Fornasini-Marchesini Local State-space Model Research Articles

Stabilization for 2-D switched T–S fuzzy systems under the state-dependent switching

This study focuses on the analysis of stability and stabilization for a class of two-dimensional (2-D) discrete-time switched systems. Owing to the inherent uncertainty and nonlinearity in real-world engineering systems, the Takagi–Sugeno (T–S) fuzzy model is employed to describe the dynamics of the 2-D switched system. The discussion encompasses two primary models for the 2-D discrete-time switched T–S fuzzy system (2DSTSFS), specifically the Roesser model and the Fornasini–Marchesini local state-space model. For 2DSTSFSs, this paper delineates sufficient stability criteria that utilize a state-dependent switching signal, facilitated by the application of the Lyapunov–Metzler inequality, ensuring that state trajectories are globally attracted. Furthermore, the paper articulates sufficient conditions for the stabilization of the 2DSTSFS. Additionally, it elucidates the transformation relationship between the two models. To corroborate the theoretical findings, a practical example is employed, demonstrating the applicability of the proposed theorems.

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.

On l 1-gain control for 2D delayed positive systems in FM LSS models: necessary and sufficient conditions

This paper presents an -gain control design for two-dimensional (2D) delayed positive systems (PSs) in the Fornasini-Marchesini local state-space model. First, for a stable 2D delayed PS, the necessary and sufficient conditions are derived for the calculation of the exact value of the -gain. Then, the controller design is given with the aid of the singular value decomposition method, which ensures that the closed-loop system is asymptotically stable at the specified -gain level. In the end, two examples are provided to identify the validity and correctness of the results of the theory.

A unified framework of convex stability conditions for 2-D switched systems with stable or unstable modes

The discrete-time two-dimensional (2-D) switched system is a powerful tool to model many practical physical processes with abrupt changes. However, the stability analysis problem for 2-D switched systems struggles to provide a unified framework for stable or unstable modes. Users have to delicately tune lots of parameters to shape the Lyapunov function for each mode. Besides, the resulting stability conditions are non-convex and thus the feasibility highly depends on the tuning parameters. In this paper, we propose a unified stability framework for 2-D switched systems, formulated by the well-known Fornasini–Marchesini local state-space model and the Roesser model. The concepts called unit switching sequence (USS) and sequence generator are presented. A novel Lyapunov function is also accordingly established which views the switching sequence as a concatenation of USSs, different from the perspective that regards the switching sequence as a combination of subsystems in previous literatures. Upon this framework, a convex stability criterion is developed, and the conservativeness can be lowered by a lifting technique at the expense of a higher off-line computational complexity. The derived theories are further extended for a standard l2-gain which is achieved under restricted switching signals for the first time. Numerical examples are provided for validity verification, conservativeness comparison, and l2-gain computation.

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.

Stabilisation for 2-D discrete-time switched nonlinear systems with mixed time-varying delays under all modes unstable

The problem of delay-dependent stability is concerned for two-dimensional switched systems in Fornasini-Marchesini local state-space model with mixed time-varying delays and with a class of generalised Lipschitz nonlinearities in this paper. The mixed time-varying delays consist of both the discrete and the distributed delays, and the time-varying delays are allowed in the states. Adopting the fast average dwell-time switching technique with Lyapunov functional, a sufficient condition of the exponential stability for two-dimensional nonlinear switched systems with all subsystems unstable is derived in terms of linear matrix inequality. Specifically, two-dimensional Abel lemma-based finite-sum inequalities approach and Jensen inequalities approach are applied to reduce the conservativeness of the result. The obtained result on stability analysis is then utilised to design a dynamic output feedback controller to stabilise the nonlinear closed-loop switched system. Finally, two numerical examples are shown to demonstrate the validity of the proposed results.

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.

Robust Filtering for 2-D Discrete-Time Switched Systems

In this article, we study the robust filtering problem for 2-D switched systems with convex-bounded uncertainties. The studied switched system is formulated by Fornasini–Marchesini local state-space model, and system matrices depend affinely on convex-bounded uncertain parameters. We first derive synthesis results for 2-D switched systems, which extends the existing results and lays a foundation for the robust filter design. To design the robust filter, we transform the 2-D switched system into an equivalent difference-algebraic representation form. Using the equivalent representation and the parameter-dependent Lyapunov function approach, sufficient conditions are established for system stability, and 2-D stationary discrete-time linear filters are designed to ensure the prescribed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{\infty }$</tex-math></inline-formula> or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_{2}$</tex-math></inline-formula> performance for all admissible uncertain parameters. Finally, a numerical example is presented to illustrate the obtained results.

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.

Non-fragile extended dissipativity-based state feedback control for 2-D Markov jump delayed systems

This work deals with the issue of extended dissipativity-based state feedback control for an array of two-dimensional Markov jump systems established upon Fornasini–Marchesini local state-space model, in which a kind of state delays is thoroughly taken into account. Meanwhile, the non-fragile control scheme is adopted to cope with the imprecisions occurring in the realization of the controller. The primary target is to devise a novel controller ensuring that the closed-loop system actualizes not merely the stochastic stability, but also meets a specified extended dissipative property. Through selecting a suitable Lyapunov–Krasovskii functional and applying a concise matrix decoupling approach, some criteria are derived to guarantee the solvability of the admissible controller. Ultimately, in order to validate the availability and practicability of the developed results, a simulation example is provided.

[formula omitted] filtering for discrete-time 2-D switched systems: An extended average dwell time approach

This paper studies the H∞ filtering problem for a class of discrete-time two-dimensional (2-D) switched systems. The 2-D systems under consideration are described by the well-known Fornasini–Marchesini local state-space model. Our attention is focused on designing a full-order filter such that the filtering error system is guaranteed to be asymptotically or exponentially stable with a prescribed H∞ disturbance attenuation level. Based on the switched quadratic Lyapunov function and the piecewise Lyapunov function approach, sufficient conditions are established to ensure the existence of the desired filters for such systems under the arbitrary switching and restricted switching signals, respectively. Furthermore, the notion of time instant for 2-D switched systems is introduced and the extended average dwell time technique is utilized under the restricted switching signal. Then the corresponding filter is designed by dealing with a convex optimization problem that can be efficiently solved via standard numerical algorithms. Finally, two numerical examples are provided to demonstrate the effectiveness of the developed filter design algorithms.

Asynchronous filtering for 2-D switched systems with missing measurements

This work is concerned with $$\mathscr {H}_{\infty }$$ filter design with missing measurements for a class of two-dimensional (2-D) switched systems represented by Fornasini–Marchesini local state-space model. The switching signal of the switched filters involve time delays, which result in the asynchronism between the filter and the system switching. The issues of asymptotic mean-square stability and $$\ell _{2}$$ -gain analysis for the 2-D switched systems are addressed firstly, based on which mode-dependent filters are designed with mode-dependent average dwell time scheme. Finally, two examples are given to demonstrate the validity of the proposed technique.

Stabilization of 2-D Switched Systems With All Modes Unstable via Switching Signal Regulation

This work is concerned with stabilization for 2-D switched systems represented by both the Fornasini-Marchesini local state-space model and the Roesser model. Without any controller, by designing a set of switching signals subject to average dwell time characteristic, the 2-D switched systems with all subsystems unstable could be stabilized. The exponential stability can be ensured for these switched systems by introducing a quasi-time-dependent Lyapunov functional. The effectiveness and merits of the proposed methods are illustrated with two examples.

1D and 2D parameter estimation of a rotary drum filter based on RLS and IV methods

ABSTRACTThis work deals with a physical one- and two-dimensional (1D and 2D) parameters estimation of a filtration process of slurry, the second stage of phosphoric acid manufacture. This study focuses on recursive least square and instrumental variable techniques applied to the (1D) and (2D) models. The model of the rotary drum filter is based on different physical laws involved in the filtration phase in order to get a simulator of the filtration process. Besides, many physical parameters rise in the system model and effect enormously the efficiency which should be modelled with precision such as permeability, porosity and viscosity. We use a constructive realization procedure for (2D) systems which may lead to a Fornasini–Marchesini local state-space model to describe the dynamic of the system states.

Finite-frequency fault detection for two-dimensional Fornasini–Marchesini dynamical systems

ABSTRACTThis paper is concerned with the fault detection problem for two-dimensional (2-D) discrete-time systems described by the Fornasini–Marchesini local state-space model. The goal of the paper is to design a fault detection filter to detect the occurrence of faults in finite-frequency domain. To this end, a finite-frequency H− index is used to describe fault sensitivity performance, and a finite-frequency H∞ index is used to describe disturbance attenuation performance. In light of the generalised Kalman–Yakubovich–Popov lemma for 2-D systems and matrix inequality techniques, convex conditions are derived for this fault detection problem. Based on these conditions, a numerical algorithm is put forward to construct a desired fault detection filter. Finally, a numerical example and an industrial example are given to illustrate the effectiveness of the proposed algorithm.

Asynchronous control for 2-D switched systems with mode-dependent average dwell time

This work is concerned with asynchronous control for a class of two-dimensional switched systems with mode-dependent average dwell time. The systems are formulated by the famous Fornasini–Marchesini local state-space model, and the switching signal of the switched controller involves time delays which lead to the asynchronism between controllers and subsystems. By extending mode-dependent average dwell time with the switched quadratic Lyapunov functional approach, the stabilization condition is established for 2-D asynchronously switched systems. Based on the stabilization result, mode-dependent state-feedback controllers are designed to guarantee H∞ performance and asymptotic stability of the corresponding closed-loop system. Finally, two examples are provided to illustrate the effectiveness of the proposed design scheme.

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.

Passive IIR deconvolution of two-dimensional digital systems subject to missing measurements

This paper deals with the deconvolution problem for two-dimensional (2-D) digital systems which are expressed by the Fornasini–Marchesini local state-space model. Since the sensor measurements are not ideal in many applications, the phenomenon of missing measurements is considered for the deconvolution. A 2-D infinite impulse response (IIR) deconvolution filter is proposed to reconstruct the original signal transmitted through the 2-D digital channel. The phenomenon of missing measurements is described by a Bernoulli process. As a stochastic variable is involved in the deconvolution filtering error system, the mean-square stability is investigated. Beside the stability, to investigate the performance of deconvolution filtering on attenuating the disturbance, the passivity and the dissipation are both studied. With the obtained results, the filter design approach is proposed and the filter gains can be obtained by solving a condition in the expression of linear matrix inequality. The performance of designed filter is demonstrated with a numerical example.

Stability analysis and stabilization of 2-D switched systems under arbitrary and restricted switchings

In this paper, the problems of stability analysis and stabilization are investigated for discrete-time two-dimensional (2-D) switched systems, which are formulated by the well-known Fornasini–Marchesini local state-space model. Firstly, by using the switched quadratic Lyapunov function approach, a sufficient stability condition is established for such systems under arbitrary switching signal. Then, the extended average dwell time technique combining with the piecewise Lyapunov function approach is developed, and it is utilized for the stability analysis of the 2-D switched systems for the restricted switching case. Based on the stability analysis results, sufficient conditions are presented to stabilize the 2-D switched systems. Finally, two examples are provided to illustrate the effectiveness of the proposed new design techniques.