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## Über dieses Buch

The purpose of Numerical Linear Algebra in Signals, Systems and Control is to present an interdisciplinary book, blending linear and numerical linear algebra with three major areas of electrical engineering: Signal and Image Processing, and Control Systems and Circuit Theory. Numerical Linear Algebra in Signals, Systems and Control will contain articles, both the state-of-the-art surveys and technical papers, on theory, computations, and applications addressing significant new developments in these areas. The goal of the volume is to provide authoritative and accessible accounts of the fast-paced developments in computational mathematics, scientific computing, and computational engineering methods, applications, and algorithms. The state-of-the-art surveys will benefit, in particular, beginning researchers, graduate students, and those contemplating to start a new direction of research in these areas. A more general goal is to foster effective communications and exchange of information between various scientific and engineering communities with mutual interests in concepts, computations, and workable, reliable practices.

## Inhaltsverzeichnis

### Chapter 1. The Anti-Reflective Transform and Regularization by Filtering

Abstract
Filtering methods are used in signal and image restoration to reconstruct an approximation of a signal or image from degraded measurements. Filtering methods rely on computing a singular value decomposition or a spectral factorization of a large structured matrix. The structure of the matrix depends in part on imposed boundary conditions. Anti-reflective boundary conditions preserve continuity of the image and its (normal) derivative at the boundary, and have been shown to produce superior reconstructions compared to other commonly used boundary conditions, such as periodic, zero and reflective. The purpose of this paper is to analyze the eigenvector structure of matrices that enforce anti-reflective boundary conditions. In particular, a new anti-reflective transform is introduced, and an efficient approach to computing filtered solutions is proposed. Numerical tests illustrate the performance of the discussed methods.
A. Aricò, M. Donatelli, J. Nagy, S. Serra-Capizzano

### Chapter 2. Classifications of Recurrence Relations via Subclasses of (H, m)-quasiseparable Matrices

Abstract
The results on characterization of orthogonal polynomials and Szegö polynomials via tridiagonal matrices and unitary Hessenberg matrices, respectively, are classical. In a recent paper we observed that tridiagonal matrices and unitary Hessenberg matrices both belong to a wider class of $$(H,1)$$-quasiseparable matrices and derived a complete characterization of the latter class via polynomials satisfying certain EGO-type recurrence relations. We also established a characterization of polynomials satisfying three-term recurrence relations via $$(H,1)$$-well-free matrices and of polynomials satisfying the Szegö-type two-term recurrence relations via $$(H,1)$$-semiseparable matrices. In this paper we generalize all of these results from $$scalar$$ (H,1) to the block (H, m) case. Specifically, we provide a complete characterization of $$(H,\,m)$$-quasiseparable matrices via polynomials satisfying $$block$$ EGO-type two-term recurrence relations. Further, $$(H,\,m)$$-semiseparable matrices are completely characterized by the polynomials obeying $$block$$ Szegö-type recurrence relations. Finally, we completely characterize polynomials satisfying m-term recurrence relations via a new class of matrices called $$(H,\,m)$$-well-free matrices.
T. Bella, V. Olshevsky, P. Zhlobich

### Chapter 3. Partial Stabilization of Descriptor Systems Using Spectral Projectors

Abstract
We consider the stabilization problem for large-scale linear descriptor systems in continuous- and discrete-time. We suggest a partial stabilization algorithm which preserves stable poles of the system while the unstable ones are moved to the left half plane using state feedback. Our algorithm involves the matrix pencil disk function method to separate the finite from the infinite generalized eigenvalues and the stable from the unstable eigenvalues. In order to stabilize the unstable poles, either the generalized Bass algorithm or an algebraic Bernoulli equation can be used. Some numerical examples demonstrate the behavior of our algorithm.
Peter Benner

### Chapter 4. Comparing Two Matrices by Means of Isometric Projections

Abstract
In this paper, we go over a number of optimization problems defined on a manifold in order to compare two matrices, possibly of different order. We consider several variants and show how these problems relate to various specific problems from the literature.
T. P. Cason, P. -A. Absil, P. Van Dooren

### Chapter 5. A Framelet-Based Algorithm for Video Enhancement

Abstract
Video clips are made up of many still frames. Most of the times, the frames are small perturbations of their neighboring frames. Recently, we proposed a framelet-based algorithm to enhance the resolution of any frames in a video clip by solving it as a super-resolution image reconstruction problem. In this paper, we extend the algorithm to video enhancement, where we compose a high-resolution video from a low-resolution one. An experimental result of our algorithm on a real video clip is given to illustrate the performance.
Raymond H. Chan, Yiqiu Dong, Zexi Wang

### Chapter 6. Perturbation Analysis of the Mixed-Type Lyapunov Equation

Abstract
This paper concerns the mixed-type Lyapunov equation $$X=A^*XB+B^*XA+Q,$$ where $$A,B,$$ and $$Q$$ are $$n\times n$$ complex matrices and $$A^*$$ the conjugate transpose of a matrix $$A.$$ A perturbation bound for the solution to this matrix equation is derived, an explicit expression of the condition number is obtained, and the backward error of an approximate solution is evaluated by using the techniques developed in Sun (Linear Algebra Appl 259:183–208, 1997), Sun and Xu (Linear Algebra Appl 362:211–228, 2003). The results are illustrated by using some numerical examples.
Mingsong Cheng, Shufang Xu

### Chapter 7. Numerical and Symbolical Methods for the GCD of Several Polynomials

Abstract
The computation of the Greatest Common Divisor (GCD) of a set of polynomials is an important issue in computational mathematics and it is linked to Control Theory very strong. In this paper we present different matrix-based methods, which are developed for the efficient computation of the GCD of several polynomials. Some of these methods are naturally developed for dealing with numerical inaccuracies in the input data and produce meaningful approximate results. Therefore, we describe and compare numerically and symbolically methods such as the ERES, the Matrix Pencil and other resultant type methods, with respect to their complexity and effectiveness. The combination of numerical and symbolic operations suggests a new approach in software mathematical computations denoted as hybrid computations. This combination offers great advantages, especially when we are interested in finding approximate solutions. Finally the notion of approximate GCD is discussed and a useful criterion estimating the $$strength$$ of a given approximate GCD is also developed.
Dimitrios Christou, Nicos Karcanias, Marilena Mitrouli, Dimitrios Triantafyllou

### Chapter 8. Numerical Computation of the Fixed Poles in Disturbance Decoupling for Descriptor Systems

Abstract
In this paper the algebraic characterizations for the fixed poles in the disturbance decoupling problem for descriptor systems are derived. These algebraic characterizations lead to a numerically reliable algorithm for computing the fixed poles. The algorithm can be implemented directly using existing numerical linear algebra tools such as LAPACK and Matlab.
Delin Chu, Y. S. Hung

### Chapter 9. Robust Control of Discrete Linear Repetitive Processes with Parameter Varying Uncertainty

Abstract
Repetitive processes propagate information in two independent directions where the duration of one of these is infinite.
They pose control problems that cannot be solved by application of results for other classes of 2D systems. This paper develops robust controller design algorithms for discrete linear processes based on the poly-quadratic stability that produce less conservative results than currently available alternatives.
Błażej Cichy, Krzysztof Gałkowski, Eric Rogers, Anton Kummert

### Chapter 10. Unique Full-Rank Solution of the Sylvester-Observer Equation and Its Application to State Estimation in Control Design

Abstract
Needs to be found, is a classical equation. There has been much study, both from theoretical and computational view points, on this equation. The results of existence and uniqueness are well-known and numerically effective algorithms have been developed in recent years (see, Datta [2]), to compute the solution.
Karabi Datta, Mohan Thapa

### Chapter 11. On Symmetric and Skew-Symmetric Solutions to a Procrustes Problem

Abstract
Using the projection theorem in a Hilbert space, the quotient singular value decomposition (QSVD) and the canonical correlation decomposition (CCD) in matrix theory for efficient tools, we obtained the explicit analytical expressions of the optimal approximation solutions for the symmetric and skew-symmetric least-squares problems of the linear matrix equation $$AXB = C$$. This can lead to new algorithms to solve such problems.
Yuan-Bei Deng, Daniel Boley

### Chapter 12. Some Inverse Eigenvalue and Pole Placement Problems for Linear and Quadratic Pencils

Abstract
Differential equation models for vibrating systems are associated with matrix eigenvalue problems. Frequently the undamped models lead to problems of the generalized eigenvalue type and damped models lead to problems of the quadratic eigenvalue type. The matrices in these systems are typically real and symmetric and are quite highly structured. The design and stabilisation of systems modelled by these equations (eg., undamped and damped vibrating systems) requires the determination of solutions to the inverse eigenvalue problems which are themselves real, symmetric and possibly have some other structural properties. In this talk we consider some pole assignment problems and inverse spectral problems for generalized and quadratic symmetric pencils, discuss some advances and point to some work that remains to be done.
Sylvan Elhay

### Chapter 13. Descent Methods for Nonnegative Matrix Factorization

Abstract
In this paper, we present several descent methods that can be applied to nonnegative matrix factorization and we analyze a recently developed fast block coordinate method called Rank-one Residue Iteration (RRI). We also give a comparison of these different methods and show that the new block coordinate method has better properties in terms of approximation error and complexity. By interpreting this method as a rank-one approximation of the residue matrix, we prove that it $$converges$$ and also extend it to the nonnegative tensor factorization and introduce some variants of the method by imposing some additional controllable constraints such as: sparsity, discreteness and smoothness.
Ngoc-Diep Ho, Paul Van Dooren, Vincent D. Blondel

### Chapter 14. A Computational Method for Symmetric Stein Matrix Equations

Abstract
In the present paper, we propose a numerical method for solving the sparse symmetric Stein equation $$AXA^T-X+BB^T=0.$$ Such problems appear in control problems, filtering and image restoration. The proposed method is a Krylov subspace method based on the global Arnoldi algorithm. We apply the global Arnoldi algorithm to extract low-rank approximate solutions to Stein matrix equations. We give some theoretical results and report some numerical experiments to show the effectiveness of the proposed method.
K. Jbilou, A. Messaoudi

### Chapter 15. Optimal Control for Linear Descriptor Systems with Variable Coefficients

Abstract
We study optimal control problems for general linear descriptor systems with variable coefficients. We derive necessary and sufficient optimality conditions for optimal solution. We also show how to solve these optimality systems via the solution of generalized Riccati-differential equations. and discussed how a modification of the cost functional leads to better solvability properties for the optimality system.
Peter Kunkel, Volker Mehrmann

### Chapter 16. Robust Pole Assignment for Ordinary and Descriptor Systems via the Schur Form

Abstract
In Chu (Syst Control Lett 56:303–314, 2007), the pole assignment problem was considered for the control system $$\dot{x} = Ax + Bu$$ with linear state-feedback $$u = Fx.$$ An algorithm using the Schur form has been proposed, producing good suboptimal solutions which can be refined further using optimization. In this paper, the algorithm is improved, incorporating the minimization of the feedback gain $$\|F\|.$$ It is also extended for the pole assignment of the descriptor system $$E\dot{x} = Ax + Bu$$ with linear state- and derivative-feedback $$u = Fx - G\dot{x}.$$ Newton refinement for the solutions is discussed and several illustrative numerical examples are presented.
Tiexiang Li, Eric King-wah Chu, Wen-Wei Lin

### Chapter 17. Synthesis of Fixed Structure Controllers for Discrete Time Systems

Abstract
In this paper, we develop a linear programming approach to the synthesis of stabilizing fixed structure controllers for a class of linear time invariant discrete-time systems. The stabilization of this class of systems requires the determination of a real controller parameter vector (or simply, a controller), $$K$$, so that a family of real polynomials, affine in the parameters of the controllers, is Schur. An attractive feature of the paper is the systematic approximation of the set of all such stabilizing controllers, $$K$$. This approximation is accomplished through the exploitation of the interlacing property of Schur polynomials and a systematic construction of sets of linear inequalities in $$K$$. The union of the feasible sets of linear inequalities provides an approximation of the set of all controllers, $$K$$, which render $$P(z,K)$$ Schur. Illustrative examples are provided to show the applicability of the proposed methodology. We also show a related result, namely, that the set of rational proper stabilizing controllers for single-input single-output linear time invariant discrete-time plants will form a bounded set in the controller parameter space if and only if the order of the stabilizing cannot be reduced any further. Moreover, if the order of the controller is increased, the set of higher order controllers will necessarily be unbounded.
Waqar A. Malik, Swaroop Darbha, S. P. Bhattacharyya

### Chapter 18. A Secant Method for Nonlinear Matrix Problems

Abstract
Nonlinear matrix equations arise in different scientific topics, such as applied statistics and control theory, among others. Standard approaches to solve them include and combine some variations of Newton’s method, matrix factorizations, and reduction to generalized eigenvalue problems. In this paper we explore the use of secant methods in the space of matrices, that represent a new approach with interesting features. For the special problem of computing the inverse or the pseudoinverse of a given matrix, we propose a specialized secant method for which we establish stability and q-superlinear convergence, and for which we also present some numerical results. In addition, for solving quadratic matrix equations, we discuss several issues, and present preliminary and encouraging numerical experiments.
Marlliny Monsalve, Marcos Raydan

### Chapter 19. On FastICA Algorithms and Some Generalisations

Abstract
The FastICA algorithm, a classical method for solving the one-unit linear ICA problem, and its generalisations are studied. Two interpretations of FastICA are provided, a scalar shifted algorithm and an approximate Newton method. Based on these two interpretations, two natural generalisations of FastICA on a full matrix are proposed to solve the parallel linear ICA problem. Specifically, these are a matrix shifted parallel ICA method and an approximate Newton-like parallel ICA method.
Hao Shen, Knut Hüper, Martin Kleinsteuber

### Chapter 20. On Computing Minimal Proper Nullspace Bases with Applications in Fault Detection

Abstract
We discuss computationally efficient and numerically reliable algorithms to compute minimal proper nullspace bases of a rational or polynomial matrix. The underlying main computational tool is the orthogonal reduction to a Kronecker-like form of the system matrix of an equivalent descriptor system realization. A new algorithm is proposed to compute a simple minimal proper nullspace basis, starting from a non-simple one. Minimal dynamic cover based computational techniques are used for this purpose. The discussed methods allow a high flexibility in addressing several fault detection related applications.
Andras Varga

### Chapter 21. Optimal Control of Switched System with Time Delay Detection of Switching Signal

Abstract
This paper deals with optimal control problems governed by switched systems with time delay detection of switching signal. We consider the switching sequence as well as the switching instants as decision variables. We present a two-level optimization method to solve it. In the first level, we fix the switching sequence, and introduce a time scaling transformation such that the switching instants are mapped into pre-assigned fixed knot points. Then, the transformed problem becomes a standard optimal parameter selection problem, and hence can be solved by many optimal control techniques and the corresponding optimal control software packages, such as MISER. In the second level, we consider the switching sequence as decision variables. We introduce a discrete filled function method to search for a global optimal switching sequence. Finally, a numerical example is presented to illustrate the efficiency of our method.
C. Z. Wu, K. L. Teo, R. Volker
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