Fornasini-Marchesini Systems Research Articles

Robust decentralized iterative learning control for large‐scale interconnected systems described by 2‐D Fornasini–Marchesini systems with iteration‐dependent uncertainties including boundary states, disturbances, and reference trajectory

SummaryThis article first investigates robust iterative learning control (ILC) problem of a class of large‐scale interconnected systems, which consist of many subsystems described by two‐dimensional linear discrete first Fornasini–Marchesini systems with iteration‐dependent uncertainties arising from not only boundary states, disturbances, but also reference trajectory. A decentralized P‐type ILC law without any information exchanges with other subsystems is proposed such that the ultimate ILC tracking error of each subsystem can converge to a bounded range, the bound of which depends continuously on the bounds of all iteration‐dependent uncertainties considered. Especially, if these iteration‐dependent uncertainties are convergent progressively along the iteration direction, perfect ILC tracking on 2‐D reference trajectory can be obtained. Additionally, a modified ILC law with compensation technique is used to a class of large‐scale interconnected systems composed of many subsystems represented by two‐dimensional linear discrete second Fornasini–Marchesini systems. Two simulation examples are used to demonstrate the effectiveness and validity of the obtained ILC results. Finally, some comparative discussions are given.

Robust high-order iterative learning control approach for two-dimensional linear discrete time-varying Fornasini–Marchesini systems with iteration-dependent reference trajectory

This paper first investigates robust tracking problem of a high-order iterative learning control (HOILC) algorithm for two classes of two-dimensional linear discrete time-varying Fornasini–Marchesini systems (2-D LDTVFMS) and 2-D LDTVFMS with input delays with iteration-dependent reference trajectory described by a high-order internal model (HOIM) operator, boundary states and disturbances. An extended high-order linear discrete inequality is proposed to guarantee the ILC tracking error of 2-D LDTVFMS converge robustly to a bounded range, the bound of which is relied on iteration-dependent uncertainties from reference trajectory, boundary states and disturbances. A simulation example on a practical thermal process is used to validate the effectiveness and feasibility of the proposed HOILC law. Additionally, it is verified by theory analysis and simulation that no matter how the ILC controller gain is selected, the convergence condition in Theorem 2 of Wan and Li [(2021). High-order internal model based iterative learning control for 2-D linear FMMI systems with iteration-varying trajectory tracking. IEEE Transactions on Systems Man and Cybernetics: Systems, 51, 1462–1472] is not satisfied. Using the proposed HOILC law, perfect tracking result in Theorem 2 of Wan and Li (2021) can be obtained.

Dissipativity-Based Two-Dimensional Control and Filtering for a Class of Switched Systems

This paper deals with the dissipative control and filtering problems for the discrete-time two-dimensional (2-D) switched Fornasini-Marchesini (FM) systems. First, the definition of 2-D TSR- ϑ-dissipativity for the FM systems is introduced. By employing the switched quadratic Lyapunov function technique and establishing multiple recursive formulas, sufficient conditions are developed to guarantee that the considered system is asymptotically stable under the arbitrary switching signal (ASS) case and exponentially stable under the restricted switching signal (RSS) case with the 2-D TSR- ϑ-dissipativity, respectively. The special cases, such as the 2-D passivity performance and the 2-D H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance, are also discussed. Based on the obtained dissipative stability conditions, the 2-D TSR- ϑ-dissipative state feedback controller and filter design problems are further studied for the addressed system. Particularly, the extended average dwell time (define this later on in text) technique is utilized for the system under the RSS case. Finally, the effectiveness of the proposed design method is illustrated by numerical examples.

Robust Iterative Learning Control for 2-D Singular Fornasini–Marchesini Systems with Iteration-Varying Boundary States

This study first investigates robust iterative learning control (ILC) issue for a class of two-dimensional linear discrete singular Fornasini–Marchesini systems (2-D LDSFM) under iteration-varying boundary states. Initially, using the singular value decomposition theory, an equivalent dynamical decomposition form of 2-D LDSFM is derived. A simple P-type ILC law is proposed such that the ILC tracking error can be driven into a residual range, the bound of which is relevant to the bound parameters of boundary states. Specially, while the boundary states of 2-D LDSFM satisfy iteration-invariant boundary states, accurate tracking on 2-D desired surface trajectory can be accomplished by using 2-D linear inequality theory. In addition, extension to 2-D LDSFM without direct transmission from inputs to outputs is presented. A numerical example is used to illustrate the effectiveness and feasibility of the designed ILC law.

Adaptive ILC of Tracking Nonrepetitive Trajectory for Two-dimensional Nonlinear Discrete Time-varying Fornasini-Marchesini Systems with Iteration-varying Boundary States

Most of adaptive iterative learning control (AILC) algorithms focus on one-dimensional (1-D) systems, rather than two-dimensional (2-D) systems. This brief is first concerned with AILC for 2-D nonlinear discrete time-varying Fornasini-Marchesini system (NDTVFMS) with nonrepetitive reference trajectory under iteration-varying boundary states. By using Lyapunov analysis method, it can guarantee that the ultimate tracking error tends to zero asymptotically, and make all identified parameters and system signals to be bounded as iteration number goes to infinity. Two illustrative examples are used to validate the effectiveness of the designed AILC approach.

Nonlinear continuous Fornasini–Marchesini model of fractional order with nonzero initial conditions

A continuous nonlinear Fornasini–Marchesini system of fractional order with nonzero initial conditions is considered. The existence, uniqueness and continuous dependence of solutions on functional parameters are studied. Some results concerning single and mixed fractional derivatives of functions of two variables are presented.

Finite-region dissipative dynamic output feedback control for 2-D FM systems with missing measurements

This paper focuses on the finite-region dissipative dynamic output feedback control problem for a class of discrete-time two-dimensional (2-D) Fornasini–Marchesini (FM) systems with missing measurements. Firstly, the definitions of finite-region boundedness (FRB) and finite-region (T, S, R)-ϑ-dissipativity are introduced for 2-D FM systems. Secondly, sufficient conditions on the FRB and finite-region (T, S, R)-ϑ-dissipativity of the resultant closed-loop 2-D systems are obtained by the construction of the Lyapunov function and the special recursive formulas. Thirdly, based on the decoupling technique, the desired dynamic output feedback controller design method for the considered 2-D systems is further provided. In the end, an example of the metal rolling process is given to show the effectiveness of the proposed design results.

Observer-based finite-region dissipative control of 2D FM systems

This paper is concerned with the problem of finite-region dissipative observer-based control for a class of discrete-time two-dimensional (2D) systems formulated by the Fornasini-Marchesini (FM) model subject to missing measurements. Firstly, the concepts of finite-region boundedness (FRB) and finite-region (T, S, R)-ϑ-dissipativity for 2D systems are introduced, respectively. Secondly, the FRB and finite-region (T, S, R)-ϑ-dissipativity are investigated for the resulting closed-loop systems, and then the corresponding conditions are derived. Based on the obtained results, the observer-based output feedback controller is designed, and the example of metal rolling process is employed to illustrate the effectiveness of the proposed design method.

Event-triggered control of discrete-time 2-D switched Fornasini–Marchesini systems

In this paper, the stability analysis and control problems for the discrete-time two-dimensional (2-D) switched systems are studied via an event-triggered scheme. The addressed 2-D system is represented by the well-known Fornasini–Marchesini (FM) local state-space model and is assumed to be operating at several modes under a switching signal. First of all, the event-triggered strategy is introduced for the 2-D switched system in order to utilize data effectively in the capacity-limited communication network. Then by constructing the index function and establishing multiple recursive formulas, the new conditions are given such that the resultant closed-loop 2-D system is asymptotically stable under the arbitrary switching signal and exponentially stable under the restricted switching signal. Particularly, the extended average dwell time (EADT) technique is employed in the case of the restricted switching signal. Based on the stability analysis results, the event-triggered state feedback controller is further developed to stabilize the corresponding 2-D switched system. Finally, the effectiveness of the proposed controller design method is demonstrated by examples.

Model transformation based sliding mode control of discrete-time two-dimensional Fornasini–Marchesini systems

This paper investigates the problem of sliding mode control (SMC) for discrete-time two-dimensional (2-D) systems subject to external disturbances. Given a 2-D Fornasini–Marchesini (FM) local state space model, attention is focused on designing the 2-D sliding surface and sliding mode controller, which guarantees the resultant closed-loop system to be asymptotically stable. Particularly, this problem is solved using the model transformation based method. First of all, sufficient conditions are formulated for the existence of a linear sliding surface guaranteeing the asymptotic stability of the equivalent sliding mode dynamics. Based on this, a sliding mode controller is synthesized to ensure that the associated 2-D FM system satisfies the reaching condition. The efficiency of the proposed 2-D SMC law design is shown by a numerical example. This paper extends the idea of model transformation to the 2-D systems and solves the SMC problem of a more general 2-D model in FM type for the first time.

Weak Markov processes as linear systems

A noncommutative Fornasini–Marchesini system (a multi-variable version of a linear system) can be realized within a weak Markov process (a model for quantum evolution). For a discrete time parameter the resulting structure is worked out systematically and some quantum mechanical interpretations are given. We introduce subprocesses and quotient processes and then the notion of a \(\gamma \)-extension for processes which leads to a complete classification of all the ways in which processes can be built from subprocesses and quotient processes. We show that within a \(\gamma \)-extension we have a cascade of noncommutative Fornasini–Marchesini systems. We study observability in this setting and as an application we gain new insights into stationary Markov chains where observability for the system is closely related to asymptotic completeness in a scattering theory for the chain.

Stability of nonlinear time-varying digital 2-D Fornasini-Marchesini system

Stability of a system described by the time-varying nonlinear 2-D Fornasini-Marchesini model is considered. There are given notions of stability of the system and theorems for stability and asymptotic stability which can be considered as the Lyapunov stability theorem extension for the system.

Triangular representation of Fornasini–Marchesini systems

We further develop the study of Fornasini---Marchesini linear systems with upper triangular state operators, addressing the problem of constructing a triangular Fornasini---Marchesini model equivalent (under a proper definition) to a given system. In particular, we are interested in the problem of determining when such a system can be constructed, without losing the information about the state of the original system.

CONTROLLABILITY OF GOURSAT-DARBOUX SYSTEMS – SOME NUMERICAL RESULTS

In the paper, we consider a linear nonautonomous Goursat-Darboux system which is a continuous version of discrete Fornasini-Marchesini system. We give some general algorithm for construction a piecewise constant control related to some density result for attainable sets connected with integrable controls and piecewise constant ones.

Two-dimensional periodically shift variant digital filters

Two-dimensional (2-D) periodically shift variant (PSV) digital filters are considered. These filters have potential applications in processing video signals with cyclostationary noise, scrambling of digital images, and in 2-D multirate signal processing. The filters are formulated in the form of the Fornasini-Marchesini (FM) state-space model with periodic coefficients. This PSV model is then represented as a new shift-invariant system which is named the "Kiok-Neon" model. This model has several advantages that include ease of analysis and reduced computations compared to the existing state-space models. An algorithm is developed that transforms any given 2-D PSV FM system to its equivalent "Kiok-Neon" model. Invertibility of this model is an important consideration, especially in image scrambling applications. A condition is established for the invertibility of the "Kiok-Neon" model of the 2-D PSV system. Also, the inverse system can be easily computed from the original. It is established that the 2-D PSV system is asymptotically stable if an equivalent shift-invariant FM system is asymptotically stable. The established results are illustrated with examples.