An extended Hessenberg form for Hamiltonian matrices

A unitary symplectic similarity transformation for a special class of Hamiltonian matrices to extended Hamiltonian Hessenberg form is presented. Whereas the classical Hessenberg form links to Krylov subspaces, the extended Hessenberg form links to extended Krylov subspaces. The presented algorithm generalizes thus the classic reduction to Hamiltonian Hessenberg form and offers more freedom in the choice of Hamiltonian condensed forms, to be used within an extended Hamiltonian QR algorithm. Theoretical results identifying the structure of the extended Hamiltonian Hessenberg form and proofs of uniqueness of the reduction process are included. Numerical experiments confirm the validity of the approach.     

Almost invariant subspaces: An approach to high gain feedback design--Part II: Almost conditionally invariant subspaces

In Part I of this paper, almost controlled invariant subspaces were studied. In this part we will consider their duals, the almost conditionally invariant subspaces. These concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem. Finally, we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if: 1) almost disturbance decoupling by state feedback, and 2) almost disturbance decoupled estimation of the to-be-controlled output are both possible.     

Almost invariant subspaces: An approach to high gain feedback design--Part I: Almost controlled invariant subspaces

In a previous paper [1] we have introduced the notion of "almost controlled invariant subspaces" which are subspaces to which one can steer the state of a linear system arbitrarily close. In the present paper we will show how these subspaces my be viewed as ordinary controlled invariant subspaces when one allows distributional inputs, or as those subspaces which can be approximated by controlled invariant subspaces. The results are applied to a number of control synthesis problems, i.e., disturbance decoupling, robustness, noisy gain stabilization, and cheap control. Part II of the paper will treat the dual theory of almost conditionally invariant subspaces.     

Three methods for refining estimates of invariant subspaces

We compare three methods for refining estimates of invariant subspaces, due to Chatelin, Dongarra/Moler/Wilkinson, and Stewart. Even though these methods all apparently solve different equations, we show by changing variables that they all solve the same equation, the Riccati equation. The benefit of this point of view is threefold. First, the same convergence theory applies to all three methods, yielding a single criterion under which the last two methods converge linearly, and a slightly stronger criterion under which the first algorithm converges quadratically. Second, it suggest a hybrid algorithm combining advantages of all three. Third, it leads to algorithms (and convergence criteria) for the generalized eigenvalue problem. These techniques are compared to techniques used in the control systems community.     

On almost invariant subspaces of structural systems anddecentralized control

Graph-theoretic conditions are obtained for a structured system to have the property that the supremal L_p-almost invariant (controllability) subspace is generically the entire state space and that the infimal ^L_p-almost conditional invariant (complementary observability) subspace is generically the zero subspace. The conditions are used to determine stabilizability of structured interconnected systems by means of decentralized feedback control. Although the obtained graph-theoretic conditions are conservative, they are considered satisfactory to the extent that the benefits of easy testability involving only binary calculations outweigh the conservativeness in the results     

A homeomorphism between observable pairs and conditioned invariant subspaces

A bijective correspondence between similarity classes of observable systems (C,A) and n-codimensional conditioned invariant subspaces of a pair (C,A) is constructed that leads to a homeomorphism of the spaces. This is applied to the parametrization of inner functions of fixed McMillan degree. Proofs using state space methods as well as using polynomial models are given.     

A coordinate atlas of the manifold of observable conditioned invariant subspaces

Given an observable system (C, A     

On correction equations and domain decomposition for computing invariant subspaces

By considering the eigenvalue problem as a system of nonlinear equations, it is possible to develop a number of solution schemes which are related to the Newton iteration. For example, to compute eigenvalues and eigenvectors of an n × n matrix A, the Davidson and the Jacobi-Davidson techniques, construct ‘good’ basis vectors by approximately solving a “correction equation” which provides a correction to be added to the current approximation of the sought eigenvector. That equation is a linear system with the residual r of the approximated eigenvector as right-hand side.One of the goals of this paper is to extend this general technique to the “block” situation, i.e., the case where a set of p approximate eigenpairs is available, in which case the residual r becomes an n × p matrix. As will be seen, solving the correction equation in block form requires solving a Sylvester system of equations. The paper will define two algorithms based on this approach. For symmetric real matrices, the first algorithm converges quadratically and the second cubically. A second goal of the paper is to consider the class of substructuring methods such as the component mode synthesis (CMS) and the automatic multi-level substructuring (AMLS) methods, and to view them from the angle of the block correction equation. In particular this viewpoint allows us to define an iterative version of well-known one-level substructuring algorithms (CMS or one-level AMLS). Experiments are reported to illustrate the convergence behavior of these methods.     

Local autocorrelation of similarities with subspaces for shift invariant scene classification

This paper presents a shift invariant scene classification method based on local autocorrelation of similarities with subspaces. Although conventional scene classification methods used bag-of-visual words for scene classification, superior accuracy of kernel principal component analysis (KPCA) of visual words to bag-of-visual words was reported. Here we also use KPCA of visual words to extract rich information for classification. In the original KPCA of visual words, all local parts mapped into subspace were integrated by summation to be robust to the order, the number, and the shift of local parts. This approach discarded the effective properties for scene classification such as the relation with neighboring regions. To use them, we use (normalized) local autocorrelation (LAC) feature of the similarities with subspaces (outputs of KPCA of visual words). The feature has both the relation with neighboring regions and the robustness to shift of objects in scenes. The proposed method is compared with conventional scene classification methods using the same database and protocol, and we demonstrate the effectiveness of the proposed method.     

Generalized invariant subspaces and parameter insensitivedisturbance-rejection problems with static output feedback

In the geometric approach some generalized invariant subspaces for uncertain linear systems are investigated, and then various parameter insensitive disturbance-rejection problems with static output feedback are formulated and necessary and sufficient conditions for the problems to be solvable are presented. Further, some sufficient conditions which are useful to check the solvability conditions are presented using the notions of generalized invariant subspaces     

A class of marked invariant subspaces with an application to algebraic Riccati equations

Invariant subspaces of a matrix A are considered which are obtained by truncation of a Jordan basis of a generalized eigenspace of A. We characterize those subspaces which are independent of the choice of the Jordan basis. An application to Hamilton matrices and algebraic Riccati equations is given.     

A repeatable inverse kinematics algorithm with linear invariant subspaces for mobile manipulators.

On the basis of a geometric characterization of repeatability we present a repeatable extended Jacobian inverse kinematics algorithm for mobile manipulators. The algorithm's dynamics have linear invariant subspaces in the configuration space. A standard Ritz approximation of platform controls results in a band-limited version of this algorithm. Computer simulations involving an RTR manipulator mounted on a kinematic car-type mobile platform are used in order to illustrate repeatability and performance of the algorithm.     

A new characterization of invariant subspaces of and applications to the optimal sensitivity problem

This paper gives a new equivalent characterization for invariant subspaces of H², when the underlying structure is specified by the so-called pseudorational transfer functions. This plays a fundamental role in computing the optimal sensitivity for a certain important class of infinite-dimensional systems, including delay systems. A closed formula, easier to compute than the well-known Zhou–Khargonekar formula, is given for optimal sensitivity for such systems. An example is given to illustrate the result.     

Iterative improvement of componentwise errorbounds for invariant subspaces belonging to a double or nearly double eigenvalue

In this paper we present a systematic method which computes bounds for invariant subspaces belonging to a double or nearly double eigenvalue. Furthermore an algorithm based on interval arithmetic tools is introduced which improves these bounds systematically.