SIAM's Advances in Design and Control series consists of texts and monographs dealing with all areas of design and control and their applications. Topics of interest include shape optimization, multidisciplinary design, trajectory optimization, feedback, and optimal control. The series focuses on the mathematical and computational aspects of engineering design and control that are usable in a wide variety of scientific and engineering disciplines.
A new rational Krylov method for the efficient solution of nonlinear eigenvalue problems, A(λ)x = 0, is proposed. This iterative method, called fully rational Krylov method for nonlinear eigenvalue problems (abbreviated as NLEIGS), is based on linear rational interpolation and generalizes the Newton rational Krylov method proposed in [R. Van Beeumen, K. Meerbergen, and W. Michiels, SIAM J. Sci. Comput., 35 (2013), pp. A327-A350]. NLEIGS utilizes a dynamically constructed rational interpolant of the nonlinear function A(λ) and a new companion-type linearization for obtaining a generalized eigenvalue problem with special structure. This structure is particularly suited for the rational Krylov method. A new approach for the computation of rational divided differences using matrix functions is presented. It is shown that NLEIGS has a computational cost comparable to the Newton rational Krylov method but converges more reliably, in particular, if the nonlinear function A(λ) has singularities nearby the target set. Moreover, NLEIGS implements an automatic scaling procedure which makes it work robustly independently of the location and shape of the target set, and it also features low-rank approximation techniques for increased computational efficiency. Small-and large-scale numerical examples are included. From the numerical experiments we can recommend two variants of the algorithm for solving the nonlinear eigenvalue problem.
The instability mechanisms, related to the implementation of distributed delay controllers in the context of finite spectrum assignment, were studied in detail in the past few years. In this note we introduce a distributed delay control law that assigns a finite closed-loop spectrum and whose implementation with a sum of point-wise delays is safe. This property is obtained by implicitly including a low-pass filter in the control loop. This leads to a closed-loop characteristic quasipolynomial of retarded type, and not one of neutral type, which was shown to be a cause of instability in previous schemes.
We propose a new uniform framework of Compact Rational Krylov (CORK) methods for solving large-scale nonlinear eigenvalue problems: A(λ)x = 0. For many years, linearizations are used for solving polynomial and rational eigenvalue problems. On the other hand, for the general nonlinear case, A(λ) can first be approximated by a (rational) matrix polynomial and then a convenient linearization is used. However, the major disadvantage of linearization based methods is the growing memory and orthogonalization costs with the iteration count, i.e., in general they are proportional to the degree of the polynomial. Therefore, the CORK family of rational Krylov methods exploits the structure of the linearization pencils by using a generalization of the compact Arnoldi decomposition. In this way, the extra memory and orthogonalization costs due to the linearization of the original eigenvalue problem are negligible for large-scale problems. Furthermore, we prove that each CORK step breaks down into an orthogonalization step of the original problem dimension and a rational Krylov step on small matrices. We also briefly discuss implicit restarting of the CORK method and how to exploit low rank structure. The CORK method is illustrated with two large-scale examples.Keywords : linearization, matrix pencil, rational Krylov, nonlinear eigenvalue problem MSC : 65F15, 15A22. COMPACT RATIONAL KRYLOV METHODS FOR NONLINEAR EIGENVALUE PROBLEMS *ROEL VAN BEEUMEN † , KARL MEERBERGEN † , AND WIM MICHIELS † Abstract. We propose a new uniform framework of Compact Rational Krylov (CORK) methods for solving large-scale nonlinear eigenvalue problems: A(λ)x = 0. For many years, linearizations are used for solving polynomial and rational eigenvalue problems. On the other hand, for the general nonlinear case, A(λ) can first be approximated by a (rational) matrix polynomial and then a convenient linearization is used. However, the major disadvantage of linearization based methods is the growing memory and orthogonalization costs with the iteration count, i.e., in general they are proportional to the degree of the polynomial. Therefore, the CORK family of rational Krylov methods exploits the structure of the linearization pencils by using a generalization of the compact Arnoldi decomposition. In this way, the extra memory and orthogonalization costs due to the linearization of the original eigenvalue problem are negligible for large-scale problems. Furthermore, we prove that each CORK step breaks down into an orthogonalization step of the original problem dimension and a rational Krylov step on small matrices. We also briefly discuss implicit restarting of the CORK method and how to exploit low rank structure. The CORK method is illustrated with two large-scale examples.
An eigenvalue based framework is developed for the stability analysis and stabilization of coupled systems with time-delays, which are naturally described by delay differential algebraic equations. The spectral properties of these equations are analyzed and a numerical method for computing characteristic roots and stability assessment is presented, thereby taking into account the effect of small delay perturbations on stability. Subsequently, the design of stabilizing controllers with a prescribed structure or order is addressed, based on a direct optimization approach. The effectiveness of the approach is illustrated with numerical examples. All algorithms have been implemented in publicly available software
The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. Our first important result is a characterization of a general nonlinear eigenvalue problem (NEP) as a standard but infinite dimensional eigenvalue problem involving an integration operator denoted B. In this paper we present a new algorithm equivalent to the Arnoldi method for the operator B. Although the abstract construction is infinite dimensional, it turns out that we can carry out the iteration in an exact way (without approximation) by using only standard linear algebra operations involving matrices (not operators). This is achieved by working with coefficients in a basis of scalar functions, typically polynomials. Due to the fact that the constructed method has a complete equivalence with the standard Arnoldi method, it also inherits many of its attractive properties. Another somewhat unexpected consequence of the construction is that the matrix of basis vectors should be expanded not only in the way done in standard Arnoldi. We expand the matrix of basis vectors not only with a column to the right, but also a block row below. We also show that the method can be interpreted as the standard Arnoldi method if applied to the generalized eigenvalue problem resulting from the spectral discretization of the operator. With this equivalence we reach a recommendation on how the scalar product should be chosen for an important class of nonlinear eigenvalue problems. Abstract The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. Our first important result is a characterization of a general nonlinear eigenvalue problem (NEP) as a standard but infinite dimensional eigenvalue problem involving an integration operator denoted B. In this paper we present a new algorithm equivalent to the Arnoldi method for the operator B. Although the abstract construction is infinite dimensional, it turns out that we can carry out the iteration in an exact way (without approximation) by using only standard linear algebra operations involving matrices (not operators). This is achieved by working with coefficients in a basis of scalar functions, typically polynomials. Due to the fact that the constructed method has a complete equivalence with the standard Arnoldi method, it also inherits many of its attractive properties. Another somewhat unexpected consequence of the construction is that the matrix of basis vectors should be expanded not only in the way done in standard Arnoldi. We expand the matrix of basis vectors not only with a column to the right, but also a block row below. We also show that the method can be interpreted as the standard Arnoldi method if applied to the generalized eigenvalue problem resulting from the spectral discretization of the operator. With this equivalence we reach a recommendation on how the scalar product should be chosen for an important class of nonli...
In this paper, we describe a stabilization method for linear time-delay systems which extends the classical pole placement method for ordinary di erential equations. Unlike methods based on ÿnite spectrum assignment, our method does not render the closed loop system, ÿnite dimensional but consists of controlling the rightmost eigenvalues. Because these are moved to the left half plane in a (quasi-)continuous way, we refer to our method as continuous pole placement. We explain the method by means of the stabilization of a linear ÿnite dimensional system in the presence of an input delay and illustrate its applicability to more general stabilization problems.
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