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In this paper, a new numerical algorithm for an eigenvalue
assignment problem, which arises from a singular system control, is
developed. The algorithm is based on orthogonal row/column compressions
which can be implemented in a numerically reliable way

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you can request a copy directly from the authors.

... In Ho (1999, 2002), numerical approaches for the IEAP are presented. The work of Chu and Ho (1999) is based on some numerical algorithms which consist of an orthogonal reduction to an upper Hessenberg form and a linear recursion deduced from 2 Â 2 Givens transformations, while the work of Chu and Ho (2002) is based on orthogonal row/ column compressions which can be implemented in a numerically stable way. It is well known that the solution to the problem of eigenvalue assignment in a linear descriptor system is generally not unique. ...

The problem of infinite eigenvalue assignment in the descriptor system
via state feedback control u = Kx is considered. The problem is related to a group of recursive equations. By proposing a general complete parametric solution to this group of recursive equations, a general complete parametric approach is presented for the proposed infinite eigenvalue assignment problem. General parametric forms of the closed-loop eigenvectors and the feedback gain matrix are given in terms of certain parameter vectors which represent the design degrees of freedom. The approach involves mainly a singular value decomposition of the matrix E and a singular value decomposition of a lower dimension matrix, and thus is very simple and requires less computational work. Moreover, it overcomes the defects of some previous works. An example is given to illustrate the effect of the approach.

... The parametrisation of controllers based on eigenvalue and eigenstructure assignment as a state-space approach has attracted significant attention in both theoretical and application points of view. Some related research studies which have been accomplished in this field can be considered as follows: improvement of the dynamic performance of power systems (Sattar 2006), stabilisation of individual generators with statefeedback-controlled SVCs through pole assignment (Zhou 2010), disturbance attenuation in multivariable linear systems (Duan, Irwin, and Liu 2000), design of reconfigurable control system (Esna Ashari, Khaki Sedigh, and Yazdanpanah 2005), extension of the state-feedback design for linear distributed parameter systems and robust stability of linear large-scale systems using eigenstructure assignment (Labibi, Lohmann, Khaki Sedigh, and Jabedar Maralani 2001;Deutscher and Harkort 2009), pole structure assignment in implicit, linear and uncontrollable systems (Loiseau and Zagalak 2009), application of symbolic algebra techniques for implementing output-feedback pole assignment algorithms for uncertain systems (Zheng, Zolotas, and Wang 2006), robust pole placement (Kautsky, Nichols, and Van Dooren 1985;Benzaouia, Mesquine, Naib, and Hmamed 2001), optimal pole assignment for discrete-time linear systems (Zhou, Li, Duan, and Wang 2009), static output feedback pole assignment (Carotenuto, Franze`, and Muraca 2001;Franze`, Carotenuto, and Muraca 2005;Bachelier, Bosche, and Mehdi 2006;Yang and Orsi 2007), numerical algorithm for an eigenvalue assignment problem which arises from a singular system (Chu and Ho 2002), pole placement of continuous linear time-invariant (LTI) systems by means of suboptimal periodic feedback in which a performance index is minimised (Lavaei, Sojoudi, and Aghdam 2010), robust pole assignment in high-order descriptor linear systems (Duan and Yu 2008), and examination of the sensitivity of the pole assignment (Higham, Konstantinov, Mehrmann, and Petkov 2004). As it is seen, in parallel with application research studies, many studies have been performed for improving the theoretical bases of these methods and also overcoming the probable drawbacks which may be encountered in some practical cases. ...

In this article, an improved method for eigenvalue assignment via state feedback in the linear time-invariant multivariable systems is proposed. This method is based on elementary similarity operations, and involves mainly utilisation of vector companion forms, and thus is very simple and easy to implement on a digital computer. In addition to the controllable systems, the proposed method can be applied for the stabilisable ones and also systems with linearly dependent inputs. Moreover, two types of state-feedback gain matrices can be achieved by this method: (1) the numerical one, which is unique, and (2) the parametric one, in which its parameters are determined in order to achieve a gain matrix with minimum Frobenius norm. The numerical examples are presented to demonstrate the advantages of the proposed method.

In this paper, we consider the partial pole assignment problem (PPAP) for high order control systems. It is shown that solving the PPAP is essentially solving a pole assignment for a linear system of a much lower order, and the robust PPAP is then concerning the robust pole assignment problem for this linear system. Based on this theory, a rather simple algorithm for solving the robust PPAP is proposed, and numerical examples show that this algorithm does lead to comparable results with earlier algorithms, but at much lower computational cost.

Der Beitrag beschreibt einen neuen Zugang zum Entwurf linearer Deskriptorsysteme durch Polvorgabe. Bei R-steuerbaren Deskriptorsystemen ermöglicht das vorgestellte Verfahren neben der Vorgabe sämtlicher endlicher Eigenwerte auch die gezielte Verschiebung der unendlichen Eigenwerte. Die resultierende Zustandsregelung liefert daher nicht nur eine vollständige Lösung des Polvorgabeproblems bei linearen Deskriptorsystemen, sondern ermöglicht gleichzeitig auch eine stabile Indexreduktion. Nach einer geeigneten Transformation werden dazu einige der Deskriptorvariablen als zusätzliche fiktive Eingangsgrößen interpretiert, die zusammen mit den realen Stellgrößen für den Entwurf einer konstanten Zustandsrückführung herangezogen werden können. Diese Rückführung wird in Anlehnung an den Entwurf einer Verkopplungsregelung anhand eines reduzierten expliziten Zustandsraummodells entworfen, wobei die Berücksichtigung der zusätzlichen algebraischen Nebenbedingungen problemlos möglich ist.

Zusammenfassung
Der Beitrag beschreibt ein neues Verfahren zur Synthese konstanter Zustandsrückführungen für lineare, zeitinvariante Deskriptorsysteme. Ausgehend von bekannten parametrischen Entwurfsmethoden für gewöhnliche Zustandsraumbeschreibungen wird eine Synthesevorschrift angegeben, welche sämtliche Freiheitsgrade beim Entwurf regulärer Deskriptorsysteme und deren Auswirkungen auf die Dynamik des geschlossenen Regelkreises sichtbar macht. Insbesondere lassen sich alle steuerbaren endlichen sowie unendlichen Eigenwerte explizit vorgeben.

We propose a new block algorithm for the generalized Sylvester-observer equation : XA - FXE = GC, where the matrices A, E, and C are given, the matrices X, F, and G need to be computed, and matrix E could be singular. The algorithm is based on an orthogonal decomposition of the triplet (A, E, C) to observer-Hessenberg-triangular form. It is a natural generalization of the widely-known observer-Hessenberg algorithm for the Sylvester-observer equation XA - FX = GC, which arises in state estimation of a standard first-order state-space control system. An application of the proposed algorithm is made to state estimation of second order control systems modeling a wide variety of vibrating structures. For dense un-structured data, the algorithm is more efficient than the recently proposed SVD-based algorithm of the authors, numerically reliable and heavily composed of Basic Linear Algebra Subprograms -Level 3 (BLAS 3) operations, which make it an ideal candidate for high-performance computing.

A direct algorithm is suggested for the computation of the linear state feedback for multi-input systems such that the resultant closed-loop system matrix has specified eigenvalues. The extra freedom can be used in different ways, for example to decrease some norm of the feedback matrix or to improve the condition of some of the eigenvalues of the closed-loop matrix. The algorithm uses unitary transformations for numerical reliability, and is based on ideas from the QR algorithm for solving the eigenproblem. The stability of the algorithm is proven by doing a backward rounding error analysis, and numerical examples are given as well.

Solutions of linear singular systems.- Time domain analysis.- Feedback control.- State observation.- Dynamic compensation for singular systems.- Structurally stable compensation in singular systems.- System analysis via transfer matrix.- to discrete-time singular systems.- Optimal control.- Some further topics.

The regular matrix pencil [sE – A –BK] is considered that characterizes a closed-loop descriptor system , u(t) = Kx(t) + ũ(t). A carefully contrived and efficient method is given for the determination of a set of matrices K such that the determinant ∣sE – A – BK∣ is a constant value independent of s. The problem is formulated as an infinite-eigenvalue assignment by finite-gain descriptor-variable feedback via a singular value decomposition of the matrix E. The result is interesting in its own right and finds application in controller and observer design.

In this paper, the infinite eigenvalue assignment problem for singular systems is studied. Necessary and sufficient conditions are presented under which there exists a state feedback such that the closed-loop system is regular and has only infinite eigenvalues. The main result is proved constructively based on some simple numerical algorithms. These numerical algorithms consist of an orthogonal reduction to an upper (block) Hessenberg form and a simple linear recursion deduced from 2×2 Givens transformations.

In this paper, the state feedback pole assignment problem of singular systems is investigated. It is well known that the pole assignment of singular systems is equivalent to the one for normal systems. The above theory is re-developed through the application of the orthogonal similarity transformation, which is numerically stable. The merit of this approach is that almost all existing methods for assigning poles in normal systems can be generalized to solve the problem in a reliable way. Parallel algorithms for this generalization procedure are also discussed.

This volume comprises a set of research papers that together will provide an up-to-date survey of the current state of the art in numerical analysis. The contributions are based on talks given at a conference in honour of Jim Wilkinson, one of the foremost pioneers in numerical analysis. The contributors were all his colleagues and collaborators and are leading figures in their respective fields. The breadth of Jim Wilkinson's research is reflected in the main themes covered: linear algebra, error analysis and computer arithmetic, algorithms, and mathematical software. Particular topics covered include analysis of the Lanczos algorithm, determining the nearest defective matrix to a given one, QR-factorizations, error propagation models, parameter estimation problems, sparse systems, and shape-preserving splines. As a whole the volume reflects the current vitality of numerical analysis and will prove an invaluable reference for all numerical analysts.

In this paper we propose a new recursive algorithm for computing the staircase form of a matrix pencil, and implicitly its Kronecker structure. The algorithm compares favorably to existing ones in terms of elegance, versatility, and complexity. In particular, the algorithm without any modification yields the structural invariants associated with a generalized state-space system and its system pencil. Two related geometric aspects are also discussed: we show that an appropriate choice of a set of nested spaces related to the pencil leads directly to the staircase form; we extend the notion of deflating subspace to the singular pencil case.

It is shown how deflation by unitary equivalence transformations
can be used to solve the eigenvalue assignment problem for descriptor
systems using state feedback. The system is first transformed to a form
that reveals its controllability. The uncontrollable part of the system,
if any, is then discarded, and the assignments are continued with the
completely controllable part of the system. To give insight into the
approach, a mathematical method for the solution is derived, without
considering numerical issues, using the idea of deflation. It is shown
how this method can be modified to derive a numerically efficient
algorithm. The numerical stability of the algorithm is proved and a
numerical example is presented

The author considers the design of observers for the discrete
singular system Ex ( k +1)= Ax ( k )+ Bu
( k ), y ( k )= Cx ( k ),
placing special emphasis on the problems of state reconstruction and
minimal-time state reconstruction. It is shown that for a singular
system, finite poles can be moved to infinity by state feedback and the
state can be reconstructed by causal observers

A general framework for pole placement of descriptor systems

- D Chu
- D Chan
- D W C Ho

D. Chu, D. Chan, and D. W. C. Ho, " A general framework for pole place-ment of descriptor systems, " Int. J. Control, vol. 67, pp. 135–152, 1997.

Computational aspects of the Jordan canonical form

- M G Cox
- S Hammarling