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A geometric approach to the theory of 2-D systems

A geometric approach to the theory of 2-D systems,10.1109/9.7251,IEEE Transactions on Automatic Control,G. Conte,A. Perdon

A geometric approach to the theory of 2-D systems
Journal: IEEE Transactions on Automatic Control - IEEE TRANS AUTOMAT CONTR , vol. 33, no. 10, pp. 946-950, 1988
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Citation Context (5)

• ...It is interesting to note that, unlike the 1-D case, the required notions of conditioned invariance are not dual of the notion of controlled invariance developed in [6], [21]...

Lorenzo Ntogramatzidis, et al. Asymptotic quotient observers for 2-D Fornasini Marchesini models

• ...Many important results have been achieved in the attempt to develop a geometric theory for 2-D systems (Conte and Perdon 1988; Conte et al. 1991; Karamancio˜glu and Lewis 1990, 1992)...
• ...In particular, a definition of controlled invariance was first proposed in Conte and Perdon (1988) for FM models...
• ...subspaces defined in Conte and Perdon (1988)...
• ...Definition 3.1 (Conte and Perdon 1988) The subspace V ⊆ R n is controlled invariant for...
• ...Nonetheless, the definition enjoys good feedback properties, as shown for the first time in Conte and Perdon (1988 ), and brief ly recalled in Lemma3.1...
• ...Algorithm 4.1 is a generalisation of a corresponding result in (Conte and Perdon 1988,...
• ...where forall i, j ∈ Z, xi, j ∈R n is the local state, ui, j ∈R m is the controlinput, wi, j ∈ R d is a disturbance to be decoupled from the output yi, j ∈R p , Ak ∈ R n×n , Bk ∈ R n×m , Hk ∈ R n×d for k = 1,2, C ∈ R p×n , D ∈ R p×m and G ∈ R p×d . The corresponding 2-D counterpart of the disturbance decoupling problem (DDP) first considered in Basile and Marro (1969), was studied and solved for FM models by Conte ...
• ...(13). A necessary condition for (40) to be satisfied is that the feedthrough matrix G be zero, and this is equivalent to condition (i) of Proposition 3.1 in Conte and Perdon (1988)...
• ...to be more interesting in the second decoupling problem considered in Conte and Perdon (1988); i.e., the measurable signal decoupling problem (MSDP), in which the disturbance w is available for measurement...
• ...Hence, this condition indeed encompasses condition (ii) of Proposition 3.1 in Conte and Perdon (1988)...
• ...Paraskevopoulos 1979; Sebek 1983 and Conte and Perdon 1988), where the model matching problem is solved via polynomial and geometric approaches, respectively...

Lorenzo Ntogramatzidis, et al. A geometric theory for 2-D systems including notions of stabilisabilit...

• ...In the last two decades, many valuable results have been achieved in the attempt to develop a geometric theory for 2-D systems, [3], [8], [9], [12]...
• ...In particular, in [3] a definition of controlled invariance was proposed for Fornasini-Marchesini (FM) models...
• ...The aim of this paper is therefore to provide a characterisation of the stability of the 2-D invariants introduced in [3]...
• ...Definition 1: (Conte and Perdon, 1988, [3])...
• ...Whereas in the 1-D case the converse is true as well, with this definition of controlled invariance the subspace of minimal dimension containing a given sequence satisfying (12) is not necessarily controlled invariant for Σ0. However, Definition 1 enjoys good feedback properties, as shown for the first time in [3], and briefly recalled in the following two lemmas...
• ...A sufficient condition for the solvability of this problem for 2-D systems without stability was first given in [3] in terms of the inclusion of certain subspaces involving V ⋆ , and the feedback matrix F solving the decoupling problem is any output-nulling friend of V ⋆ ...

Lorenzo Ntogramatzidis, et al. A geometric approach with stability for two-dimensional systems

• ...In the last two decades, many efforts have been focused on extending the concept of controlled invariance – which lies at the heart of the geometric approach – to Fornasini-Marchesini and to Roesser state-space models, [2], [5], [7]...
• ...The first pioneering contribution on this matter was the paper by Conte and Perdon in 1988, [2], where a definition of controlled invariance is presented for 2-D Fornasini-Marchesini models...
• ...Indeed, in [2] solvability conditions are given for the decoupling of i) non accessible input functions via static local state feedback and ii) measurable signals via a mixed local state feedback and static feedforward action...
• ...In this paper, the definitions of controlled invariant and output-nulling subspaces given in [2] are taken into consideration and...
• ...By straightforwardly extending the definitions of [2] to the case where the feedthrough matrix D may differ from the zero matrix, we say that an output-nulling subspace VΣ of Σ is a subspace of R n satisfying the inclusion ⎡...
• ...Σ . Clearly, when D = 0, such definition reduces to that of a controlled invariant subspace contained in the null-space of C, [2]...
• ...It was shown in [2] that this last implication does not hold in the 2-D case...
• ...Unlike the 1-D case, the converse of Proposition 2 is not true in general, as pointed out in [2] and in [5]...
• ...In the following theorem, the classic algorithm for the computation of the largest output-nulling subspace of Σ presented in [2] is extended to 2-D non-strictly proper systems...
• ...The proof of the former is standard, and can be carried out along the same lines of that of Proposition 2.7 in [2]...
• ...A classic problem, that was studied and solved for the first time for Fornasini-Marchesini strictly proper models by Conte and Perdon in [2], is that of finding conditions ensuring that a static local state feedback input function ui,j = Fxi,j exists such that the output function is not affected by the disturbance w. That is, such that when the boundary conditions on the local state are at zero the output is identically zero...
• ...Hence, such a condition is equivalent to condition i) of Proposition 3.1 in [2]...
• ...The presence of the feedthrough matrices D and G appears to be more interesting in the second decoupling problem tackled in [2], i.e., the measurable signal decoupling problem, in which the disturbance w is supposed to be available for measurement...
• ...which is the natural extension to non-strictly proper systems of condition ii) of Proposition 3.1 in [2]...
• ...Hence, this condition indeed encompasses condition ii) of Proposition 3.1 in [2]...
• ...[10], [12] and [2], where the model matching problem is solved via polynomial and geometric approaches, respectively...

Lorenzo Ntogramatzidis, et al. New Results and Applications of the Geometric Approach to Two-Dimensio...

• ...Since then, these models have been analyzed extensively [2],[6],[9],[12]...
• ...The results here may be compared to [lo] which considers 1-D implicit systems, and to [2] which considers state-space 2-D systems...

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