## Krylov space methods on state-space control models (1994)

Venue: | Circuits, Systems, and Signal Processing |

Citations: | 43 - 4 self |

### BibTeX

@INPROCEEDINGS{Boley94krylovspace,

author = {Daniel L. Boley},

title = {Krylov space methods on state-space control models},

booktitle = {Circuits, Systems, and Signal Processing},

year = {1994},

pages = {733--758}

}

### Years of Citing Articles

### OpenURL

### Abstract

We give an overview of various Lanczos/Krylov space methods and how they are being used for solving certain problems in Control Systems Theory based on state-space models. The matrix methods used are based on Krylov sequences and are closely related to modern iterative methods for standard matrix problems such as sets of linear equations and eigenvalue calculations. We show how these methods can be applied to problems in Control Theory such as controllability, observability and model reduction. All the methods are based on the use of state-space models, which may be very sparse and of high dimensionality. For example, we show how one may compute an approximate solution to a Lyapunov equation arising from discrete-time linear dynamic system with a large sparse system matrix by the use of the Arnoldi Algorithm, and so obtain an approximate Grammian matrix. This has applications in model reduction. The close relation between the matrix Lanczos algorithm and the algebraic structure of linear control systems is also explored. 1

### Citations

1323 |
GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems
- Saad, Schultz
- 1986
(Show Context)
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1087 | The Algebraic Eigenvalue Problem - Wilkinson |

764 |
The Symmetric Eigenvalue Problem
- Parlett
- 1980
(Show Context)
Citation Context ...atrices. The idea was to reduce a general matrix to tridiagonal form, from which the eigenvalues could be easily determined. For symmetric matrices, the Lanczos Algorithm has been studied extensively =-=[17, 53]-=-. In that case, the convergence of the algorithm, when used to compute eigenvalues, has been extensively analyzed in [43, 52, 57, 61], [69, p270ff]. This algorithm is particularly suited for large spa... |

613 | Matrix Perturbation Theory
- Stewart, Sun
- 1990
(Show Context)
Citation Context ... k = 1; 2; 3; \Delta \Delta \Delta kA k k 2s`ffi k : Furthermore, if A is diagonalizable, then the above also holds for ffi = ae(A) ! 1. Proof: One can find a consistent matrix norm such that kAksffi =-=[63]-=-. Since all norms on a finite dimensional space are equivalent, one can find a constant ` such that kMk 2s`kMksfor all matrices M . The result follows from kA k k 2s`kA k ks`kAk ks`ffi k : If A is dia... |

572 | The theory of matrices - Gantmacher - 1959 |

377 | An iteration method for the solution of the eigenvalue problem of linear differential and integral
- Lanczos
- 1950
(Show Context)
Citation Context ...m, such as tridiagonal, and represents the projection of the original matrix operator A onto the Krylov Space, typically of smaller dimension. The Lanczos Algorithm was originally proposed by Lanczos =-=[47]-=- as a method for the computation of eigenvalues of symmetric and nonsymmetric matrices. The idea was to reduce a general matrix to tridiagonal form, from which the eigenvalues could be easily determin... |

369 |
Principal component analysis in linear systems: Controllability, observability and model reduction
- Moore
- 1981
(Show Context)
Citation Context ... of D r , and e 1 = ( 1 0 \Delta \Delta \Delta 0 ) T denotes the initial coordinate unit vector of appropriate dimensions. -- 11 -- 4 Model Reduction via Balanced Realization The balanced realization =-=[51]-=- is a method for balancing the gains between inputs and states with those between states and outputs, and isolating the states with small gains. These small gain states can be truncated away without d... |

277 |
All Optimal Hankel Norm Approximations of Linear Multivariable Systems and Their Lm Error Bounds
- Glover
(Show Context)
Citation Context ...e gains between inputs and states with those between states and outputs, and isolating the states with small gains. These small gain states can be truncated away without disturbing the system by much =-=[2, 22, 27]-=-. The Lanczos-type algorithms can be used to compute approximate balanced realizations for systems of very large dimensionality, whereas most other methods are restricted to systems of only modest dim... |

267 |
The principle of minimized iterations in the solution of the matrix eigenvalue problem
- Arnoldi
- 1951
(Show Context)
Citation Context ...situation has not been addressed for the block case. The Lanczos Algorithm [47] is an example of a method that generates bases for Krylov subspaces starting with a given vector. The Arnoldi Algorithm =-=[4] can be th-=-ought of as a "one-sided" method, which generates one sequence of vectors that span the reachable space. 2.1 Arnoldi Algorithm The first algorithm we will describe is the Arnoldi Algorithm [... |

149 | An Implementation of the Look-Ahead Lanczos Algorithm for Non-Hermitian Matrices
- Freund, Gutknecht, et al.
- 1993
(Show Context)
Citation Context ..., p388ff]. More recently, several modifications allowing the Lanczos process to continue after such breakdowns have been proposed in [35, 34, 55], and a numerical implementation has been developed in =-=[23, 24]-=-. The close connection between the modified Non-symmetric Lanczos Algorithm and orthogonal polynomials with respect to indefinite inner products is discussed in [7, 8, 28]. Recently, in [10, 54] the c... |

103 | A LookAhead Lanczos Algorithm for Unsymmetric Matrices - Parlett, Taylor, et al. - 1985 |

98 | On the partial stochastic realization problem
- Byrnes, Lindquist
- 1997
(Show Context)
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87 | Numerical solution of the stable, non-negative definite Lyapunov equation
- Hammarling
- 1982
(Show Context)
Citation Context ...m (18) [51]. Effective methods for solving (27), (28) directly for the Cholesky factors without forming W c , W o and for finding the SVD of a matrix product without forming the product were given in =-=[37]-=- and [39, 5], respectively, thus enhancing the numerical accuracy of the results. Partition the matrices in (30) conformally as b A = ` b A 11 b A 12 b A 21 b A 22 ' ; b B = ` b B 1 b B 2 ' ; b C = ( ... |

80 |
The block Lanczos methods for computing eigenvalues
- Golub, Underwood
- 1977
(Show Context)
Citation Context ...alues, has been extensively analyzed in [43, 52, 57, 61], [69, p270ff]. This algorithm is particularly suited for large sparse matrix problems. A block Lanczos analog has been studied and analyzed in =-=[29, 17, 53]-=-. However, until recently, the nonsymmetric Lanczos Algorithm has received much less attention. Some recent computational experience with this algorithm can be found in [16]. Besides some numerical st... |

70 |
Analytic properties of Schmidt pairs for a Hankel operator and the generalized Schur-Takagi problem
- Adamjan, Arov, et al.
- 1971
(Show Context)
Citation Context ...expansions) exist, and many are equivalent to the Lanczos Process [11, 13, 32, 31, 54, and refs. therein], and many use related algorithms on related Hankel matrices, though based on different theory =-=[1, 36, 66]-=-. The methods presented in this paper are distinctive in that they are based on the use of state-space descriptions, which facilitate the generalization of many SISO methods to MIMO systems, and which... |

64 | Padé-Type Approximation and General Orthogonal Polynomials - Brezinski - 1980 |

61 |
Model reduction using a projection formulation
- Villemagne, Skelton
- 1987
(Show Context)
Citation Context ...hich yields a transfer function b F (s) = 1 X i=0 b F i s i+1 (38) for which b F i = F i for i = 0; \Delta \Delta \Delta ; k for some given k. The parameters F i are called the high frequency moments =-=[68]-=- or Markov Parameters [40, 62]. It is a simple consequence of the matrix identity (Neumann Series) (I \Gamma M) \Gamma1 = 1 X i=0 M i whenever ae(M) ! 1: that the transfer function for (18) and (19) s... |

58 |
On the rates of convergence of the Lanczos and the block-Lanczos methods
- Saad
- 1980
(Show Context)
Citation Context ...For symmetric matrices, the Lanczos Algorithm has been studied extensively [17, 53]. In that case, the convergence of the algorithm, when used to compute eigenvalues, has been extensively analyzed in =-=[43, 52, 57, 61]-=-, [69, p270ff]. This algorithm is particularly suited for large sparse matrix problems. A block Lanczos analog has been studied and analyzed in [29, 17, 53]. However, until recently, the nonsymmetric ... |

55 |
The computation of eigenvalues and eigenvectors of very large sparse matrices
- Paige
- 1971
(Show Context)
Citation Context ...For symmetric matrices, the Lanczos Algorithm has been studied extensively [17, 53]. In that case, the convergence of the algorithm, when used to compute eigenvalues, has been extensively analyzed in =-=[43, 52, 57, 61]-=-, [69, p270ff]. This algorithm is particularly suited for large sparse matrix problems. A block Lanczos analog has been studied and analyzed in [29, 17, 53]. However, until recently, the nonsymmetric ... |

46 |
new identification and model reduction algorithm via singular value decomposition
- Kung, “A
- 1978
(Show Context)
Citation Context ..., expressed as a rational function of polynomials (see e.g. [25, Ch. 15 x10], [18, 32, 42]). Several authors have also looked at the MIMO realization problem (see e.g. [67] and refs. therein, such as =-=[3, 14, 46, 70]-=-. Many of the algorithms involved are recursive algorithms which are intimately related to the Clustered Lanczos Algorithm (see e.g. [11, 32, 54]). We describe two approaches for the MIMO case propose... |

44 |
Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms
- Laub, Heath, et al.
- 1987
(Show Context)
Citation Context ...be transformed by contragredient (or congruence) transformations into c W c = T \Gamma1 W c T \GammaT ; c W o = T T W o T : (31) The balanced realization may be computed by the following prescription =-=[48, 50]-=-. Define L c L T c , L o L T o as the Cholesky factorizations of W c , W o , respectively. Compute the SVD of the product L T o L c to obtain the factorization U \SigmaV T = L T o L c , where U; V are... |

44 |
Krylov-subspace methods for the Sylvester equation
- Hu, Reichel
- 1992
(Show Context)
Citation Context ...us and is treated in [59]. In addition, one can obtain an error bound for the discrete time solution. Similar techniques have been proposed for the more general Sylvester equation AX \Gamma XB = C in =-=[56]-=-, in which the approximate solutions can be recursively generated as the Arnoldi process advances. We show how the Arnoldi Algorithm yields an approximate solution to (32), then we show that it satisf... |

43 |
Estimates for some computational techniques in linear algebra
- Kaniel
- 1966
(Show Context)
Citation Context ...For symmetric matrices, the Lanczos Algorithm has been studied extensively [17, 53]. In that case, the convergence of the algorithm, when used to compute eigenvalues, has been extensively analyzed in =-=[43, 52, 57, 61]-=-, [69, p270ff]. This algorithm is particularly suited for large sparse matrix problems. A block Lanczos analog has been studied and analyzed in [29, 17, 53]. However, until recently, the nonsymmetric ... |

41 |
Lanczos Algorithms for Large Symmetric
- Cullum, Willoughby
- 1985
(Show Context)
Citation Context ...atrices. The idea was to reduce a general matrix to tridiagonal form, from which the eigenvalues could be easily determined. For symmetric matrices, the Lanczos Algorithm has been studied extensively =-=[17, 53]-=-. In that case, the convergence of the algorithm, when used to compute eigenvalues, has been extensively analyzed in [43, 52, 57, 61], [69, p270ff]. This algorithm is particularly suited for large spa... |

40 |
Model reduction with balanced realizations: An error bound and frequency weighted generalizations
- Enns
- 1984
(Show Context)
Citation Context ...e gains between inputs and states with those between states and outputs, and isolating the states with small gains. These small gain states can be truncated away without disturbing the system by much =-=[2, 22, 27]-=-. The Lanczos-type algorithms can be used to compute approximate balanced realizations for systems of very large dimensionality, whereas most other methods are restricted to systems of only modest dim... |

40 | The Padé table and its relation to certain algorithms of numerical analysis - Gragg - 1972 |

25 |
Approximate linear realizations of given dimension via ho’s algorithm
- Zeiger, McEwen
- 1974
(Show Context)
Citation Context ..., expressed as a rational function of polynomials (see e.g. [25, Ch. 15 x10], [18, 32, 42]). Several authors have also looked at the MIMO realization problem (see e.g. [67] and refs. therein, such as =-=[3, 14, 46, 70]-=-. Many of the algorithms involved are recursive algorithms which are intimately related to the Clustered Lanczos Algorithm (see e.g. [11, 32, 54]). We describe two approaches for the MIMO case propose... |

23 | The unsymmetric Lanczos algorithms and their relations to Padé approximation, continued fractions, and the qd algorithm
- Gutknecht
- 1990
(Show Context)
Citation Context ...hole process from the beginning with different starting vectors [69, p388ff]. More recently, several modifications allowing the Lanczos process to continue after such breakdowns have been proposed in =-=[35, 34, 55]-=-, and a numerical implementation has been developed in [23, 24]. The close connection between the modified Non-symmetric Lanczos Algorithm and orthogonal polynomials with respect to indefinite inner p... |

20 |
The nonsymmetric Lanczos algorithm and controllability
- Boley, Golub
- 1991
(Show Context)
Citation Context ...] can be thought of as a "one-sided" method, which generates one sequence of vectors that span the reachable space. 2.1 Arnoldi Algorithm The first algorithm we will describe is the Arnoldi =-=Algorithm [4, 9, 69]-=-, which is a recursive way to generate an orthonormal basis for the Krylov space generated by a given matrix A and vectors X. It will be seen that it is also a way to reduce a given matrix to block up... |

18 |
A generalized nonsymmetric Lanczos procedure
- Cullum, Kerner, et al.
- 1989
(Show Context)
Citation Context ...ed and analyzed in [29, 17, 53]. However, until recently, the nonsymmetric Lanczos Algorithm has received much less attention. Some recent computational experience with this algorithm can be found in =-=[16]. Besides -=-some numerical stability problems, the method suffered from the possibility of an incurable breakdown from which the only way to "recover" was to restart the whole process from the beginning... |

14 |
Nonsymmetric Lanczos and Finding Orthogonal Polynomials Associated with Indefinite Weights, Numerical Analysis report NA-90-09
- Boley, Elhay, et al.
- 1990
(Show Context)
Citation Context ...entation has been developed in [23, 24]. The close connection between the modified Non-symmetric Lanczos Algorithm and orthogonal polynomials with respect to indefinite inner products is discussed in =-=[7, 8, 28]-=-. Recently, in [10, 54] the close relation was observed independently between the Lanczos Algorithm and the controllability- observability structure of dynamical systems. All the above papers address ... |

12 |
Modified moments for indefinite weight functions
- Golub, Gutknecht
- 1989
(Show Context)
Citation Context ...entation has been developed in [23, 24]. The close connection between the modified Non-symmetric Lanczos Algorithm and orthogonal polynomials with respect to indefinite inner products is discussed in =-=[7, 8, 28]-=-. Recently, in [10, 54] the close relation was observed independently between the Lanczos Algorithm and the controllability- observability structure of dynamical systems. All the above papers address ... |

11 | Rational Chebyshev approximation on the unit disk
- TREFETHEN
- 1981
(Show Context)
Citation Context ...expansions) exist, and many are equivalent to the Lanczos Process [11, 13, 32, 31, 54, and refs. therein], and many use related algorithms on related Hankel matrices, though based on different theory =-=[1, 36, 66]-=-. The methods presented in this paper are distinctive in that they are based on the use of state-space descriptions, which facilitate the generalization of many SISO methods to MIMO systems, and which... |

9 |
Controllability and observability in multivariable control systems
- Gilbert
- 1963
(Show Context)
Citation Context ... S o as the complements of S c and S o , respectively, as well as their mutual intersections: S co = S c " S o ; S co = S c " S o ; S co = S c " S o ; S co = S c " S o : (21) The K=-=alman Decomposition [26, 41]-=- of (18), (19) is obtained by applying a similarity transformation T = ( T co ; T co ; T co ; T co ) where each T ab is a basis for the corresponding S ab . When T is applied, we obtain the new system... |

9 |
Computing the SVD of a product of two matrices
- Heath, Laub, et al.
- 1986
(Show Context)
Citation Context ...1]. Effective methods for solving (27), (28) directly for the Cholesky factors without forming W c , W o and for finding the SVD of a matrix product without forming the product were given in [37] and =-=[39, 5]-=-, respectively, thus enhancing the numerical accuracy of the results. Partition the matrices in (30) conformally as b A = ` b A 11 b A 12 b A 21 b A 22 ' ; b B = ` b B 1 b B 2 ' ; b C = ( b C 1 b C 2 ... |

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- Boley, Lee, et al.
- 1992
(Show Context)
Citation Context ...n problem (see e.g. [67] and refs. therein, such as [3, 14, 46, 70]. Many of the algorithms involved are recursive algorithms which are intimately related to the Clustered Lanczos Algorithm (see e.g. =-=[11, 32, 54]-=-). We describe two approaches for the MIMO case proposed recently for constructing such a smaller state-space model, both based on the use of a large state-space model. The first approach was proposed... |

8 |
Computation of Balancing Transformations
- Laub
- 1980
(Show Context)
Citation Context ...he grammians will be transformed by contragredient (or congruence) transformations into �Wc = T −1 WcT −T , � Wo = T T WoT. (31) The balanced realization may be computed by the following prescription =-=[48, 50]-=-. Define LcL T c , LoL T o as the Cholesky factorizations of Wc, Wo, respectively. Compute the SVD of the product L T o Lc to obtain the factorization UΣV T = L T o Lc, where U,V are orthogonal matric... |

7 |
An error bound for a discrete reduced order model of a linear multivariable system
- Al-Saggaf, Franklin
- 1987
(Show Context)
Citation Context ...e gains between inputs and states with those between states and outputs, and isolating the states with small gains. These small gain states can be truncated away without disturbing the system by much =-=[2, 22, 27]-=-. The Lanczos-type algorithms can be used to compute approximate balanced realizations for systems of very large dimensionality, whereas most other methods are restricted to systems of only modest dim... |

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- Gutknecht, Trefethen
- 1982
(Show Context)
Citation Context ...expansions) exist, and many are equivalent to the Lanczos Process [11, 13, 32, 31, 54, and refs. therein], and many use related algorithms on related Hankel matrices, though based on different theory =-=[1, 36, 66]-=-. The methods presented in this paper are distinctive in that they are based on the use of state-space descriptions, which facilitate the generalization of many SISO methods to MIMO systems, and which... |

5 |
Block-Krylov Component Synthesis Method for Structural Model Reduction
- Craig, Hale
- 1988
(Show Context)
Citation Context ...ained by the oblique projection satisfies (CT )(L T AT ) i (L T B) = CA i B for i = \Gammap \Gamma s \Gamma 1; \Delta \Delta \Delta ; +q + t + 1: This approach has been exploited very successfully in =-=[15, 64, 65]-=- to applications in large flexible space structures. One problem with this method is that the block Nonsymmetric Lanczos Algorithm may break down, if the matrix in step 7 of the algorithm is singular.... |

5 |
Approximation of infinitedimensional systems
- Gu, Khargonekar, et al.
- 1989
(Show Context)
Citation Context ...nd (19) are state-space realizations of the transfer function (39). In applications, F (s) may be either the usual case of a finite dimensional system of large order or an infinite dimensional system =-=[33]. In eithe-=-r case, the model reduction problem is "solved" by constructing (realizing) a lower order model of the form (18) such that b C b A i b B = F i for i = 0; \Delta \Delta \Delta ; k, for a give... |

5 |
Mathematical description of linear systems
- Kalman
- 1963
(Show Context)
Citation Context ... S o as the complements of S c and S o , respectively, as well as their mutual intersections: S co = S c " S o ; S co = S c " S o ; S co = S c " S o ; S co = S c " S o : (21) The K=-=alman Decomposition [26, 41]-=- of (18), (19) is obtained by applying a similarity transformation T = ( T co ; T co ; T co ; T co ) where each T ab is a basis for the corresponding S ab . When T is applied, we obtain the new system... |

5 |
A Decentralized Linear Quadratic Control Design Method for Flexible Structures
- Su
- 1989
(Show Context)
Citation Context ...K(A \Gamma1 ; A \Gamma1 B; p), respectively [68], where M \GammaT j (M T ) \Gamma1 j (M \Gamma1 ) T . In order to create models matching both the high frequency moments and the low frequency moments, =-=[64]-=- propose combining the vectors generated from the Block Lanczos method using A, B, C T with those from the Lanczos method using A \Gamma1 , A \Gamma1 B, C T . This is based on the following theorem, w... |

5 |
Dooren, Numerical aspects of system and control algorithms
- Van
- 1989
(Show Context)
Citation Context ...n of a transfer function b F (s), expressed as a rational function of polynomials (see e.g. [25, Ch. 15 x10], [18, 32, 42]). Several authors have also looked at the MIMO realization problem (see e.g. =-=[67]-=- and refs. therein, such as [3, 14, 46, 70]. Many of the algorithms involved are recursive algorithms which are intimately related to the Clustered Lanczos Algorithm (see e.g. [11, 32, 54]). We descri... |

4 |
A Second Course on Linear Systems
- Desoer
- 1969
(Show Context)
Citation Context ...he state to zero, and the unobservable space S o as the subspace of the x-space which one cannot detect using the outputs. A classical algebraic characterization of these spaces is given by Theorem 2 =-=[19]-=-. S c j K(A; B; 1); S o j nullspace n ( K(A T ; C T ; 1) ) T o j K(A T ; C T ; 1) ? ; (20) -- 9 -- where S ? denotes the orthogonal complement of the set S. One may also define the uncontrollable spac... |

3 |
Model reduction for linear multivariable systems
- Hickin, Sinha
- 1980
(Show Context)
Citation Context ...ction b F (s) = 1 X i=0 b F i s i+1 (38) for which b F i = F i for i = 0; \Delta \Delta \Delta ; k for some given k. The parameters F i are called the high frequency moments [68] or Markov Parameters =-=[40, 62]-=-. It is a simple consequence of the matrix identity (Neumann Series) (I \Gamma M) \Gamma1 = 1 X i=0 M i whenever ae(M) ! 1: that the transfer function for (18) and (19) satisfies the identity F (s) = ... |

3 |
Computation of "Balancing" Transformations
- Laub
- 1980
(Show Context)
Citation Context ...be transformed by contragredient (or congruence) transformations into c W c = T \Gamma1 W c T \GammaT ; c W o = T T W o T : (31) The balanced realization may be computed by the following prescription =-=[48, 50]-=-. Define L c L T c , L o L T o as the Cholesky factorizations of W c , W o , respectively. Compute the SVD of the product L T o L c to obtain the factorization U \SigmaV T = L T o L c , where U; V are... |

2 |
New results on the algebraic theory of linear systems: The solution of the cover problems
- Antoulas
- 1983
(Show Context)
Citation Context ..., expressed as a rational function of polynomials (see e.g. [25, Ch. 15 x10], [18, 32, 42]). Several authors have also looked at the MIMO realization problem (see e.g. [67] and refs. therein, such as =-=[3, 14, 46, 70]-=-. Many of the algorithms involved are recursive algorithms which are intimately related to the Clustered Lanczos Algorithm (see e.g. [11, 32, 54]). We describe two approaches for the MIMO case propose... |

2 |
Recursive algorithms for the matrix Pad'e problem
- Bultheel
- 1980
(Show Context)
Citation Context |

2 |
partial realizations, transfer functions and canonical forms
- On
- 1979
(Show Context)
Citation Context ... a given k. In the SISO case, such a reduced order realization may be found via computation of a transfer function b F (s), expressed as a rational function of polynomials (see e.g. [25, Ch. 15 x10], =-=[18, 32, 42]-=-). Several authors have also looked at the MIMO realization problem (see e.g. [67] and refs. therein, such as [3, 14, 46, 70]. Many of the algorithms involved are recursive algorithms which are intima... |