Sparsity-Promoting Optimal Control of Distributed Systems

Transcription

1 Sparsity-Promoting Optimal Control of Distributed Systems Mihailo Jovanović mihailo joint work with: Makan Fardad Fu Lin LIDS Seminar, MIT; Nov 6, 2012

2 wind farms raft APPLICATIONS: micro-cantilevers aircraft formations satellite constellations Large dynamic networks 1 OF INCREASING IMPORTANCE IN MODERN TECHNOLOGY INTERACTIONS CAUSE COMPLEX BEHAVIOR cannot be predicted by analyzing isolated subsystems SPECIAL STRUCTURE every subsystem has sensors and actuators

3 memoryless structured controller Structured distributed control Blue layer: distributed plant and its interaction links 2 KEY CHALLENGE: identification of a signal exchange network performance vs. sparsity

4 MEMORYLESS CONTROLLER: u(t) = [ F p F v ] [ p(t) v(t) Objective: design [ F p F v ] to minimize steady-state variance of p, v, u ] Example: Mass-spring system 3 u 1 (t) u 2 (t) u 3 (t) u 4 (t) OPTIMAL CONTROLLER = LINEAR QUADRATIC REGULATOR } {{ } F p p 1 (t) p 2 (t) p 3 (t) p 4 (t) } {{ } F v v 1 (t) v 2 (t) v 3 (t) v 4 (t)

5 position feedback matrix: position gains for middle mass: Structure of optimal controller 4 OBSERVATIONS Diagonals almost constant (modulo edges) Off-diagonal decay of a centralized gain Bamieh, Paganini, Dahleh, IEEE TAC 02 Motee & Jadbabaie, IEEE TAC 08

6 position feedback matrix: position gains for middle mass: Structure of optimal controller 5 OBSERVATIONS Diagonals almost constant (modulo edges) Off-diagonal decay of a centralized gain Bamieh, Paganini, Dahleh, IEEE TAC 02 Motee & Jadbabaie, IEEE TAC 08

7 Enforcing localization? One approach: truncating centralized controller 6 POSSIBLE DANGERS Performance degradation Instability

8 ❶ SPARSITY-PROMOTING OPTIMAL CONTROL Design of sparse and block sparse feedback gains Tools from control theory, optimization, and compressive sensing ❷ ALGORITHM Alternating direction method of multipliers Outline 7 ❸ EXAMPLES ❹ SUMMARY AND OUTLOOK

9 SPARSITY-PROMOTING OPTIMAL CONTROL 8

10 ẋ = A x + B 1 d + B 2 u [ ] [ Q 1/2 0 z = x + 0 u = F x R 1/2 CLOSED-LOOP H 2 NORM ( J(F ) = trace e (A B 2F ) T t ( Q + F T RF ) e (A B 2F )t dt B 1 B1 T 0 ] u ) State-feedback H 2 problem 9 no structural constraints globally optimal controller: A T P + P A P B 2 R 1 B T 2 P + Q = 0 F c = R 1 B T 2 P

11 minimize J(F ) + γ card (F ) variance amplification sparsity-promoting penalty function Sparsity-promoting H 2 problem 10 card (F ) number of non-zero elements of F F = card (F ) = 8 γ > 0 performance vs. sparsity tradeoff Lin, Fardad, Jovanović, IEEE TAC 12 (submitted; arxiv: v2)

12 l 1 norm: weighted l 1 norm: F ij i, j W ij F ij, W ij 0 i, j separable: sum of element-wise functions CARDINALITY VS. WEIGHTED l 1 NORM {W ij = 1/ F ij, F ij 0} card (F ) = i, j W ij F ij Convex relaxations of card (F ) 11 RE-WEIGHTED SCHEME Use feedback gains from previous iteration to form weights W + ij = 1 F ij + ε Candès, Wakin, Boyd, J. Fourier Anal. Appl. 08

13 sum-of-logs: i,j ( log 1 + F ) ij, 0 < ε 1 ε A non-convex relaxation of card (F ) 12 Candès, Wakin, Boyd, J. Fourier Anal. Appl. 08

14 original problem: minimize card (F ) subject to J(F ) σ relaxation: minimize g(f ) subject to J(F ) σ l 1 weighted l 1 sum-of-logs Sparsity-promoting penalty functions 13

15 A CLASS OF CONVEX PROBLEMS 14

16 Consensus by distributed computation 15 Definition 1: (balan called balanced if P j6¼i In a balanced dig entering a node and lea nodes. The most impo is that w ¼ 1 is also a (or 1 T L ¼ 0). Theorem 1: Conside ogy G applying the foll _x i ðtþ ¼ X j2n i a i RELATIVE INFORMATION EXCHANGE WITH NEIGHBORS Fig. 2. Examples of networks with n ¼ 20 nodes: (a) a regular network with 80 links and (b) a random network with 45 links. simple distributed averaging algorithm ẋ i (t) = ( ) x i (t) x j (t) is strongly connected (SC) j N i if there is a directed path connecting any two arbitrary nodes s; t of the graph. 6 connected network convergence to the average value Lemma 2: (spectral localization) Let G be a strongly connected digraph on n nodes. Then rankðlþ ¼n 1and all nontrivial eigenvalues of L have positive real parts. Suppose G is a strongl Laplacian of G with a satisfying T L ¼ 0. Th i) a consensus initial states; ii) the algorithm s the linear fun group decision iii) if the digraph asymptotically Proof: The conve

17 Consensus with stochastic disturbances ẋ i (t) = Average mode: x(t) = 1 N j N i N i = 1 ( ) x i (t) x j (t) + d i (t) x i (t) : undergoes random walk 16 If other modes are stable, x i (t) fluctuates around x(t) deviation from average: steady-state variance: x i (t) = x i (t) x(t) lim E ( x T (t) x(t) ) t

18 minimize trace ( Q 1/2 (F + 11 T /N) 1 Q 1/2 + R F ) + γ W F l1 subject to F 1 = 0 F + 11 T /N 0 can be formulated as SDP: minimize trace (X + R F ) + γ 1 T Y 1 subject to [ X Q 1/2 Q 1/2 F + 11 T /N ] 0 Design of undirected networks of single integrators 17 Convex optimization problem Y W F Y F 1 = 0

19 UNDIRECTED NETWORK WITH STOCHASTIC DISTURBANCES local performance graph: identified communication graph: Sparsity-promoting consensus algorithm 18 z = Q 1/2 loc x ( ) I 1 N 11T x u card (F ) /card (F c ) = 7% (J J c ) /J c = 14%

20 F (γ) := arg min (J(F ) + γ g(f )) F Parameterized family of feedback gains 19

21 minimize J(F ) + γ g(f ) Step 1: introduce additional variable/constraint minimize J(F ) + γ g(g) subject to F G = 0 benefit: decouples J and g Alternating direction method of multipliers 20 Step 2: introduce augmented Lagrangian L ρ (F, G, Λ) = J(F ) + γ g(g) + trace ( Λ T (F G) ) + ρ 2 F G 2 F

22 L ρ (F, G, Λ) = J(F ) + γ g(g) + trace ( Λ T (F G) ) + ρ 2 F G 2 F ADMM: F k+1 := arg min F G k+1 := arg min G L ρ (F, G k, Λ k ) L ρ (F k+1, G, Λ k ) Λ k+1 := Λ k + ρ (F k+1 G k+1 ) Step 3: use ADMM for augmented Lagrangian minimization 21 MANY MODERN APPLICATIONS distributed computing distributed signal processing image denoising machine learning Eckstein & Bertsekas, 92; Boyd et al., 11

23 ADMM raft { identifies sparsity patterns provides good initial condition for structured design NECESSARY CONDITIONS FOR OPTIMALITY OF THE STRUCTURED PROBLEM (A B 2 F ) T P + P (A B 2 F ) = ( Q + F T R F ) (A B 2 F ) L + L (A B 2 F ) T = B 1 B T 1 [( R F B T 2 P ) L ] I S = 0 22 Step 4: Polishing back to structured optimal design Newton s method + conjugate gradient F = I S - structural identity I S =

24 EXAMPLES 23 mihailo/software/lqrsp/

25 diag (F v ): Mass-spring system 24 γ = 10 4 γ = 0.03 γ = 0.1

26 (J J c ) /J c : 25 Performance comparison: sparse vs. centralized γ card (F ) /card (F c ) (J J c ) /J c 10% 0.75% 6% 2.4% 2% 7.8%

27 Each vehicle: relative information exchange u i = j i F ij (x i x j ), i {2,..., N 1} Formation of vehicles 26 Leaders: equipped with GPS devices u 1 = j 1 F 1j (x 1 x j ) F 11 x 1 u N = j N F Nj (x N x j ) F NN x N

28 IDENTIFIED COMMUNICATION ARCHITECTURES: γ = 0 γ = 0.01 γ = all-to-all nearest neighbors + leaders-to-all nearest neighbors + leaders-to-some

29 Network with 100 nodes 28 [ ṗi v i ] = [ ] [ ] 1 1 pi 1 2 v }{{ i } unstable dynamics + [ ] e α(i,j) pj v j j i }{{} coupling + [ 0 1 ] (d i + u i ) α(i, j): Euclidean distance between nodes i and j Motee & Jadbabaie, IEEE TAC 08

30 (J J c ) /J c : (J J c ) /J c : 29 Performance comparison: sparse vs. centralized γ card (F ) /card (F c )

31 30 identified communication graph: γ = 5 card (F ) /card (F c ) = 8.8% (J J c ) /J c = 24.6%

32 31 identified communication graph: γ = 11 card (F ) /card (F c ) = 5.1% (J J c ) /J c = 40.9%

33 32 identified communication graph: γ = 18 card (F ) /card (F c ) = 3.4% (J J c ) /J c = 48.7%

34 33 identified communication graph: γ = 30 card (F ) /card (F c ) = 2.4% (J J c ) /J c = 54.8%

35 communication graph of a truncated centralized gain: 34 card (F ) = 7380 (36.9%) non-stabilizing

36 card b (F ) number of non-zero blocks of F card b (F ) = i,j card ( F ij F ) Extension: Block sparsity 35 PENALTY FUNCTIONS THAT PROMOTE BLOCK SPARSITY generalized l 1, weighted l 1, sum-of-logs

37 M. JOVANOVIĆ, U OF M raft Cyclic biochemical An example from networks biochemical with inhibitory reactionsfeedback Cyclic Biochemical Networks with Inhibitory Feedback CELLULAR SIGNALING CYCLIC INTERCONNECTION STRUCTURE Kholodenko Cyclic 00; Shvartsman Biochemical et al. Networks 01 with Inhibitory Feedback Cellular Signaling: Kholodenko (2000), Shvartsman et al. (2001), and others Cellular Signaling: Kinase Kholodenko (2000), Shvartsman et al. (2001), and others M3 M3 Kinase Phosphatase M3 P Phosphatase M3 P M2 M2 Phosphatase Phosphatase M2 P M2 Gene Regulation: Jacob & Monod ( 61), Goodwin ( 65), Elowitz Phosphatase GENE REGULATION & Leibler (2000) Jacob & Monod 61; Goodwin ẋ Gene Regulation: Jacob i = [A] & Monod 65; ii x i Elowitz + [B ( 61), 1 ] ii Goodwin & d i Leibler + [B ( 65), 2 ] ii 00 u i Elowitz & Leibler (2000) P M1 M1 Phosphatase M1 P M1 P : DNA : mrna : enzyme : endproduct [A] ii = 3 1 0, [B 1 ] ii = , [B 2 ] ii = 1 0 Metabolic Pathways: 0: DNA 3 Morales 1 : mrna and McKay : ( 67), enzyme 0 Stephanopoulos 0 1 : endproductet al. ( 98) 0 Metabolic Pathways: Morales and McKay ( 67), Stephanopoulos et al. ( 98) METABOLIC PATHWAYS Morales & McKay 67; Stephanopoulos et al. 98

38 37 NONSYMMETRIC BIDIRECTIONAL COUPLING ẋ i = [A] ii x i N (i j) (x i x j ) + [B 1 ] ii d i + [B 2 ] ii u i j = 1

39 block sparse design: sparse design: Block sparse vs. sparse 38 small difference in performance: J bs J s J s < 1%

40 ALGORITHM: DETAILS 39

41 weighted l 1 : sum-of-logs: raft minimize G minimize G ij minimize G ij γ g(g) + ρ 2 G V 2 F V := F k+1 + (1/ρ)Λ k (γ W ij G ij + ρ 2 (G ij V ij ) 2) i, j ( ( γ log 1 + G ) ij ε i, j + ρ 2 (G ij V ij ) 2 ) Separability of G-minimization problem 40 cardinality: minimize G ij (γ card (G ij ) + ρ 2 (G ij V ij ) 2) i, j separability element-wise analytical solution

42 weighted l 1 : shrinkage a = (γ/ρ)w ij cardinality: truncation b = 2γ/ρ Solution to G-minimization problem 41 sum-of-logs (with ρ = 100, ε = 0.1): γ = 0.1 γ = 1 γ = 10

43 minimize F U := G k J(F ) + ρ 2 F U 2 F (1/ρ)Λ k NECESSARY CONDITIONS FOR OPTIMALITY: (A B 2 F )L + L(A B 2 F ) T = B 1 B T 1 (A B 2 F ) T P + P (A B 2 F ) = (Q + F T RF ) F L + ρ(2r) 1 F = R 1 B T 2 P L + ρ(2r) 1 U Solution to F -minimization problem 42 ITERATIVE SCHEME Given F 0 solve for {L 1, P 1 } F 1 {L 2, P 2 } F 2 descent direction + line search convergence

44 SPARSITY-PROMOTING OPTIMAL CONTROL Performance vs. sparsity tradeoff RELATED WORK Lin, Fardad, Jovanović, IEEE TAC 12 (submitted; arxiv: v2) Leader selection in large dynamic networks Lin, Fardad, Jovanović, IEEE TAC 12 (submitted) Summary and outlook 43 Optimal synchronization of sparse oscillator networks Optimal dissemination of information in social networks Fardad, Lin, Jovanović, ACC 12 Fardad, Zhang, Lin, Jovanović, CDC 12 (to appear) Wide-area control of power systems Dörfler, Jovanović, Chertkov, Bullo, ACC 13 (submitted)

45 SOFTWARE raft mihailo/software/lqrsp/ >> solpath = lqrsp(a,b1,b2,q,r,options); ONGOING RESEARCH Design of sparse optimal estimators Observer-based sparse optimal design Identification of sparse dynamics 44 Finite horizon problems WISH LIST Performance bounds on structured feedback design Distributed implementation of ADMM Identification of convex problems

46 TEAM: SUPPORT: raft Makan Fardad Fu Lin (Syracuse University) (U of M) Acknowledgments 45 NSF CAREER Award CMMI NSF Award CMMI DISCUSSIONS: Stephen Boyd Roland Glowinski