Realization Of Linear Systems Research Articles

LINEAR SYSTEMS ON GRAPHS WITH A REAL STRUCTURE

A degeneration of a smooth projective curve to a strongly stable curve gives rise to a specialization map from divisors on curves to divisors on graphs. In this paper we show that this specialization behaves well under the presence of real structures. In particular we study real linear systems on graphs with a real structure and we prove results on them comparable to results in the classical theory of real curves. We also consider generalizations to metric graphs and tropical curves

On HSS and AHSS iteration methods for nonsymmetric positive definite Toeplitz systems

Two iteration methods are proposed to solve real nonsymmetric positive definite Toeplitz systems of linear equations. These methods are based on Hermitian and skew-Hermitian splitting (HSS) and accelerated Hermitian and skew-Hermitian splitting (AHSS). By constructing an orthogonal matrix and using a similarity transformation, the real Toeplitz linear system is transformed into a generalized saddle point problem. Then the structured HSS and the structured AHSS iteration methods are established by applying the HSS and the AHSS iteration methods to the generalized saddle point problem. We discuss efficient implementations and demonstrate that the structured HSS and the structured AHSS iteration methods have better behavior than the HSS iteration method in terms of both computational complexity and convergence speed. Moreover, the structured AHSS iteration method outperforms the HSS and the structured HSS iteration methods. The structured AHSS iteration method also converges unconditionally to the unique solution of the Toeplitz linear system. In addition, an upper bound for the contraction factor of the structured AHSS iteration method is derived. Numerical experiments are used to illustrate the effectiveness of the structured AHSS iteration method.

Computation of positive realization of MIMO hybrid linear systems in the form of second Fornasini-Marchesini model

Computation of positive realization of MIMO hybrid linear systems in the form of second Fornasini-Marchesini modelThe realization problem for positive multi-input and multi-output (MIMO) linear hybrid systems with the form of second Fornasini-Marchesini model is formulated and a method based on the state variable diagram for finding a positive realization of a given proper transfer matrix is proposed. Sufficient conditions for the existence of the positive realization of a given proper transfer matrix are established. A procedure for computation of a positive realization is proposed and illustrated by a numerical example.

RATIONAL KRYLOV FOR REAL PENCILS WITH COMPLEX EIGENVALUES

A rational Krylov algorithm for eigenvalue computation is described. It is usable on a real matrix pencil with complex eigenvalues and builds up a real basis. The main purpose is to get real reduced models of a real linear dynamic system. Two variants are described, one where two real vectors are added to the Krylov space in each step and another where just one real vector is added in each step. Results are reported from one small example that has been used earlier and where the solution is known, and one more realistic example, a linear descriptor system from a computational fluid dynamics application.

Identification of Multivariable Canonical State-Space Representations

The paper introduces a method to estimate a linear state-space model from its input-output data. Its starting point is the conversion of the state-space model into an input-output ARMAX-like model. In order to conveniently estimate the parameters of this ARMAX model, it is necessary, given the structural characteristics of the initial state-space model, to assume that the dynamic state matrix A is nonderogatory. Under this condition, it is shown that the input-output model and the state-space model from which it is derived have the same order. Based on this important result in linear system realization theory, a canonical state-space representation is finally obtained through a simple construction technique. The developed method can be regarded as an alternative to the standard subspace methods particularly in situations where the state coordinates basis need to be maintained constant with respect to the input-output realization used for the estimation.

A notion of possible controllability for uncertain linear systems with structured uncertainty

This paper introduces a notion of possible controllability for a class of uncertain linear systems with structured uncertainty described by averaged integral quadratic constraints. This notion relates to the question of when a state is controllable for some possible value of the uncertainty. The notion of possible controllability is motivated by a desire to extend the theory of minimal realization for linear time invariant systems to the case of uncertain systems with structured uncertainty.

A Period-Specific Realization of Linear Continuous-Time Systems

It is a well-known fact that a weighting pattern matrix is realizable as a linear periodic system if and only if the matrix is separable and periodic. This fact, however, can not cope with a reasonable question when a weighting pattern matrix can be realized as a linear periodic system with a specific period of time. This paper answers the question by constructing two types of period-specific realizations. Moreover, this paper describes in detail how the lowest dimension of period-specific realizations is identified.

非線形系の重み付き平衡実現とモデル低次元化

This paper proposes a weighted balanced realization and model order reduction for nonlinear systems. This is an extension of a nonlinear balanced realization, which is based on singular value analysis of the nonlinear Hankel operator, to include input and output weights. The proposed method is a natural nonlinear extension of weighted balanced realization for linear systems. Furthermore, it is proved that a reduced order model derived by the proposed procedure is also stable if the original model is so. Finally, the effectiveness of the method is shown by a numerical simulation.

Comment on �Precision Free-Inertial Navigation with Gravity Compensation by an Onboard Gradiometer�

An apparent weakness in the arguments within the derivation of [1, Appendix] is identified using explicit numerical examples which further demonstrate that the results are of limited benefit. Our prior experience in specifying linear system realizations [2], [3] alerted us to an apparent problem with the new alternative procedure offered in [1, Appendix], as a potentially more straightforward way to achieve a multi-input multi-output (MIMO) linear system realization from a matrix power spectrum or, equivalently, from a given matrix correlation function by explicitly delineating both the structure and parameter values of an underlying white noise-driven Linear Time Invariant (LTI) state variable model that provides such a vector random process as its output. The arguments for the derivation of the matrix Lyapunov equation (that the variance/cross-covariance matrix satisfies ) [5, pp. 222-226] are quite familiar to many analysts but fall short in Ref. [1]’s attempt to extend them in its overly concise but appealing result, where Eq. A4 equates a function of one variable on the left hand side 2 to a function of two variables on the right hand side as an obvious impossibility. Among the beneficial results offered in [2], [3] is a non degenerate statistically stationary 2-channel numerical example: 3 Ryy(τ ) =   1 6 e−2|τ| + 1 6 e−|τ| .. 1 4 e−2|τ| · · · · · · · · · 1 4e −2|τ| .. 12e −|τ|   ⇔ Syy(s) = [ 2−s (4−s2)(1−s2) 1 (4−s2) 1 (4−s2) 1 (1−s2) ] = W (−s)W (s), (1) where these second order statistics correspond to a demonstrable closed-form solution both for the intermediate (non-unique [3]) matrix spectral factorization (MSF): W (s) ≡ [ −s−(√7/2) (2+s)(1+s) −1/2 (2+s)(1+s) −s−(√7/2) (2+s)(1+s) 3/2 (2+s)(1+s) ] = H(sI − F )−1G (2) (where details of accomplishing the MSF here are provided in [2, Appendix B]) and for the resulting associated linear system realization: d dt x(t) = F1x(t) +G1w′(t) =   0 1 0 0 −1 −3 0 0 0 0 0 1 0 0 −2 −3   x(t) +   −1 0 (6−√7)/2 −1/2 −1 0 (6−√7)/2 3/2  w′(t), (3) ∗Research funded by TeK Associates’ IR&D Contract No. 07-101, 9 Meriam St., Suite 7-R, Lexington, MA 02420-5336, USA. Tel./Fax: (781) 862-5870, thomas h kerr @ msn.com, www.TeKAssociates.biz 1Precise regularity conditions guranteeing that the steady-state constant symmetric positive definite matrix solution Px can be obtained from Eq. A7 with dPx(t)/dt ≡ 0 is that F have only eigenvalues with real parts strictly negative and that (F,L) be a controllable pair, where Q = LLT (and where L is a factor resulting from a Choleski decomposition of Q) [4]. 2In using this form, at this point in the derivation in [1], the assumption of stationarity had not yet been invoked as a slight mis-step of prematurely using an assumption that is invoked later. 3The power spectrum matrix depicted here on the right hand side of Eq. 1 consists of elements represented in the frequency domain by the bilateral Laplace transform, which relates to the Fourier transform via the substitution s = ω.

Balanced realizations of regime-switching linear systems

In this work, we establish a framework for balanced realization of linear systems subject to regime switching modulated by a continuous-time Markov chain with a finite state space. First, a definition of balanced realization is given. Then a ρ-balanced realization is developed to approximate the system of balancing equations, which is a system of time-varying algebraic equations. When the state space of the Markov chain is large, the computational effort becomes a real concern. To resolve this problem, we introduce a two-time-scale formulation and use decomposition/aggregation and averaging techniques to reduce the computational complexity. Based on the two-time-scale formulation, further approximation procedures are developed. Numerical examples are also presented for demonstration.

Global SCD algorithm for real positive definite linear systems with multiple right-hand sides

In the present paper, we present the left conjugate direction (LCD) method for linear systems with multiple right-hand sides. We define the method by using left and right conjugate direction matrices and Petrov–Galerkin condition. The method has no breakdown for real positive definite systems. The method reduces to the usual CG-type method when A is symmetric positive definite. We also show how to apply these methods for solving the Lyapunov matrix equation. Finally a numerical experiment is given to illustrate the result.

Solving Real Linear Systems with the Complex Schur Decomposition

If the complex Schur decomposition is used to solve a real linear system, then the computed solution generally has a complex component because of roundoff error. We show that the real part of the computed solution that is obtained in this way solves a nearby real linear system. Thus, it is “numerically safe” to obtain real solutions to real linear systems via the complex Schur decomposition. This result is useful in certain Kronecker product situations where fast linear equation solving is made possible by reducing the involved matrices to their complex Schur form. This is critical because in these applications one cannot work with the real Schur form without greatly increasing the volume of work.

Robust [formula omitted] performance analysis and synthesis of linear polytopic discrete-time periodic systems via LMIs

A particular class of uncertain linear discrete-time periodic systems is considered. The problem of robust stabilization of real polytopic linear discrete-time periodic systems via a periodic state-feedback control law is tackled here, along with H 2 performance optimization. Using additional slack variables and the periodic Lyapunov lemma, an extended sufficient condition of robust H 2 stabilization is proposed. Based on periodic parameter-dependent Lyapunov functions, this last condition is shown to be always less conservative than the more classic one based on the quadratic stability framework. This is illustrated on a numerical example from the literature.

A frontal solver for the 21st century

SUMMARY In recent years there have been a number of important developments in frontal algorithms for solving the large sparse linear systems of equations that arise fromnite-element problems. We report on the design of a new fully portable and ecient frontal solver for large-scale real and complex unsymmetric linear systems fromnite-element problems that incorporates these developments. The new package oers both aexible reverse communication interface and a simple to use all-in-one interface, which is designed to make the package more accessible to new users. Other key features include automatic element ordering using a state-of-the-art hybrid multilevel spectral algorithm, minimal main memory requirements, the use of high-level BLAS, and facilities to allow the solver to be used as part of a parallel multiple front solver. The performance of the new solver, which is written in Fortran 95, is illustrated using a range of problems from practical applications. The solver is available as package HSL MA42 ELEMENT within the HSL mathematical software library and, for element problems, supersedes the well-known MA42 package. Copyright ? 2006 John Wiley & Sons, Ltd.

A condition for the superiority of the (2, 2)-step methods over the related Chebyshev method

The (2, 2)-step iterative methods related to an optimal Chebyshev method for solving a real and nonsymmetric linear system A x = b are studied. A condition under which the asymptotic rate of convergence of the optimal Chebyshev method can be improved by a related (2, 2)-step method is derived. The condition depends not only on the location of the extreme eigenvalues of T but also on whether the ratio of the minor axis to the major axis of the optimal ellipse is greater than the golden ratio. Two numerical examples are given to illustrate our results.

Solving Linear Systems Whose Elements Are Nonlinear Functions of Intervals

The paper addresses the problem of solving linear algebraic systems the elements of which are, in the general case, nonlinear functions of a given set of independent parameters taking on their values within prescribed intervals. Three kinds of solutions are considered: (i) outer solution, (ii) interval hull solution, and (iii) inner solution. A simple direct method for computing a tight outer solution to such systems is suggested. It reduces, essentially, to inverting a real matrix and solving a system of real linear equations whose size n is the size of the original system. The interval hull solution (which is a NP-hard problem) can be easily determined if certain monotonicity conditions are fulfilled. The resulting method involves solving n+1 interval outer solution problems as well as 2n real linear systems of size n. A simple iterative method for computing an inner solution is also given. A numerical example illustrating the applicability of the methods suggested is solved.

The superiority of a new type (2,2)-step iterative method over the related Chebyshev method

A new type (2,2)-step iterative method related to an optimal Chebyshev method is developed for solving real and nonsymmetric linear systems of the form Ax= b. It is an extension of the (2,2)-step iterative method introduced in [Numer. Linear Algebra Appl. 7 (2000) 169]. The superiority of the new type (2,2)-step iterative method over the optimal Chebyshev method is derived in the case where the known (2,2)-step iterative method may not improve the asymptotic rate of convergence. Two numerical examples are given to illustrate the results.

A Comparison of Two Equivalent Real Formulations for Complex-Valued Linear Systems Part 2: Results

Many iterative linear solver packages focus on real-valued systems and do not deal well with complex-valued systems, even though preconditioned iterative methods typically apply to both real and complex-valued linear systems. Instead, commonly available packages such as PETSc and Aztec tend to focus on the real-valued systems, while complex-valued systems are seen as a late addition. At the same time, by changing the complex problem into an equivalent real formulation (ERF), a real valued solver can be used. In this paper we consider two ERF’s that can be used to solve complex-valued linear systems. We investigate the spectral properties of each and show how each can be preconditioned to move eigenvalues in a cloud around the point (1,0) in the complex plane. Finally, we consider an interleaved formulation, combining each of the previously mentioned approaches, and show that the interleaved form achieves a better outcome than either separate ERF. [This article is the second part of a sequence of reports. See the December 2002 issue for Part 1—Editor.]

Solution Methods for $\mathbb R$-Linear Problems in $\mathbb C^n $

We consider methods, both direct and iterative, for solving an ${\mathbb R}$-linear system $Mz+ M_{\#}\overline{z}= b$ in ${\mathbb C}^n$ with a pair of matrices $M, M_{\#} \in {\mathbb C}^{n \times n}$ and a vector $b\in {\mathbb C}^n$. Algorithms that avoid formulating the problem as an equivalent real linear system in ${\mathbb R}^{2n}$ are introduced. Conversely, this implies that real linear systems in ${\mathbb R}^{2n}$ can be solved with the methods proposed in this paper. Our study is motivated by Krylov subspace iterations, which when used with the real formulation can be disastrous in the standard linear case. Related matrix analysis and spectral theory are developed.

On the solution sets of particular classes of linear interval systems

We characterize the solution set S of real linear systems Ax= b by a set of inequalities if b lies between some given bounds b ̄ , b ̄ and if the n× n coefficient matrix A varies similarly between two bounds A̱ and Ā. In addition, we restrict A to a particular class of matrices, for instance the class of the symmetric, the skew-symmetric, the persymmetric, the Toeplitz, and the Hankel matrices, respectively. In this way, we generalize the famous Oettli–Prager criterion (Numer. Math. 6 (1964) 405), results by Hartfiel (Numer. Math. 35 (1980) 355) and the contents of the papers (in: R.B. Kearfott, V. Kreinovich (Eds.), Applications of Interval Computations, Kluwer, Boston, MA, 1996, pp. 61–79) and (SIAM J. Matrix Anal. Appl. 18 (1997) 693).