Real Schur Form Research Articles

COMPARING ONE-SHOT AND MULTI-SHOT METHODS FOR SOLVING PERIODIC RICCATI DIFFERENTIAL EQUATIONS

One-shot methods and recently proposed multi-shot methods for computing stabilizing solutions of continuous-time periodic Riccati differential equations are examined and evaluated on two test problems: (i) a stabilization problem for an artificially constructed time-varying linear system for which the exact solution is known; (ii) a nonlinear stabilization problem for a devil stick juggling model along a periodic trajectory. The numerical comparisons are performed using both general purpose and symplectic integration methods for solving the associated Hamiltonian differential systems. In the multi-shot method a stable subspace is determined using recent algorithms for computing a reordered periodic real Schur form. The results show the increased accuracy achievable by combining multi-shot methods with structure preserving (symplectic) integration techniques.

Direct Eigenvalue Reordering in a Product of Matrices in Periodic Schur Form

A direct method for eigenvalue reordering in a product of a K-periodic matrix sequence in periodic or extended periodic real Schur form is presented and analyzed. Each reordering of two adjacent sequences of diagonal blocks is performed tentatively to guarantee backward stability and involves solving a K-periodic Sylvester equation (PSE) and constructing a K-periodic sequence of orthogonal transformation matrices. An error analysis of the direct reordering method is presented, and results from computational experiments confirm the stability and accuracy of the method for well-conditioned as well as ill-conditioned problems. These include matrix sequences with fixed and time-varying dimensions, and sequences of small and large periodicity.

Computation of transfer function matrices of periodic systems

We present a numerical approach to evaluate the transfer function matrices of a periodic system corresponding to lifted state-space representations as constant systems. The proposed pole-zero method determines each entry of the transfer function matrix in a minimal zeros-poles-gain representation. A basic computation is the minimal realization of special single-input single-output periodic systems, for which both balancing-related as well as orthogonal periodic Kalman forms based algorithms can be employed. The main computational ingredient to compute poles is the extended periodic real Schur form of a periodic matrix. This form also underlies the solution of periodic Lyapunov equations when computing minimal realizations via balancing-related techniques. To compute zeros and gains, numerically stable fast algorithms are proposed, which are specially tailored to particular single-input single-output periodic systems. The new method relies exclusively on reliable numerical computations and is well suited for robust software implementations. Numerical examples computed with MATLAB-based implementations show the applicability of the proposed method to high-order periodic systems.

Matlab code for sorting real Schur forms

AbstractIn Matlab 6, there exists a command to generate a real Schur form, wheras another transforms a real Schur form into a complex one. There do not exist commands to prescribe the order in which the eigenvalues appear on the diagonal of the upper (quasi‐) triangular factor T. For the complex case, a routine is sketched in Golub and Van Loan (Matrix Computations (3rd edn). The John Hopkins University Press: Baltimore and London, 1996), that orders the diagonal of T according to their distance to a target value τ. In this technical note, we give a Matlab routine to sort real Schur forms in Matlab. It is based on a block‐swapping procedure by Bai and Demmel (Linear Algebra and Its Applications 1993; 186: 73) We also describe how to compute a partial real Schur form (see Saad (Numerical methods for large eigenvalue problems. Manchester University Press: Manchester, 1992.)) in case the matrix A is very large. Sorting real Schur forms, both partially and completely, has important applications in the computation of real invariant subspaces. Copyright © 2002 by John Wiley & Sons, Ltd.

Design of linear-phase IIR digital filters using singular perturbational model reduction

A new technique for the design of a passband linear-phase infinite impulse response (IIR) digital filter is presented. This technique is based on computing orthonormal bases, which span the left and right invariant subspaces associated with the large and small eigenvalues of the cross-Gramian matrix W/sub c0/, which tends to be symmetric and easily constructed. The orthonormal bases are computed by using the ordered real Schur form decomposition, which has been proved to be robust and numerically stable. The proposed technique is illustrated by several examples.

On the cubic convergence of the Paardekooper method

We show a simple way how asymptotic convergence results can be conveyed from a simple Jacobi method to a block Jacobi method. Our pilot methods are the well known symmetric Jacobi method and the Paardekooper method for reducing a skew-symmetric matrix to the real Schur form. We show resemblance in the quadratic and cubic convergence estimates, but also discrepances in the asymptotic assumptions. By numerical tests we confirm that our asymptotic assumptions for the Paardekooper method are most general.

The real two-zero algorithm: a parallel algorithm to reduce a real matrix to a real Schur form

We introduce a new method to reduce a real matrix to a real Schur form by a sequence of similarity transformations that are 3D orthogonal transformations. Two significant features of this method are that: all the transformed matrices and all the computations are done in the real field; and it can be easily parallelized. We call the algorithm that uses this method the real two-zero (RTZ) algorithm. We describe both serial and parallel implementations of the RTZ algorithm. Our tests indicate that the rate of convergence to a real Schur form is quadratic for real near-normal matrices with real distinct eigenvalues. Suppose n is the order of a real matrix A. In order to choose a sequence of 3D orthogonal transformations on A, we need to determine some ordering on triples in T={(k,l,m)|1/spl les/k<l<m/spl les/n}, where (k,l,m) defines the three coordinates under the 3D transformation. We show how the ordering of the triples used in our implementations can be generated cyclically in an algorithm.<<ETX>>

Jacobi’s Method for Skew-Symmetric Matrices

A Jacobi-type method is given for reducing real skew-symmetric matrices to real Schur form. $4 \times 4$ rotations are used to reduce the “off diagonal” part of the matrix. Asymptotic quadratic convergence is established for the special cyclic method, provided the eigenvalues are distinct.

On swapping diagonal blocks in real Schur form

We discuss a new version of an existing algorithm for reordering the eigenvalues on the diagonal of a matrix in real Schur form by performing an orthogonal similarity transformation. A detailed error analysis and software description are presented. Numerical examples show the superiority of our algorithm over previous algorithms.

A modified Schur method for solving algebraic Riccati equations

A modification to the Schur method for solving algebraic Riccati equations is presented. It is shown that only half rows of the orthogonal matrix which brings the associated Hamiltonian matrix into real Schur form is necessary to solve the equation. The required storage and computation is reduced somewhat.

A real Schur form method for modeling singularly perturbed systems

A method is presented for modeling a two-time-scale system in the singularly perturbed form. The method uses an ordered real Schur decomposition, which can be efficiently computed using standard subroutines from EISPACK. Three results are given. First, it is shown that any two-time-scale system can be modeled in the singularly perturbed form by a transformation into an ordered real Schur form, followed by balancing. Second, under some conditions on the ordered real Schur decomposition, a procedure is given to achieve the modeling task with all fast variables chosen from the original state variables. Third, necessary and sufficient conditions are given to achieve modeling by permutation of the original state variables. >

Pole assignment in real Schur form

Abstract In this paper an algorithm for pole assignment in real Schur form by gain state feedback is given. The proposed algorithm shifts, at each iteration, 1 or 2 poles in the last diagonal block of the closed-loop system. Only the last few columns in real Schur form are altered and therefore the Schur form is preserved at each iteration. After the completion of each iteration, the orthogonal similarity transformation is applied so that a new block is moved to the last diagonal block and the iterations continue. The desired feedback matrix K at the end of the algorithm is found to be the sum of the component matrices, each of which assign one or two poles.

On optimal pole assignment in a specified region

In this paper, the problem of pole assignment and linear quadratic design have been amalgamated. A design method that has the merits of both approaches is presented, and the problem of assigning the poles in a specified region is solved. The characterizations for quadratic cost functions have been studied and these results used to assign poles in real Schur form in a region so that the optimality is maintained. Designed procedures have been elaborated with examples.

An efficient Schur method to solve the stabilizing problem

An efficient solution to the stabilizing problem for both continuous and discrete constant linear systems is presented. The real Schur form reduction of the system state matrix is used.