Fornasini-Marchesini Local State-space Model Research Articles

Expected Power Bound for Two-Dimensional Digital Filters in the Fornasini-Marchesini Local State-Space Model

This letter examines the expected power bound (EPB) for two-dimensional (2-D) digital filters in the Fornasini-Marchesini local state-space (FMLSS) model with Wiener process noise. The goal of this letter is to establish a new criterion whereby 2-D digital filters in the FMLSS form have a 2-D EPB. The criterion also ensures asymptotic stability without Wiener process noise. A numerical example demonstrates the usefulness of the proposed result. The criterion in the letter and that in provide a systematic framework for EPB of 2-D digital filters.

Finite-Frequency Model Reduction of Two-Dimensional Digital Filters

This technical note is concerned with the model reduction problem of two-dimensional (2-D) digital filters over finite-frequency ranges. The 2-D digital filter is described by the Fornasini-Marchesini local state-space (FM LSS) model. With the aid of the generalized Kalman-Yakubovich-Popov (GKYP) lemma for 2-D systems, sufficient conditions for the finite-frequency model reduction problem are derived. Compared with full-frequency methods, the proposed finite-frequency method can get a better approximation performance over finite-frequency ranges. An example is given to demonstrate the effectiveness of the proposed method.

Model approximation for two-dimensional Markovian jump systems with state-delays and imperfect mode information

This paper is concerned with the problem of $$\mathscr {H}_{\infty }$$H� model approximation for a class of two-dimensional (2-D) discrete-time Markovian jump linear systems with state-delays and imperfect mode information. The 2-D system is described by the well-known Fornasini-Marchesini local state-space model, and the imperfect mode information in the Markov chain simultaneously involves the exactly known, partially unknown and uncertain transition probabilities. By using the characteristics of the transition probability matrices, together with the convexification of uncertain domains, a new $$\mathscr {H}_{\infty }$$H� performance analysis criterion for the underlying system is firstly derived, and then two approaches, namely, the convex linearisation approach and iterative approach, to the $$\mathscr {H}_{\infty }$$H� model approximation synthesis are developed. The solutions to the problem are formulated in terms of strict linear matrix inequalities (LMIs) or a sequential minimization problem subject to LMI constraints. Finally, simulation studies are provided to illustrate the effectiveness of the proposed design methods.

Filtering design for two-dimensional Markovian jump systems with state-delays and deficient mode information

This paper is concerned with the problem of H∞ filtering for a class of two-dimensional Markovian jump linear systems described by the Fornasini–Marchesini local state-space model. The systems under consideration are subject to state-delays and deficient mode information in the Markov chain. The description of deficient mode information is comprehensive that simultaneously includes the exactly known, partially unknown and uncertain transition probabilities. By invoking the properties of the transition probability matrix, together with the convexification of uncertain domains, a new H∞ performance analysis criterion for the filtering error system is firstly derived. Then, via some matrix inequality linearisation procedures, two approaches for the filter synthesis are proposed. It is shown that both the full-order and reduced-order filters can be constructed by solving a set of linear matrix inequalities. Finally, simulation studies are provided to illustrate the effectiveness of the proposed design methods.

Regular unknown input functional observers for 2-D singular systems

This paper concerns with asymptotic regular unknown input functional observers (UIFOs) for two-dimensional (2-D) acceptable singular systems described by the Fornasini-Marchesini local state-space second model. A sufficient condition for the existence of an asymptotic regular UIFO is first presented in terms of a rank condition on the given system matrices. Based on this, an asymptotic 2-D regular UIFO is constructed using a linear matrix inequality (LMI) technique, while a new method is also given for designing regular functional observers for 2-D acceptable singular systems. Two illustrative examples are provided to demonstrate the feasibility and effectiveness of the proposed method.

Robust stabilisation for a class of stochastic two‐dimensional non‐linear systems with time‐varying delays

This study addresses the robust stabilisation problem for the two-dimensional (2D) discrete non-linear stochastic systems with time-varying delays in states. Based on the well-known Fornasini–Marchesini local state-space model, a more general 2D discrete non-linear system is considered with both stochastic disturbances and time-varying delays. Our attention is focused on the design of a delay-dependent state-feedback controller so that the 2D stochastic non-linear system is guaranteed to be globally asymptotically stabilisable in the mean square. By using the stochastic analysis and some inequality techniques, sufficient criteria ensuring the existence of such controllers are derived in terms of matrix inequalities, which can be effectively solved by resorting to some standard convex optimisation algorithms. Furthermore, such stabilisation results are extended to more general cases where the system matrices contain either polytopic or norm-bounded parameter uncertainties. Finally, a numerical example is provided to demonstrate the applicability of the proposed controller design approach.

ℋ∞Control for Two-Dimensional Markovian Jump Systems with State-Delays and Defective Mode Information

This paper investigates the problem ofℋ∞state-feedback control for a class of two-dimensional (2D) discrete-time Markovian jump linear time-delay systems with defective mode information. The mathematical model of the 2D system is established based on the well-known Fornasini-Marchesini local state-space model, and the defective mode information simultaneously consists of the exactly known, partially unknown, and uncertain transition probabilities. By carefully analyzing the features of the transition probability matrices, together with the convexification of uncertain domains, a newℋ∞performance analysis criterion for the underlying system is firstly derived, and then theℋ∞state-feedback controller synthesis is developed via a linearisation technique. It is shown that the controller gains can be constructed by solving a set of linear matrix inequalities. Finally, an illustrative example is provided to verify the effectiveness of the proposed design method.

Robust Non-Fragile Control of 2-D Discrete Uncertain Systems: An LMI Approach

This paper considers the problem of robust non-fragile control for a class of two-dimensional (2-D) discrete uncertain systems described by the Fornasini-Marchesini second local state-space (FMSLSS) model under controller gain variations. The parameter uncertainty is assumed to be norm-bounded. The problem to be addressed is the design of non-fragile robust controllers via state feedback such that the resulting closed-loop system is asymptotically stable for all admissible parameter uncertainties and controller gain variations. A sufficient condition for the existence of such controllers is derived based on the linear matrix inequality (LMI) approach combined with the Lyapunov method. Finally, a numerical example is illustrated to show the contribution of the main result.

LMI Approach to Suboptimal Guaranteed Cost Control for 2-D Discrete Uncertain Systems

This paper studies the problem of the guaranteed cost control via static-state feedback controllers for a class of two-dimensional (2-D) discrete systems described by the Fornasini-Marchesini second local state-space (FMSLSS) model with norm bounded uncertainties. A convex optimization problem with linear matrix inequality (LMI) constraints is formulated to design the suboptimal guaranteed cost controller which ensures the quadratic stability of the closed-loop system and minimizes the associated closed-loop cost function. Application of the proposed controller design method is illustrated with the help of one example.

Realization of multidimensional systems in Fornasini-Marchesini state-space model

This paper presents a constructive procedure that can generate a Fornasini-Marchesini (local) state-space model realization for a given n-D system with lower order than the existing procedure given by Alpay and Dubi. It is clarified that the method of Alpay and Dubi usually generates a realization with unnecessarily high order even for a simple rational transfer function, and, by taking into account of the structural characteristic of the given transfer function or transfer matrix, it is possible to construct a realization with much lower order. Nontrivial examples are given to illustrate the basic idea as well as the effectiveness of the proposed approach.

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.

Robust H∞ Filtering for 2D Stochastic Systems

This paper investigates the problem of H∞ filter design for two-dimensional (2D) stochastic systems. The stochastic perturbation is first introduced into the well-known Fornasini-Marchesini local state-space model. Our attention is focused on the design of full-order and reduced-order filters, which guarantee the filtering error system to be mean-square asymptotically stable and to have a prescribed H∞ disturbance attenuation performance. Sufficient conditions for the existence of such filters are established in terms of linear matrix inequalities, and the corresponding filter design is cast into a convex optimization problem, which can be efficiently handled by using available numerical software. In addition, the obtained results are 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, are taken into consideration. A numerical example is provided to illustrate the usefulness of the proposed filter design procedures.

Positive real control for uncertain two-dimensional systems

This brief deals with the problem of positive real control for uncertain two-dimensional (2-D) discrete systems described by the Fornasini-Marchesini local state-space model. The parameter uncertainty is time-invariant and norm-bounded. The problem we address is the design of a state feedback controller that robustly stabilizes the uncertain system and achieves the extended strictly positive realness of the resulting closed-loop system for all admissible uncertainties. A version of positive realness for 2-D discrete systems is established. Based on this, a condition for the solvability of the positive real control problem is derived in terms of a linear matrix inequality. Furthermore,the solution of a desired state feedback controller is also given. Finally, we provide a numerical example 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.

Synthesis of 2-D state-space digital filters with low sensitivity based on the Fornasini-Marchesini model

The balanced realization and low-sensitivity structure of two-dimensional recursive digital filters are considered in the framework of the Fornasini-Marchesini local state-space model. A procedure is introduced for the balanced realization of 2-D recursive digital filters. The sensitivities of a 2-D transfer function are investigated with respect to the coefficients in the local state-space model. The overall sensitivity is evaluated using the 2-D controllability and observability Gramians. The filter structure reducing the overall sensitivity is synthesized for two cases: one free from l/sub 2/ scaling constraints on the state variables and the other under the scaling constraints. These filter structures are shown to be closely related to the balanced realization. An example is given to illustrate the utility of the proposed technique.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

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>