Doubly-indexed dynamical systems: State-space models and structural properties

Fornasini, Ettore; Marchesini, G.

doi:10.1007/bf01776566

Cited by 919 publications

(348 citation statements)

References 17 publications

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“…Marchesini 1978/1979;Kaczorek 1985;Klamka 1991), which is well known in the theory of discrete multidimensional systems. It should be underlined that such discrete systems play an important role in the theory of automatic control (cf.…”

Section: F T ϕ (T) ϕ (T) ψ (T) ψ (T) Dt + G ϕ (0) ϕ (0) ψ mentioning

confidence: 99%

On the existence of an optimal solution of the Mayer problem governed by 2D continuous counterpart of the Fornasini-Marchesini model

Bors

Majewski

2012

Multidim Syst Sign Process

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In the paper the optimization problem described by some nonlinear hyperbolic equation being continuous counterpart of the Fornasini-Marchesini model is considered. A theorem on the existence of at least one solution to this hyperbolic PDE is proved and some properties of the set of all solutions are established. The existence of a solution to an optimization problem under appropriate assumptions is the main result of this paper. Some application of the obtained results to the process of gas filtration is also presented.Keywords Mayer problem · Continuous counterpart of the Fornasini-Marchesini model · Existence of optimal solutions

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Section: F T ϕ (T) ϕ (T) ψ (T) ψ (T) Dt + G ϕ (0) ϕ (0) ψ mentioning

confidence: 99%

On the existence of an optimal solution of the Mayer problem governed by 2D continuous counterpart of the Fornasini-Marchesini model

Bors

Majewski

2012

Multidim Syst Sign Process

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“…Consequently links exist with 2D systems described by the Roesser (Roesser, 1975) and Fornasini-Marchesini (Fornasini and Marchesini, 1978) state-space models, which evolve of the complete upper-right quadrant of the 2D plane. In previous work, see, e.g, (Rogers et al, 2007) and the relevant cited references, it has been established that some, but not all, systems theoretic properties of repetitive processes can be addressed by application of known methods for Roesser and Fornasini-Marchesini systems.…”

Section: Introductionmentioning

confidence: 99%

Pass profile exponential and asymptotic stability of nonlinear repetitive processes

et al. 2017

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This paper considers discrete and differential nonlinear repetitive processes using the state-space model setting. These processes are a particular class of 2D systems that have their origins in the modeling of physical processes. Their distinguishing characteristic is that one of the two independent variables needed to describe the dynamics evolves over a finite interval and therefore they are defined over a subset of the upper-right quadrant of the 2D plane. The current stability theory for nonlinear dynamics assumes that they operate over the complete upper-right quadrant and this property may be too strong for physical applications, particulary in terms of control law design. With applications in mind, the contribution of this paper is the use of vector Lyapunov functions to characterize a new property termed pass profile exponential stability.

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“…Representations for the dynamics of these systems include the Roesser (Roesser, 1975) and Fornasini-Marchesini state-space model (Fornasini and Marchesini, 1978). The Roesser model, originally proposed for image processing applications, partitions the state vector into two sub-vectors, commonly termed horizontal and vertical respectively.…”

Section: Introductionmentioning

confidence: 99%

“…The Roesser model, originally proposed for image processing applications, partitions the state vector into two sub-vectors, commonly termed horizontal and vertical respectively. In Fornasini-Marchesini state-space model (Fornasini and Marchesini, 1978)) one state vector is used.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of 2D Roesser systems via vector Lyapunov functions

et al. 2017

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The paper gives new results that contribute to the development of a stability theory for 2D nonlinear discrete and differential systems described by a state-space model of the Roesser form using an extension of Lyapunov's method. One of the main difficulties in using such an approach is that the full derivative or its discrete counterpart along the trajectories cannot be obtained without explicitly finding the solution of the system under consideration. This has led to the use of a vector Lyapunov function and its divergence or its discrete counterpart along the system trajectories. Using this approach, new conditions for asymptotic stability are derived in terms of the properties of two vector Lyapunov functions. The properties of asymptotic stability in the horizontal and vertical dynamics, respectively, are introduced and analyzed. This new properties arise naturally for repetitive processes where one of the two independent variables is defined over a finite interval. Sufficient conditions for exponential stability in terms of the properties of one vector Lyapunov function are also given as a natural follow on from the asymptotic stability analysis.

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Doubly-indexed dynamical systems: State-space models and structural properties

Abstract: Abstract. Doubly-indexed dynamical systems provide a state space realization of two-dimensional filters which includes previous state models. Algebraic criteria for testing structural properties (reachability, observability, internal stability) are introduced.

Cited by 919 publications

References 17 publications

On the existence of an optimal solution of the Mayer problem governed by 2D continuous counterpart of the Fornasini-Marchesini model

On the existence of an optimal solution of the Mayer problem governed by 2D continuous counterpart of the Fornasini-Marchesini model

Pass profile exponential and asymptotic stability of nonlinear repetitive processes

Stability analysis of 2D Roesser systems via vector Lyapunov functions

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