Balancing Skew-Hamiltonian/Hamiltonian Pencils With Applications in Control Engineering Vasile Sima National Institute for Research & Development in Informatics, 8–10 Bd. Mares¸al Averescu, Bucharest, Romania Keywords: Deflating Subspaces, Generalized Eigenvalues, Numerical Methods, Skew-Hamiltonian/Hamiltonian Matrix Pencil, Software, Structure-preservation. Abstract: Badly-scaled matrix pencils could reduce the reliability and accuracy of computed results for many numeri- cal problems, including computation of eigenvalues and deflating subspaces, which are needed in many key procedures for optimal and H∞ control, model reduction, spectral factorization, and so on. Standard balancing techniques can improve the results in many cases, but there are situations when the solution of the scaled problem is much worse than that for the unscaled problem. This paper presents a new structure-preserving balancing technique for skew-Hamiltonian/Hamiltonian matrix pencils, and illustrates its good performance in solving eigenvalue problems and algebraic Riccati equations for large sets of examples from well-known benchmark collections with difficult examples. 1 INTRODUCTION Jiang and Voigt, 2013), and the references therein. Quite often, the pencil matrices have large norms Computing eigenvalues and bases of certain associ- and elements with highly different magnitude. Such ated invariant or deflating subspaces of matrices or pencils imply potential numerical difficulties for matrix pencils is central to various numerical tech- software implementations of eigensolvers, with niques in control engineering and other domains. negative consequences on the reliability and accuracy When the corresponding eigenproblems have special of the results, see, e.g., (Sima and Benner, 2015a). structure, implying structural properties of their spec- Balancing procedures can be used to improve the tra, it is important to use structure-preserving and/or numerical behavior. Ward (1981) proposed a bal- structure-exploiting algorithms. General numerical ancing technique for general matrix pencils, which algorithms cannot ensure that the theoretical prop- has been incorporated in state-of-the-art software erties are preserved during computations (Paige and packages, such as LAPACK (Anderson et al., 1999). Van Loan, 1981; Van Loan, 1984; Kressner, 2005). (This will be referred below as standard balancing.) Common special structures are Hamiltonian and sym- The data matrices are preprocessed by equivalence plectic matrices or matrix pencils, which are encoun- transformations, in two optional stages: the first tered in optimal or H∞ control (e.g., solution of alge- stage uses permutations to find isolated eigenvalues braic Riccati equations, or evaluation of the H∞- and (which are available by inspection, with no rounding L∞-norms of linear dynamic systems), spectral factor- errors), and the second stage uses diagonal scaling ization, model reduction, etc., see, for instance, (Bru- transformations to make the row and corresponding insma and Steinbuch, 1990; Laub, 1979; Mehrmann, column 1-norms as close as possible. Balancing 1991; Sima, 1996). While structured matrices are en- may reduce the 1-norm of the scaled matrices, but countered for standard linear dynamic systems, gen- there is no guarantee in general. This is the reason eralized and descriptor systems may involve struc- why full balancing is either avoided or provided as tured matrix pencils. Often such pencils can be re- an option in the LAPACK subroutines; some expert cast as skew-Hamiltonian/Hamiltonian pencils, for driver routines, such as DGEES, DGEESX, DGGES, which dedicated algorithms have been developed, see DGGESX, and DGGEV use permutations only, while e.g., (Benner et al., 2007; Benner et al., 2002; Ben- other drivers, e.g., DGEEVX and DGGEVX, have an ner et al., 2005; Benner et al., 2012a; Benner et al., input argument allowing either permutations, scaling, 2012b; Benner et al., 2013a; Benner et al., 2013b; both permutations and scaling, or no balancing at all. 177 Sima, V. Balancing Skew-Hamiltonian/Hamiltonian Pencils - With Applications in Control Engineering. DOI: 10.5220/0005981201770184 In Proceedings of the 13th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2016) - Volume 1, pages 177-184 ISBN: 978-989-758-198-4 Copyright c 2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved ICINCO 2016 - 13th International Conference on Informatics in Control, Automation and Robotics Note that U is related to a basis, U, of the stable right zero imaginary parts, but possibly nonzero real parts. deflating subspace of the original pencil, αS βH ,by The implementation takes care of the compact storage 1 1 − T T the transformatione U = bl diag(R− ,L− )U, hence scheme when computing P DP,..., P WP. T T T U1 U2 := U = bl diag(R,L)U. Therefore, Consider a structured pencil, (S,H ), and assume the stabilizing solutione of the original ARE can be that S = I2n. The order of magnitude of the dif- computed as follows e ferences in size of the nonzero elements in the H 1 1 1 matrix can be huge. For instance, for the CM6 X = U2U − = LU2U − R− . (13) 1 1 and CM6 IS examples from the COMPleib collec- Formula (13) allows to represent and use the solution tion (Leibfritz and Lipinski, 2003), the maximum X in a factored form, whiche maye be useful for nu- and minimum absolute values of the nonzero en- 5 324 merical reasons. If balancing also involves symplec- tries are about 2.53 10 and 4.9407 10− , hence tic permutations of the form (5) only, then the right their ratio is not representable· in the· usual double 1 1 T precision representation, and it is evaluated as Inf hand side in (13) becomes PLU2U1− R− P , where P denotes here the product of the ℓ 1 permutations (∞). The standard balancing algorithm implemented − performed. Such factorization, is,e however,e not possi- in the LAPACK Library subroutine DGGBAL (for ma- ble if J -permutations are also needed, in which case trix pencils) returns scaling factors covering a very it is necessary to apply the balancing transformations large range of magnitudes, namely with maximum 159 47 before solving the set of linear systems giving X. and minimum values 10 and 10− , respectively, If S is a general matrix, the order of S and H is for both left and right scaling factors. No eigenvalue 2(n+ p), where n is the dimension of the state vector, can be isolated. The scaling transformation matrices, e 206 and p may be chosen as p = m/2 (i.e., p = m/2, if L and R , have condition numbers with values 10 . m is even, and p = (m + 1)/2,⌈ otherwise),⌉ with m the The 1-norms of scaled matrices, H and S, are about number of system inputs. In this case, the computa- 10228 and 10184, respectively, while the 1-norms of tions are similar, but U and U , i = 1,2, refer to the the original H and S are 6.9123e105 ande 1. Com- i i · lines 1 : n and n + p + 1:2n + p of the matrix bases puting the eigenvalues or deflating subspaces using U and U, respectively.e the balanced matrices would return results very far from the true ones. CM6 and CM6 IS are not the only e examples in the COMPleib collection which produce numerical troubles for eigenproblem-related compu- 3 IMPLEMENTATION ISSUES tations. The CM5 and CM5 IS examples have also a huge ratio, 10279, between the magnitudes of their The developed balancing algorithm operates only on elements. Other very large ratios are 10141, for CM4 the matrices A, D, E, C, V, and W, and preserves and CM4 IS, 1072, for CM3 and CM3 IS, or 1037, for the pencil structure. Moreover, the pairs of skew- CM2 and CM2 IS. Hermitian matrices, D and E, and Hermitian matri- ces, V and W , are stored compactly in two n (n+1) The proposed algorithm uses an adaptation of arrays, DE and VW. Specifically, the lower triangle× of the basic LAPACK procedure for finding the scaling E and the upper triangle of D are concatenated along factors, but optionally limits the range of their their main diagonals, and similarly for W and V : variation, possibly via an outer loop. Specifically, the user can set a threshold value, τ; if τ 0, DEi j = ei j, j = 1 : n, i j, ≥ ≥ the entries whose absolute values are smaller than DEi, j+1 = di j, j = 1 : n, i j, τM0, where M0 = max( H (s,s) 1, S(s,s) 1), with ≤ s := ℓ : n n + ℓ : 2n,k are consideredk k negligible,k VWi j = wi j, j = 1 : n, i j, ≥ and do not∪ count for computing the scaling factors. VWi, j+1 = vi j, j = 1 : n, i j. (14) ≤ (For the CM6 and CM6 IS examples, ℓ = 1, and 20 This way, the storage requirements are reduced by setting τ = 10− , all entries with magnitude less 2 15 4n 2n memory locations, in comparison with the than about 7 10− will be taken as zero by the general− algorithm in (Anderson et al., 1999), which procedure.) If·τ < 0 on entry, an outer loop over a set 2 needs 8n storage space for S and H . Note that in of values τi > 0 will enable to select a set of scaling the real case, the diagonal elements of D and E, dii factors which, if possible, ensure the reduction of a and eii, i = 1 : n, which should be zero, by definition, desired norm-related measure for the scaled matri- are not stored and not used. However, in the complex ces. For τ = 1, this measure is the minimum of − case, dii and eii, i = 1 : n, should have zero real parts, maxi( Hi(s,s) 1/ Si(s,s) 1, Si(s,s) 1/ Hi(s,s) 1) k k k k k k k k while the imaginary parts might be nonzero; more- where Hi(s,s) and Si(s,s) are the scaled submatrices over, the diagonal elements of V and W should have corresponding to the threshold τi.