Two-dimensional Linear Systems Research Articles

Blind phase retrieval for aberrated linear shift-invariant imaging systems

We develop a means to reconstruct an input complex coherent scalar wavefield, given a through focal series (TFS) of three intensity images output from a two-dimensional (2D) linear shift-invariant optical imaging system with unknown aberrations. This blind phase retrieval technique unites two methods, namely (i) TFS phase retrieval and (ii) iterative blind deconvolution. The efficacy of our blind phase retrieval procedure has been demonstrated using simulated data, for a variety of Poisson noise levels.

Novel state-space self-tuning control for two-dimensional linear discrete-time stochastic systems

Novel state-space self-tuning control for two-dimensional linear discrete-time stochastic systems

The qualitative analysis of a class of planar Filippov systems

We use the theory of differential inclusions, Filippov transformations and some appropriate Poincaré maps to discuss the special case of two-dimensional discontinuous piecewise linear differential systems with two zones. This analysis applies to uniqueness and non-uniqueness for the initial value problem, stability of stationary points, sliding motion solutions, number of closed trajectories, existence of heteroclinic trajectories connecting two saddle points forming a heteroclinic cycle and existence of the homoclinic trajectory

Exponential Dichotomy for Asymptotically Hyperbolic Two-Dimensional Linear Systems

We consider the problem of determining the existence of exponential dichotomy for a class of linear nonautonomous ODEs. An approach is explored that combines numerical techniques with rigorous perturbation theory. It is applicable to a given problem within the class we consider. Numerical results illustrate the utility of the approach.

Stability analysis for a class of discrete-time two-dimensional nonlinear systems

Stability analysis for a class of discrete-time two-dimensional nonlinear systems is addressed in this paper. A linear matrix inequality based sufficient condition for asymptotic stability is proposed. This condition is an analogous counterpart for two-dimensional linear systems which is shown to be easily verifiable and suitable for design problems. The result is illustrated by a numerical example.

Integral conditions on the uniform asymptotic stability for two-dimensional linear systems with time-varying coefficients

This paper is concerned with the uniform asymptotic stability of the zero solution of the linear systemx′=A(t)x\mathbf {x}’ = A(t)\mathbf {x}withA(t)A(t)being a2×22\times 2matrix. Our result can be used without knowledge about a fundamental matrix of the system.

Reachability of positive 2D fractional linear systems

Two new classes of positive two-dimensional (2D) fractional linear systems are introduced in this work. Two definitions of fractional differences of 2D function are proposed. The solutions of fractional 2D state equations of the linear systems are given. The classical Cayley–Hamilton theorem is extended to 2D fractional systems. The necessary and sufficient conditions for the positivity of 2D fractional systems are established. The notion of reachability is introduced for the positive fractional systems and the necessary and sufficient conditions for the reachability are established. The considerations are illustrated with numerical examples.

Attractivity for two-dimensional linear systems whose anti-diagonal coefficients are periodic

This paper deals with the linear systemx′=A(t)x\mathbf {x}’ = A(t)\mathbf {x}withA(t)A(t)being a2×22\times 2matrix. The anti-diagonal components ofA(t)A(t)are assumed to be periodic, but the diagonal components are not necessarily periodic. Our concern is to establish sufficient conditions for the zero solution to be attractive. Floquet theory is of no use in solving our problem, because not all components are periodic. Another approach is adopted. Some simple examples are included to illustrate the main result.

Strong practical stability and stabilization of discrete linear repetitive processes

This paper considers two-dimensional (2D) discrete linear systems recursive over the upper right quadrant described by well known state-space models. Included are discrete linear repetitive processes that evolve over subset of this quadrant. A stability theory exists for these processes based on a bounded-input bounded-output approach and there has also been work on the design of stabilizing control laws, elements of which have led to the assertion that this stability theory is too strong in many cases of applications interest. This paper develops so-called strong practical stability as an alternative in such cases. The analysis includes computationally efficient tests that lead directly to the design of stabilizing control laws, including the case when there is uncertainty associated with the process model. The results are illustrated by application to a linear model approximation of the dynamics of a metal rolling process.

Beurling–Borg Type Theorem for Two-Dimensional Linear Differential Systems

Abstract For the two-dimensional linear differential system with Lebesgue integrable coefficients 𝑝𝑖𝑘 : [𝑎, 𝑏] → ℝ (𝑖 = 1, 2), a Beurling–Borg type theorem is proved on an upper estimate of the number of zeros of an arbitrary non-trivial solution.

Competition Between Transients in the Rate of Approach to a Fixed Point

The goal of this paper is to provide and apply tools for analyzing a specific aspect of transient dynamics not covered by previous theory. The question we address is whether one component of a perturbed solution to a system of differential equations can overtake the corresponding component of a reference solution as both converge to a stable node at the origin, given that the perturbed solution was initially farther away and that both solutions are nonnegative for all time. We call this phenomenon tolerance, for its relation to a biological effect. We show using geometric arguments that tolerance will exist in generic linear systems with a complete set of eigenvectors and in excitable nonlinear systems. We also define a notion of inhibition that may constrain the regions in phase space where the possibility of tolerance arises in general systems. However, these general existence theorems do not not yield an assessment of tolerance for specific initial conditions. To address that issue, we develop some analytical tools for determining if particular perturbed and reference solution initial conditions will exhibit tolerance.

Anomaly of fractal dimensions observed in stochastically switched systems

We studied an anomaly in fractal dimensions measured from the attractors of dynamical systems driven by stochastically switched inputs. We calculated the dimensions for different switching time lengths in two-dimensional linear dynamical systems, and found that changes in the dimensions due to the switching time length had a singular point when the system matrix had two different real eigenvalues. Using partial dimensions along each eigenvector, we explicitly derived a generalized dimension D(q) and a multifractal spectrum f(alpha) to explain this anomalous property. The results from numerical calculations agreed well with those from analytical equations. We found that this anomaly is caused by linear independence, inhomogeneity of eigenvalues, and overlapping conditions. The mechanism for the anomaly could be identified for various inhomogeneous systems including nonlinear ones, and this reminded us of anomalies in some physical values observed in critical phenomena.

Minimal subsets of projective flows

We study the minimal subsets of the projective flow defined by a two-dimensional linear differential system with almost periodic coefficients. We show that such a minimal set may exhibit Li-Yorke chaos and discuss specific examples in which this phenomenon is present. We then give a classification of these minimal sets, and use it to discuss the bounded mean motion property relative to the projective flow.

Parameter Estimation of Two-Dimensional Linear Differential Systems via Fourier Based Modulation Function

In this paper, parameter identification of two-dimensional (2-D) linear differential systems via two-dimensional modulating functions is proposed. In this method, a partial differential equation on the finite time intervals converts into an algebraic equation linear in parameters. Then the parameters of the system can be estimated using the least squares algorithm. The underlying computations utilize a 2-D fast Fourier transform algorithm on polynomials of the data without the need for estimating unknown initial or/and boundary conditions at the beginning of each finite time interval. Numerical simulations are presented to confirm the theoretical results.

Constrained Optimal Control Theory for Differential Linear Repetitive Processes

Differential repetitive processes are a distinct class of continuous-discrete two-dimensional linear systems of both systems theoretic and applications interest. These processes complete a series of sweeps termed passes through a set of dynamics defined over a finite duration known as the pass length, and once the end is reached the process is reset to its starting position before the next pass begins. Moreover the output or pass profile produced on each pass explicitly contributes to the dynamics of the next one. Applications areas include iterative learning control and iterative solution algorithms, for classes of dynamic nonlinear optimal control problems based on the maximum principle, and the modeling of numerous industrial processes such as metal rolling, longwall cutting, etc. In this paper we develop substantial new results on optimal control of these processes in the presence of constraints where the cost function and constraints are motivated by practical application of iterative learning control to robotic manipulators and other electromechanical systems. The analysis is based on generalizing the well-known maximum and ?-maximum principles to them.

Phase contrast, phase retrieval and aberration balancing in shift-invariant linear imaging systems

We treat the problems of phase-contrast image formation, deterministic phase retrieval and aberration balancing, in the imaging of weak objects using two-dimensional shift-invariant linear imaging systems. Three classes of model sample are considered: weak phase objects, weak phase-amplitude objects and single-material weak phase-amplitude objects. For each class of sample we show how the various aberration coefficients, which characterise a given imaging system, contribute to the structure of the associated phase-contrast image. The corresponding inverse problem, of obtaining a closed-form expression for the input wave-field given one or more aberrated phase-contrast images of the same, is then examined. Two sample applications are considered: analyser-crystal phase-contrast imaging of weak objects using hard X-rays, and Zernike-type phase-contrast imaging. We close with a discussion of how coherent and incoherent aberrations may be “balanced” against one another, briefly mentioning the applications of this idea to both “deblur by defocus” and proximity-corrected X-ray lithography.

Stabilizability of two-dimensional linear systems via switched output feedback

The problem of stabilizing a second-order SISO LTI system of the form x ˙ = Ax + Bu , y = Cx with feedback of the form u ( x ) = v ( x ) Cx is considered, where v ( x ) is real-valued and has domain which is all of R 2 . It is shown that, when stabilization is possible, v ( x ) can be chosen to take on no more than two values throughout the entire state-space (i.e., v ( x ) ∈ { k 1 , k 2 } for all x and for some k 1 , k 2 ), and an algorithm for finding a specific choice of v ( x ) is presented. It is also shown that the classical root locus of the corresponding transfer function C ( sI - A ) - 1 B has a strong connection to this stabilization problem, and its utility is demonstrated through examples.

Exact finite-difference schemes for two-dimensional linear systems with constant coefficients

An exact finite-difference scheme for a system of two linear differential equations with constant coefficients, ( d / d t ) x ( t ) = A x ( t ) , is proposed. The scheme is different from what was proposed by Mickens [Nonstandard Finite Difference Models of Differential Equations, World Scientific, New Jersey, 1994, p. 147], in which the derivatives of the two equations are formed differently. Our exact scheme is in the form of ( 1 / φ ( h ) ) ( x k + 1 - x k ) = A [ θ x k + 1 + ( 1 - θ ) x k ] ; both derivatives are in the same form of ( x k + 1 - x k ) / φ ( h ) .

Determinantal representations of smooth cubic surfaces

For every smooth (irreducible) cubic surface S we give an explicit construction of a representative for each of the 72 equivalence classes of determinantal representations. Equivalence classes (under GL3 × GL3 action by left and right multiplication) of determinantal representations are in one to one correspondence with the sets of six mutually skew lines on S and with the 72 (two-dimensional) linear systems of twisted cubic curves on S. Moreover, if a determinantal representation M corresponds to lines (a 1,...,a 6) then its transpose M t corresponds to lines (b 1,...,b 6) which together form a Schläfli’s double-six \(a_1\ldots a_6 \choose b_1\ldots b_6\) . We also discuss the existence of self-adjoint and definite determinantal representation for smooth real cubic surfaces. The number of these representations depends on the Segre type F i . We show that a surface of type F i , i = 1,2,3,4 has exactly 2(i−1) nonequivalent self-adjoint determinantal representations none of which is definite, while a surface of type F 5 has 24 nonequivalent self-adjoint determinantal representations, 16 of which are definite.

On the Kneser-Type Solutions for Two-Dimensional Linear Differential Systems with Deviating Arguments

For the differential system,,, where,,, we get necessary and sufficient conditions that this system does not have solutions satisfying the condition for. Note one of our results obtained for this system with constant coefficients and delays (, where and). The inequality is necessary and sufficient for nonexistence of solutions satisfying this condition.