Derivation and reduction of the singular Fornasini–Marchesini state‐space model for a class of multidimensional systems

Admissible transformation approach to Roesser state‐space model realization of singular multidimensional systems

“…The dimension of $$\bm{x}(i_1,\ldots ,i_n)$$, i.e. r , is called the order of the SNDRM [5], while $$r_i$$ is called the partial order w.r.t. $$\bm{x}_{i}(i_1,\ldots ,i_n)$$, for $$i=1,\ldots ,n$$, respectively.…”

“…An n ‐D rational transfer matrix $$H(z)$$ is said to be causal if all its entries are causal [9]. Similarly, an n ‐D rational transfer matrix $$H(z)$$ is said to be non‐causal if there exists at least one non‐causal its entry [5].…”

“…Problem Given an non‐causal n ‐D transfer matrix $$H(z)$$, find unknown matrices A , B , C , D , E , and $$\bm{r} = (r_1, \ldots , r_n)$$ such that Equation (3) holds true. This model $$(A, B, C, D, E)$$ is called an SNDRM realization for the given n ‐D transfer matrix $$H(z)$$ [5, 19, 24]. …”

“…Singular multidimensional ( n ‐D) systems theory has been receiving considerable attention in the past few decades [1–3] due to the potential value and extensive applications in thermal processes [4], spatially interconnected systems [5], including electrical, and mechanical ones. A fundamental problem in the field of singular n ‐D systems is the construction of a realization for a given non‐causal multivariate transfer matrix by a certain kind of an n ‐D state‐space model, typically by Roesser model [6] or Fornasini–Marchesini (F–M) model [5]. On the other hand, modelling the plant as a generalized linear fractional representation (GLFR) of the uncertainty matrix plays an important role in the robustness analysis and synthesis of uncertain systems[6, 7], such as DC‐DC converters [8].…”

“…However, there are only few results on the realization of singular n ‐D systems in the literature. For example, a constructive realization method has been proposed in [5] for the singular n ‐D F–M models by using the realization methods of the regular n ‐D F–M models. A generalized linear fractional representation approach has been provided in ref.…”