Optimal Input Elimination

Transcription

1 SICE Journal of Control, Measurement, and System Integration, Vol. 11, No. 2, pp , March 2018 Optimal Input Elimination Kazuhiro SATO Abstract : This paper studies an optimal inputs elimination problem in large-scale network systems. We first solve an H 2 optimization problem of the difference between the transfer functions of the original system and the system after the input variables elimination. It is shown that the problem can be rigorously solved by calculating the gradient and Hessian of this objective function. The solution means that, when the input variables to be eliminated were fixed, the H 2 optimal inputs elimination is achieved by simply eliminating input variables without changing the driver nodes, which are state variables that are directly affected by an input signal. We next solve a finite combinatorial optimization problem to decide input variables to be eliminated. The objective function is defined by using the solution to the H 2 optimization problem. It is shown that a greedy algorithm gives the global optimal solution to the finite combinatorial problem within a practical time. The algorithm can be understood that we eliminate input variables in ascending order of the average controllability centralities which assign relative importances to each node within a network. Finally, we demonstrate how to use the results in this paper by a simple example. Key Words : input elimination, H 2 optimization, large-scale network systems. 1. Introduction Large scale network systems such as smart grids and social networks have received increased attention in recent years. For such complex networks, it is not clear how we should select the driver nodes, which are state variables directly affected by an input signal. This is because although the selection problem can be formulated as a finite combinatorial optimization problem, a brute force approach for solving the problem quickly becomes intractable. For this reason, there are many previous works on input selection problems [1] [14]. Reference [4] has shown that sparse inhomogeneous networks are difficult to control, while dense homogeneous networks can be controlled by using a few driver nodes. Furthermore, it has been found that the minimum number of driver nodes to make a system controllable is mainly determined by the degree distribution of the network. However, reference [5] has proved that finding the driver nodes is non-deterministic polynomial time (NP) hard. To solve the computationally hard problem, references [6],[11] [14] have introduced energy performance indices on the choice of the optimal input set, and have solved it by efficient approximation algorithms. Moreover, references [2],[7] [10] have characterized all possible solutions to problems related with that in [5] by using graph theory, and have given efficient numerical algorithms for solving the problems. In this paper, we study an optimal inputs elimination problem related with the input selection problems. This problem is based on a different viewpoint from the previous works in [1] [14]. In fact, all the previous studies add the new input variable to enhance the controllability in a network, while we eliminate redundant input variables, which do not almost affect the output. Our problem is practically important, because we School of Regional Innovation and Social Design Engineering, Kitami Institute of Technology, Hokkaido , Japan ksato@mail.kitami-it.ac.jp (Received July 26, 2017) (Revised October 2, 2017) better eliminate the variable to reduce cost when some input variable does not almost affect each output in a network, i.e., when the elimination does not almost reduce the system performance. To formulate the problem, we consider the linear system ẋ = Ax + Bu, (1) y = Cx, where x R n, u R m,andy R p are the state, input, and output variables, respectively, and A R n n, B R n m,and C R p n are constant matrices. The system (1) can express a network system. In fact, in the terminology of graph theory, the matrix A induces a graph G = (V, E) of the network in which the nodes correspond to states, i.e., V = {1, 2,...,n} and the edges correspond to nonzero entries of A, i.e., (i, j) E whenever a ji 0. The nonzero entries of the matrix B describe how each actuator affects the nodes in the network; i.e., we can consider a situation where the matrix B determines the driver nodes in the network system (1). Furthermore, we assume that the system after the input variables elimination is expressed as ẋ = Ax + Bu J, (2) ȳ = Cx, where B R n (m r) and u J R m r is the input constructed by eliminating input variables in the original input u. The elimination is specified by a given set J which contains r elements. Then, we first consider an H 2 optimization problem of the difference between the transfer functions of the original system (1) and the system which is equivalent to the reduced system (2) subject to the fixed set J. Next, to decide the optimal set J, we solve a finite combinatorial optimization problem, the objective function of which is defined by using the solution of the H 2 optimization problem. The contributions of this paper are as follows. 1) We prove that the optimal solution to the H 2 optimization JCMSI 0002/18/ c 2017 SICE

2 SICE JCMSI, Vol. 11, No. 2, March problem is essentially given by B = B J, where B J is defined as the matrix reconstructed by eliminating the column vectors, which is specified by the set J, inthe matrix B. To this end, we calculate the gradient and Hessian of the objective function. The optimal solution means that the H 2 optimal elimination is achieved by simply eliminating input variables without changing the driver nodes when the input variables to be eliminated were determined. Furthermore, we prove that, if the system (1) is observable, then the solution is the unique global optimal solution. 2) We prove that the optimal set J can be obtained by a greedy algorithm within a polynomial time. This is non-trivial because finite combinatorial problems are NP-hard in general. The proof is achieved by using the results of [13]. The algorithm can be understood that we eliminate input variables in ascending order of an average controllability centrality which assigns a relative importance to each node within a network. This paper is organized as follows. In Section 2, we formulate the H 2 optimization problem and finite combinatorial optimization problem to be solved in this paper. In Section 3, we rigorously solve the H 2 optimization problem by calculating the gradient and Hessian of the objective function. In Section 4, we show that a greedy algorithm solves the finite combinatorial optimization problem. In Section 5, we demonstrate how to use the results in this paper by a simple example. The conclusion is presented in Section 6. Notation: The sets of real and complex numbers are denoted by R and C, respectively. The identity matrix of size n is denoted as I n. The symbol 0 n R n is a vector with only zero entries. Given a vector v C n, v denotes the the Euclidean norm. The Hilbert space L 2 (R n )isdefinedby { L 2 (R n ):= f :[0, ) R } n f (t) 2 dt <. Given a measurable function f :[0, ) R n, f L 2 and f L denote the L 2 and L norms of f, respectively, i.e., f L 2 := f (t) 2 dt, 0 f L := sup f (t). t 0 Given a matrix A C m n, A and A F denote the induced norm and the Frobenius norm, respectively, i.e., A := max v C n \{0} Av, v A F := tr(a A), where the superscript denotes the Hermitian conjugation and tr(a A) is the trace of A A, i.e., the sum of the diagonal elements of A A. For a matrix function G(s) C m n, G H 2 and G H denote the H 2 and H norms of G, respectively, i.e., 1 G H 2 := G(iω) 2 F 2π dω, G H := sup σ(g(iω)), ω R where i denotes the imaginary unit and σ(g(iω)) denotes the maximum singular value of G(iω) Problem Setup This section formulates two problems to be solved. Let us consider the linear system (1) as an original network system. The transfer function of the system (1) is defined as G(s) := C(sI n A) 1 B for s C. To rigorously formulate our problem, we also consider x = A x + Bu, (3) ỹ = C x, where B S J R n m, and the transfer function is given by G(s) := C(sI n A) 1 B. Here, J is composed of the numbers which specify which columns are replaced by 0 n,ands J is defined as the set of free real matrices except for the column vectors specified by J; i.e., S J is a subspace of R n m. Note that the system (3) is equivalent to the form (2). That is, the system (3) is a system after the input variables elimination. In fact, for example, if the number of inputs m is 6 and J = {1, 2, 5}, S J = {( 0 n 0 n B 1 0 n B 2 ) B 1 R n 2, B 2 R n}, then (3) is expressed as x = A x + ( ) B 1 B 2 ỹ = C x. u 3 u 4 u 6, This means that the input variables u 1, u 2,andu 5 were eliminated from the original system. Thus, the system (4) has the form (2). In this paper, we want to clarify the system (3) which best approximates the original system (1) in the sense of the difference between the outputs of the systems (1) and (3). To this end, we assume that the matrix A is stable; i.e., the real parts of all eigenvalues of the matrix A are negative. This is because if the matrix A is not stable, ỹ is usually quite different from y for any B S J. We thus consider the following optimal inputs elimination problem. Problem 1: minimize g(j) subject to J {1, 2,...,m}, J = r. Here, g(j) := f ( B ), and J denotes the number of elements in J, where (4) f ( B) := G G 2 H 2, (5) and B S J is a solution to the following optimization problem.

3 102 Problem 2: SICE JCMSI, Vol. 11, No. 2, March 2018 where B = ( ) B b i 1 b i 2 b i k R n (m+k) and u = ( ) u T u 1 u 2 u T k R m+k. That is, the matrix B is the augmented matrix of B, and the vector u is the augmented vector of u. In Problem 3, b 1, b 2,...,b N are arbitrary and h(s ) depends on them. In Problem 1, g(j) depends on B because g(j) := f ( B ). However, B is not arbitrary because B is a so- lution to Problem 2. Thus, the main novelty of this paper is to solve Problem 2. minimize f ( B) Remark 2 Our key idea for formulating Problem 2 is to use subject to B S J. the transfer function G of the system (3) instead of the transfer function Problems 1 and 2 mean that we determine the driver nodes of the system after the inputs elimination such that G best approximates G in the sense of the H 2 norm. The difference between Ḡ = C(sI n A) 1 B of the system (2). This is based on the observation that since our problem and the problem in [13] is explained in Remark the size of Ḡ is different from that of G, we cannot calculate the 1. By solving Problems 1 and 2, we can eliminate redundant difference of the two transfer functions; i.e., G Ḡ cannot be input variables which do not affect the output. This serves to defined in the usual sense. In contrast, we can calculate G G reduce cost when the inputs elimination does not almost reduce because the size of G is the same with that of G. the system performance, i.e., when G G H 2 is sufficiently small. The aim of Problem 2 is to eliminate input variables such that y ỹ L becomes as small as possible when the set J was given. In fact, since the matrix A is stable, y ỹ L G G H 2 u L 2 for any u L 2 (R m ). The proof is similar to that in Appendix A in [15]. Hence, if f ( B) issufficiently small, we can expect that y ỹ L becomes small. As a result, Problem 1 means that we select the set J such that y ỹ L is as small as possible. Furthermore, note that we have y ỹ L 2 G G H u L 2 for any u L 2 (R m ). However, the modified problem replaced f ( B) with G G H is difficult to solve rigorously, as explained in Remark 3. Since Problem 1 is a finite combinatorial optimization problem, we can solve it by brute force by enumerating all possible subsets of size r, evaluating g for all of these subsets, and picking the best subset if we obtained the solution to Problem 2. However, when we consider a large-scale network system, the number of possible subsets increases factorially as J becomes larger. As the result, the brute force approach quickly becomes intractable. To solve Problem 1, we need to obtain the solution to Problem 2. Hence, we first give a global optimal solution to Problem 2 in the next section. An efficient method for solving Problem 1 is proposed in Section 4. Remark 1 Although Problem 1 is similar to that of the input selection problem in [13], Problem 2 is novel. In fact, [13] has considered the following problem. Problem 3: maximize h(s ) subject to S {b 1, b 2,...,b N }, S = k. Here, the function h(s ) means a magnitude of the controllability in the system ẋ = Ax + B u, y = Cx, Remark 3 The objective function f ( B)isdifferentiable, as discussed in Section 3. However, the function G G H is not differentiable; i.e., we cannot calculate the gradient of G G H. Thus, if we replace the objective function f ( B) ofproblem2 to G G H, we cannot expect that we can obtain the solution to the new problem. For this reason, we adopt f ( B) asthe objective function of Problem 2. Furthermore, note that it is possible to bound the H norm from above by a constant multiple of the H 2 norm if the pole structure of the transfer function is known [16],[17]. Hence, we can expect that the solution to Problem 2 becomes a near-optimal solution to the modified problem in some cases. 3. A Global Optimal Solution to Problem 2 This section proves the following theorem. Theorem 1 A global optimal solution B S J to Problem 2 is given by B = (B) J elimination, (6) where (B) J elimination denotes the same matrix B except for the column vectors specified by the set J and the all exceptional column vectors are replaced by 0 n. Moreover, if the system (1) is observable, then (6) is the unique global optimal solution to Problem 2. Theorem 1 means that, if the set J was determined, we can achieve the optimal elimination by simply eliminating input variables specified by J. That is, the system after the H 2 optimal inputs elimination specified by J is given by x = A x + B J u J, (7) ỹ = C x, where B J R n (m r) is defined as the matrix reconstructed by eliminating the column vectors, which are specified by J, of B and u J R m r denotes the inputs constructed by eliminating the inputs, which are specified by J, ofu. To prove Theorem 1, we note that f ( B) can be written as f ( B) = tr(cw c C T ) = tr((b B) T W o (B B)), (8) because the matrix A is stable, where W c and W o are the controllability and observability Gramians, respectively, which are the solutions to the Lyapunov equations

4 SICE JCMSI, Vol. 11, No. 2, March AW c + W c A T + (B B)(B B) T = 0, (9) A T W o + W o A + C T C = 0. (10) The proof of (8) is similar to that in Appendix B in [15]. From now on, we derive the gradient and Hessian of the objective function f.letf denote the extension of the objective function f to the ambient Euclidean space R n m. The directional derivative of f at B in the direction B can be calculated as D f ( B)[ B ] = tr( B T ( 2W o (B B))). (11) Since the gradient f ( B) satisfies D f ( B)[ B ] = tr( B T ( f ( B))), (11) implies that f ( B) = 2W o (B B). Since ( f ( B)) J elimination is the projection of f ( B) onto the subspace S J, the gradient grad f ( B)for B S J is given by grad f ( B) = ( f ( B)) J elimination = 2W o ( (B B) J elimination ) = 2W o ( (B)J elimination B ). (12) The Hessian Hess f ( B) atany B S J is given by Hess f ( B)[ B ] = ( Dgrad f ( B)[ B ] ) J elimination = 2W o B, (13) where B T BS J,andwhereT BS J is the tangent space of S J at the point B. Here, note that T BS J can be identified with S J, because S J is a vector space. For a detailed explanation of the concept of the Hessian, see [18]. Thus, B, Hess f ( B)[ B ] := tr( B T Hess f ( B)[ B ]) = 2tr( B T W o B ). (14) Since the observability Gramian W o is symmetric positive semidefinite, (14) implies that B, Hess f ( B)[ B ] 0forany 0 B T BS J and any B S J. Hence, the objective function f is convex on S J [19]. If (6) holds, (12) yields grad f ( B) = 0; i.e., (6) is a local optimal solution at least. Actually, (6) is a global optimal solution to Problem 2, because the function f is convex on S J. Moreover, if the system (1) is observable, then W o is a symmetric positive definite matrix. Hence, then (14) leads us to B, Hess f ( B)[ B ] > 0forany0 B T BS J and any B S J ; i.e., the objective function f is strictly convex on S J [19]. This completes the proof of Theorem An Efficient Method for Solving Problem 1 This section shows that the global optimal solution to Problem 1 is given by Algorithm 1. As explained below, this algorithm resolves the computational difficulty of the brute force approach. To see this, we note that g(j) = tr(cw c (J)C T ), (15) where W c (J) satisfies the Lyapunov equation AW c (J) + W c (J)A T + B J B T J = 0 (16) for B J := B (B) J elimination. This is because g(j) = f ((B) J elimination ) from Theorem 1. By a direct calculation, (15) can be rewritten as g(j) = g({ j}). (17) j J Note that this has been proved in Theorem 4 in [13]. In a discrete time setting, [6] has proved a similar result. It follows from (15) that g({ j}) = tr(cw c ({ j})c T ), (18) where, from (16), W c ({ j}) satisfies the Lyapunov equation AW c ({ j}) + W c ({ j})a T + b j b T j = 0. (19) Here, b j denotes the j-th column vector of the matrix B. Algorithm 1 Greedy algorithm for solving Problem 1. 1: J 2: for k = 1, 2,...,r do 3: J J arg min g({ j}) j {1,2,...,m}\J 4: end for Using the relations (17) and (18), Problem 1 can be exactly solved by Algorithm 1; i.e., we can obtain the global optimal solution to Problem 1. Before we perform Algorithm 1, we can calculate g({ j}) forall j {1, 2,...,m} in advance. Hence, the number of calculations of the Lyapunov equation (19) is m times. To solve the Lyapunov equation efficiently, we can use an effective method for solving (19) such as the Bartels- Stewart-algorithm [20] even if the matrix A is a dense matrix. For sparse cases, we can use more effective methods. For example, see [21]. Thus, we can perform the optimal inputs elimination within a practical time. The function g({ j}) in (18) can be related to a dynamic network centrality measure which assigns a relative importance to each node within a network. In fact, if C = I n and b j = e j, where e j has 1 in the j-th entry and zeros elsewhere, (18) is rewritten as g({ j}) = tr(w c ( j)), (20) where W c ({ j}) satisfies the Lyapunov equation AW c ({ j}) + W c ({ j})a T + e j e T j = 0. The value tr(w c ( j)) in (20) is called the average controllability centrality for the node j [13]. Hence, if C = I n and b j = e j, Algorithm 1 means that we eliminate r input variables in ascending order of the average controllability centralities, while a proposed algorithm in [13] adds input variables in descending order of those values. 5. Numerical Example This section demonstrates how to use the results in this paper by a simple example. Note that we can also apply the results for large-scale network systems as explained in Section 1, in thesameway. The system matrices of the plant (1) are given by

5 A := , ( ) B := , C :=, and thus the eigenvalues of the matrix A are 1, 1, 1, 3, 4; i.e., A is stable. From Theorem 1 and the discussion of Section 4, the optimal solution to Problem 1 can be examined by calculating g({1}), g({2}), g({3}), and g({4}), where g({ j}) ( j = 1, 2, 3, 4) are defined as (18). By a direct calculation, we have g({1}) = , g({2}) = , g({3}) = , g({4}) = Hence, if r = 1 in Problem 1, then the optimal solution to Problem 1 is J = {3}. Furthermore, if r = 2, then the optimal solution to Problem 1 is J = {1, 3}. Moreover, if r = 3, then the optimal solution to Problem 1 is J = {1, 2, 3}. 6. Conclusion We have studied an optimal inputs elimination problem in large-scale network systems. We first rigorously solved the H 2 optimizationproblem of the difference between the original and the reduced transfer functions. The solution means that the H 2 optimal elimination is achieved by simply eliminating input variables without changing the driver nodes when the input variables to be eliminated were fixed. The selection problem of the input variables has been formulated as a finite combinatorial optimization problem. It has been shown that the proposed algorithm gives the global optimal solution to the finite combinatorial problem within a polynomial time. In future work, we will develop a hybrid method based on our results and the results in [13] to enhance controllability in large-scale network systems. Acknowledgments The author thanks to Mr. Yusuke Fujimoto for information on this subject. References [1] N. Bof, G. Baggio, and S. Zampieri: On the role of network centrality in the controllability of complex networks, IEEE Transactions on Control of Network Systems, Vol. 4, No. 3, pp , [2] J.F. Carvalho, S. Pequito, A.P. Aguiar, S. Kar, and K.H. Johansson: Composability and controllability of structural linear time-invariant systems: Distributed verification, Automatica, Vol.78, pp , [3] C. Commault and J.-M. Dion: Input addition and leader selection for the controllability of graph-based systems, Automatica, Vol. 49, No. 11, pp , [4] Y.Y. Liu, J.J. Slotine, and A. Barabási: Controllability of complex networks, Nature, Vol. 473, No. 7346, pp , [5] A. Olshevsky: Minimal controllability problems, IEEE Transactions on Control of Network Systems, Vol. 1, No. 3, pp , SICE JCMSI, Vol. 11, No. 2, March 2018 [6] F. Pasqualetti, S. Zampieri, and F. Bullo: Controllability metrics, limitations and algorithms for complex networks, IEEE Transactions on Control of Network Systems, Vol. 1, No. 1, pp , [7] S. Pequito, S. Kar, and A.P. Aguiar: A framework for structural input/output and control configuration selection in large-scale systems, IEEE Transactions on Automatic Control, Vol. 61, No. 2, pp , [8] S. Pequito, S. Kar, and A.P. Aguiar: Minimum cost input/output design for large-scale linear structural systems, Automatica, Vol. 68, pp , [9] S. Pequito and G.J. Pappas: Structural minimum controllability problem for switched linear continuous-time systems, Automatica, Vol. 78, pp , [10] S. Pequito, V.M. Preciado, A.-L. Barabási, and G.J. Pappas: Trade-offs between driving nodes and time-to-control in complex networks, Scientific Reports, Vol. 7, 39978, [11] T. Summers: Actuator placement in networks using optimal control performance metrics, 55th IEEE Conference on Decision and Control (CDC), pp , [12] T. Summers and I. Shames: Convex relaxations and gramian rank constraints for sensor and actuator selection in networks, IEEE International Symposium on Intelligent Control (ISIC), pp. 1 6, [13] T.H. Summers, F.L. Cortesi, and J. Lygeros: On submodularity and controllability in complex dynamical networks, IEEE Transactions on Control of Network Systems, Vol. 3, No. 1, pp , [14] V. Tzoumas, M.A. Rahimian, G.J. Pappas, and A. Jadbabaie: Minimal actuator placement with bounds on control effort, IEEE Transactions on Control of Network Systems, Vol. 3, No. 1, pp , [15] K. Sato: Riemannian optimal control and model matching of linear port-hamiltonian systems, IEEE Transactions on Automatic Control, Vol. 62, No. 12, pp , [16] S. Hara, B.D.O. Anderson, and H. Fujioka: Relating H 2 - and H -norm bounds for sampled-data systems, IEEE Transactions on Automatic Control, Vol. 42, No. 6, pp , [17] T. Ivanov, B.D.O. Anderson, P.-A. Absil, and M. Gevers: New relations between norms of system transfer functions, Systems & Control Letters, Vol. 60, No. 3, pp , [18] P.-A. Absil, R. Mahony, and R. Sepulchre: Optimization Algorithms on Matrix Manifolds, Princeton University Press, [19] S. Boyd and L. Vandenberghe: Convex Optimization, Cambridge University Press, [20] R.H. Bartels and G.W. Stewart: Solution of the matrix equation AX + XB = C, Communications of the ACM, Vol. 15, No. 9, pp , [21] T. Penzl: A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM Journal on Scientific Computing, Vol. 21, No. 4, pp , Kazuhiro SATO (Member) He received his B.S., M.S., and Ph.D. degrees from Kyoto University, Japan, in 2009, 2011, and 2014, respectively. He was a Post Doctoral Fellow at Kyoto University from 2014 to He is currently an Assistant Professor at Kitami Institute of Technology. He likes applied mathematics. He is a member of IEEE.