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

Venue: | Circuits, Systems, and Signal Processing |

Citations: | 48 - 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

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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... |

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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... |

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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... |

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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 ... |

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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 ... |

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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... |

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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... |

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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... |

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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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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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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 |
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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... |

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6 |
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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.... |

6 |
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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 |
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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 |
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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 |
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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 |
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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 |
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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... |

3 |
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(Show Context)
Citation Context |

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(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 |
partial realizations, transfer functions and canonical forms
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(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... |