Hamiltonian Pencil Research Articles

Matrix Pencils with Coefficients that have Positive Semidefinite Hermitian Parts

We analyze when an arbitrary matrix pencil is strictly equivalent to a dissipative Hamiltonian pencil and show that this heavily restricts the spectral properties. In order to relax the spectral properties, we introduce matrix pencils with coefficients that have positive semidefinite Hermitian parts. We relate the Kronecker structure of these pencils to that of an underlying skew-Hermitian pencil and discuss their regularity, index, numerical range, and location of eigenvalues. Further, we study matrix polynomials with positive semidefinite Hermitian coefficients and use linearizations with positive semidefinite Hermitian parts to derive sufficient conditions for having a spectrum in the left half plane and to derive bounds on the index.

Estimation of Structured Distances to Singularity for Matrix Pencils with Symmetry Structures: A Linear Algebra--Based Approach

We study the structured distance to singularity for a given regular matrix pencil $A+sE$, where $(A,E)\in \mathbb S \subseteq (\mathbb C^{n,n})^2$. This includes Hermitian, skew-Hermitian, $*$-even, $*$-odd, $*$-palindromic, T-palindromic, and dissipative Hamiltonian pencils. We present a purely linear algebra-based approach to derive explicit computable formulas for the distance to the nearest structured pencil $(A-\Delta_A)+s(E-\Delta_E)$ such that $A-\Delta_A$ and $E-\Delta_E$ have a common null vector. We then obtain a family of computable lower bounds for the unstructured and structured distances to singularity. Numerical experiments suggest that in many cases, there is a significant difference between structured and unstructured distances. This approach extends to structured matrix polynomials with higher degrees.

Hamiltonian evolutions of twisted polygons in

In this paper we find a discrete moving frame and their associated invariants along projective polygons in , and we use them to describe invariant evolutions of projective N-gons. We then apply a reduction process to obtain a natural Hamiltonian structure on the space of projective invariants for polygons, establishing a close relationship between the projective N-gon invariant evolutions and the Hamiltonian evolutions on the invariants of the flow. We prove that any Hamiltonian evolution is induced on invariants by an invariant evolution of N-gons—what we call a projective realization—and both evolutions are connected explicitly in a very simple way. Finally, we provide a completely integrable evolution (the Boussinesq lattice related to the lattice W3-algebra), its projective realization in and its Hamiltonian pencil. We generalize both structures to n-dimensions and we prove that they are Poisson, defining explicitly the n-dimensional generalization of the planar evolution (a discretization of the Wn-algebra). We prove that the generalization is completely integrable, and we also give its projective realization, which turns out to be very simple.

Numerical algorithms and existence results on LQ control of descriptor systems with conditions on x(0−) and stability

We present a numerical algorithm, based on using orthogonal matrices, and sufficient existence conditions for an LQ control problem of descriptor systems, which generalise the existing results. The associated Hamiltonian pencil may be singular and may have infinite generalised eigenvalues (IGEs) of multiplicity > 1. Besides on x(0+), we present optimality conditions on x(0−). We present sufficient conditions for stability of the optimal closed-loop descriptor system, as well as a stabilisation algorithm.

A characterization of solutions of the discrete-time algebraic Riccati equation based on quadratic difference forms

This paper is concerned with a characterization of all symmetric solutions to the discrete-time algebraic Riccati equation (DARE). Dissipation theory and quadratic difference forms from the behavioral approach play a central role in this paper. Along the line of the continuous-time results due to Trentelman and Rapisarda [H.L. Trentelman, P. Rapisarda, Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation, SIAM J. Contr. Optim. 40 (3) (2001) 969–991], we show that the solvability of the DARE is equivalent to a certain dissipativity of the associated discrete-time state space system. As a main result, we characterize all unmixed solutions of the DARE using the Pick matrix obtained from the quadratic difference forms. This characterization leads to a necessary and sufficient condition for the existence of a non-negative definite solution. It should be noted that, when we study the DARE and the dissipativity of the discrete-time system, there exist two difficulties which are not seen in the continuous-time case. One is the existence of a storage function which is not a quadratic function of state. Another is the cancellation between the zero and infinite singularities of the dipolynomial spectral matrix associated with the DARE, due to the infinite generalized eigenvalues of the associated Hamiltonian pencil. One of the main contributions of this paper is to demonstrate how to resolve these difficulties.

A Numerical Method for a Generalized Algebraic Riccati Equation

In this paper we develop a numerical method for computing the semistabilizing solution of a generalized algebraic Riccati equation (GARE). The semistabilizing solution of such a GARE has been used to characterize the solvability of the $(J, J^\prime)$‐spectral factorization problem for general rational matrices which have poles and zeros on the extended imaginary axis. The main difficulty for solving such a GARE is that its associated skew‐Hamiltonian/Hamiltonian pencil has eigenvalues on the extended imaginary axis; consequently, it is not clear which eigenspace of the associated skew‐Hamiltonian/Hamiltonian pencil can characterize the desired semistabilizing solution; i.e., it is not clear which eigenvectors and principal vectors corresponding to the eigenvalues on the extended imaginary axis should be contained in the eigenspace that we wish to compute, and hence the well‐known generalized eigenvalue approach for the classical algebraic Riccati equations cannot be directly employed for it. Our proposed method consists of computations of the eigendecomposition of the system pencil corresponding to the eigenvalues on the extended imaginary axis and the stable eigenspace of an augmented matrix pencil; hence, it is a generalization of the generalized eigenvalue approach for the classical algebraic Riccati equations.

ハミルトンペンシルの固有構造に着目した特異スペクトル分解

ハミルトンペンシルの固有構造に着目した特異スペクトル分解

Numerically Reliable Computation of Optimal Performance in Singular $H_{\inf}$ Control

A numerically stable algorithm is described for the computation of the optimal $H_{\inf}$ performance $\gamma_{{\rm opt}}$ when the feedthrough matrices D12 or D21 are rank-deficient. Using only orthogonal transformations, the singular $H_{\inf}$ problem is reduced to a regular subproblem and a standard Riccati-based $\gamma$-iteration is applied to this subproblem to compute $\gamma_{{\rm opt}}$. Various interpretations of this scheme are given in terms of the infinite zero structure of the plant and the deflation of Hamiltonian pencils. The implementation of this algorithm is straightforward and its performance and reliability are confirmed by extensive numerical testing.

Generalized Discrete-Time Riccati Theory

A Riccati-like equation, termed the generalized (discrete-time) algebraic Riccati equation, which incorporates as special cases both the standard and the constrained discrete-time algebraic Riccati equations, is introduced and investigated under the weakest possible assumptions imposed on the initial data. A complete characterization of the conditions under which such an equation of general form has a stabilizing solution is presented in terms of the so-called proper deflating subspace of the extended Hamiltonian pencil. An evaluation of an associated quadratic index along constrained stable trajectories is given in terms of the stabilizing solution to the generalized Riccati equation. Possible applications of the developed theory range from nonstandard spectral and inner-outer factorizations to $H^2 $ and $H^\infty $ control in singular cases. The results exposed in the present paper are the discrete-time counterpart of those stated in the authors’ previous paper concerning the generalized (singular) continuous-time Riccati theory. The results could be also seen as an extension to singular cases of the usual discrete-time algebraic Riccati equation theory (of indefinite sign).

Generalized continuous-time Riccati theory

A Riccati-like equation which incorporates as special cases the standard continuous-time algebraic Riccati equation and the constrained continuous-time algebraic Riccati equation is introduced and studied. A characterization of the conditions under which an equation of general form has a stabilizing solution are investigated in terms of the so-called proper deflating subspace of the extended Hamiltonian pencil.