition. Submitted to Automatica. 19/19