Fornasini Marchesini Research Articles

Robust Stabilization of Uncertain 2-D Discrete Delayed Systems

This paper is concerned with the stabilization of a class of two-dimensional discrete state-delayed systems described by the Fornasini–Marchesini second local state-space model subject to polytopic uncertainties. A delay-dependent stability criterion is derived for the existence of a memoryless state feedback controller such that the closed-loop system is robustly stable. To match practical issues the influence of finite wordlength nonlinearities in control implementation, delays and parameter uncertainties belonging to a known convex polytope are also discussed. The presented criteria are obtained by employing parameter-dependent Lyapunov–Krasovskii function and slack variables that augments the searching space allowing less conservative solutions. Finally, examples are provided to show the effectiveness of the proposal.

Graph-theoretic approach for structural controllability of two-dimensional linear systems

Abstract The aim of this work is to present some criteria of structural-controllability relative to a certain structural two-dimensional linear systems. The most important property of two-dimensional systems is that the information is spread in two independent directions. To study their structural-controllability, we considered a new structural digraph associated with the different structural two-dimensional linear systems. The construction of this structural digraph is based on the disjoint-union graph notion, which translates the independence of the two dynamics, and consider separately the horizontal and vertical part of the state space model representation of two-dimensional systems. We define the two-dimensional disjoint-union digraph of structural Givone–Roesser model, and determine the two-dimensional disjoint-union digraph of structural Fornasini–Marchesini model, finally we derive a criteria based on the two-dimensional disjoint-union digraph.

Series concatenation of 2D convolutional codes by means of input-state-output representations

ABSTRACTIn this paper, we investigate the properties of two-dimensional (2D) convolutional codes which are obtained from series concatenation of two 2D convolutional codes. For this purpose, we confine ourselves to dealing with finite-support 2D convolutional codes and make use of the Fornasini–Marchesini input-state-output (ISO) model representations. Within these ISO representations, we study when the structural properties of modal reachability and modal observability of the two given ISO representations carry over to the resulting 2D convolutional code. Moreover, we provide necessary conditions for obtaining a systematic concatenated convolutional code. Finally, we present a lower bound on its free distance.

Switched mechanisms for stability and l1-gain analysis of T-S fuzzy positive systems described by the F-M second model

In this paper, the Takagi–Sugeno fuzzy positive system is analyzed under the switched mechanisms, which is described by the discrete-time Fornasini–Marchesini Second model. By utilizing the co-positive Lyapunov function and the novel average dwell time method, sufficient conditions are derived under which the proposed system is positive, exponentially stable and has l1-gain bound. Furthermore, the criteria obtained are presented in the form of matrix inequalities which can be solved by the standard software. Finally, a numerical example is given to demonstrate the effectiveness of the approach outlined in the paper.

Stability and constrained control for positive two-dimensional systems with delays in the second FM model

Abstract This paper addresses the stability and control problem of linear positive two-dimensional discrete-time systems with multiple delays in the second Fornasini–Marchesini model. The contribution lies in three aspects. First, a novel proof is provided to establish necessary and sufficient conditions of asymptotic stability for positive two-dimensional delayed systems. Then, a state-feedback controller is designed to ensure the non-negativity and stability of the closed-loop systems. Finally, a sufficient condition for the existence of constrained controllers is developed under the additional constraint of bounded control, which means that the control inputs and the states of the closed-loop systems are bounded. Two examples are given to validate the proposed methods.

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.

Robust iterative learning control for multi-phase batch processes: an average dwell-time method with 2D convergence indexes

ABSTRACTIn order to cope with system disturbances in multi-phase batch processes with different dimensions, a hybrid robust control scheme of iterative learning control combined with feedback control is proposed in this paper. First, with a hybrid iterative learning control law designed by introducing the state error, the tracking error and the extended information, the multi-phase batch process is converted into a two-dimensional Fornasini–Marchesini (2D-FM) switched system with different dimensions. Second, a switching signal is designed using the average dwell-time method integrated with the related switching conditions to give sufficient conditions ensuring stable running for the system. Finally, the minimum running time of the subsystems and the control law gains are calculated by solving the linear matrix inequalities. Meanwhile, a compound 2D controller with robust performance is obtained, which includes a robust extended feedback control for ensuring the steady-state tracking error to converge rapidly. The application on an injection molding process displays the effectiveness and superiority of the proposed strategy.

Simultaneous estimation of multiple channel faults for two-dimensional linear systems

ABSTRACTThis study focuses on the problem of simultaneous estimation of multiple channel faults for two-dimensional linear systems, which are described by Fornasini–Marchesini second (FM-II) model, and the faults that exist in state equation and measurement equation. By transforming the fault in the measurement equation as augmented state, the FM-II model with faults in the state equation and measurement equation can be rewritten into a singular system. Hence, several observers are proposed for the singular systems, and then the estimation of the faults in the state equation and measurement equation can be obtained. Using Lyapunov stability theory, sufficient conditions for the existence of the asymptotically stable observer and uniformly ultimately bounded observer are derived in the context of time domain. For the bounded observer, the upper bound of estimation error can be provided referring to the fault bound. Numerical and practical examples are given to demonstrate the effectiveness of the proposed method.

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.

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.

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.

Robust Optimal $$H_\infty $$ H ∞ Control for 2-D Discrete Systems Using Asymmetric Lyapunov Matrix

This paper is concerned with the design of optimal \(H_\infty \) static-state feedback controller for an uncertain 2-D discrete system described by the Fornasini–Marchesini second model. The parametric uncertainties are assumed to be norm bounded. The main contribution of the paper lies in the application of the bounded real lemma derived using a more generalized asymmetric Lyapunov matrix. Based on the derived bounded real lemma, a linear matrix inequality (LMI)-based sufficient condition for the existence of \(H_\infty \) static-state feedback controller is established. Furthermore, convex optimization problem with LMI constraints is formulated to design the optimal \(H_\infty \) static-state feedback controller which not only guarantees the asymptotic stability of the closed-loop system but also achieves optimized noise attenuation level \(\gamma \) for all admissible parametric uncertainties. Finally, with the help of various examples, it is demonstrated that the optimized noise attenuation level \(\gamma \) is better achievable by using asymmetric Lyapunov matrix than using symmetric Lyapunov Matrix.

Robust ${\mathscr {H}}_{\infty }$ Filtering for a Class of Two-Dimensional Uncertain Fuzzy Systems With Randomly Occurring Mixed Delays

This paper is concerned with the robust ${\mathscr {H}}_{\infty }$ filtering problem for a class of 2-D uncertain fuzzy systems with randomly occurring mixed delays (ROMDs). The underlying 2-D systems are described by the Fornasini–Marchesini model, and the uncertainty is expressed in a linear fraction form. An improved Takagi–Sugeno (T–S) fuzzy model corresponding to the spatial promise variables is adopted to represent the complicated 2-D nonlinear system. The mixed delays consisting of both discrete and distributed delays are allowed to appear in a random manner governed by two sets of Bernoulli distributed white sequences with known probability. A full-order fuzzy filter is constructed to estimate the output signal such that, in the presence of parameter uncertainties and ROMDs, the dynamics of the estimation errors is asymptotically stable with a prescribed ${\mathscr {H}}_{\infty }$ disturbance attenuation level. Based on the stochastic analysis technique and the Lyapunov-like functional, sufficient conditions are established to ensure the existence of the desired filters, and the explicit expressions of such filters are derived by means of the solution to a class of convex optimization problems that can be solved via standard software packages. A numerical example is provided to demonstrate the effectiveness of the developed filter design algorithms, and the filter performances with and without fuzzy rules are also compared.

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.

1D and 2D modelling of a chemical process using the FM-II model: Rotary drum filter

This work treats the physical one-dimensional (1D) and two-dimensional (2D) modelling of a chemical process of filtration of slurry, which is the second step of phosphoric acid manufacture. This work focuses on the different physical laws involved in the filtration stage in order to obtain a simulator of the filter. In this work, we are interested in the 1D modelling of a rotary filter using all parameters and physical phenomena established in the filtration phase. Then, a 2D model emerged from the previous model by choosing both a temporal and a spatial variable. These two parameters are quite necessary for the construction of a 2D model based on the method of Fornasini–Marchesini (FM-II) to describe the dynamics of the system.

State-space formulation of 2-D frequency transformation in Fornasini–Marchesini second model

This paper presents a state-space formulation for the two-dimensional (2-D) frequency transformation in the Fornasini–Marchesini second (FM II) model. Specifically, by introducing some new state vectors, an equivalent description of the FM II model will be first established. Then based on this new description, a state-space formulation of the 2-D frequency transformation in the FM II model is derived by revealing the substantial input and output relations among the state vectors of the prototype filter, all-pass filters and the transformed filter. The resultant formulation owns an elegant and general expression, and can be viewed as a natural extension of its counterpart in the Roesser model. Furthermore, as one of the various possible applications of the proposed formulation, a 2-D zero-phase IIR filters design procedure will be shown, by which the desired 2-D zero-phase IIR filters can be constructed in a more flexible way to avoid the kind of image distortions caused by the nonlinear phase of the used filter. Three typical image processing examples as well as the computational complexity analysis will be given to show the effectiveness and efficiency of the proposed procedure.

Fault detection and isolation of two-dimensional (2D) systems

In this work, we develop a novel fault detection and isolation (FDI) scheme for discrete-time two-dimensional (2D) systems that are represented by the Fornasini–Marchesini model II (FMII). This is accomplished by generalizing the basic invariant subspaces including unobservable, conditioned invariant and unobservability subspaces of 1D systems to 2D models. These extensions have been achieved and facilitated by representing a 2D model as an infinite dimensional (Inf-D) system on a Banach vector space, and by particularly constructing algorithms that compute these subspaces in a finite and known number of steps. By utilizing the introduced subspaces, the FDI problem is formulated and necessary and sufficient conditions for its solvability are provided. Sufficient conditions for solvability of the FDI problem for 2D systems using both deadbeat and LMI-based filters have also been developed. Moreover, the capabilities and advantages of our proposed approach are demonstrated by performing an analytical comparison with the currently available 2D geometric methods in the literature.

Input–output finite-region stability and stabilization for discrete 2-D Fornasini–Marchesini models

This paper establishes input–output finite-region stability (I–O FRS) and stabilization analysis methods to investigate the transient behavior of output signals for discrete two-dimensional (2-D) Fornasini–Marchesini (FM) models. For 2-D systems with two classes of input signals, by building special recursive formulas, sufficient I–O FRS conditions are formulated in terms of solvability of linear matrix inequalities (LMIs). Furthermore, aiming at designing static state feedback controllers, I–O finite-region stabilization issues are analyzed for the corresponding 2-D systems and sufficient conditions are reported to guarantee I–O FRS of the closed-loop systems. To conclude the paper, we provide some numerical examples to confirm the validity of the proposed methods.

Reduced-Order $$H_{\infty }$$ H ∞ Filtering with Intermittent Measurements for a Class of 2D Systems

The problem of \(H_{\infty }\) filter design for a class of 2D systems is solved here in the presence of intermittent measurements. Data dropouts are characterized using a stochastic variable satisfying the Bernoulli random binary distribution. Our attention is focused on the design of reduced-order \(H_{\infty }\) filters such that the filtering error 2D stochastic system is robust mean-square asymptotically stable and fulfills a given \(H_{\infty }\) disturbance attenuation level. We use a new formulation for a class of 2D system Fornasini–Marchesini (FM) models. A sufficient condition is established by means of the linear matrix inequalities (LMI) technique. The efficiency and viability of the proposed techniques and tools are demonstrated through a set of numerical examples.

Novel Results on Generalized Dissipativity of Two-Dimensional Digital Filters

Existing results on the generalized dissipativity of digital filters are limited to 1-D cases. This brief presents some novel results on the generalized dissipativity of 2-D filters in the discrete Fornasini–Marchesini second (FMS) form. More specifically, we propose a criterion to ascertain that single 2-D filters in the discrete FMS form are 2-D generalized dissipative. Using this result, a condition is established such that the interconnected 2-D filters in the discrete FMS form are 2-D generalized dissipative. Furthermore, the asymptotic stability of unforced interconnected 2-D filters is examined. It is shown that the results developed in our recent work for 1-D filters and in this brief for 2-D filters serve as an overall framework for the generalized dissipativity of digital filters.