Finite-dimensional Matrix Research Articles

Output feedback control of saturated discrete-time linear systems using parameter-dependent Lyapunov functions

In this paper, we present a new output feedback control approach for discrete-time linear systems subject to actuator saturations using parameter-dependent Lyapunov functions. The saturation level indicator serves as a scheduling parameter. The resulting nonlinear controller is expressed in a quasi-LPV (linear parameter-varying) form, and the stabilization and disturbance attenuation problems are formulated and solved as finite-dimensional linear matrix inequality (LMI) optimization problems. Our approach is less conservative than a single quadratic Lyapunov function method. Specifically, the proposed output feedback control law asymptotically stabilizes the open loop system with a larger domain of attraction and achieves better disturbance attenuation under energy and magnitude bounded disturbances.

Infinite-dimensional Lie algebras in 4D conformal quantum field theory**Lecture at the workshops ‘Lie Theory and Its Applications in Physics’, 18–24 June 2007, Varna, Bulgaria; ‘Infinite- Dimensional Algebras and Quantum Integrable Systems’, 23–27 July, 2007, Faro, Portugal; and ‘Supersymmetries and Quantum Symmetries’, 30 July–4 August, 2007, Dubna, Russia.

The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of two-dimensional chiral conformal field theory, to a higher (even) dimensional spacetime. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields, VM(x, y), where the M span a finite dimensional real matrix algebra closed under transposition. The associative algebra is irreducible iff its commutant coincides with one of the three real division rings. The Lie algebra of (the modes of) the bilocal fields is in each case an infinite-dimensional Lie algebra: a central extension of corresponding to the field of reals, of u(∞, ∞) associated with the field of complex numbers, and of so*(4∞) related to the algebra of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups O(N), U(N) and , respectively.

Trace Formulae of Characteristic Polynomial and Cayley–Hamilton's Theorem, and Applications to Chiral Perturbation Theory and General Relativity

By using combinatorics, we give a new proof for the recurrence relations of the characteristic polynomial coefficients, and we further obtain an explicit expression for the generic term of the coefficient sequence, which yields the trace formulae of the Cayley–Hamilton's theorem with all coefficients explicitly given. This implies a byproduct, a complete expression for the determinant of any finite-dimensional matrix in terms of the traces of its successive powers. And we discuss some of their applications to chiral perturbation theory and general relativity.

On the approximation of spectra of linear operators on Hilbert spaces

We present several new techniques for approximating spectra of linear operators (not necessarily bounded) on an infinite-dimensional, separable Hilbert space. Our approach is to take well-known techniques from finite-dimensional matrix analysis and show how they can be generalized to an infinite-dimensional setting to provide approximations of spectra of elements in a large class of operators. We conclude by proposing a solution to the general problem of approximating the spectrum of an arbitrary bounded operator by introducing the n-pseudospectrum and argue how that can be used as an approximation to the spectrum.

Optimal Output Regulation for Discrete-Time Switched and Markovian Jump Linear Systems

First, three different but related output regulation performance criteria for the linear time-varying system are defined in the discrete-time domain, namely, the peak impulse response, peak output variance, and average output variance per unit time. Then they are extended for switched linear systems and Markovian jump linear systems, and characterized by an increasing union of finite-dimensional linear matrix inequality conditions. Finally, the infinite-horizon suboptimal LQG control problem, which aims to maintain the average output variance below a given level subject to the uniform exponential stability of the closed-loop system, is solved for both switched linear systems and Markovian jump linear systems; the solution is given by a dynamic linear output feedback controller that not only perfectly observes the present mode but also recalls a finite number of past modes.

A PROPOSAL FOR A NONPERTURBATIVE REGULARIZATION OF $\mathcal{N} = 2 SUSY 4D GAUGE THEORY

In this letter we show that supersymmetry (like geometry) can be approximated using finite-dimensional matrix models and fuzzy manifolds. In particular, we propose a nonperturbative regularization of [Formula: see text] supersymmetric U(n) gauge action in 4D. In some planar large-N limits we recover exact SUSY together with the smooth geometry of [Formula: see text].

The rotating Morse potential model for diatomic molecules in the tridiagonal J-matrix representation: I. Bound states

This is the first in a series of articles in which we study the rotating Morse potential model for diatomic molecules in the tridiagonal J-matrix representation. Here, we compute the bound-state energy spectrum by diagonalizing the finite-dimensional Hamiltonian matrix of H2, LiH, HCl and CO molecules for arbitrary angular momentum. The calculation was performed using the J-matrix basis that supports a tridiagonal matrix representation for the reference Hamiltonian. Our results for these diatomic molecules have been compared with available numerical data satisfactorily. The proposed method is handy, very efficient, and it enhances accuracy by combining analytic power with a convergent and stable numerical technique.

Extreme points of the set of density matrices with positive partial transpose

We present a necessary and sufficient condition for a finite-dimensional density matrix to be an extreme point of the convex set of density matrices with positive partial transpose with respect to a subsystem. We also give an algorithm for finding such extreme points and illustrate this by some examples.

Dynamic output feedback control of quasi-LPV mechanical systems

A methodology for controlling nonlinear mechanical systems expressed in non-singular parametric descriptor quasi-linear parameter varying form is developed. For mechanical systems, this form is naturally obtained through the Lagrangian formulation. Use of this form avoids the explicit inversion of the mass matrix, thus eliminating the additional resulting linear fractional parameter dependence complexity. Parametric Lyapunov functions are used for the design. Sufficient conditions for the stability and dynamic output feedback control with guaranteed induced ${\cal L}_2$-norms from the disturbance to the controlled output are given. The conditions for the reduction of the solvability conditions to finite-dimensional linear matrix inequalities are provided. The results are shown on a two-link flexible manipulator example.

Act-and-wait concept for continuous-time control systems with feedback delay

The act-and-wait control concept is introduced for continuous-time control systems with feedback delay associated with infinite poles. The point of the method is that the feedback is periodically switched on (act) and off (wait) during the control. It is shown that if the duration of waiting (when the control is switched off) is larger than the feedback delay, then the system can be represented by a finite dimensional monodromy matrix, and a finite number of eigenvalues describe stability. This way, the infinite dimensional pole placement problem is reduced to a finite dimensional one. The efficiency of the method is demonstrated on a case study

AF Embeddings and the Numerical Computation of Spectra in Irrational Rotation Algebras

A natural problem at the interface of operator theory and numerical analysis is that of finding a (finite dimensional) matrix whose eigenvalues approximate the spectrum of a given (infinite dimensional) operator. It is well-known that classical work of Pimsner and Voiculescu produces explicit matrix models for an interesting class of nontrivial examples (e.g., many discretized one-dimensional Schrödinger operators). In this paper, we observe that the spectra of their models (often) converge in the strongest possible sense—in the Hausdorff metric—and demonstrate that the rate of convergence is, in general, best possible.

Finitely many Dirac-delta interactions on Riemannian manifolds

This work is intended as an attempt to study the nonperturbative renormalization of bound state problem of finitely many Dirac-delta interactions on Riemannian manifolds, S2, H2, and H3. We formulate the problem in terms of a finite dimensional matrix, called the characteristic matrix Φ. The bound state energies can be found from the characteristic equation Φ(−ν2)A=0. The characteristic matrix can be found after a regularization and renormalization by using a sharp cut-off in the eigenvalue spectrum of the Laplacian, as it is done in the flat space, or using the heat kernel method. These two approaches are equivalent in the case of compact manifolds. The heat kernel method has a general advantage to find lower bounds on the spectrum even for compact manifolds as shown in the case of S2. The heat kernels for H2 and H3 are known explicitly, thus we can calculate the characteristic matrix Φ. Using the result, we give lower bound estimates of the discrete spectrum.

Absolute Stability via Boundary Control of a Semilinear Parabolic PDE

We consider the problem of achieving global absolute stability of an unstable equilibrium solution of a semilinear dissipative parabolic partial differential equation (PDE) through boundary control. The state space of the system is extended in order to write the action of the boundary control as an unbounded operator in an abstract evolution equation. Absolute stability via boundary control is accomplished by analyzing a control Lyapunov function based on the infinite-dimensional dynamics and applying a finite-dimensional linear quadratic regulator (LQR) controller. Sufficient conditions for absolute stability of the infinite-dimensional system are established by the feasibility of two finite-dimensional linear matrix inequalities (LMIs). Numerical results are presented for a Dirichlet boundary controlled system, however the analysis in this work applies to Nuemann and Robin type boundary controllers as well.

Revisiting matrix games: The concept of neighborhood invader strategies

We extend the concept of neighborhood invader strategy (NIS) to finite-dimensional matrix games and compare this concept to the evolutionarily stable strategy (ESS) concept. We show that these two concepts are not equivalent in general. Just as ESS's may not be unique, NIS's may also not be unique. However, if there is an ESS and a NIS then these strategies must be the same. We show that an ESNIS (an ESS and NIS) for any matrix game is unique and that a mixed ESS with full support is a NIS. Thus a mixed ESS with full support is not invadable by any pure or mixed strategy and it can invade any pure or mixed strategy. An ESS which is an ESNIS, therefore, has better chance of being established evolutionarily through dynamic selection.

SUSY anomalies break Script N = 2 to Script N = 1: the supersphere and the fuzzy supersphere

The N=1 SUSY on S^2 and its fuzzy finite-dimensional matrix version are known. The latter regulates quantum field theories, and seems suitable for numerical work and capable of higher dimensional generalizations. In this paper, we study their instanton sectors. They are SUSY generalizations of U(1) bundles on S^2 and their fuzzy versions, and can be characterized by $k\in\mathbb{Z}$, the SUSY Chern numbers. In the no-instanton sector (k=0), N=2 SUSY can be chirally realized, the 3 new N=2 generators anticommuting with the ``Dirac'' operator defining the free action. If $k\neq 0$, the Dirac operator has zero modes which form an N=1 supermultiplet and an atypical representation of N=2 SUSY. They break the chiral SUSY generators by the Fujikawa mechanism. We have not found this mechanism for SUSY breakdown in the literature. All these phenomena occur also on the supersphere SUSY, the graded commutative limit of the fuzzy model. We plan to discuss that as well in a later work.

Input design via LMIs admitting frequency-wise model specifications in confidence regions

A framework for reformulating input design problems in prediction error identification as convex optimization problems is presented. For linear time-invariant single input/single output systems, this framework unifies and extends existing results on open-loop input design that are based on the finite dimensional asymptotic covariance matrix of the parameter estimates. Basic methods for parametrizing the input spectrum are provided and conditions on these parametrizations that guarantee that all possible covariance matrices for the asymptotic distribution of the parameter estimates can be generated are provided. A wide range of model quality constraints can be handled. In particular, different frequency-by-frequency constraints can be used. This opens up new applications of input design in areas such as robust control. Furthermore, quality specifications can be imposed on all models in a confidence region. Thus, allowing for statements such as with at least 99% probability the model quality specifications will be satisfied.

Description of the electron–hydrogen collision by the Coulomb Fourier transform method

A recently developed Coulomb Fourier Transform method is applied to the system containing one heavy ion and two electrons. The transformed Hamiltonian is described with a controlled accuracy in an effective finite basis set as a finite dimensional operator matrix. The kernels of interaction are formulated in terms of the so called Nordsieck integrals.

The finite section method and problems in frame theory

The finite section method is a convenient tool for approximation of the inverse of certain operators using finite-dimensional matrix techniques. In this paper we demonstrate that the method is very useful in frame theory: it leads to an efficient approximation of the inverse frame operator and also solves related computational problems in frame theory. In the case of a frame which is localized w.r.t. an orthonormal basis we are able to estimate the rate of approximation. The results are applied to the reproducing kernel frame appearing in the theory for shift-invariant spaces generated by a Riesz basis.

STABILITY SWITCHES AND REVERSALS OF LINEAR SYSTEMS WITH COMMENSURATE DELAYS: A MATRIX PENCIL CHARACTERIZATION

This paper addresses the problem of asymptotic stability of linear time-delay systems including commensurate delays. More precisely we focus on the characterization of stability switches and reversals using a matrix pencil approach. The proposed approach makes use of the generalized eigenvalue distribution with respect to the unit circle of some appropriate finite-dimensional matrix pencils. Classical problems, as for example, hyperbolicity and delay-independent/delay-dependent stability characterizations are reconsidered, and simple computational conditions are derived.

A Meshless Method for Computational Stochastic Mechanics

This paper presents a stochastic meshless method for probabilistic analysis of linear-elastic structures with spatially varying random material properties. Using Karhunen-Loève (K-L) expansion, the homogeneous random field representing material properties was discretized by a set of orthonormal eigenfunctions and uncorrelated random variables. Two numerical methods were developed for solving the integral eigenvalue problem associated with K-L expansion. In the first method, the eigenfunctions were approximated as linear sums of wavelets and the integral eigenvalue problem was converted to a finite-dimensional matrix eigenvalue problem that can be easily solved. In the second method, a Galerkin-based approach in conjunction with meshless discretization was developed in which the integral eigenvalue problem was also converted to a matrix eigenvalue problem. The second method is more general than the first, and can solve problems involving a multi-dimensional random field with arbitrary covariance functions. In conjunction with meshless discretization, the classical Neumann expansion method was applied to predict second-moment characteristics of the structural response. Several numerical examples are presented to examine the accuracy and convergence of the stochastic meshless method. A good agreement is obtained between the results of the proposed method and the Monte Carlo simulation. Since mesh generation of complex structures can be far more time-consuming and costly than the solution of a discrete set of equations, the meshless method provides an attractive alternative to the finite element method for solving stochastic-mechanics problems.