2D Linear Systems Research Articles

STABILIZATION OF DISCRETE 2D LINEAR SYSTEMS IN FORNASINI-MARCHESINI MODEL

This paper is concerned with the stabilization problem of discrete 2D linear systems described by the second Fornasini-Marchesini model. A necessary and sufficient condition involving the characteristic polynomial is first quoted by which the unforced system is structurally or exponentially stable. On the basis of the derived stability condition, a tractable condition is formulated in the form of linear matrix inequality for obtaining the controller gain of a desired stabilizing state-feedback controller.

Iteration learning control for 2D linear discrete systems with randomly varying trail lengths

This paper considers the problem of randomly varying trial lengths of a class of two-dimensional (2D) linear systems represented by Roesser model, and two new iterative learning control (ILC) schemes are proposed. One is a conventional P-type control rule with a modified tracking error; the other is an ILC law with an iteration-average operator. Both learning control schemes are developed by a 2D stochastic variable to describe varying trial lengths. Under the Bernoulli distribution assumption and the initial state conditions, the convergence analysis is performed rigorously in the probability sense. Finally, illustrative examples have been provided to verify the theoretical results.

Finite difference based iterative learning control with initial state learning for a class of fractional order two‐dimensional continuous‐discrete linear systems

SummaryThis article presents a PI‐type iterative learning control (ILC) law with initial state learning for a class of () fractional order two‐dimensional (2D) linear systems. First, by using backward difference method, the finite difference approximation of the fractional order derivative is obtained, which leads to globally order accuracy. Then, a PI‐ILC law is constructed at the nodes and the convergence analysis of the iterative scheme is proved. A linear matrix inequality‐based sufficient condition is derived to guarantee that the tracking error is asymptotically convergent. The obtained convergence condition is fractional order dependent. Most of the classical ILC conditions for fractional order one‐dimensional linear systems fall into the special cases of this article. Finally, the simulation results show the effectiveness of the proposed control method.

An Explicit Method for Stability Analysis of 2D Systems Described by Transfer Function

In this paper, we propose a method to test the stability of two-dimensional (2D) linear discrete systems described by transfer function. And a complete region of disturbance parameters is solved to ensure the considered system with perturbations stability. Different from any other traditional algebraic methods, the algorithm of this method is explicit, which a large number of higher-order polynomial iterative operation doesn’t need to be carried out. By the fractional linear transformations, new condition is given in Hurwitz theorem, which doesn’t involve fractional calculation. The method is non-conservative for stability analysis and solving stable parameter region of 2D systems. It simplifies some existing methods. The computational cost is reduced. And it is better to solve the stability problems for 2D systems with disturbances. Examples illustrate the efficiency for testing the stability of higher-order 2D system and solving the robust stability problem of uncertain 2D systems.

Uncertainty principle for the quaternion Fourier transform

The two-sided quaternion Fourier transform was introduced for the analysis of 2D linear time-invariant partial-differential systems. It has been shown to be a powerful tool in image processing. In this paper, several uncertainty inequalities for the two-sided quaternion Fourier transform are given with optimal constants, including the Pitt's inequality, logarithmic uncertainty inequality, Hausdorff–Young inequality, Hirschman's entropy inequality, generalized Heisenberg inequality, local uncertainty principle and qualitative uncertainty principle.

Serial concatenation of a block code and a 2D convolutional code

In this paper we study two different concatenation schemes of twodimensional (2D) convolutional codes. We consider Fornasini–Marchesini state space representation of 2D linear systems to describe our concatenated codes. Also we present upper and lower bounds on the distance of the proposed concatenated codes.

Structure-Preserving Finite Volume Method for 2D Linear and Non-Linear Port-Hamiltonian Systems

In this work we extend the results of a high order finite volume semi-discretization for port-Hamiltonian system 1D linear case (Kotyczka (2016)) to the 2D linear case, worked on the wave equation. The existing pHs discretization methods deal only with the geometric part, in this paper we perform an adapted symplectic time stepping to get the fully discrete scheme in order to preserve both the geometrical properties and the energy aspects. We also show that staggered finite volume method carry over to a non-linear problem, the 2D irrotational shallow water equations. However, due to the non linearity and the non separability of the Hamiltonian, some difficulties arise both for the high order accuracy in the spatial discretization, and also for the symplecticity of the time integration.

Solution to the Fractional-Order 2D Continuous Systems Described by the Second Fornasini-Marchesini Model

In this article the fractional-orders continuous 2D (two-dimensional) linear systems described by the state-space equations of the second Fornasini-Marchesini model structure are considered. The state-space equation contains 2D and 1D (one-dimensional) partial fractional-order derivatives described by the Caputo type definition. For this model the general solution formula is derived using 2D Laplace transform method. Finally the internal positivity property of such systems is investigated. The theoretical considerations are illustrated by numerical simulations.

A novel approach to exponential stability of continuous-time Roesser systems with directional time-varying delays

This paper addresses the problem of exponential stability analysis of two-dimensional (2D) linear continuous-time systems with directional time-varying delays. An abstract Lyapunov-like theorem which ensures that a 2D linear system with delays is exponentially stable for a prescribed decay rate is exploited for the first time. In light of the abstract theorem, and by utilizing new 2D weighted integral inequalities proposed in this paper, new delay-dependent exponential stability conditions are derived in terms of tractable matrix inequalities which can be solved by various computational tools to obtain maximum allowable bound of delays and exponential decay rate. Two numerical examples are given to illustrate the effectiveness of the obtained results.

Integrated state/disturbance observers for two‐dimensional linear systems

In this study, the problem of simultaneous estimation of system states and disturbances is addressed for two-dimensional (2D) linear systems. The considered 2D systems can be described as Fornasini–Marchesini second local state–space model coupled with unknown disturbances in the measurement equation. Asymptotically stable and uniformly ultimately bounded integrated state/disturbance observers are proposed as solutions to the simultaneous estimation problem under different system conditions. For the asymptotically stable integrated state/disturbance observer, a necessary and sufficient condition for its existence is presented and proved. For the uniformly ultimately bounded integrated state/disturbance observer, a sufficient condition for its existence is given and proved, and it can be designed such that its estimation error has arbitrarily small upper bound. Moreover, some constructive design methods are given for the proposed integrated state/disturbance observers. Finally, a thermal process plant and another numerical example are provided to illustrate the effectiveness of the proposed methods.

Computer aided methods for stability analysis of 2D linear systems described by the first Fornasini-Marchesini model

Computer aided methods for stability analysis of 2D linear systems described by the first Fornasini-Marchesini model

Discussion on: “Positive Switched 2D Linear Systems Described by the Roesser Models”

Discussion on: “Positive Switched 2D Linear Systems Described by the Roesser Models”

Positive Switched 2D Linear Systems Described by the Roesser Models

The positive switched 2D linear systems described by the Roesser models are addressed. Necessary and sufficient conditions for the asymptotic stability of the positive switched system are established for any switching. The considerations are illustrated by numerical examples.

New stability conditions for positive continuous-discrete 2D linear systems

New stability conditions for positive continuous-discrete 2D linear systemsNew necessary and sufficient conditions for asymptotic stability of positive continuous-discrete 2D linear systems are established. Necessary conditions for the stability are also given. The stability tests are demonstrated on numerical examples.

New Stability Tests of Positive Standard and Fractional Linear Systems

New tests for checking asymptotic stability of positive 1D continuous-time and discrete-time linear systems without and with delays and of positive 2D linear systems described by the general and the Roesser models are proposed. Checking of the asymptotic stability of positive 2D linear systems is reduced to checking of suitable corresponding 1D positive linear systems. It is shown that the stability tests can be also applied to checking the asymptotic stability of fractional discrete-time linear systems with delays. Effectiveness of the tests is shown on numerical examples.

Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems

Similarity transformation of matrices to one common canonical form and its applications to 2D linear systemsThe notion of a common canonical form for a sequence of square matrices is introduced. Necessary and sufficient conditions for the existence of a similarity transformation reducing the sequence of matrices to the common canonical form are established. It is shown that (i) using a suitable state vector linear transformation it is possible to decompose a linear 2D system into two linear 2D subsystems such that the dynamics of the second subsystem are independent of those of the first one, (ii) the reduced 2D system is positive if and only if the linear transformation matrix is monomial. Necessary and sufficient conditions are established for the existence of a gain matrix such that the matrices of the closed-loop system can be reduced to the common canonical form.

Positivity and stabilization of fractional 2D linear systems described by the Roesser model

Positivity and stabilization of fractional 2D linear systems described by the Roesser modelA new class of fractional 2D linear discrete-time systems is introduced. The fractional difference definition is applied to each dimension of a 2D Roesser model. Solutions of these systems are derived using a 2DZ-transform. The classical Cayley-Hamilton theorem is extended to 2D fractional systems described by the Roesser model. Necessary and sufficient conditions for the positivity and stabilization by the state-feedback of fractional 2D linear systems are established. A procedure for the computation of a gain matrix is proposed and illustrated by a numerical example.

Positivity and stabilization of fractional 2D linear systems described by the Roesser model

Positivity and stabilization of fractional 2D linear systems described by the Roesser modelA new class of fractional 2D linear discrete-time systems is introduced. The fractional difference definit...

Practical stability and asymptotic stability of positive fractional 2D linear systems

AbstractA new concept of the practical stability of the positive fractional 2D linear systems is proposed. Necessary and sufficient conditions for the practical stability and the asymptotic stability of the positive fractional 2D systems are established. It is shown that the checking of the practical stability and the asymptotic stability of positive fractional 2D linear systems can be reduced to testing the stability of corresponding 1D positive linear systems. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Practical stability of positive fractional 2D linear systems

A new concept (notion) of the practical stability of the positive fractional 2D linear systems is proposed. Necessary and sufficient conditions for the practical stability of the positive fractional 2D systems are established. It is shown that the positive fractional 2D systems is practically unstable 1) if a corresponding positive 2D system is asymptotically unstable, 2) if some matrices of the 2D system are nonnegative.