Optimal Linear Iterations for Distributed Agreement

Transcription

1 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 WeA02.5 Optimal Linear Iterations for Distributed Agreement Qing Hui and Haopeng Zhang Abstract A new optimal distributed linear averaging (ODLA) problem is presented in this paper. This problem is motivated from the distributed averaging problem which arises in the context of distributed algorithms in computer science and coordination of groups of autonomous agents in engineering. The aim of the ODLA problem is to compute the average of the initial values at nodes of a graph through an optimal distributed algorithm in which the nodes in the graph can only communicate with their neighbors. Optimality is given by a minimization problem of a quadratic cost functional under infinite horizon. We show that this problem has a very close relationship with the notion of semistability. By developing new necessary and sufficient conditions for semistability of linear discrete-time systems, we convert the original ODLA problem into two equivalent optimization problems. These two problems are convex optimization problems and can be solved by using some numerical techniques. I. INTRODUCTION We consider a network characterized by a connected graph G (V, E) consisting of the set of nodes V {1,...,q} and the set of edges E V V, where each edge {i, j} E is an unordered pair of distinct nodes. The set of neighbors of node i is denoted by N i {j V : {i, j} E}. Finally, we denote the value of the node i {1,...,q} at time t by x i (t) R. Each node i holds an initial value on the network x i (0) R. The network gives the allowed communication between two nodes if and only of they are neighbors. We are interested in computing the average of the initial values, (1/q) q i1 x i(0), via an optimal distributed algorithm in which the nodes only communicate with their neighbors. The distributed averaging problem arises in the context of coordination of networks of autonomous agents, and in particular, the consensus or agreement problem among the agents. Distributed consensus problems have been studied extensively in the computer science literature 1 4. Recently it has found a wide range of applications, in areas such as formation control of underwater autonomous vehicles 5, coordination of mobile robots 6, 7, and sensor networks 8, 9. Even though many consensus protocol algorithms have been developed over the last several years in the literature (see and the numerous references therein), optimality properties of these algorithms have been largely ignored. Optimality here refers to minimization of certain cost functional for coordination algorithms subject to stability and net- This work was supported in part by a research grant from Texas Tech University. Q. Hui and H. Zhang are with the Department of Mechanical Engineering, Texas Tech University, Lubbock, TX , USA (qing.hui@ttu.edu; haopeng.zhang@ttu.edu). work connectivity constraints. In this paper, we address the equivalent formulation of optimal semistable linear iteration algorithms with a quadratic infinite horizon cost functional, which is named for the optimal distributed linear averaging (ODLA) problem. To this end, we develop necessary and sufficient conditions for semistability of linear discrete-time systems characterized by Lyapunov equations and the notion of weak semiobservability. Built on these relevant results, we present two equivalent optimization problems regarding the ODLA problem and suggest possible ways to solve them numerically. The organization of this paper is as follows. Section II gives the formulation of the ODLA problem motivated from the distributed averaging problem. Section III explores necessary conditions for optimality and semistability of the ODLA problem. Based on these results, necessary and sufficient conditions for semistability of the ODLA problem are developed by introducing the notion of weak semiobservability in Section IV. These necessary and sufficient conditions turn out to be the bridge establishing the equivalent ODLA problems. The new equivalent formulation is shown in Section V in which two equivalent ODLA problems are proposed. Finally, some remarks on numerical approaches to solve the equivalent problems are presented in Section VI. II. PROBLEM FORMULATION The notion we use in this paper is fairly standard. Specifically, R (resp., C) denotes the set of real (resp., complex) numbers, Z + denotes the set of nonnegative integers, R n (resp., C n ) denotes the set of n 1 real (resp., complex) column vectors, R n m (resp., C n m ) denotes the set of n m real (resp., complex) matrices, ( ) T denotes transpose, ( ) denotes complex conjugate transpose, ( ) # denotes the group generalized inverse, and I n or I denotes the n n identity matrix. Furthermore, we write for the Euclidean vector norm, R(A) and N(A) for the range space and the null space of a matrix A, spec(a) for the spectrum of the square matrix A, tr( ) for the trace operator, E for the expectation operator, and A 0 (resp., A > 0) to denote the fact that the Hermitian matrix A is nonnegative (resp., positive) definite. Finally, we write B ε (x), x R n, ε > 0, for the open ball with radius ε and center x, for the Kronecker product, and vec( ) for the column stacking operator. In this paper, we consider distributed linear iterations given by the form x i (t + 1) W (i,i) x i (t) + j N i W (i,j) x j (t), i 1,...,q, t Z +, (1) /10/$ AACC 81

2 where W (i,j) denotes the weight on x j at node i. Letting W (i,j) 0 for j N i, this iteration can be rewritten as a compact form x(t + 1) Wx(t), t Z +, (2) where x(t) x 1 (t),..., x q (t) T R q. The constraint on the matrix W can be expressed as W W, where W {W R q q : W (i,j) 0 if {i, j} E and i j}. (3) Recall from 20 that a matrix A R q q is (discretetime) semistable if spec(a) {s C : s < 1} {1} and, if 1 spec(a), then 1 is semisimple. Lemma 2.1: Consider the linear iteration (2). Then lim t x(t) exists if and only if W is semistable. If W is semistable, then lim t W t I q (W I q )(W I q ) #. The linear iteration (2) implies that x(t) W t x(0) for t Z +. We want to choose the weight matrix W so that for any initial value x(0), lim t x(t) α1 and the cost functional J(W, x(0)) (x(t) α1) T Q(x(t) α1) +(Wx(t) αw 1) T R(Wx(t) αw 1) Q 1/2 (x(t) α1) 2 + R 1/2 (Wx(t) αw 1) 2 (4) is minimized, where α R, 1 1,...,1 T R q, Q Q T 0, and R R T > 0. The cost functional (4) is motivated by the quadratic cost functional in the LQR control theory whereas the control input u(t) can be viewed as Wx(t). Lemma 2.2: For any x(0) R q, lim t x(t) α1 if and only if 1 T W 1 T, (5) W 1 1, (6) W is semistable. (7) Lemma 2.3: If for any x(0) R q, lim t x(t) α1, then α 1 q 1T x(0) 1 q x i (0). (8) q i1 Hence, the optimal distributed linear averaging (ODLA) problem can be formulated as minimize J(W, x(0)) subject to W W, 1 T W 1 T, W 1 1, W is semistable, x(t + 1) Wx(t). The ODLA problem given above considers optimization on infinite horizon. This problem can also be formulated under finite horizon as follows. minimize J N (W, x(0)) N (x(t) α1) T Q(x(t) α1) + (Wx(t) αw 1) T R(Wx(t) αw 1) subject to W W, 1 T W 1 T, W 1 1, W is semistable, x(t + 1) Wx(t). It should be pointed out that the ODLA problem is a well defined problem for a large class of graphs. The following example shows wellposedness for q 2. Example 2.1: Consider network system given by x 1 (t + 1) u 1 (t), (9) x 2 (t + 1) u 2 (t). (10) Assume that the distributed algorithms are given by u 1 (t) k 11 x 1 (t) + k 12 x 2 (t), (11) u 2 (t) k 21 x 1 (t) + k 22 x 2 (t), (12) where k ij R, i, j 1, 2, k 12, k The closed-loop system is given by x1 (t + 1) k11 k 12 x1 (t). (13) x 2 (t + 1) k 21 k 22 x 2 (t) To reach distributed agreement, we must have Hence, x1 (t + 1) x 2 (t + 1) and Then we have x(t) and k 11 + k 12 1, (14) k 21 + k 22 1, (15) k 11 + k 22 1 < 1. (16) 1 k12 k 12 x1 (t) 1 k 21 x 2 (t) k 21 (17) 0 < k 12 + k 21 < 2. (18) k21+k 12(1 k 12 k 21) t k 21 k 21(1 k 12 k 21) t k 12 k 12(1 k 12 k 21) t (1 k 12 k 21) t x(0) (19) lim x(t) t k 21 k 21 k 12 k 12 x(0). (20) Next, the cost functional is chosen to be J(K, x(0)) (u 1 (t) u e1 ) 2 + (u 2 (t) u e2 ) 2, (21) where u ei lim t u i (t), i 1, 2. Substituting (19) and (20) into (21) yields J(K, x(0)) x T a11 a (0) 12 x(0), (22) a 21 a 22 82

3 where Thus, (k k 2 a 11 21)(1 k 12 k 21 ) 2 (k 12 + k 21 ) 2 1 (1 k 12 k 21 ) 2, a 12 a 11, a 21 a 12, a 22 a 11. (23) J(K, x(0)) a 11 (x 1 (0) x 2 (0)) 2. (24) Note that a 11 can be rewritten as a 11 k k (k 12 + k 21 ) 2 1 (1 k 12 k 21 ) 2 1. (25) Since (k k2 21 )/(k 12 + k 21 ) 2 1/2 and 1/1 (1 k 12 k 21 ) 2 1 0, it follows that a 11 0, where the equality holds if and only if k 12 k 21 1/2. Hence, min K J(K, x(0)) 0 and the optimal distributed algorithm K is unique and given by K (26) Alternatively, if we choose the cost functional given by J(K, x(0)) (x 1 (t) x e1 ) 2 + (x 2 (t) x e2 ) 2 +(u 1 (t) u e1 ) 2 + (u 2 (t) u e2 ) 2, (27) where x ei lim t x i (t), i 1, 2, then J(K, x(0)) is given by (24) with a 11 (k k2 21 )1 + (1 k 12 k 21 ) 2 (k 12 + k 21 ) 2 1 (1 k 12 k 21 ) 2. (28) Note that a 11 can be rewritten as a 11 k k (k 12 + k 21 ) 2 1 (1 k 12 k 21 ) 2 1. (29) Since (k k2 21 )/(k 12 + k 21 ) 2 1/2 and 2/1 (1 k 12 k 21 ) 2 1 1, it follows that a 11 1/2, where the equality holds if and only if k 12 k 21 1/2. Hence, min K J(K, x(0)) (1/2)(x 1 (0) x 2 (0)) 2 and the optimal distributed algorithm K is unique and given by (26). It can be seen from this example that direct computation of W is quite tedious even for the lowest dimension q 2. It becomes almost impossible to obtain W for higher dimension q by using this method. Hence, it is worth developing an optimization-based algorithm to solve the ODLA problem numerically. III. NECESSARY CONDITIONS FOR OPTIMALITY Suppose the linear iteration (2) is semistable. Then it follows from 20, p. 447 that lim t x(t) x e, where x e I q (I q W)(I q W) # x(0). Now, it follows that J(u, x 0 ) x(t) x e T (Q + W T RW)x(t) x e x(0) x e T (W t ) T RW t x(0) x e x T (0)(I q W)(I q W) # T (W t ) T R W t (I q W)(I q W) # x(0) tr x T (0) (I q W)(I q W) # T (W t ) T RW t (I q W)(I q W) # x(0) tr (I q W)(I q W) # T (W t ) T t RW (I q W)(I q W) # x(0)x T (0) tr PV, (30) where we assume that the initial state x 0 is a random variable such that Ex 0 0 and Ex 0 x T 0 V, we used the fact that x(t) x e W t x(0) x e, R Q + W T RW, and P (I q W)(I q W) # T (W t ) T RW t (I q W)(I q W) #. (31) Lemma 3.1 (21): Consider the linear iteration (2). If (2) is semistable, then, for every q q nonnegative definite matrix R, (x(t) x e ) T R(x(t) x e ) <, (32) where x e I q (I q W)(I q W) # x(0). Lemma 3.2: If (2) is semistable, then P given by (31) is well defined, that is, 0 P <. Lemma 3.3: If (2) is semistable, then P given by (31) satisfies (I q W) T P(I q W) (I q W) T (W T PW + R) (I q W). (33) Equation (33) is a Lyapunov equation for semistability of (2). We call (33) the (discrete-time) semistable Lyapunov equation. Note that if (2) is (discrete-time) asymptotically stable, then the semistable Lyapunov equation (33) reduces to the standard Lyapunov equation P W T PW + R. The following result is immediate. Lemma 3.4: If (2) is semistable, then for every nonnegative-definite matrix R R T R q q, there exists a nonnegative-definite matrix P P T R q q such that (33) holds. 83

4 IV. NECESSARY AND SUFFICIENT CONDITIONS FOR SEMISTABILITY Definition 4.1: Let A R n n and C R m n. The pair (A, C) is weakly semiobservable if n N(C(A I n ) k ) N(A I n ). (34) k1 Recall from Definition 2.3 of 21 that the pair (A, C) is semiobservable if and only if n N(C(A I n ) k 1 ) N(A I n ). (35) k1 It is easy to see from the definitions of semiobservability and weak semiobservability that (A, C) is weakly semiobservable if and only if (A, CA) is semiobservable. Motivated by Lemma 2.1 of 21, we have the following lemma. Lemma 4.1: If there exist matrices P 0 and R 0 in R q q such that (33) holds and the pair (W, R) is weakly semiobservable, then i) N(P(W I q )) N(W I q ) N( R(W I q )) and ii) N(W I q ) R(W I q ) {0}. Proof: Note that (W, R) is weakly semiobservable if and only if N( R(W I q ) k ) N(W I q ). (36) k1 Since N((I q W) T R(Iq W)(I q W) k 1 ) N( R(I q W)(I q W) k 1 ) N( R(I q W) k ), it follows that N((I q W) T R(Iq W) k ) k1 N( R(I q W) k ) k1 N(I q W). (37) Let P (I q W) T P(I q W) and Q (I q W) T R(Iq W). Then (33) becomes P W T PW + Q (38) and (37) becomes N( Q(I q W) k 1 ) N(I q W). (39) k1 Now the result directly follows from Lemma 2.1 of 21 by noting that N( P) N(P(I q W)) and N( Q) N( R(I q W)). The part of the converse result for Lemma 3.4 can be stated as follows. Lemma 4.2: Assume that for every nonnegative-definite matrix R R T R q q satisfying (W, R) is weakly semiobservable, there exists a nonnegative-definite matrix P P T R q q such that (33) holds. Then (2) is semistable. Lemma 4.3: Consider the linear iteration (2). Then (2) is semistable if and only if for every weakly semiobservable pair (W, R) with nonnegative-definite R, there exists a q q matrix P 0 such that (33) holds. Lemma 4.4 (21): Let A R n n and B R m m. If A and B are semistable, then A B is semistable. Lemma 4.5: Let x R n and A R n n, and assume A is discrete-time semistable. Then At x exists if and only if x R(A I n ). In this case, At x (A I n ) # x. Theorem 4.1: Consider the linear iteration (2). Then (2) is semistable if and only if for every weakly semiobservable pair (W, R) with nonnegative-definite R, there exists a q q matrix ˆP > 0 such that ˆP W T ˆPW + (Iq W) T R(Iq W). (40) Such a ˆP is not unique. Furthermore, if (W, R) is weakly semiobservable and ˆP satisfies (40), then ˆP (W t ) T (I q W) T R(Iq W)W t + αxx T, x N(I q W T ), α > 0. (41) Proof: Suppose (W, R) is weakly semiobservable. Then it follows from Lemma 4.3 that there exists a q q matrix P 0 such that (33) holds. Since, by Lemma 4.1, N(W I q ) R(W I q ) {0}, it follows from 22, p. 119 that W I q is group invertible. Thus, let L I q (W I q )(W I q ) # and note that L 2 L. Hence, L is the unique q q matrix satisfying N(L) R(W I q ), R(L) N(W I q ), and Lx x for all x N(W I q ). Now, define ˆP (W I q ) T P(W I q ) + L T L. (42) Next, we show that ˆP is positive definite. Consider the function V (x) x T ˆPx, x R q. If V (x) 0 for some x R q, then P(W I q )x 0 and Lx 0. It follows from i) of Lemma 4.1 that x N(W I q ), and Lx 0 implies that x R(W I q ). Now, it follows from ii) of Lemma 4.1 that x 0. Hence, ˆP is positive definite. Next, since L(W I q ) W I q (W I q )(W I q ) # (W I q ) 0, it follows that ˆP W T ˆPW (W Iq ) T P(W I q ) + L T L W T (W I q ) T P(W I q )W W T L T LW (W I q ) T P(W I q ) + L T L (W I q ) T W T PW(W I q ) L T L (I q W) T R(Iq W). (43) Conversely, if there exists ˆP > 0 such that (40) holds, consider the function U(x) x T ˆPx, x R q. Then U(x(t+ 1)) U(x(t)) x T (t)(i q W) T R(Iq W)x(t) 0, t Z +, and {x R q : x T (I q W) T R(Iq W)x 0} N( R(I q W)). To obtain the largest invariant set M contained in N( R(I q W)), consider a solution x(t) of (2) such that R(Iq W)x(t) 0 for all t Z +. Then, R(Iq W)x(t + 1) R(I q W)x(t) 0, that is, R(I q W) 2 x(t) 0. This implies R(I q W) i x(t) 0 for all t Z + and i 1, 2,.... Now, it follows from (36) that x(t) N(I q W) for all t Z +. Thus, M N(I q W). Since N(I q W) consists of only equilibrium points, it 84

5 follows that M N(I q W). For x e N(I q W), Lyapunov stability of x e now follows by considering the Lyapunov function U(x x e ). For any ˆP > 0 satisfying (40) and M 0, let P ˆP + L T ML. Clearly, P ˆP > 0. It is easy to verify that P is a solution to (40), and hence, such a ˆP is not unique. Finally, since W is semistable, it follows from the above result that there exists an q q positive-definite matrix ˆP such that (40) holds or, equivalently, vec ˆP (W W) T vec ˆP + vec(i q W) T R(Iq W), that is, I q 2 (W W) T vec ˆP vec(i q W) T R(Iq W). Hence, vec(i q W) T R(Iq W) R(I q 2 (W W) T ). Next, it follows from Lemma 4.4 that W W is semistable, and hence, by Lemma 4.5, ( ) vec 1 (I q 2 (W W) T ) # vec(i q W) T R(Iq W) vec 1( ((W W) T ) t vec(i q W) T ) R(I q W) vec 1( ((W t ) T (W t ) T )vec(i q W) T ) R(I q W) (W t ) T (I q W) T R(Iq W)W t, (44) where in (44) we used the facts that (X Y ) T X T Y T, (X Y )(Z W) XZ Y W, and vec(xy Z) (Z T X)vecY 20, Chapter 7. Hence, ˆP (W t ) T (I q W) T R(Iq W)W t + vec 1 (z), (45) where z satisfies z NI q 2 (W W) T and vec 1 (z) (vec 1 (z)) T 0. Since (W W) T is discrete-time semistable, it follows that I q 2 (W W) T is semistable. Hence, the general solution to the equation (W W) T z z is given by z λx x, where x N(I q W T ) and λ R. Hence, vec 1 (z) vec 1 (αx x) αxx T, α > 0, where we used the fact that xy T vec 1 (y x) (Proposition of 20). V. A NEW FORMULATION FOR THE ODLA PROBLEM Theorem 5.1: Consider the linear iteration (2). Assume (W, R) is weakly semiobservable for nonnegative definite R. Let S min be a solution to the minimization problem { min tr ((I q W) # ) T S(I q W) # V : S > 0 } and S W T SW + (I q W) T R(Iq W). (46) Then for P given by (31), tr((i q W) # ) T S min (I q W) # V trpv. (47) Furthermore, such a S min is not unique. Finally, all the solutions to the minimization problem (46) can be parametrized as P min S min + αxx T, x N(I q W T ) N(((I q W) # ) T ), α > 0. (48) Theorem 5.2: Consider the linear iteration (2). Assume (W, R) is weakly semiobservable for nonnegative definite R. Let S min be a solution to the minimization problem (46) and Z min be a solution to the minimization problem { min trzv : Z > 0 and Z W T ZW } +(I q W) T R(Iq W). (49) Then tr ((I q W) # ) T S min (I q W) # V tr ((I q W) # ) T Z min (I q W) # V. (50) Example 5.1: To illustrate Theorem 5.2, consider the linear iteration (2). Assume Q 0 and R I q. Now, it follows from Theorem 5.2 that the OLDA problem can be recast as the minimization problem (49). In this case, Z W T ZW +(I q W) T W T W(I q W). To simplify our discussion, we assume that W is idempotent and symmetric. Then it follows that W is semistable and (5) and (6) hold. Furthermore, Z W T ZW, or equivalently, Z W W T (Z W)W. Since W is semistable, it follows from Theorem 4.1 that Z W xx T, where x N(I q W T ) and W + xx T > 0. Hence, Z min W + xx T. Note that (I q W) 2 I q 2W + W 2 I q W. Then it follows from iii) of Proposition of 20, p. 229 that (I q W) # I q W, and hence, the cost functional is given by tr(i q W)Z min (I q W)V 0. For q 2, we solve W 2 W W T to obtain three nonzero matrices 1 0 a a(1 a) W 1, W 2 a(1 a) 1 a W a a(1 a) a(1 a) 1 a where 0 < a < 1. The second matrix W 2 with a 1/2 corresponds to distributed semistable state agreement design, which is consistent with the result in Example 2.1. In summary, a new formulation for the OLDA problem based on Theorem 5.2 is given by where minimize x T (0)Zx(0), subject to W W S, 1 T W 1 T, W 1 1, Z > 0, Z W T ZW + (I q W) T S (Q + W T RW)(I q W), {W R q q : (W, Q + W T RW) is weakly semiobservable}. (51) It follows from the above description that this new formulation of the OLDA problem needs to consider the fourthorder matrix equation and the set S, which are not easy to compute in practice. Next, we further boil down this formulation by imposing some simply assumptions. 85

6 Lemma 5.1: Let Q > 0 and W 2 W. Then (W, Q + W T RW) is weakly semiobservable. Lemma 5.2: Let Q > 0 and W 2 W. Then Z W T ZW + (I q W) T (Q + W T RW)(I q W) always has a solution Z 0. Furthermore, Z W T W + rr T, where r N(I q W T ), and tr((i q W) # ) T Z min (I q W) # V 0. Note that it follows from Lemma 5.2 that x T (0)Zx(0) x T (0)W T Wx(0) + x T (0)rr T x(0) x T (0)W T Wx(0). (52) Then we have the following formulation. Proposition 5.1: The solution to the following problem gives a solution to the ODLA problem: minimize Wx(0) 2 subject to W W, 1 T W 1 T, W 1 1, W 2 W. (53) VI. CONCLUDING REMARKS By stating that solving the optimization problem (49) is equivalent to the ODLA problem according to Theorem 5.2, one could possibly use some computational methods such as the interior-point method or the subgradient method to solve the optimization problem (49) on a large-scale graph, but problems with more than a few thousand edges are probably beyond the capabilities of current interior-point semidefinite programming solvers and the subgradient method has no simple stopping criterion that guarantees a certain level of suboptimality. Hence, as the future research, we need to consider alternatives such as the stochastic optimization methods 23. The stochastic optimization methods have attracted much attention in recent years since their algorithms do not require properties such as linearity, differentiability, convexity, separability or nonexistence of constraints. In particular, we will explore the Particle Swarm Optimization (PSO) algorithm introduced in 24 to solve the optimization problem (49) in this paper. We choose the PSO algorithm to solve the equivalent ODLA problem for its efficiency, fewer parameters to adjust, and the ability to escape local maxima. The efficiency factor is especially important to our problem due to the large parameter space to search. The PSO algorithm is one of the techniques based on swarm intelligence 25 which simulates the social behavior of bird flocks or fish schools. The PSO algorithm starts with a set of random solutions, or particles, and they can communicate with each other. The particles then move through the solution space based on evaluation of these particles according to a fitness criterion. As the algorithm progresses, the particles will accelerate towards those particles with better fitness values. 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