P. Van Dooren

Robust stability and stabilization for singular systems with state delay and parameter uncertainty

Considers the problems of robust stability and stabilization for uncertain continuous singular systems with state delay. The parametric uncertainty is assumed to be norm bounded. The purpose of the robust stability problem is to give conditions such that the uncertain singular system is regular, impulse free, and stable for all admissible uncertainties, while the purpose of the robust stabilization is to design a state feedback control law such that the resulting closed-loop system is robustly stable. These problems are solved via the notions of generalized quadratic stability and generalized quadratic stabilization, respectively. Necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are derived. A strict linear matrix inequality (LMI) design approach is developed. An explicit expression for the desired robust state feedback control law is also given. Finally, a numerical example is provided to demonstrate the application of the proposed method.

The Computation of Kronecker's Canonical Form of a Singular Pencil

We develop stable algorithms for the computation of the Kronecker structure of an arbitrary pencil. This problem can be viewed as a generalization of the wellknown eigenvalue problem of pencils of the type LambdaI- A. We first show that the elementary divisors (Lambda- a)(f) of a regular pencil LambdaB - A can be retrieved with a deflation algorithm acting on the expansion (Lambda - alpha)B -(A - alphaB). This method is a straightforward generalization of Kublanovskaya’s algorithm for the determination of the Jordan structure of a constant matrix. We also show how to use this method to determine the structure of the infinite elementary divisors of Lambda-B. In the case of singular pencils, the occurrence of Kronecker indices-containing the singularity of the pencil-somewhat complicates the problem. Yet our algorithm retrieves these indices with no additional effort, when determining the elementary divisors of the pencil. The present ideas can also be used to separate from an arbitrary pencil a smaller regular pencil containing only the finite elementary divisors of the original one. This is shown to be an effective tool when used together with the QZ algorithm.

Model Reduction of MIMO Systems via Tangential Interpolation

In this paper, we address the problem of constructing a reduced order system of minimal McMillan degree that satisfies a set of tangential interpolation conditions with respect to the original system under some mild conditions. The resulting reduced order transfer function appears to be generically unique and we present a simple and efficient technique to construct this interpolating reduced order system. This is a generalization of the multipoint Pade technique which is particularly suited to handle multiinput multioutput systems.

Optimization Problems over Positive Pseudopolynomial Matrices

The Nesterov characterizations of positive pseudopolynomials on the real line, the imaginary axis, and the unit circle are extended to the matrix case. With the help of these characterizations, a class of optimization problems over the space of positive pseudopolynomial matrices is considered. These problems can be solved in an efficient manner due to the inherent block Toeplitz or block Hankel structure induced by the characterization in question. The efficient implementation of the resulting algorithms is discussed in detail. In particular, the real line setting of the problem leads naturally to ill-conditioned numerical systems. However, adopting a Chebyshev basis instead of the natural basis for describing the polynomial matrix space yields a restatement of the problem and of its solution approach with much better numerical properties.

A Grassmann--Rayleigh Quotient Iteration for Computing Invariant Subspaces

The classical Rayleigh quotient iteration (RQI) allows one to compute a one-dimensional invariant subspace of a symmetric matrix A. Here we propose a generalization of the RQI which computes a p-dimensional invariant subspace of A. Cubic convergence is preserved and the cost per iteration is low compared to other methods proposed in the literature.

High performance algorithms for Toeplitz and block Toeplitz matrices

In this paper, we present several high performance variants of the classical Schur algorithm to factor various Toeplitz matrices. For positive definite block Toeplitz matrices, we show how hyperbolic Householder transformations may be blocked to yield a block Schur algorithm. This algorithm uses BLAS3 primitives and makes efficient use of a memory hierarchy. We present three algorithms for indefinite Toeplitz matrices. Two of these are based on look-ahead strategies and produce an exact factorization of the Toeplitz matrix. The third produces an inexact factorization via perturbations of singular principal miners. We also present an analysis of the numerical behavior of the third algorithm and derive a bound for the number of iterations to improve the accuracy of the solution. For rank-deficient Toeplitz least-squares problems, we present a variant of the generalized Schur algorithm that avoids breakdown due to an exact rank-deficiency. In the presence of a near rank-deficiency, an approximate rank factorization of the Toeplitz matrix is produced. Finally, we suggest an algorithm to solve the normal equations resulting from a real Toeplitz least-squares problem based on transforming to Cauchy-like matrices. This algorithm exploits both realness and symmetry in the normal equations.

Rational and polynomial matrix factorizations via recursive pole-zero cancellation

We develop a recursive algorithm for obtaining factorizations of the type R(λ)=R1(λ)R2(λ) where all three matrices are rational and R1(λ) is nonsingular. Moreover the factors R1(λ) and R2(λ) are such that either the poles of [R1(λ)]-1 and R2(λ) are in a prescribed region Γ of the complex plane, or their zeros. Such factorizations cover the specific cases of coprime factorization, inner-outer factorization, GCD extraction, and many more. The algorithm works on the state-space (or generalized state-space) realization of R(λ) and derives in a recursive fashion the corresponding realizations of the factors.

An improved algorithm for the computation of structural invariants of a system pencil and related geometric aspects

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

Cubically Convergent Iterations for Invariant Subspace Computation

We propose a Newton-like iteration that evolves on the set of fixed dimensional subspaces of

Model reduction via truncation: an interpolation point of view

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