Fornasini-Marchesini Local State-space Research Articles

Stability analysis of 2-D linear discrete systems based on the Fornasini–Marchesini second model: Stability with asymmetric Lyapunov matrix

The stability of two-dimensional (2-D) linear discrete systems based on the Fornasini–Marchesini local state-space (LSS) model is considered. A stability criterion using the asymmetric Lyapunov matrix P is presented. A special case of the criterion is discussed.

Output Feedback H ∞ Stabilization of 2D Discrete Switched Systems in FM LSS Model

This paper investigates the dynamic output feedback H∞ stabilization problem for a class of discrete-time 2D (two-dimensional) switched systems represented by a model of FM LSS (Fornasini–Marchesini local state space) model. First, sufficient conditions for the exponential stability and weighted H∞ disturbance attenuation performance of the underlying system are derived via the average dwell time approach. Then, based on the obtained results, dynamic output feedback controller is proposed to guarantee that the resulting closed-loop system is exponentially stable and has a prescribed disturbance attenuation level γ. Finally, two examples are provided to verify the effectiveness of the proposed method.

Robust finite frequency [formula omitted] filtering for uncertain 2-D systems: The FM model case

This paper is concerned with the problem of robust finite frequency (FF) H∞ filtering for uncertain two-dimensional (2-D) discrete-time systems in the Fornasini–Marchesini local state-space (FM LSS) model with polytopic uncertain parameters. The goal of the paper is to design filters such that the FF H∞ norm of the filtering error system has a specified upper bound for all uncertainties. In light of a recently developed generalized bounded real lemma, a linear matrix inequality-based approach is proposed for robust FF H∞ filter analysis and design. It is demonstrated that the presented approach to robust FF H∞ filter design covers the latest standard H∞ filtering result. Moreover, it is shown that the existing results specialized for the Roesser model, when applied to the FM LSS model through a model transformation, are much more restrictive than the proposed results in the paper, further justifying this work.

Delay-dependent robust stability and stabilisation of uncertain two-dimensional discrete systems with time-varying delays

The problems of delay-dependent robust stability and stabilisation for a class of uncertain two-dimensional (2-D) discrete systems with time-varying delays described by the second Fornasini-Marchesini local state space (FM LSS) model are investigated in this study. By employing improved free-weighting matrix approach, delay-dependent stability criterion for the nominal 2-D state-delayed system is first obtained and formulated in terms of linear matrix inequalities (LMIs); and then, a stabilisation design criterion using static state feedback controller is derived based on the stability result. Since the design criterion is not expressed strictly in terms of LMI, an improved cone complementary linearisation algorithm is employed to solve the non-convex feasibility problem. Furthermore, the stability and stabilisation results are extended to robust stability and stabilisation of the uncertain 2-D state-delayed system. Numerical examples are provided to demonstrate the effectiveness and advantage of the proposed method.

Filtering for uncertain 2-D discrete systems with state delays

This paper is concerned with the problem of robust H ∞ filtering for two-dimensional (2-D) discrete systems with time-delays in states. The 2-D systems under consideration are described in terms of the well-known Fornasini–Marchesini local state-space (FMLSS) models with time-delays. Our attention is focused on the design of a full-order filter such that the filtering error system is guaranteed to be asymptotically stable with a prescribed H ∞ disturbance attenuation performance. Sufficient conditions for the existence of desired filters are established by using a linear matrix inequality (LMI) approach, and the corresponding filter design problem is then cast into a convex optimization problem that can be efficiently solved by resorting to some standard numerical software. Furthermore, the obtained results are extended to more general cases where the system matrices contain either polytopic or norm-bounded parameter uncertainties. A simulation example is provided to illustrate the effectiveness of the proposed design method.

ℋ︁∞ and l2–l∞ filtering for two‐dimensional linear parameter‐varying systems

AbstractIn this paper, the ℋ︁∞ and l2–l∞ filtering problem is investigated for two‐dimensional (2‐D) discrete‐time linear parameter‐varying (LPV) systems. Based on the well‐known Fornasini–Marchesini local state‐space (FMLSS) model, the mathematical model of 2‐D systems under consideration is established by incorporating the parameter‐varying phenomenon. The purpose of the problem addressed is to design full‐order ℋ︁∞ and l2–l∞ filters such that the filtering error dynamics is asymptotic stable and the prescribed noise attenuation levels in ℋ︁∞ and l2–l∞ senses can be achieved, respectively. Sufficient conditions are derived for existence of such filters in terms of parameterized linear matrix inequalities (PLMIs), and the corresponding filter synthesis problem is then transformed into a convex optimization problem that can be efficiently solved by using standard software packages. A simulation example is exploited to demonstrate the usefulness and effectiveness of the proposed design method. Copyright © 2007 John Wiley & Sons, Ltd.

Hamiltonian (script capital H)]∞ mode reduction for two-dimensional discrete state-delayed systems

The problem of ℋ∞ model reduction for two-dimensional (2-D) discrete systems with delay in state is considered. The mathematical model of 2-D systems is established on the basis of the well-known Fornasini–Marchesini local state-space. First, conditions are established to guarantee the asymptotic stability and a prescribed noise attenuation level in the ℋ∞ sense for the underlying system. For a given stable system, attention is focused on the construction of a reduced-order model, which approximates the original system well in an ℋ∞ norm sense. Sufficient conditions are proposed for the existence of admissible reduced-order solutions. Since these obtained conditions are not expressed as strict linear matrix inequalities (LMIs), the cone complementary linearisation method is exploited to cast them into sequential minimisation problems subject to LMI constraints, which can be readily solved using standard numerical software. These obtained results are further extended to more general cases whose system states contain multiple delays. Two numerical examples are provided to demonstrate the effectiveness of the proposed techniques.

ℋ∞ model reduction for uncertain two-dimensional discrete systems

This paper investigates the problem of ℋ∞ model reduction for two-dimensional (2-D) discrete systems with parameter uncertainties residing in a polytope. For a given robustly stable system, our attention is focused on the construction of a reduced-order model, which also resides in a polytope and approximates the original system well in an ℋ∞ norm sense. Both Fornasini–Marchesini local state-space (FMLSS) and Roesser models are considered through parameter-dependent approaches, with sufficient conditions obtained for the existence of admissible reduced-order solutions. Since these obtained conditions are not expressed as strict linear matrix inequalities (LMIs), the cone complementary linearization method is exploited to cast them into sequential minimization problems subject to LMI constraints, which can be readily solved using standard numerical software. In addition, the development of zeroth order models is also presented. Two numerical examples are provided to show the effectiveness of the proposed theories. Copyright © 2005 John Wiley & Sons, Ltd.

Stability and Stabilization of Uncertain 2-D Discrete Systems with Stochastic Perturbation

This paper is concerned with the problem of stability analysis and stabilization of two-dimensional (2-D) discrete systems with stochastic perturbation. The 2-D stochastic system model is first established based on the Fornasini-Marchesini local state-space (FMLSS) model, and mean-square asymptotic stability is derived by means of linear matrix inequality (LMI) technique. This stability result is further extended to more general cases where the system matrices also contain uncertain parameters. The most frequently used ways of dealing with parameter uncertainties, including polytopic and norm-bounded characterizations, have been taken into consideration. Based on this, the robust stabilization problem for 2-D systems with both deterministic and stochastic uncertainties is addressed, with sufficient LMI conditions obtained for the existence of stabilizing controllers, which can be solved via efficient numerical algorithms. An illustrative example is provided to demonstrate the applicability of the proposed approach.

H∞ deconvolution filtering of 2-d digital systems

This paper deals with the problem of two-dimensional (2-D) digital system deconvolution under the H∞ performance specification. The 2-D digital system under consideration is expressed by the Fornasini-Marchesini local state-space (FM LSS) model. For a given 2-D digital system, a 2-D deconvolution filter is designed to reconstruct the 2-D signal to meet a prescribed H∞ performance specification. A key analytical means for the deconvolution filter design is the 2-D bounded realness property derived in this paper. Applying this property, the 2-D H∞ deconvolution filter design is formulated into a convex optimization problem characterized by linear matrix inequalities (LMIs). A procedure for the design of an H∞ deconvolution filter for a given 2-D digital system with measurement noise is developed first. The LMI approach to the 2-D H∞ deconvolution filtering problem is further extended to the 2-D H∞ robust deconvolution filtering problem with polytopic modeling uncertainties in the 2-D system. The obtained results on the 2-D deconvolution filtering and robust deconvolution filtering problems are demonstrated with examples.

On frequency weighted minimal L/sub 2/ sensitivity of 2-D systems using Fornasini-Marchesini LSS model

The optimal state-space realization design problem is investigated in terms of minimizing a frequency weighted L/sub 2/-sensitivity measure for two-dimensional (2-D) linear digital systems represented with the Fornasini-Marchesini local state-space (LSS) model. An expression is derived for the transfer function error due to finite wordlength (FWL) effects on parameters. The optimal realization problem is then formulated, which is shown to be equivalent to minimizing the frequency weighted L/sub 2/-sensitivity measure. The problem is solved and the optimal design procedures are illustrated with an example.

A generalized on the study synthesis of 2-D state-space digital filters with minimum roundoff noise

The Fornasini-Marchesini local state-space (LSS) model is used as the basis for a novel expression for the output-noise variance due to roundoff together with an I/sub 2/ scaling on the state variables. An optimal Fornasini-Marchesii LSS model structure is then synthesized that minimizes the output noise due to roundoff, subject to an l/sub 2/ scaling constraint. The synthesis utilizes a 2-D similarity transformation matrix that is not block-diagonal, but general. This requires solving only one optimization problem. Some constraints imposed on the Fornasini-Marchesini LSS model and the 2-D similarity transformation yield the results obtained with the Roesser LSS model. The proposed synthesis theory is therefore quite general and simple. An example is given to illustrate its utility.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>