Roesser State-space Model Research Articles

On the reduction of repetitive processes into singular and non-singular Roesser models

In this paper we present new methods for the reduction of a polynomial system matrix describing a discrete linear repetitive process, to equivalent singular and non-singular 2-D state space representations. Particularly, a zero coprime system equivalence transformation resulting in a singular Roesser state space model, preserving the core algebraic structure of the original system matrix, is proposed. As a second step utilizing the singular Roesser model introduced, we further reduce the system to a non-singular, zero coprime system equivalent Roesser model. Both models are constructed by inspection or by applying elementary matrix manipulations and have significantly smaller dimensions compared to similar reductions found in the literature.

Constructing the Singular Roesser State-Space Model Description of 3D Spatio-Temporal Dynamics From the Polynomial System Matrix

This paper considers systems theoretic properties of linear systems defined in terms of spatial and temporal indeterminates. These include physical applications where one of the indeterminates is of finite duration. In some cases, a singular Roesser state-space model representation of the dynamics has found use in characterizing systems theoretic properties. The representation of the dynamics of many linear systems is obtained in terms of transform variables and a polynomial system matrix representation. This paper develops a direct method for constructing the singular Roesser state-space realization from the system matrix description for 3D systems such that relevant properties are retained. Since this method developed relies on basic linear algebra operations, it may be highly effective from the computational standpoint. In particular, spatially interconnected systems of the form of the ladder circuits are considered as the example. This application confirms the usefulness and effectiveness of the proposed method independently of the system spatial order.

Reliable Control for Two-Dimensional Systems Subject to Extended Dissipativity

The problem of reliable controller design for two-dimensional (2-D) systems subject to extended dissipativity is investigated in this paper. Considering the wide usage of Roesser state-space model, a discrete-time Roesser model is introduced to describe 2-D systems. To reinforce the reliability of the system under consideration, actuator-failure model which may severely degrade performance, or even destabilize the system is introduced. Finally, the sufficient condition for the existence of reliable controller is presented to guarantee the mean square asymptotic stability and 2-D extended dissipativity of the closed-loop system under admissible actuator failures. The gains of 2-D controller are derived by means of the convex optimization method. A simulation result is exploited to verify the effectiveness and merits of the theoretical findings.

Common eigenvector approach to exact order reduction for Roesser state-space models of multidimensional systems

The well-known Popov–Belevitch–Hautus (PBH) tests play an important role in the Kalman decomposition of 1-D systems and reveal the relationship among the eigenvalues, the eigenvectors and the reducibility of a given 1-D state-space model. This paper is to try to generalize the PBH tests to the n-D case for the exact reducibility of n-D Roesser models by exploiting the so-called common eigenvectors. Specifically, the notion of constrained common eigenvectors is introduced, for the first time, which provides insight into the relationship between reducibility and multiple eigenvalues. Based on this result, new reducibility conditions and the corresponding reduction procedure are developed for n-D Roesser models. It will be shown that this common eigenvector approach is applicable to a larger class of Roesser models for which the existing approaches may not be applied to do further order reduction. A Gröbner basis approach is proposed to compute such a constrained common eigenvector, which also leads to an equivalent reducibility condition. Moreover, a generalization to the state delay case is also given so that the eigenvalues of both the system matrix and the state-delay system matrix can be treated simultaneously.

Equivalence of wave linear repetitive processes and the singular 2-D Roesser state-space model

This paper develops a direct method for transforming a polynomial system matrix describing a discrete wave linear repetitive process to a 2-D singular state-space Roesser model description where all relevant properties, including the zero coprimeness properties of the system matrix, are retained. It is shown that the transformation is zero coprime system equivalence. The structure of the resulting system matrix in singular form and the transformation are also established.

Finite frequency H∞ filtering for uncertain two-dimensional continuous systems

This paper investigates the problem of robust finite frequency (FF) [Formula: see text] filtering for two-dimensional (2D) continuous systems described by the Roesser state-space model with norm-bounded uncertainties. A further generalized Kalman–Yakubivich–Popov (KYP) lemma for 2D continuous Roesser systems is presented in a unified form. By the given generalized KYP lemma, the problem of standard [Formula: see text] filtering for uncertain 2D continuous Roesser systems is extended to the FF case. Finally, an illustrative example is provided to validate the effectiveness of the proposed method.

Stabilization of Two-Dimensional Nonlinear Systems Described by Fornasini--Marchesini and Roesser Models

The paper considers nonlinear two-dimensional systems described by the Fornasini--Marchesini or Roesser state-space models. Conditions for such systems to have a physically motivated exponential stability property are derived using vector Lyapunov functions. A form of passivity, termed exponential passivity, is introduced and used, together with a vector storage function, to develop a feedback based control law that guarantees exponential stability of the controlled system. For cases where noise is present, stochastic dissipativity in the second moment is defined and then a particular case of this property, termed passivity in the mean square, is used, together with a vector storage function, to develop a feedback based control law such that the controlled system also has this property. Two physically motivated particular cases, a system with nonlinear actuator dynamics and additive noise and a linear system with state-dependent noise, respectively, are also considered to demonstrate the effectiveness of the new results.

Eigenvalue trim approach to exact order reduction for roesser state-space model of multidimensional systems

In this paper, a new notion of eigenvalue trim or co-trim for n-D Roesser (state-space) model is first introduced, which reveals the internal connection between the eigenvalues of the system matrix and the reducibility of the considered Roesser model. Then, new reducibility conditions and the corresponding order reduction algorithms based on eigenvalue trim or co-trim are proposed for exact order reduction of a given n-D Roesser model, and it will be shown that this eigenvalue trim approach can be applied even to those systems for which the existing approaches cannot do any further order reduction. Furthermore, a new transformation for n-D Roesser models, by swapping certain rows and columns and interchanging certain entries that belong to different blocks corresponding to different variables, will be established, which can transform an n-D Roesser model whose order cannot be reduced any more by the proposed approach to another equivalent Roesser model with the same order so that this transformed Roesser model can still be reduced further. Examples are given to illustrate the details as well as the effectiveness of the proposed approach.

Lattice Structure Realization for The Design of 2-D Digital Allpass Filters With General Causality

This paper presents a lattice structure for the realization of two-dimensional (2-D) recursive digital allpass filters (DAFs) with general causality. We employ four basic lattice sections to realize 2-D recursive DAFs with wedge-shaped coefficient support region like a nonsymmetric half-plane (NSHP) support region. The theory and transfer functions of the realized 2-D lattice DAFs are derived. Some variations of the 2-D lattice structure are also presented. We use the Roesser state space model to verify the minimal realization of the proposed 2-D recursive lattice DAF. We present a least-squares design technique and a minimax design technique to solve the nonlinear optimization problems of the proposed 2-D lattice DAF structure. The novelty of the presented lattice structure is that it not only inherits the desirable attributes of 1-D Gray-Markel lattice allpass structure but also possesses the advantage of better performance over the existing 2-D lattice allpass structures. Finally, several design examples are provided for conducting illustration and comparison.

A novel elementary operation approach with Jordan transformation to order reduction for Roesser state-space model

This paper proposes a novel elementary operation approach to order reduction for the Roesser state-space model of multidimensional (n-D) systems by introducing a new kind of transformation, i.e., the Jordan transformation, which guarantees the establishment of an objective matrix with more general structure than the existing one. Then two basic order reduction techniques are developed which can overcome the difficulty encountered by the existing methods and reveal, for the first time, the fact that the order reduction is still possible even when the column (or row) blocks in the related n-D polynomial matrix are full rank. Furthermore, based on the Jordan transformation, an equivalence relationship between two Roesser models after using the elementary operations among the different blocks will be clarified. Although these operations do not directly lower the total order of the model, the partial orders can be changed so that it may nevertheless yield a possibility for further order reduction. It turns out that this new approach includes our previous elementary operation order reduction approach just as a special case. Examples are given to illustrate the details as well as the effectiveness of the proposed approach.

On elimination of overflow oscillations in linear time-varying 2-D digital filters represented by a Roesser model

This paper proposes a novel linear matrix inequality (LMI)-based stability criterion to avoid overflow oscillations in 2-D digital filters described by a Roesser state-space model with saturation-type overflow characteristics. In contrast to the conventional approaches, the proposed overflow oscillation elimination methodology can be applied to linear parameter varying time-variant digital filters under overflow subject to both state and output disturbances. For such filters, the proposed criterion ensures the zero-input overflow stability as well as robustness against interferences with a guaranteed H∞ performance bound. A numerical example is presented to demonstrate the effectiveness of the proposed approach.

Reliable [formula omitted] control of 2-D continuous nonlinear systems with time varying delays

This paper investigates the reliable H∞ stabilization problem for a class of two-dimensional (2-D) continuous nonlinear state-delayed systems represented by the Roesser state-space model, where the nonlinear function satisfies the sector bounded condition. By choosing an appropriate Lyapunov–Krasovskii functional, sufficient conditions for asymptotical stability with H∞ performance of the given system are derived. Then, a reliable controller is proposed such that the resulting closed-loop system is asymptotically stable and has a prescribed H∞ performance level γ in the presence of actuator failures. Finally, an example is given to illustrate the effectiveness of the proposed method.

Elementary Operation Approach to Order Reduction for Roesser State-Space Model of Multidimensional Systems

This paper proposes a new order reduction approach for Roesser state-space model of multidimensional ( n-D) systems based on elementary operations, by inversely applying the basic idea adopted in the new elementary operation approach to the Roesser model realization of n-D systems. It will be shown first that the n-D order reduction problem can be formulated into an elementary operation problem of an n-D polynomial matrix obtained from the coefficient matrices of the given Roesser model. Based on this problem formulation, a basic order reduction procedure and three techniques are presented, by which the intrinsic relationship among the blocks with respect to different variables can be investigated to achieve a further possible order reduction. It turns out that the new proposed approach is applicable to a wider class of Roesser models than the existing reduction approaches and provides a possible way to explore the equivalence between two systems. Examples are given to illustrate the main idea as well as the effectiveness of the proposed approach.

Robust [formula omitted] control for a class of 2-D discrete delayed systems

In this paper, we deal with the problem of robust H∞ control for a class of 2-D discrete uncertain systems with delayed perturbations described by the Roesser state-space model (RM). The problem to be addressed is the design of robust controllers via state feedback such that the stability of the resulting closed-loop system is guaranteed and a prescribed H∞ performance level is ensured for all delayed perturbations. By utilizing the Lyapunov method and some results, H∞ controllers are given. The results are delay-dependent and can be expressed in terms of linear matrix inequalities (LMIs). Finally, some numerical examples are given to illustrate the effectiveness of the proposed results.

Robust $$H_{\infty }$$ H ∞ filtering for uncertain two-dimensional continuous systems with time-varying delays

This paper deals with the problem of delay-dependent robust \(H_{\infty }\) filtering for uncertain two-dimensional (2-D) continuous systems described by Roesser state space model with time-varying delays, with the uncertain parameters assumed to be of polytopic type. A sufficient condition for \(H_{\infty }\) noise attenuation is derived in terms of linear matrix inequalities, so a robust \(H_{\infty }\) filter can be obtained by solving a convex optimization problem. Finally, some examples are provided to illustrate the effectiveness of the proposed methodology.

Gain-scheduled control of two-dimensional discrete-time linear parameter-varying systems in the Roesser model

This paper is concerned with gain-scheduled control of two-dimensional discrete-time linear parameter-varying systems described by a Roesser state-space model with matrices depending affinely on time-varying scheduling parameters. The parameter admissible values and variations are assumed to belong to given intervals. Linear matrix inequality based methods are devised for designing static state feedback gain-scheduled controllers with either an H∞ or quadratic regulator-type performance. The control designs build on quadratically parameter-dependent Lyapunov functions and allow for incorporating information on available bounds on the parameters variation. The proposed controller gain can be independent, affine, quadratic, or a matrix fraction of quadratic polynomial matrices in the scheduling parameters.

An Iterative Procedure for Multidimensional Realization By LFT Techniques

In this paper, by utilizing Linear Fractional Transformation (LFT) techniques, an iterative realization procedure resulting in multidimensional (n-D) Roesser state-space model is proposed. The philosophy employed here is to achieve an overall n-D realization by realizing relatively lower-dimensional systems with respect to (w.r.t.) less-number of variables iteratively. Symbolic examples are presented to illustrate the basic idea as well as the effectiveness of the proposed procedure.

$${{\mathcal H}_{\infty}}$$ control of linear multidimensional discrete systems

This paper presents a comprehensive investigation on the $${{\mathcal H}_{\infty}}$$ control problem of linear multidimensional (nD) discrete systems described by the nD Roesser (local) state-space model. A Bounded Real Lemma consisting of a series of conditions is first established for general nD systems. The proposed nD conditions directly reduce to their 1D counterparts when n = 1, and besides several sufficient conditions which include the existing 2D results as special cases, some necessary and sufficient conditions are also shown to explore further insights to the considered problem. By applying a linear matrix inequality (LMI) condition of the nD Bounded Real Lemma, the nD $${{\mathcal H}_{\infty}}$$ control problem is then considered for three kinds of control laws, namely, static state feedback (SSF) control, dynamic output feedback (DOF) control and static output feedback (SOF) control, respectively. The nD $${{\mathcal H}_{\infty}}$$ SSF and DOF control problems are formulated in terms of an LMI and LMIs, respectively, and thus tractable by using any available LMI solvers. In contrast, the solution condition of the nD $${{\mathcal H}_{\infty}}$$ SOF controller is not strictly in terms of LMIs, therefore an iterative algorithm is proposed to solve this nonconvex problem. Finally, numerical examples are presented to demonstrate the application of these different kinds of nD $${{\mathcal H}_{\infty}}$$ control solutions to practical nD processes as well as the effectiveness of the proposed methods.

Gain-Scheduled H∞ Controller Synthesis for 2-D Discrete-Time Linear Parameter-Varying Systems

AbstractThis paper proposes a gain-scheduled H∞ control design method for 2-dimensional discrete-time linear parameter-varying systems. The system is described by a Roesser state-space model with matrices depending affinely on time-varying parameters whose admissible values are assumed to belong to a given convex bounded polyhedral domain. A convex optimization approach in terms of linear matrix inequalities is devised for designing a gain-scheduled H∞ control via static state feedback. The controller design is based on a parameter-dependent Lyapunov function and the controller gain is a matrix fraction of quadratic polynomial matrices on the scheduling parameters.

A new elementary operation approach to multidimensional realization and LFR uncertainty modeling: the SISO case

In this paper, a new elementary operation approach is proposed to the problem of multidimensional (n-D) Roesser state-space model realization or linear fractional representation (LFR) uncertainty modeling. The new approach can overcome the main difficulties encountered by Galkowski's approach and always generates a standard or regular realization for a given causal n-D transfer function. In particular, the n-D realization problem is formulated as an elementary operation problem of a certain n-D polynomial matrix in a way totally different from Galkowski's approach so that the singularity problem can be completely avoided. A general constructive realization procedure and an algorithm for evaluating the realization order associated with this procedure will be proposed, which can be easily implemented by symbolic software in, e.g., MATLAB or Maple. Some further techniques for a realization with lower order will also be shown. Symbolic and numerical examples will be presented throughout the paper to illustrate the basic ideas as well as the effectiveness of the proposed approach.