Fornasini-Marchesini Local State-space Research Articles

Zonotopic state bounding for 2-D systems with dynamic event-triggered mechanism

This paper investigates the problem of zonotope-based state estimation for 2-D systems represented by both the Fornasini–Marchesini local state–space (FMLSS) model and the Roesser model with dynamic event-triggered mechanism (ETM) where the exogenous disturbance and measurement noise are assumed to be unknown-but-bounded (UBB). First, considering the different characteristics of these two models, we design different dynamic ETMs for both FMLSS type and Roesser type respectively. A zonotopic set-membership estimation approach is proposed for the considered 2-D system to obtain the estimation interval. Then, three optimal observer design methods are proposed for different scenarios. In addition, the relationship of the proposed criteria is further discussed between the 2-D augmented error FMLSS system and the 2-D augmented error Roesser system considered in this paper. Finally, simulation results are given to illustrate the effectiveness of the proposed methods.

Zonotope-Based Interval Estimation for 2-D FMLSS Systems Using an Event-Triggered Mechanism

This study considers the problem of interval estimation for event-triggered 2-D systems in Fornasini–Marchesini local state-space (FMLSS) model. In such systems, the general hypothesis for practical applications is that exogenous disturbances and measurement noise are unknown but bounded. First, an observer design criterion is deduced from the defined zonotopic radius and weighted radius of the FMLSS model. Subsequently, to economize the restricted communication resources, an event-driven data transmission strategy is employed for the observer. Then, a sufficient criterion is established to guarantee the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {L}_\infty$</tex-math></inline-formula> performance for the considered 2-D FMLSS system with the event-triggered scheme. From the obtained observer, the estimation interval of the 2-D system is derived by using the zonotopic state estimation approach. Finally, the validity is confirmed by simulations.

Event-triggered dynamic output quantized control for 2-D switched systems

It is well known that designing mode-dependent event-triggered control (MDETC) brings challenging difficulties to theoretical analysis, especially for two-dimensional (2-D) switched systems. Therefore, for 2-D switched Fornasini–Marchesini local state-space (FMLSS) systems, this paper designs a MDETC to investigate global exponential stabilization almost surely (GES a.s.). A MDETC based on dynamic output quantization control scheme is designed, which not only has a wide range of practicability, but also greatly saves network bandwidth resources. By constructing mode-dependent Lyapunov functions that include two time directions, some novel sufficient conditions are provided such that the switched FMLSS system achieves GES a.s. Unlike most previous results, our results do not require each mode to be stable, not even after adding control. Finally, numerical experiments are provided to verify the validity of our main theoretical results.

Stability Analysis and Synthesis for 2-D Switched Systems with Random Disturbance

In this paper, the stability analysis and controller design of two-dimensional (2-D) switched system with random disturbances are studied. First, based on the Fornasini–Marchesini local state space model, a random discrete 2-D switched system model is proposed. Secondly, based on the method of switching quadratic Lyapunov function, Schur complement and linear matrix inequality the criterion of the sufficiency of the asymptotically stable under any switching signal is established. The extended average residence time method is used to obtain a sufficient criterion of the mean square exponential stability of the random discrete 2-D switched system under restricted switching conditions. According to the stability analysis results, the conditions for the existence of the state feedback controller are given and the corresponding state feedback controller is designed. Finally, two examples are given to illustrate the effectiveness of the proposed method.

Improved filtering H∞ finite frequency of Takagi-Sugeno fuzzy systems

The work treats the filter H∞ finite frequency (FF) in Takagi-Sugeno (T-S) two dimensional (2-D) systems described by Fornasini-Marchesini local state-space (FM LSS)models. The goal of this work is to find an FF H∞ T-S fuzzy filter model design in such a way that the error system is stable and has a reduced FF H∞ performance over FF area swith noise is established as aprerequisite. Via the use of the generalized Kalman Yakubovich Popov (gKYP) lemma, Lyapunov functions approach, Finsler’s lemma, and parameterize slack matrices, new design conditions guaranteeing the FF H∞ T-S fuzzy filter method of FM LSS models are developed by solving linear matrix inequalities (LMIs). At last, the simulation results are provided to show the effectiveness and the validity of the proposed FF T-S fuzzy of FM LSS models strategy by a practical application has been made.

Expected power bound and stability of two-dimensional digital filters with multiplicative noise in the FMLSS model

This paper studies the two-dimensional (2-D) expected power bound (EPB) for 2-D digital filters with multiplicative noise in the Fornasini-Marchesini local state-space (FMLSS) model. By virtue of a class of linear matrix inequalities (LMIs), the novel existence criterion of 2-D EPB for 2-D digital filters in the considered model is obtained. The corresponding stabilization controller design method is presented based on the above criterion. Finally, two examples are presented to verify the effectiveness of our results.

Finite frequency model reduction for 2-D fuzzy systems in FM model

This paper deals with the problem of H∞ model reduction for two-dimensional (2D) discrete Takagi-Sugeno (T-S) fuzzy systems described by Fornasini-Marchesini local state-space (FM LSS) models, over finite frequency (FF) domain. New design conditions guaranteeing the FF H∞ model reduction are established in terms of Linear Matrix Inequalities (LMIs). To highlight the effectiveness of the proposed H∞ model reduction design, a numerical example is given.

Finite-region stabilization and H∞ control for 2-D FMLSS asynchronously switched system

In this paper, dwell time method incorporated with multiple Lyapunov-like functions are utilized to investigate the finite-region asynchronously switched control problems for 2-D switched system in the Fornasini-Marchesini local state-space (FMLSS) model. The finite-region stability (FRS) conditions for 2-D asynchronously switched FMLSS model are attained when there is no external disturbances. Then, the sufficient conditions in terms of dwell time scheme for the finite-region H∞ control of 2-D asynchronously switched FMLSS model are derived. Based on the established conditions, the sufficient solvability conditions for designing the corresponding finite-region mode-dependent state feedback controllers are given according to linear matrix inequalities (LMIs). Two numerical examples are employed to show the effectiveness of the provided design methods.

Dissipative Control of 2-D Switched Discrete System Via Dwell-Time-Dependent Approach

This paper studies the dissipative control problem for two-dimensional (2-D) switched discrete-time linear system represented by Fornasini–Marchesini local state-space (FMLSS) model via dwell-time-dependent Lyapunov function (DTDLF) approach. Consider the definition of (Q, S, R)- $$\alpha $$ -dissipativity for 2-D switched discrete FMLSS model in triangular region and based on the DTDLF approach, we first give the linear matrix inequality (LMI)-based sufficient condition on asymptotic stability for the 2-D switched discrete FMLSS model and then propose the LMI-based sufficient condition which can guarantee the given 2-D switched system to be asymptotically stable and strictly (Q, S, R)- $$\alpha $$ -dissipative. Furthermore, the dwell-time-dependent dissipative state-feedback controller is designed, and the LMI-based sufficient condition for designing such controller is proposed. Besides, dwell-time-dependent passive state-feedback controller and dwell-time-dependent $$H_\infty $$ state-feedback controller could be derived as the special cases of dissipative control from the established result. Finally, illustrative examples are depicted to verify the efficiency and superiority of the proposed techniques.

Quasi-time-dependent stabilisation for 2-D switched systems with persistent dwell-time

ABSTRACTThis paper is concerned with the stabilisation problem for a class of discrete-time two-dimensional (2-D) switched systems with persistent dwell-time (PDT). The systems are described by the well-known Fornasini-Marchesini local state space (FMLSS) model. The concept of PDT switching signals is introduced herein, and each stage consists of a dwell-time period in which no switching occurs and a persistent period an arbitrary switching occurs. Based on a proper Lyapunov function suitable to the PDT switching, which is both quasi-time-dependent (QTD) and mode-dependent, the QTD state feedback controller is designed to ensure the closed-loop system is exponentially stable. Compared with time-independent criteria, new results are more general and flexible, and have less conservativeness. Finally, two examples are provided to show the effectiveness and potential of our proposed methods.

Non-weighted H∞ performance for 2-D FMLSS switched system with maximum and minimum dwell time

This paper investigates the H∞ performance of two-dimensional (2-D) switched system represented by Fornasini–Marchesini local state-space (FMLSS) model with maximum and minimum dwell time approach. By using the multiple Lyapunov function approach, and designing a set of switching signals subject to maximum and minimum dwell time characteristic, respectively, for all stable subsystems or both stable and unstable subsystems exist, we give the sufficient condition on exponential stability of the given switched system, and propose the sufficient condition which can guarantee that the given switched system is exponentially stable and has a specified H∞ disturbance attenuation level γ. All the results obtained are on normal noise attenuation index of strictly non-weighted form, which are better than the existing results on weak one of weighted form from the physical point of view. Finally, numerical examples are presented to display the effectiveness of the proposed results.

Filter design for discrete-time two-dimensional T–S fuzzy systems with finite frequency specification

ABSTRACTThis paper deals with the filter design problem of two-dimensional (2-D) discrete-time nonlinear systems described by Fornasini–Marchesini local state–space (FM LSS) model under Takagi–Sugeno (T–S) fuzzy rules. The frequency of disturbance input is assumed to be known and to reside in a finite frequency (FF) range. A novel so-called FF gain is defined for 2-D discrete-time systems, which extends the standard gain. The aim of this paper is to design filters such that the filtering error system is asymptotically stable and has the disturbance attenuation performance in sense of FF gain. Sufficient conditions for the existence of a desired fuzzy filter are established in terms of linear matrix inequalities (LMIs). Simulation examples demonstrate the technique and its advantage.

An LMI approach to design robust &lt;i&gt;H&lt;/i&gt;&lt;SUB align="right"&gt;&amp;infin; controller for 2-D systems with delays

This paper focuses on H∞ controller design for two-dimensional (2-D) discrete state delay systems described by the second Fornasini-Marchesini local state space (FM LSS) model with polytopic uncertainties. The objective is to design a robust state feedback controller such that the corresponding closed-loop system achieves robust asymptotic stability and guarantees a prescribed H∞ performance index for all admissible uncertainties. Based on parameter-dependent Lyapunov functions and Finsler's lemma, a new sufficient condition for H∞ performance analysis of 2-D discrete systems with delays is developed. Then a stabilising state feedback controller is designed. A simulation example is exploited to show the effectiveness of the proposed approach.

Two-Dimensional Peak-to-Peak Filtering for Stochastic Fornasini–Marchesini Systems

This paper is concerned with the problem of two-dimensional (2-D) stochastic peak-to-peak filter design for Fornasini–Marchesini local state-space (FM LSS) systems with state-multiplicative noise. First, a novel sufficient condition is established, which guarantees that the underlying FM LSS system is stochastically mean-square stable with 2-D stochastic peak-to-peak performance. A deterministic FM LSS system case is also discussed. Subsequently, based on the result, a new 2-D stochastic peak-to-peak filter of general form is designed for FM LSS systems with state-multiplicative noise. Finally, a numerical example is presented to show the effectiveness of the proposed 2-D stochastic peak-to-peak filter.

Quasi-time-dependent control for 2-D switched systems with actuator saturation

This paper is concerned with controller design for a class of two-dimensional (2-D) switched systems with actuator saturation, which are formulated by the well-known Fornasini–Marchesini local state-space (FMLSS) model. By constructing a quasi-time-dependent Lyapunov functional, the stability and ℓ2-gain analysis for 2-D switched systems are addressed firstly. Then, a set of state feedback controllers is designed, which is both quasi-time-dependent and mode-dependent. Compared with the time-independent criteria, the new results lead to less conservatism. The effectiveness and potential of our proposed methods are illustrated by a numerical example.

Dissipative stability analysis and control of two-dimensional Fornasini–Marchesini local state-space model

ABSTRACTThis paper is concerned with the problems of dissipative stability analysis and control of the two-dimensional (2-D) Fornasini–Marchesini local state-space (FM LSS) model. Based on the characteristics of the system model, a novel definition of 2-D FM LSS (Q, S, R)-α-dissipativity is given first, and then a sufficient condition in terms of linear matrix inequality (LMI) is proposed to guarantee the asymptotical stability and 2-D (Q, S, R)-α-dissipativity of the systems. As its special cases, 2-D passivity performance and 2-D H∞ performance are also discussed. Furthermore, by use of this dissipative stability condition and projection lemma technique, 2-D (Q, S, R)-α-dissipative state-feedback control problem is solved as well. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

Passivity and Finite-Gain Performance for Two-Dimensional Digital Filters: The FM LSS Model Case

This brief considers the two-dimensional (2-D) passivity performance analysis problem of 2-D digital filters in the Fornasini–Marchesini local state–space (FM LSS) form. The purpose is to establish a new criterion in terms of linear matrix inequality such that the 2-D digital filters in the FM LSS form ensure 2-D passivity performance with a storage function. Moreover, this brief reveals that the 2-D digital filters exhibit the 2-D finite-gain performance under the proposed criterion. Without interference, the criterion guarantees the asymptotic stability of 2-D digital filters. An application example to a 2-D low-pass filter is given to demonstrate the usefulness of the presented passivity criterion. The criteria in this brief and in a previous work of one of the authors, as a whole, give methodical results for the 2-D passive removal of the overflow oscillation of 2-D digital filters with external inferences.

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.

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.