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

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. 1. Introduction Two-dimensional systems have been the subject of much research, due to the fact that several phenomena related to digital technology, image processing, geophysics and robotics, can be represented through the theory of two-dimensional systems. In the 1970’s several extensions were proposed for two-dimensional systems (see e.g. Fornasini & Marchesini, 1976 and Givone & Roesser, 1972); there has also been a great deal of synthesis work in recent decades (see e.g. Kaczorek, 1985, Bose, 2003, Gao et al., 2004, Rogers & Owens, 1997 and Gutlan, 1987). The fundamental property of these systems is that they propagate information in two independent directions or by two elements $$ z^{- 1}_{1} $$ and $$ z^{- 1}_{2} $$ in circuits theory. Among the recent work on two-dimensional systems we can cite Pal & Negi (2017) where the authors study the problem of stability analysis for a class of two-dimensional discrete systems in the presence of actuator saturation and interval-like time varying state delay, in Lin et al. (2010) M. H. Lin and co-authors define a state-space self-tuning control for two-dimensional multi-input multi-output linear discrete-time stochastic systems, Xu et al. (2005) where Li Xu and co-authors propound a constructive method based on algorithms for the realization of local Fornasini–Marchesini discrete time model, Chen & Lin (2012) investigate in their study the problem of output feedback repetitive control for uncertain discrete-time systems and formulating the problem by using certain class of two-dimensional systems and Ahn & Bas (2015) treat in thier study the asymptotic stability, dissipative control and filtering for Roesser model (RM) using a linear matrix inequality approach. One of the methods of analysis is the extension of the techniques that exist in the one-dimensional case. The analysis of the systems can be studied through the state spaces. We consider here discrete time systems. There are three discrete-time classical two-dimensional state space models, Givone–Roesser Model, Fornasini–Marchesini Model (FMM) and Attasi Model. They introduced a description of these systems by linear state space models that allowed the design of tests for controllability, observability, attainability and stability. On the other hand, the structural dynamic systems theory has recently undergone significant developments in various field relating to the engineering sciences, including underwater and space exploration. The main challenge in this area is the design of simple and manageable structural control criteria, we can cite a few recent works of C. Commault, J. M. Dion and their co-authors. In the study by Dion & Commault (2013), they study graphs associated with dynamic systems that are well suited to the analysis of structural controllability by adding inputs. In the study by Kibangou & Commault (2014) the authors study the observability of linear systems modeled with highly regular graphs, and bipartite graphs that capture the observability properties; Commault & Dion (2015), the authors examine the problem of minimal control, more precisely, the problem of controlling a linear systems with an input vector having a minimum non-zero input, and analyse the model of sparsity of the input vector; and in Commault et al. (2017) Commault, Dion and Boukhobza consider interconnected networks to show that the controllable subspace can have a part that will be present for almost all values of the free parameters. We describe a characterization of the fixed controllable subspace using the representation by the structural graph and the study by Maza et al. (2012). In this context the graph theory appears as an extremely suitable tool to define such conditions. The first work about the graph theoretic approach for the structural controllability appears in the studies by Reinschke (1988), Murota (1987), Andrei (1985) and Lin (1974) and E. Fornasini et al. analyse in Fornasini & Valcher (1997) and Fornasini & Valcher (1998) a digraph associated with a pair of matrices to study irreducible matrix pairs and primitivity of positive matrix pairs; Pereira et al. (2013) defined a superposition digraph to characterizate the global reachability of two-dimensional systems. To study the structural controllability to the two-dimensional linear systems, we define the structural Givone–Roesser model, and structural FMM, then we introduce a graph associated with the structural two-dimensional model that preserves the fundamental properties which is the independence of the two dynamics, which is based on the notion of disjoint union of two sub-graphs, and expresses the independence property of the two dynamics, thus the already existing results can be exploited through the partial analysis of graphs, and can introduce the notion of horizontal and vertical structural controllability; then, we establish the link between the Givone–Roesser structural graphs and Fornasini–Marchesini structural model. Finally, we synthesize structural controllability criteria for structural two-dimensional linear systems, and illustrate our results with illustrative examples. 2. Preliminaries 2.1. Two-dimensional Discrete Time Models There are several state-space models for two-dimensional systems introduced by Roesser (1975), Fornasini & Marchesini (1997), Attasi (1973) and Kurek (1985) which have been generalized to the nD models later on. These models have been commonly used to describe two-dimensional systems and to investigate their several properties. Here, we will only concentrate on the most common state-space models i.e. RM and FMM. 2.1.1. Givone–Roesser Model Givone & Roesser (1972) have introduced the first state space model for the theory of iterative linear circuits. An iterative circuit is a combination of individual cells. The input and output equations are   \begin{align} \begin{cases} \left( \begin{array}{@{}cccc@{}} x^{h}_{n_{1}}(i_{1}+1,i_{2}) \\ x^{v}_{n_{2}}(i_{1},i_{2}+1) \end{array}\!\!\right) = \left( \begin{array}{@{}cccc@{}} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\!\!\right) \left( \begin{array}{@{}cccc@{}} x^{h}_{n_{1}}(i_{1},i_{2}) \\ x^{v}_{n_{2}}(i_{1},i_{2}) \end{array}\!\!\right) + \left( \begin{array}{@{}cccc@{}} B_{1} \\ B_{2} \end{array}\!\!\right) u(i_{1},i_{2}) \\[12pt] y(i_{1},i_{2})= \left( \begin{array}{@{}rrrr@{}} C_{1} & C_{2} \end{array}\!\!\right) \left( \begin{array}{@{}cccc@{}} x^{h}_{n_{1}}(i_{1},i_{2}) \\ x^{v}_{n_{2}}(i_{1},i_{2}) \end{array}\!\!\right) \end{cases} \end{align} (2.1) where, $$x^{h}_{n_{1}}(i_{1},i_{2}) \in R^{n_{1}}$$ is the horizontal state vector, $$x^{v}_{n_{2}}(i_{1},i_{2}) \in R^{n_{2}}$$ is the vertical state vector, y(i1, i2)∈ Rp is the output vector and u(i1, i2) ∈ Rm is the input vector, and we denote by Ri the set of real vectors. A11, A12, A21, A22, B1, B2, C1 and C2 are real matrices of appropriate dimensions. In this model i and j are the positive integer valued horizontal and vertical coefficients. 2.2. Fornasini---Marchesini model Fornasini and Marchesini have generalized the model of S. Attasi in Fornasini & Marchesini (1976), and propose the following model, in its compact form   \begin{align} \begin{cases} x(i_{1}+1,i_{2}+1)=A_{1}x(i_{1}+1,i_{2}) + A_{2}x(i_{1},i_{2}+1)+ A_{0}x(i_{1},i_{2}) + B_{1}u(i_{1}+1,i_{2}) + B_{2}u(i_{1},i_{2}+1) \\ y(i_{1},i_{2}) = Cx(i_{1},i_{2})\end{cases} \end{align} (2.2) where x(i1, i2) is the state vector of the model, u(i1, i2) is the input vector and y(i1, i2) is the output vector of the model. A1, A2, A0, B1, B2 and C are real matrices of appropriate size. 3. Graph associated with a structural linear system The associated graph with the structural is defined in Reinschke (1988), Andrei (1985) and Murota (1987). The generic properties of the system can then often be characterized very simply in terms of properties of the associated graph. This makes some results very intuitive. This modeling has the following characteristics. To such a system, one can easily associate a graph G = (V, W) the set of vertices is V = U ∪ X ∪ Y where U, X and Y are inputs, states and outputs, respectively, given by $$u= \left \{u_{1},u_{2},\ldots ,u_{m} \right \}$$, $$x= \left \{x_{1},x_{2},\ldots ,x_{n} \right \}$$ and $$y= \{y_{1},y_{2},\ldots ,y_{p}\}$$. The set of arcs $$W = \left \{ (u_{i}, x_{j}) | B_{ji} \neq 0 \right \} \cup \left \{ (x_{i}, x_{j}) | A_{ji} \neq 0 \right \} \cup \left \{(x_{i}, y_{j}) | C_{ji} \neq 0 \right \}$$, where Aji (resp. Bji, Cji) is the element (j, i) of the matrix A (resp. B, C), the direction of the arcs is from ui to xj, xi to xj and xi to yj. 3.1. Structural-Controllability In this subsection the results are taken from the studies by Reinschke (1988) and Andrei (1985), and the fundamental criteria of structural controllability are exposed. Definition 3.1 The elements of a structure matrix [Q] are either fixed at zero or indeterminate values which are assumed to be independent of one another. Definition 3.2 A numerically matrix Q is called an admissible numerical realization if it can be obtained by fixing all indeterminate entries of [Q] to some particular values. Example 3.1 This example illustrate the Definition 3.1  \begin{align} A = \left( \begin{array}{@{}cccc@{}} 0 & 0 & a_{13} & 0 \\ a_{21} & 0 & 0 & a_{24} \\ 0 & 0 & a_{33} & 0 \\ 0 & a_{24} & 0 &0 \end{array}\right)\!, \end{align} (3.1)  \begin{align}\ B = \left( \begin{array}{@{}cc@{}} 0 & 0 \\ b_{21} & 0 \\ 0 & b_{32} \\ 0 & b_{24} \end{array}\right) \end{align} (3.2) and   \begin{align} C = \left( \begin{array}{@{}cccc@{}} 0 & c_{12} & 0 & 0 \\ c_{21} & 0 & 0 & c_{24} \end{array}\right) \!. \end{align} (3.3) The corresponding structural matrices are   \begin{align} [A] = \left( \begin{array}{@{}cccc@{}} 0 & 0 & l_{1} & 0 \\ l_{2} & 0 & 0 & l_{3} \\ 0 & 0 & l_{4} & 0 \\ 0 & l_{5} & 0 & 0 \end{array}\right)\!, \end{align} (3.4)  \begin{align} [B] = \left( \begin{array}{@{}cc@{}} 0 & 0 \\ l_{6} & 0 \\ 0 & l_{7} \\ 0 & l_{8} \end{array}\right) \end{align} (3.5) and   \begin{align} [C] = \left( \begin{array}{@{}cccc@{}} 0 & l_{9} & 0 & 0 \\ l_{10} & 0 & 0 & l_{11} \end{array}\right)\!.\end{align} (3.6) If   \begin{align} Q = \left( \begin{array}{@{}cc@{}} A & B \\ C & 0 \end{array}\right) \end{align} (3.7) then the corresponding structure matrix is   \begin{align} [Q] = \left[ \begin{array}{@{}cccccc@{}} 0 & 0 & l_{1} & 0 & 0 & 0 \\ l_{2} & 0 & 0 & l_{3} & l_{6} & 0 \\ 0 & 0 & l_{4} & 0 & 0 & l_{7} \\ 0 & l_{5} & 0 & 0 & 0 & l_{8} \\ 0 & l_{9} & 0 & 0 & 0 & 0 \\ l_{10} & 0 & 0 & l_{11} & 0 & 0 \end{array}\right] \end{align} (3.8) where the vanishing elements are fixed at zero while the other elements have unknown real value li. Definition 3.3 A class of systems given by its structure matrix pair [A, B] is said to be Structurally Controllable if there exists at least one admissible realization (A, B) ∈ [A, B] being Controllable in the usual numerical sense i.e. rank[B, AB, …, An−1B] = n. Remark 1 Reinschke (1988) The structural rank of [Q] is defined as S-rank [Q] = maxQ∈[Q] rank Q. Remark 2 We assume that in static state feedback we have u = Ex, with E a real matrix of appropriate size, and in the context of controllability investigation the following square bloc matrix will prove to be most appropriate   \begin{align} Q_{1} = \left( \begin{array}{@{}cc@{}} A & B \\ E & 0 \end{array}\right) \text{of appropriate size.} \end{align} (3.9) Definition 3.4 Reinschke (1988) A class of systems is said to be Input-connectable if in the digraph G([Q]) there is, for each state vertex, a path from at least one of the input vertices to the chosen state vertex. We illustrate the Definition 3.4 with the following example. Example 3.2 Consider a structural system with three state vertices, three input vertices and two output vertices with   \begin{align} [A] = \left( \begin{array}{@{}ccc@{}} 0 & l_{4} & 0 \\ 0 & 0 & l_{5} \\ 0 & 0 & 0 \end{array}\right)\!, \end{align} (3.10)  \begin{align} [B] = \left( \begin{array}{@{}ccc@{}} l_{1} & 0 & 0 \\ 0 & l_{2} & 0 \\ 0 & 0 & l_{3} \end{array}\right) \end{align} (3.11) and   \begin{align} [C] = \left( \begin{array}{@{}ccc@{}} l_{6} & 0 & 0 \\ 0 & 0 & l_{7} \end{array}\right) \!.\end{align} (3.12) This system is input-connectable. Indeed, for the vertex x1 we have one path $$\left \{u_{1},x_{1} \right \}$$, for the vertex x2 we have two paths $$\left \{u_{2},x_{2} \right \}$$ and $$\left \{u_{1},x_{1},x_{2} \right \}$$ and for the vertex x3 we have three paths $$\left \{u_{3},x_{3} \right \}$$, $$\left \{u_{1},x_{1},x_{2},x_{3} \right \}$$ and $$\left \{u_{2},x_{2},x_{3} \right \}$$ as shown in Fig. 1. Fig. 1. View largeDownload slide Illustrative diagraph of input-connectable system. Fig. 1. View largeDownload slide Illustrative diagraph of input-connectable system. Remark 3 Reinschke (1988) A cycle family corresponds to a non-vanishing term of the determinant $$det \overline{Q}$$. Hence the matrix $$\overline{Q}$$ has full rank in the structural sense i.e. an admissible realization of [Q]. Definition 3.5 Reinschke (1988) A given cycle family in G([Q]) is said to be of width w if this cycle family touches exactly w state vertices. 3.1.1. Criteria of Structural Controllability Theorem 3.6 A class of systems characterized by the n × (n + m) structure matrix pair [A, B] is structurally-controllable if and only if It is input-connectable. S-rank [A, B] = n. Theorem 3.7 A class of systems characterized by the n × (n + m) structure matrix [A, B] is structurally-controllable if and only if the digraph G([Q1]) meets both the following conditions. For each state vertex in G([Q1]) there is at least one path from one of the m input vertices to the chosen state vertex. There is at least one cycle family of width G([Q1]). Corollary 1 A class of system characterized by the structure matrix pair [A, B] is strongly structurally-controllable if and only if the G([Q1]) meets both the following conditions. Input-connectability. There is exactly one cycle family of width n in G([Q1]). 4. Main results 4.1. Structural Two-Dimensional Discrete Time Models 4.1.1. Structural Givone–Roesser Model The structural Givone–Roesser model is   \begin{align} \begin{cases} \left( \begin{array}{@{}c@{}} x^{h}_{n_{1}}\left[(i_{1}+1,i_{2})\right] \\ x^{v}_{n_{2}}\left[(i_{1},i_{2}+1)\right] \end{array}\right) = \left[ \begin{array}{@{}cc@{}} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right] \left( \begin{array}{@{}c@{}} x^{h}_{n_{1}}\left[ i_{1},i_{2}\right] \\ x^{v}_{n_{2}}\left[ i_{1},i_{2}\right] \end{array}\right) + \left[ \begin{array}{@{}c@{}} B_{1} \\ B_{2} \end{array}\right] u\left[ i_{1},i_{2}\right] \\[12pt] y\left[i_{1},i_{2}\right]= \left[ \begin{array}{@{}cc@{}} C_{1} & C_{2} \end{array}\right] \left( \begin{array}{@{}c@{}} x^{h}_{n_{1}}\left[ i_{1},i_{2}\right] \\ x^{v}_{n_{2}}\left[ i_{1},i_{2}\right] \end{array}\right)\!.\end{cases} \end{align} (4.1) In the following, we assume that the matrix $$A= \left ( {A_{11} \atop A_{21}} \quad {A_{12} \atop A_{22}} \right )$$ is block diagonalizable, as well $$\tilde{A}=\left ( {A^{1} \atop 0} \quad {0 \atop A^{2}}\right )$$, and then we obtain an equivalent system to (4.1), concerning the methods of diagonalization and block diagonalization, we can see, Eisenfeld (1973) and Wirth (2009). 4.2. Structural FMM In the following we propose the compact form of Structural Fornasini and Marchesini model as   \begin{align} \begin{cases} x\left[i_{1}+1,i_{2}+1\right] = \left[A_{1} \right] x\left[i_{1}+1,i_{2} \right] + \left[A_{2} \right] x\left[i_{1},i_{2}+1 \right] \\ \qquad\qquad\qquad\qquad + \left[A_{0} \right] x\left[i_{1},i_{2}\right] +\left[B_{1}\right] u\left[i_{1}+1,i_{2} \right] + \left[B_{2} \right] u\left[i_{1},i_{2}+1 \right] \\ y\left[i_{1},i_{2}\right] = \left[C \right] x\left[i_{1},i_{2}\right] \end{cases} \end{align} (4.2) and also we give the disjoint-union digraph which is associate with a structural two-dimensional linear system, its main property is that the state vector consists of a horizontal part and a vertical part so we introduce an orthogonal graph which is a disjoint-union digraph, where the set of nodes becomes V = (Vh, Vv). Let G1 and G2 two graphs, we will note G1 + G2 the disjoint union of graphs Gi, i = 1, 2. It corresponds to the graph (V 1 ∪ V 2, W1 ∪ W2), this means that the vertices and edges of G1 and G2 are considered separately. 4.2.1. The Disjoint-Union Digraph associated with the Structural Givone–Roesser Model The graph associated with the structural Givone–Roesser model is (GGR) = (V, W) such as V = U ∪ X ∪ Y, where   \begin{align*} U=&\,\left\{ u_{1}(i_{1},i_{2}),u_{2}(i_{1},i_{2}),\ldots,u_{m}(i_{1},i_{2}) \right\}, \nonumber\\ X =&\, X^{h}(i_{1},i_{2}) \cup X^{v}(i_{1},i_{2}) = \left\{{x^{h}_{1}}(i_{1},i_{2}),{x^{h}_{2}}(i_{1},i_{2}),\ldots,x^{h}_{n_{1}}(i_{1},i_{2}) \right\} \nonumber\\ \quad&\cup \left\{{x^{v}_{1}}(i_{1},i_{2}),{x^{v}_{2}}(i_{1},i_{2}),\ldots,x^{v}_{n_{2}}(i_{1},i_{2}) \right\} \nonumber \nonumber \end{align*} and   \begin{align} Y =&\, \left\{y_{1}(i_{1},i_{2}),y_{2}(i_{1},i_{2}),\ldots,y_{p}(i_{1},i_{2}) \right\} \end{align} (4.3) and arcs are defined as follows   \begin{align} W =&\, \left\{ \big(u_{j}(i_{1},i_{2}), x^{h}_{k_{h}}(i_{1},i_{2})\big) \big | B^{1}_{kj} \neq 0 \right\} \ j=\overline{1,m_{1}} \,\,\textrm{and}\,\, k_{h}=\overline{1,n_{1}} \nonumber\\ \cup&\, \left\{ \big(u_{j}(i_{1},i_{2}), x^{v}_{k_{v}}(i_{1},i_{2})\big) \big| B^{2}_{kj} \neq 0 \right\} \ j=\overline{1,m_{2}}\,\, \textrm{and}\,\, k_{v}=\overline{1,n_{2}} \nonumber\\ \cup&\, \left\{ \big(x^{h}_{j_{h}}(i_{1},i_{2}), x^{h}_{k_{h}}(i_{1},i_{2})\big) \big| A^{1}_{ji} \neq 0 \right\} \ j_{h}=\overline{1,n_{1}} \ \textrm{and}\,\, k_{h}=\overline{1,n_{1}} \nonumber\\ \cup&\, \left\{ \big(x^{v}_{j_{v}}(i_{1},i_{2}), x^{v}_{k_{v}}(i_{1},i_{2})\big) \big| A^{2}_{ji} \neq 0 \right\} \ j_{v}=\overline{1,n_{2}} \,\,\textrm{and}\,\, k_{v}=\overline{1,n_{2}} \nonumber\\ \cup&\, \left\{\big(x^{h}_{j_{h}}(i_{1},i_{2}), y_{k}(i_{1},i_{2})\big) \big| C^{1}_{kj} \neq 0 \right\} \ j_{h}=\overline{1,n_{1}} \,\,\textrm{and}\,\, k=\overline{1,p_{1}} \nonumber\\ \cup&\, \left\{\big(x^{v}_{j_{v}}(i_{1},i_{2}), y_{k}(i_{1},i_{2})\big) \big| C^{2}_{kj} \neq 0 \right\} \ j_{v}=\overline{1,n_{2}} \,\,\textrm{and}\,\, k=\overline{1,p_{2}}\end{align} (4.4) with m1 + m1 = m, p1 + p1 = p and   \begin{align} \begin{array}{l} \ A = \left( \begin{array}{@{}cc@{}} A^{1} & 0 \\ 0 & A^{2} \end{array}\right), \ B = \left( \begin{array}{@{}c@{}} B^{1} \\ B^{2} \end{array}\right) \quad \textrm{and} \quad C = \left( \begin{array}{@{}cc@{}} C^{1} & C^{2} \end{array}\right) \end{array}\!.\end{align} (4.5) Example 4.1 The following digraph associated with this structural two-dimensional Givone–Roesser model illustrates the disjoint-union digraph, which is presented in Fig. 2.   \begin{cases} \left( \begin{array}{@{}c@{}} x^{h}_{{1}}\left[(i_{1}+1,i_{2})\right] \\ x^{h}_{{2}}\left[(i_{1}+1,i_{2})\right] \\{x^{v}_{1}}\left[(i_{1},i_{2}+1)\right] \end{array}\right) = \left[ \begin{array}{@{}ccc@{}} 0 & l_{4} & 0 \\ l_{5} & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \left( \begin{array}{@{}c@{}} {x^{h}_{1}}\left[ i_{1},i_{2}\right] \\{x^{h}_{2}}\left[ i_{1},i_{2}\right] \\{x^{v}_{1}}\left[ i_{1},i_{2}\right] \end{array}\right) + \left[ \begin{array}{@{}ccc@{}} l_{1} & 0 & 0 \\ 0 & l_{2} & 0 \\ 0 & 0 & l_{3} \end{array}\right] \left( \begin{array}{@{}c@{}} {u^{h}_{1}}\left[ i_{1},i_{2}\right] \\{u^{h}_{2}}\left[ i_{1},i_{2}\right] \\{u^{v}_{1}}\left[ i_{1},i_{2}\right] \end{array}\right) \\[20pt] \left( \begin{array}{@{}c@{}} y^{h}\left[i_{1},i_{2}\right] \\ y^{v}\left[i_{1},i_{2}\right] \end{array}\right)= \left[ \begin{array}{@{}ccc@{}} 0 & l_{6} & 0 \\ 0 & 0 & l_{7} \end{array}\right] \left( \begin{array}{@{}c@{}} {x^{h}_{1}}\left[ i_{1},i_{2}\right] \\{x^{h}_{2}}\left[ i_{1},i_{2}\right] \\{x^{v}_{1}}\left[ i_{1},i_{2}\right] \end{array} \right)\end{cases} Similarly with the structural two-dimensional Givone–Roesser model, the first part of the graph corresponds to the horizontal sub-graph, and the second part corresponds to the vertical sub-graph, the non-zero elements of the matrices are the free parameters l = (l1, l2, …, l7); we can see that there exists a set of state-intput paths disjoint vertices $$\left ({u^{h}_{1}}(i_{1},i_{2}),{x^{h}_{1}}(i_{1},i_{2}),{x^{h}_{2}}(i_{1},i_{2}) \right ) $$, $$\left ({u^{h}_{2}}(i_{1},i_{2}),{x^{h}_{2}}(i_{1},i_{2}),{x^{h}_{1}}(i_{1},i_{2}) \right ) $$ and $$\left ({u^{v}_{1}}(i_{1},i_{2}),{x^{v}_{1}}(i_{1},i_{2})\right )$$. Fig. 2. View largeDownload slide Illustrative diagraph of two-dimensional Givone–Roesser model. Fig. 2. View largeDownload slide Illustrative diagraph of two-dimensional Givone–Roesser model. 4.3. Detemination of Fornasini–Marchesini Disjoint-Union Digraph Using the following variable change μ[(i1, i2)] = x[(i1, i2 + 1)] − A2x[(i1, i2)], we have   \begin{align*} \mu [(i_{1},i_{2})]&=A_{0}x[(i_{1},i_{2})]+A_{1}x[(i_{1},i_{2}+1)]+Bu[(i_{1},i_{2})] \\ &= A_{0}x[(i_{1},i_{2})]+A_{1} (\mu [(i_{1},i_{2})]+A_{2}x[(i_{1},i_{2})]) + Bu[(i_{1},i_{2})] \\ &= A_{1}\mu [(i_{1},i_{2})]+(A_{0}+A_{1}A_{2})x[(i_{1},i_{2}+1)] + Bu[(i_{1},i_{2})] \end{align*} the Fornasini–Marchesini disjoint-union digraph can be defined as a Givone–Roesser disjoint-union digraph, by V = U ∪ X ∪ Y, where   \begin{align*} X &= X^{h}(i_{1},i_{2}) \cup X^{v}(i_{1},i_{2})\\ &= \left\{\mu_{1}(i_{1},i_{2}),\mu_{2}(i_{1},i_{2}),\ldots,\mu_{n_{1}}(i_{1},i_{2}) \right\} \\ &\cup \left\{x_{1}(i_{1},i_{2}),x_{2}(i_{1},i_{2}),\ldots,x_{n_{2}}(i_{1},i_{2}) \right\} \end{align*} and where Xh(i1, i2) = M(i1, i2), Xv(i1, i2) = X(i1, i2) and arcs became   \begin{align} \begin{array}{l} W = \left\{ (u_{j}(i_{1},i_{2}), \mu_{k}(i_{1},i_{2})) \big| B_{kj} \neq 0 \right\} \\[3pt] \qquad \ \left\{ (\mu_{j}(i_{1},i_{2}), \mu_{k}(i_{1},i_{2}))\big | A^{1}_{ji} \neq 0 \right\} \cup \\[6pt] \qquad \ \left\{ (x_{j}(i_{1},i_{2}), x_{k}(i_{1},i_{2}))\big | A^{2}_{ji} \neq 0 \right\} \cup \\[6pt] \qquad \ \left\{(x_{j}(i_{1},i_{2}), y_{k}(i_{1},i_{2}))\big | C_{kj} \neq 0 \right\} \end{array} \end{align} (4.6) with   \begin{align*} A=& \left[ \begin{array}{@{}cc@{}} A_{1} & A_{0}+A_{1}A_{2} \\ I & A_{2} \end{array}\right] \textrm{where} \left[ \begin{array}{@{}cc@{}} A^{1} & 0 \\ 0 & A^{2} \end{array}\right] \quad \textrm{is the diagonal bloc matrix of A} \\ & \left[ \begin{array}{@{}c@{}} B_{1} \\ B_{2} \end{array}\right] = \left[ \begin{array}{@{}c@{}} B \\ 0 \end{array}\right] \ \textrm{and} \ \left[ \begin{array}{@{}cc@{}} C_{1} & C_{2} \end{array}\right] = \left[ \begin{array}{@{}cc@{}} 0 & C \end{array}\right]\!. \end{align*} 4.4. Main Properties of Two-Dimensional Disjoint-Union Digraph associated with Structural Two-Dimensional System Since the two-dimensional digraph is the disjoint union of two sub-digraphs (see Figs. 3 and 4), we can then use this property for studying the partial structural controllability of the two-dimensional systems, where the sub-digraph G(h) corresponds to the horizontal part of two-dimensional system respectively the sub-digraph G(v) corresponds to the vertical part of two-dimensional system. If the conditions of the structural-controllability are satisfied for only G(h) (respectively G(v)), we obtain a partial horizontal structural-controllability respectively partial vertical structural-controllability. Example 4.2 We consider the example of Givone–Roesser bi-dimensional graph, This example illustrates that we can locally analyse the Givone–Roesser two-dimensional graph i.e. we can study only the structural controllability of horizontal sub-graph respectively vertical sub-graph. Fig. 3. View largeDownload slide Horizontal sub-graph. Fig. 3. View largeDownload slide Horizontal sub-graph. Fig. 4. View largeDownload slide Vertical sub-graph. Fig. 4. View largeDownload slide Vertical sub-graph. Lemma 4.1 G is the disjoint union of Gh and Gv, and is input-connectable if and only if Gh and Gv are input connectable. Proof. Necessary The states vertex are either horizontal or vertical, if the state vertex is horizontal, $${x^{h}_{n}} \in G^{h}$$, and given that Gh is input connectables, there is a path from at least one of the input vertices to $${x^{h}_{n}} \in G^{h}$$, in the same way if the state vertex is vertical, $${x^{v}_{m}} \in G^{h}$$, there is a path from at least one of the input vertices to $${x^{v}_{m}} \in G^{v}$$. Sufficient Obviously, if G is input connectable, then Gh and Gv are by definition input connectable, because G = Gh ∪ Gv. Lemma 4.2 Knowing that $$G(\left [Q_{1} \right ]) = G^{h}(\left [Q_{1} \right ]) \cup G^{v}(\left [Q_{1} \right ])$$. If for each horizontal state vertex in $$G^{h}(\left [Q_{1} \right ])$$ and for each vertical state vertex in $$G^{v}(\left [Q_{1} \right ])$$ there is at least one path from one of the m1 input vertices to $${x^{h}_{n}}$$, and there is at least one path from one of the m2 input vertices to $${x^{v}_{m}}$$, then there exist respectively for each state vertex in $$G(\left [Q_{1} \right ])$$ there is at least one path from one of the mi for i = 1, 2, with m = m1 + m2 input vertices to the chosen state vertex. Proof. The states vertex are either horizontal or vertical, if the state vertex is horizontal, $${x^{h}_{i}} \in G^{h}(\left [Q_{1} \right ])$$, with $$i \in \overline{1,m_{1}}$$, then, there is at least a path from one of the m1 input vertices to $${x^{h}_{i}}$$, in the same way if the state vertex is vertical, $${x^{v}_{i}} \in G^{v}(\left [Q_{1} \right ])$$, then, in $$G(\left [Q_{1} \right ])$$ there is at least one path from one of the mi for i = 1, 2, with m = m1 + m2 input vertices to the chosen state vertex. Lemma 4.3 If there is at least one cycle family of width n1 in $$G^{h}(\left [Q_{1} \right ])$$, and least one cycle family of width n2 in $$G^{v}(\left [Q_{1} \right ])$$, then there exist respectively at least one cycle family of width ni for i = 1, 2, with n = n1 + n2 in $$G(\left [Q_{1} \right ])=G^{h}(\left [Q_{1} \right ]) \cup G^{v}(\left [Q_{1} \right ])$$. Proof. Knowing that the number of state vertices in $$G^{h}(\left [Q_{1} \right ])$$ is n1 and the number of state vertices in $$G^{v}(\left [Q_{1} \right ])$$ is n2. If there is at least one cycle family of width n1 respectively n2, then, by definition the cycle family is in $$G^{h}(\left [Q_{1} \right ])$$ respectively in $$G^{v}(\left [Q_{1} \right ])$$, so there exist respectively at least one cycle family of width ni for i = 1, 2, with n = n1 + n2 in $$G(\left [Q_{1} \right ])$$. Remark 4 The fundamental notion that is used in this lemmas is the principal property of the two-dimensional digraph, who is the disjoint union of two sub-graph; the proof of lemma 4.1, 4.2 and 4.3 is founded on the idea that there exists a horizontal (respectively vertical) sub-graph. 4.5. Generalization of the Criteria of Structural-Controllability for the Structural Two-dimensional Systems The previous three lemmas are necessary to prove the criteria of structural controllability of the two-dimensional linear systems. Theorem 4.4 A class of system characterized by the n × (n + m) structure matrix pair [A, B] is structural-controllable if and only if the structure matrix pair [A1, B1] and [A2, B2] are structural-controllable. Proof. Necessary If for each state vertex in $$G([{Q^{h}_{1}}])$$ respectively $$G([{Q^{v}_{1}}])$$ there is at least one path from one of the m1 input respectively m2 input vertices to the chosen state vertices, then, according to the Lemma 4.2, there exist for each state vertex in $$G([Q_{1}])= G([{Q^{h}_{1}}]) \cup G([{Q^{v}_{1}}])$$, at least one path from one of the m = m1 + m2 input vertices to the chosen state vertex. If there is at least one cycle family of width n1 in $$G([{Q^{h}_{1}}])$$ and n2 in $$G([{Q^{v}_{1}}])$$, then, according to the Lemma 4.3, there exist at least one family of width n = n1 + n2 in $$G([Q_{1}])= G([{Q^{h}_{1}}]) \cup G([{Q^{v}_{1}}])$$. Sufficient It is evident that if the system is structural controllable, then for each sub-system characterized by the ni + (ni + mi) structure matrix pair [Ai, Bi] is structural controllable, especially for the [A1, B1] and [A2, B2]. Theorem 4.5 A class of system characterized by the n × (n + m) structure matrix pair [A, B] is structural-controllable if and only if the structure matrix [A1, B1] corresponding to Gh([Q1]) and [A2, B2] corresponding to Gv([Q1]) are structural-controllable. Proof. Necessary If [A1, B1] and [A2, B2] are structural-controllable then, Gh([Q1]) and Gv([Q1]) are input-connectable, by using the lemma 4.1, G([Q1]) is input-connectable. If [A1, B1] and [A2, B2] are structural-controllable, then, s-rank [A1, B1] = n1 and s-rank [A2, B2] = n2, and given that the bi-dimensional graph is defined as a disjoint union of two sub-graph s-rank [A, B] = n1 + n2 = n. Sufficient It is evident that if the system is structural-controllable, then for each sub system characterized by the ni + (ni + mi) structure matrix the pair [Ai, Bi] is structural controllable, especially for the [A1, B1] and [A2, B2]. We finely deduce the following result: Corollary 2 A class of system characterized by the structure matrix pair [A, B] is strongly structurally controllable if and only if [A1, B1]. and [A2, B2]. are strongly structurally controllable. 5. Illustrative examples We illustrate previous results concerning structural-controllability of two-dimensional linear models by some examples. Example 5.1 We consider a linear two-dimensional structured model with two horizontal states, a vertical state, two horizontal inputs, a vertical input, a horizontal output and a vertical output.   $$ A = \left[ \begin{array}{@{}ccc@{}} 0 & 0 & 0 \\ l_{1} & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\!, \,\, B = \left[ \begin{array}{@{}ccc@{}} l_{3} & 0 & 0 \\ 0 & l_{4} & 0 \\ 0 & 0 & l_{5} \end{array}\right]\!,\ \textrm{and}\ C = \left[ \begin{array}{@{}ccc@{}} 0 & l_{6} & 0 \\ 0 & 0 & l_{7} \end{array}\right] $$ This Structural two-dimensional model given by Fig. 5 is structurally controllable by applying the Theorem 4.4. In fact, the conditions of structural controllability characterized by Lemma 4.2 and Lemma 4.3 are satisfied in the horizontal part and vertical part of the graph. If we put $$ E = \left [ \begin{array}{@{}ccc@{}} l_{7} & 0 & 0 \\ 0 & l_{8} & 0 \\ 0 & 0 & l_{9} \end{array}\right ]$$ then, its G[Q1] graph is characterized by Fig. 6 as follows. In this graph (Fig. 7) we consider just input vertex and the state vertex by Theorem 4.4 the structural two-dimensional model is structurally controllable. Fig. 5. View largeDownload slide Illustrative diagraph of structural-controllability of a two-dimensional linear model. Fig. 5. View largeDownload slide Illustrative diagraph of structural-controllability of a two-dimensional linear model. Fig. 6. View largeDownload slide Illustrative diagraph of structural-controllability of a two-dimensional linear model with static state feedback. Fig. 6. View largeDownload slide Illustrative diagraph of structural-controllability of a two-dimensional linear model with static state feedback. Fig. 7. View largeDownload slide Illustrative diagraph of partial structural-controllability of a two-dimensional linear model. Fig. 7. View largeDownload slide Illustrative diagraph of partial structural-controllability of a two-dimensional linear model. Example 5.2 This example illustrates the partial structural controllability, we consider here the structural graph of certain structural two-dimensional linear system. We remark that the vertical sub-graph is not structural controllable, but the horizontal sub-graph is structural controllable, then we can conclude that the two-dimensional linear model is horizontal-partial structural controllable as defined in the sub-section 4.4. 6. Conclusion In the studies by Fornasini & Valcher (1997) and Fornasini & Valcher (1998) E. Fornasini et al. consider the two-dimensional directed graph of pair matrix (A, B) such as x(i1 + 1, i2 + 1) = Ax(i1 + 1, i2) + Bx(i1, i2 + 1), and study characteristic polynomials of irreducible matrix pairs (A, B), respectively, premitivity of positivity matrix pair (A, B) among other by the graph theoretic description. They define the associated directed graph D*(A, B) with arcs of two different kinds, namely, A-arcs and B-arcs; there are A-arcs from vertex vi to vertex vj if (vi, vj) is in A, and B-arcs if (vi, vj) is in B. To study the characterizations of global reachability of two-dimensional structured systems Pereira et al. (2013). introduce the shift operators σ1x(i1, i2) = x(i1 + 1, i2) and σ2x(i1, i2) = x(i1, i2 + 1) to rewrite the Fornasini–Marchesini model as one-dimensional model, and define the directed graph as the superposition of two sub-graphs and eliminating the repeated edges to use results that exist in one dimension. In this paper, we have studied the structural approach of two-dimensional systems by focusing on a classical problem. The principal idea is the disjoint union graph which represents the structural two-dimensional system as a whole, with the same approach, it is possible to solve notably other problems of control and stability, using the structural graph associated with the two-dimensional system and derivative of the criteria. Our current interest is focused on the structural analysis of the observability of the multidimensional systems which will be studied in a separate paper. References Ahn, C. K. & Bas, M. V. ( 2015) Two-dimensional dissipative control and filtering for Roesser model. IEEE Trans. Automatic Control , 60. Andrei, N. ( 1985) Sparse Systems. Digraph Approach of Large-Scale Linear Systems Theory . Koln: TUV Rheinland. Attasi, S. ( 1973) Systèmes linéaires homogènes à deux indices. Tech. Report Laboria, 31. Bose, T. S. ( 2003) Multidimensional Systems Theory and Applications (Second Edition) . The Netherlands: Dordrecht, Kluwer. Chen, W. & Lin, Y. ( 2012) 2D system approach based output feedback repetitive control for uncertain discrete-time systems. Internat. J. Control, Automation Syst. , 10, 257-- 264. Google Scholar CrossRef Search ADS   Commault, C. & Dion, J. M. ( 2015) The single-input minimal controllability problem for structured systems. Syst. Control Lett., Elsevier , 80, 50-- 55. Google Scholar CrossRef Search ADS   Commault, C., Dion, J. M. & Boukhobza, T. ( 2017) On the fixed controllable subspace in linear structured systems. Syst. Control Lett. , 102, 42-- 47. Google Scholar CrossRef Search ADS   Dion, J. M. & Commault, C. ( 2013) Input addition for the controllability of graph-based systems. 10th IEEE Int. Conf. Control Automation . Eisenfeld, J. ( 1973) Block diagonalization of partitioned matrix operators. Linear Algebra Appl. , 6, 183-- 191. Google Scholar CrossRef Search ADS   Fornasini, E. & Marchesini, G. ( 1976) State-space realization theory of two-dimensional filters. IEEE Trans. Autum. Contr. , Vol. AC-21, 484-- 491. Google Scholar CrossRef Search ADS   Fornasini, E. & Valcher, E. ( 1997) Directed graphs, 2D state models, and characteristic polynomials of irreducible matrix pairs. Linear Algebra Appl. , 263, 275-- 310. Google Scholar CrossRef Search ADS   Fornasini, E. & Valcher, E. ( 1998) Primitivity of positive matrix pairs: algebraic characterization, graph theoretic description and 2D systems interpretation. SIAM. J. Matrix Anal. & Appl. , 71-- 88. Gao, H., Lam, J., Xu, S. & Wang, C. ( 2004) Stabilization and H $$\infty $$ control of two-dimensional Markovian jump systems. IMA J. Math. Cont. Inf . Givone, D. & Roesser, R. P. ( 1972) Multidimensional linear iterative circuits general properties. IEEE Trans. On Computers , C-21, 1067-- 1073. Google Scholar CrossRef Search ADS   Gutlan, S. ( 1987) State-space stability of two-dimensional systems. IMA J. Math. Cont. Inf . Kaczorek, T. ( 1985) Two-Dimensional Linear Systems . Berlin, Germany: Springer. Kibangou, A. Y. & Commault, C. ( 2014) Observability in connected strongly regular graphs and distance regular graphs. IEEE Trans. Cont. Network Syst.,  1, 360-- 369. Google Scholar CrossRef Search ADS   Kurek, J. ( 1985) The general state-space model for a two-dimensional linear digital systems. IEEE Trans. Autom. Contr. , AC-30, 600-- 601. Google Scholar CrossRef Search ADS   Lin, C. ( 1974) Structural controllability. IEEE Trans. Automat. Cont.  19, 201-- 208. Google Scholar CrossRef Search ADS   Lin, M. H., Tsai, J. S. H., Chen, C. W. & Shieh, L. S. ( 2010) Novel state-space self-tuning control for two dimensional linear discrete-time stochastic systems. IMA J. Math. Cont. Inf . Maza, S., Simon, C. & Boukhobza, T. ( 2012) Impact of the actuator failures on the structural controllability of linear systems: a graph theoretical approach. IET Cont. Theory Appl., Institution Eng. Technol , 6, 412-- 419. Google Scholar CrossRef Search ADS   Murota, K. ( 1987) Systems Analyses by Graphs and Matroids. Structural Solvability and Controllability . New York: Spinger Berlin Heidelberg. Google Scholar CrossRef Search ADS   Pal, V. C. & Negi, R. ( 2017) Based anti-windup controller for two-dimensional discrete delayed systems in presence of actuator saturation. IMA J Math. Cont. Inf . Pereira, R., Rocha, P. & Simoes, R. ( 2013) Characterizations of global reachability of 2D structured systems. Multidimensional Syst. Signal Process. , 24, 51-- 64. Google Scholar CrossRef Search ADS   Reinschke, K. J. ( 1988) Multivariable Control a Graph-Theoretic Approach . New York: Spinger Berlin Heidelberg. Google Scholar CrossRef Search ADS   Roesser, P. R. ( 1975) A discrete state-space model for linear image processing. IEEE Trans. Automat. Control , AC-20, 1-- 10. Google Scholar CrossRef Search ADS   Rogers, E. & Owens, D. H. ( 1997) Stability theory and performance bounds for a class of two-dimensional linear systems with interpass smoothing effects. IMA J. Math. Cont. Inf . Wirth, J. ( 2009) Block-diagonalisation of matrices and operators. Linear Algebra Appl. , 431, 895-- 902. Google Scholar CrossRef Search ADS   Xu, L., Wu, L., Wu, Q., Lin, Z. & Xiao, Y. ( 2005) On realization of 2D discrete systems by Fornasini-Marchesini model. Int. J. Cont., Automation, Syst. , 3 631-- 639. © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

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

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Abstract

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. 1. Introduction Two-dimensional systems have been the subject of much research, due to the fact that several phenomena related to digital technology, image processing, geophysics and robotics, can be represented through the theory of two-dimensional systems. In the 1970’s several extensions were proposed for two-dimensional systems (see e.g. Fornasini & Marchesini, 1976 and Givone & Roesser, 1972); there has also been a great deal of synthesis work in recent decades (see e.g. Kaczorek, 1985, Bose, 2003, Gao et al., 2004, Rogers & Owens, 1997 and Gutlan, 1987). The fundamental property of these systems is that they propagate information in two independent directions or by two elements $$ z^{- 1}_{1} $$ and $$ z^{- 1}_{2} $$ in circuits theory. Among the recent work on two-dimensional systems we can cite Pal & Negi (2017) where the authors study the problem of stability analysis for a class of two-dimensional discrete systems in the presence of actuator saturation and interval-like time varying state delay, in Lin et al. (2010) M. H. Lin and co-authors define a state-space self-tuning control for two-dimensional multi-input multi-output linear discrete-time stochastic systems, Xu et al. (2005) where Li Xu and co-authors propound a constructive method based on algorithms for the realization of local Fornasini–Marchesini discrete time model, Chen & Lin (2012) investigate in their study the problem of output feedback repetitive control for uncertain discrete-time systems and formulating the problem by using certain class of two-dimensional systems and Ahn & Bas (2015) treat in thier study the asymptotic stability, dissipative control and filtering for Roesser model (RM) using a linear matrix inequality approach. One of the methods of analysis is the extension of the techniques that exist in the one-dimensional case. The analysis of the systems can be studied through the state spaces. We consider here discrete time systems. There are three discrete-time classical two-dimensional state space models, Givone–Roesser Model, Fornasini–Marchesini Model (FMM) and Attasi Model. They introduced a description of these systems by linear state space models that allowed the design of tests for controllability, observability, attainability and stability. On the other hand, the structural dynamic systems theory has recently undergone significant developments in various field relating to the engineering sciences, including underwater and space exploration. The main challenge in this area is the design of simple and manageable structural control criteria, we can cite a few recent works of C. Commault, J. M. Dion and their co-authors. In the study by Dion & Commault (2013), they study graphs associated with dynamic systems that are well suited to the analysis of structural controllability by adding inputs. In the study by Kibangou & Commault (2014) the authors study the observability of linear systems modeled with highly regular graphs, and bipartite graphs that capture the observability properties; Commault & Dion (2015), the authors examine the problem of minimal control, more precisely, the problem of controlling a linear systems with an input vector having a minimum non-zero input, and analyse the model of sparsity of the input vector; and in Commault et al. (2017) Commault, Dion and Boukhobza consider interconnected networks to show that the controllable subspace can have a part that will be present for almost all values of the free parameters. We describe a characterization of the fixed controllable subspace using the representation by the structural graph and the study by Maza et al. (2012). In this context the graph theory appears as an extremely suitable tool to define such conditions. The first work about the graph theoretic approach for the structural controllability appears in the studies by Reinschke (1988), Murota (1987), Andrei (1985) and Lin (1974) and E. Fornasini et al. analyse in Fornasini & Valcher (1997) and Fornasini & Valcher (1998) a digraph associated with a pair of matrices to study irreducible matrix pairs and primitivity of positive matrix pairs; Pereira et al. (2013) defined a superposition digraph to characterizate the global reachability of two-dimensional systems. To study the structural controllability to the two-dimensional linear systems, we define the structural Givone–Roesser model, and structural FMM, then we introduce a graph associated with the structural two-dimensional model that preserves the fundamental properties which is the independence of the two dynamics, which is based on the notion of disjoint union of two sub-graphs, and expresses the independence property of the two dynamics, thus the already existing results can be exploited through the partial analysis of graphs, and can introduce the notion of horizontal and vertical structural controllability; then, we establish the link between the Givone–Roesser structural graphs and Fornasini–Marchesini structural model. Finally, we synthesize structural controllability criteria for structural two-dimensional linear systems, and illustrate our results with illustrative examples. 2. Preliminaries 2.1. Two-dimensional Discrete Time Models There are several state-space models for two-dimensional systems introduced by Roesser (1975), Fornasini & Marchesini (1997), Attasi (1973) and Kurek (1985) which have been generalized to the nD models later on. These models have been commonly used to describe two-dimensional systems and to investigate their several properties. Here, we will only concentrate on the most common state-space models i.e. RM and FMM. 2.1.1. Givone–Roesser Model Givone & Roesser (1972) have introduced the first state space model for the theory of iterative linear circuits. An iterative circuit is a combination of individual cells. The input and output equations are   \begin{align} \begin{cases} \left( \begin{array}{@{}cccc@{}} x^{h}_{n_{1}}(i_{1}+1,i_{2}) \\ x^{v}_{n_{2}}(i_{1},i_{2}+1) \end{array}\!\!\right) = \left( \begin{array}{@{}cccc@{}} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\!\!\right) \left( \begin{array}{@{}cccc@{}} x^{h}_{n_{1}}(i_{1},i_{2}) \\ x^{v}_{n_{2}}(i_{1},i_{2}) \end{array}\!\!\right) + \left( \begin{array}{@{}cccc@{}} B_{1} \\ B_{2} \end{array}\!\!\right) u(i_{1},i_{2}) \\[12pt] y(i_{1},i_{2})= \left( \begin{array}{@{}rrrr@{}} C_{1} & C_{2} \end{array}\!\!\right) \left( \begin{array}{@{}cccc@{}} x^{h}_{n_{1}}(i_{1},i_{2}) \\ x^{v}_{n_{2}}(i_{1},i_{2}) \end{array}\!\!\right) \end{cases} \end{align} (2.1) where, $$x^{h}_{n_{1}}(i_{1},i_{2}) \in R^{n_{1}}$$ is the horizontal state vector, $$x^{v}_{n_{2}}(i_{1},i_{2}) \in R^{n_{2}}$$ is the vertical state vector, y(i1, i2)∈ Rp is the output vector and u(i1, i2) ∈ Rm is the input vector, and we denote by Ri the set of real vectors. A11, A12, A21, A22, B1, B2, C1 and C2 are real matrices of appropriate dimensions. In this model i and j are the positive integer valued horizontal and vertical coefficients. 2.2. Fornasini---Marchesini model Fornasini and Marchesini have generalized the model of S. Attasi in Fornasini & Marchesini (1976), and propose the following model, in its compact form   \begin{align} \begin{cases} x(i_{1}+1,i_{2}+1)=A_{1}x(i_{1}+1,i_{2}) + A_{2}x(i_{1},i_{2}+1)+ A_{0}x(i_{1},i_{2}) + B_{1}u(i_{1}+1,i_{2}) + B_{2}u(i_{1},i_{2}+1) \\ y(i_{1},i_{2}) = Cx(i_{1},i_{2})\end{cases} \end{align} (2.2) where x(i1, i2) is the state vector of the model, u(i1, i2) is the input vector and y(i1, i2) is the output vector of the model. A1, A2, A0, B1, B2 and C are real matrices of appropriate size. 3. Graph associated with a structural linear system The associated graph with the structural is defined in Reinschke (1988), Andrei (1985) and Murota (1987). The generic properties of the system can then often be characterized very simply in terms of properties of the associated graph. This makes some results very intuitive. This modeling has the following characteristics. To such a system, one can easily associate a graph G = (V, W) the set of vertices is V = U ∪ X ∪ Y where U, X and Y are inputs, states and outputs, respectively, given by $$u= \left \{u_{1},u_{2},\ldots ,u_{m} \right \}$$, $$x= \left \{x_{1},x_{2},\ldots ,x_{n} \right \}$$ and $$y= \{y_{1},y_{2},\ldots ,y_{p}\}$$. The set of arcs $$W = \left \{ (u_{i}, x_{j}) | B_{ji} \neq 0 \right \} \cup \left \{ (x_{i}, x_{j}) | A_{ji} \neq 0 \right \} \cup \left \{(x_{i}, y_{j}) | C_{ji} \neq 0 \right \}$$, where Aji (resp. Bji, Cji) is the element (j, i) of the matrix A (resp. B, C), the direction of the arcs is from ui to xj, xi to xj and xi to yj. 3.1. Structural-Controllability In this subsection the results are taken from the studies by Reinschke (1988) and Andrei (1985), and the fundamental criteria of structural controllability are exposed. Definition 3.1 The elements of a structure matrix [Q] are either fixed at zero or indeterminate values which are assumed to be independent of one another. Definition 3.2 A numerically matrix Q is called an admissible numerical realization if it can be obtained by fixing all indeterminate entries of [Q] to some particular values. Example 3.1 This example illustrate the Definition 3.1  \begin{align} A = \left( \begin{array}{@{}cccc@{}} 0 & 0 & a_{13} & 0 \\ a_{21} & 0 & 0 & a_{24} \\ 0 & 0 & a_{33} & 0 \\ 0 & a_{24} & 0 &0 \end{array}\right)\!, \end{align} (3.1)  \begin{align}\ B = \left( \begin{array}{@{}cc@{}} 0 & 0 \\ b_{21} & 0 \\ 0 & b_{32} \\ 0 & b_{24} \end{array}\right) \end{align} (3.2) and   \begin{align} C = \left( \begin{array}{@{}cccc@{}} 0 & c_{12} & 0 & 0 \\ c_{21} & 0 & 0 & c_{24} \end{array}\right) \!. \end{align} (3.3) The corresponding structural matrices are   \begin{align} [A] = \left( \begin{array}{@{}cccc@{}} 0 & 0 & l_{1} & 0 \\ l_{2} & 0 & 0 & l_{3} \\ 0 & 0 & l_{4} & 0 \\ 0 & l_{5} & 0 & 0 \end{array}\right)\!, \end{align} (3.4)  \begin{align} [B] = \left( \begin{array}{@{}cc@{}} 0 & 0 \\ l_{6} & 0 \\ 0 & l_{7} \\ 0 & l_{8} \end{array}\right) \end{align} (3.5) and   \begin{align} [C] = \left( \begin{array}{@{}cccc@{}} 0 & l_{9} & 0 & 0 \\ l_{10} & 0 & 0 & l_{11} \end{array}\right)\!.\end{align} (3.6) If   \begin{align} Q = \left( \begin{array}{@{}cc@{}} A & B \\ C & 0 \end{array}\right) \end{align} (3.7) then the corresponding structure matrix is   \begin{align} [Q] = \left[ \begin{array}{@{}cccccc@{}} 0 & 0 & l_{1} & 0 & 0 & 0 \\ l_{2} & 0 & 0 & l_{3} & l_{6} & 0 \\ 0 & 0 & l_{4} & 0 & 0 & l_{7} \\ 0 & l_{5} & 0 & 0 & 0 & l_{8} \\ 0 & l_{9} & 0 & 0 & 0 & 0 \\ l_{10} & 0 & 0 & l_{11} & 0 & 0 \end{array}\right] \end{align} (3.8) where the vanishing elements are fixed at zero while the other elements have unknown real value li. Definition 3.3 A class of systems given by its structure matrix pair [A, B] is said to be Structurally Controllable if there exists at least one admissible realization (A, B) ∈ [A, B] being Controllable in the usual numerical sense i.e. rank[B, AB, …, An−1B] = n. Remark 1 Reinschke (1988) The structural rank of [Q] is defined as S-rank [Q] = maxQ∈[Q] rank Q. Remark 2 We assume that in static state feedback we have u = Ex, with E a real matrix of appropriate size, and in the context of controllability investigation the following square bloc matrix will prove to be most appropriate   \begin{align} Q_{1} = \left( \begin{array}{@{}cc@{}} A & B \\ E & 0 \end{array}\right) \text{of appropriate size.} \end{align} (3.9) Definition 3.4 Reinschke (1988) A class of systems is said to be Input-connectable if in the digraph G([Q]) there is, for each state vertex, a path from at least one of the input vertices to the chosen state vertex. We illustrate the Definition 3.4 with the following example. Example 3.2 Consider a structural system with three state vertices, three input vertices and two output vertices with   \begin{align} [A] = \left( \begin{array}{@{}ccc@{}} 0 & l_{4} & 0 \\ 0 & 0 & l_{5} \\ 0 & 0 & 0 \end{array}\right)\!, \end{align} (3.10)  \begin{align} [B] = \left( \begin{array}{@{}ccc@{}} l_{1} & 0 & 0 \\ 0 & l_{2} & 0 \\ 0 & 0 & l_{3} \end{array}\right) \end{align} (3.11) and   \begin{align} [C] = \left( \begin{array}{@{}ccc@{}} l_{6} & 0 & 0 \\ 0 & 0 & l_{7} \end{array}\right) \!.\end{align} (3.12) This system is input-connectable. Indeed, for the vertex x1 we have one path $$\left \{u_{1},x_{1} \right \}$$, for the vertex x2 we have two paths $$\left \{u_{2},x_{2} \right \}$$ and $$\left \{u_{1},x_{1},x_{2} \right \}$$ and for the vertex x3 we have three paths $$\left \{u_{3},x_{3} \right \}$$, $$\left \{u_{1},x_{1},x_{2},x_{3} \right \}$$ and $$\left \{u_{2},x_{2},x_{3} \right \}$$ as shown in Fig. 1. Fig. 1. View largeDownload slide Illustrative diagraph of input-connectable system. Fig. 1. View largeDownload slide Illustrative diagraph of input-connectable system. Remark 3 Reinschke (1988) A cycle family corresponds to a non-vanishing term of the determinant $$det \overline{Q}$$. Hence the matrix $$\overline{Q}$$ has full rank in the structural sense i.e. an admissible realization of [Q]. Definition 3.5 Reinschke (1988) A given cycle family in G([Q]) is said to be of width w if this cycle family touches exactly w state vertices. 3.1.1. Criteria of Structural Controllability Theorem 3.6 A class of systems characterized by the n × (n + m) structure matrix pair [A, B] is structurally-controllable if and only if It is input-connectable. S-rank [A, B] = n. Theorem 3.7 A class of systems characterized by the n × (n + m) structure matrix [A, B] is structurally-controllable if and only if the digraph G([Q1]) meets both the following conditions. For each state vertex in G([Q1]) there is at least one path from one of the m input vertices to the chosen state vertex. There is at least one cycle family of width G([Q1]). Corollary 1 A class of system characterized by the structure matrix pair [A, B] is strongly structurally-controllable if and only if the G([Q1]) meets both the following conditions. Input-connectability. There is exactly one cycle family of width n in G([Q1]). 4. Main results 4.1. Structural Two-Dimensional Discrete Time Models 4.1.1. Structural Givone–Roesser Model The structural Givone–Roesser model is   \begin{align} \begin{cases} \left( \begin{array}{@{}c@{}} x^{h}_{n_{1}}\left[(i_{1}+1,i_{2})\right] \\ x^{v}_{n_{2}}\left[(i_{1},i_{2}+1)\right] \end{array}\right) = \left[ \begin{array}{@{}cc@{}} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right] \left( \begin{array}{@{}c@{}} x^{h}_{n_{1}}\left[ i_{1},i_{2}\right] \\ x^{v}_{n_{2}}\left[ i_{1},i_{2}\right] \end{array}\right) + \left[ \begin{array}{@{}c@{}} B_{1} \\ B_{2} \end{array}\right] u\left[ i_{1},i_{2}\right] \\[12pt] y\left[i_{1},i_{2}\right]= \left[ \begin{array}{@{}cc@{}} C_{1} & C_{2} \end{array}\right] \left( \begin{array}{@{}c@{}} x^{h}_{n_{1}}\left[ i_{1},i_{2}\right] \\ x^{v}_{n_{2}}\left[ i_{1},i_{2}\right] \end{array}\right)\!.\end{cases} \end{align} (4.1) In the following, we assume that the matrix $$A= \left ( {A_{11} \atop A_{21}} \quad {A_{12} \atop A_{22}} \right )$$ is block diagonalizable, as well $$\tilde{A}=\left ( {A^{1} \atop 0} \quad {0 \atop A^{2}}\right )$$, and then we obtain an equivalent system to (4.1), concerning the methods of diagonalization and block diagonalization, we can see, Eisenfeld (1973) and Wirth (2009). 4.2. Structural FMM In the following we propose the compact form of Structural Fornasini and Marchesini model as   \begin{align} \begin{cases} x\left[i_{1}+1,i_{2}+1\right] = \left[A_{1} \right] x\left[i_{1}+1,i_{2} \right] + \left[A_{2} \right] x\left[i_{1},i_{2}+1 \right] \\ \qquad\qquad\qquad\qquad + \left[A_{0} \right] x\left[i_{1},i_{2}\right] +\left[B_{1}\right] u\left[i_{1}+1,i_{2} \right] + \left[B_{2} \right] u\left[i_{1},i_{2}+1 \right] \\ y\left[i_{1},i_{2}\right] = \left[C \right] x\left[i_{1},i_{2}\right] \end{cases} \end{align} (4.2) and also we give the disjoint-union digraph which is associate with a structural two-dimensional linear system, its main property is that the state vector consists of a horizontal part and a vertical part so we introduce an orthogonal graph which is a disjoint-union digraph, where the set of nodes becomes V = (Vh, Vv). Let G1 and G2 two graphs, we will note G1 + G2 the disjoint union of graphs Gi, i = 1, 2. It corresponds to the graph (V 1 ∪ V 2, W1 ∪ W2), this means that the vertices and edges of G1 and G2 are considered separately. 4.2.1. The Disjoint-Union Digraph associated with the Structural Givone–Roesser Model The graph associated with the structural Givone–Roesser model is (GGR) = (V, W) such as V = U ∪ X ∪ Y, where   \begin{align*} U=&\,\left\{ u_{1}(i_{1},i_{2}),u_{2}(i_{1},i_{2}),\ldots,u_{m}(i_{1},i_{2}) \right\}, \nonumber\\ X =&\, X^{h}(i_{1},i_{2}) \cup X^{v}(i_{1},i_{2}) = \left\{{x^{h}_{1}}(i_{1},i_{2}),{x^{h}_{2}}(i_{1},i_{2}),\ldots,x^{h}_{n_{1}}(i_{1},i_{2}) \right\} \nonumber\\ \quad&\cup \left\{{x^{v}_{1}}(i_{1},i_{2}),{x^{v}_{2}}(i_{1},i_{2}),\ldots,x^{v}_{n_{2}}(i_{1},i_{2}) \right\} \nonumber \nonumber \end{align*} and   \begin{align} Y =&\, \left\{y_{1}(i_{1},i_{2}),y_{2}(i_{1},i_{2}),\ldots,y_{p}(i_{1},i_{2}) \right\} \end{align} (4.3) and arcs are defined as follows   \begin{align} W =&\, \left\{ \big(u_{j}(i_{1},i_{2}), x^{h}_{k_{h}}(i_{1},i_{2})\big) \big | B^{1}_{kj} \neq 0 \right\} \ j=\overline{1,m_{1}} \,\,\textrm{and}\,\, k_{h}=\overline{1,n_{1}} \nonumber\\ \cup&\, \left\{ \big(u_{j}(i_{1},i_{2}), x^{v}_{k_{v}}(i_{1},i_{2})\big) \big| B^{2}_{kj} \neq 0 \right\} \ j=\overline{1,m_{2}}\,\, \textrm{and}\,\, k_{v}=\overline{1,n_{2}} \nonumber\\ \cup&\, \left\{ \big(x^{h}_{j_{h}}(i_{1},i_{2}), x^{h}_{k_{h}}(i_{1},i_{2})\big) \big| A^{1}_{ji} \neq 0 \right\} \ j_{h}=\overline{1,n_{1}} \ \textrm{and}\,\, k_{h}=\overline{1,n_{1}} \nonumber\\ \cup&\, \left\{ \big(x^{v}_{j_{v}}(i_{1},i_{2}), x^{v}_{k_{v}}(i_{1},i_{2})\big) \big| A^{2}_{ji} \neq 0 \right\} \ j_{v}=\overline{1,n_{2}} \,\,\textrm{and}\,\, k_{v}=\overline{1,n_{2}} \nonumber\\ \cup&\, \left\{\big(x^{h}_{j_{h}}(i_{1},i_{2}), y_{k}(i_{1},i_{2})\big) \big| C^{1}_{kj} \neq 0 \right\} \ j_{h}=\overline{1,n_{1}} \,\,\textrm{and}\,\, k=\overline{1,p_{1}} \nonumber\\ \cup&\, \left\{\big(x^{v}_{j_{v}}(i_{1},i_{2}), y_{k}(i_{1},i_{2})\big) \big| C^{2}_{kj} \neq 0 \right\} \ j_{v}=\overline{1,n_{2}} \,\,\textrm{and}\,\, k=\overline{1,p_{2}}\end{align} (4.4) with m1 + m1 = m, p1 + p1 = p and   \begin{align} \begin{array}{l} \ A = \left( \begin{array}{@{}cc@{}} A^{1} & 0 \\ 0 & A^{2} \end{array}\right), \ B = \left( \begin{array}{@{}c@{}} B^{1} \\ B^{2} \end{array}\right) \quad \textrm{and} \quad C = \left( \begin{array}{@{}cc@{}} C^{1} & C^{2} \end{array}\right) \end{array}\!.\end{align} (4.5) Example 4.1 The following digraph associated with this structural two-dimensional Givone–Roesser model illustrates the disjoint-union digraph, which is presented in Fig. 2.   \begin{cases} \left( \begin{array}{@{}c@{}} x^{h}_{{1}}\left[(i_{1}+1,i_{2})\right] \\ x^{h}_{{2}}\left[(i_{1}+1,i_{2})\right] \\{x^{v}_{1}}\left[(i_{1},i_{2}+1)\right] \end{array}\right) = \left[ \begin{array}{@{}ccc@{}} 0 & l_{4} & 0 \\ l_{5} & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \left( \begin{array}{@{}c@{}} {x^{h}_{1}}\left[ i_{1},i_{2}\right] \\{x^{h}_{2}}\left[ i_{1},i_{2}\right] \\{x^{v}_{1}}\left[ i_{1},i_{2}\right] \end{array}\right) + \left[ \begin{array}{@{}ccc@{}} l_{1} & 0 & 0 \\ 0 & l_{2} & 0 \\ 0 & 0 & l_{3} \end{array}\right] \left( \begin{array}{@{}c@{}} {u^{h}_{1}}\left[ i_{1},i_{2}\right] \\{u^{h}_{2}}\left[ i_{1},i_{2}\right] \\{u^{v}_{1}}\left[ i_{1},i_{2}\right] \end{array}\right) \\[20pt] \left( \begin{array}{@{}c@{}} y^{h}\left[i_{1},i_{2}\right] \\ y^{v}\left[i_{1},i_{2}\right] \end{array}\right)= \left[ \begin{array}{@{}ccc@{}} 0 & l_{6} & 0 \\ 0 & 0 & l_{7} \end{array}\right] \left( \begin{array}{@{}c@{}} {x^{h}_{1}}\left[ i_{1},i_{2}\right] \\{x^{h}_{2}}\left[ i_{1},i_{2}\right] \\{x^{v}_{1}}\left[ i_{1},i_{2}\right] \end{array} \right)\end{cases} Similarly with the structural two-dimensional Givone–Roesser model, the first part of the graph corresponds to the horizontal sub-graph, and the second part corresponds to the vertical sub-graph, the non-zero elements of the matrices are the free parameters l = (l1, l2, …, l7); we can see that there exists a set of state-intput paths disjoint vertices $$\left ({u^{h}_{1}}(i_{1},i_{2}),{x^{h}_{1}}(i_{1},i_{2}),{x^{h}_{2}}(i_{1},i_{2}) \right ) $$, $$\left ({u^{h}_{2}}(i_{1},i_{2}),{x^{h}_{2}}(i_{1},i_{2}),{x^{h}_{1}}(i_{1},i_{2}) \right ) $$ and $$\left ({u^{v}_{1}}(i_{1},i_{2}),{x^{v}_{1}}(i_{1},i_{2})\right )$$. Fig. 2. View largeDownload slide Illustrative diagraph of two-dimensional Givone–Roesser model. Fig. 2. View largeDownload slide Illustrative diagraph of two-dimensional Givone–Roesser model. 4.3. Detemination of Fornasini–Marchesini Disjoint-Union Digraph Using the following variable change μ[(i1, i2)] = x[(i1, i2 + 1)] − A2x[(i1, i2)], we have   \begin{align*} \mu [(i_{1},i_{2})]&=A_{0}x[(i_{1},i_{2})]+A_{1}x[(i_{1},i_{2}+1)]+Bu[(i_{1},i_{2})] \\ &= A_{0}x[(i_{1},i_{2})]+A_{1} (\mu [(i_{1},i_{2})]+A_{2}x[(i_{1},i_{2})]) + Bu[(i_{1},i_{2})] \\ &= A_{1}\mu [(i_{1},i_{2})]+(A_{0}+A_{1}A_{2})x[(i_{1},i_{2}+1)] + Bu[(i_{1},i_{2})] \end{align*} the Fornasini–Marchesini disjoint-union digraph can be defined as a Givone–Roesser disjoint-union digraph, by V = U ∪ X ∪ Y, where   \begin{align*} X &= X^{h}(i_{1},i_{2}) \cup X^{v}(i_{1},i_{2})\\ &= \left\{\mu_{1}(i_{1},i_{2}),\mu_{2}(i_{1},i_{2}),\ldots,\mu_{n_{1}}(i_{1},i_{2}) \right\} \\ &\cup \left\{x_{1}(i_{1},i_{2}),x_{2}(i_{1},i_{2}),\ldots,x_{n_{2}}(i_{1},i_{2}) \right\} \end{align*} and where Xh(i1, i2) = M(i1, i2), Xv(i1, i2) = X(i1, i2) and arcs became   \begin{align} \begin{array}{l} W = \left\{ (u_{j}(i_{1},i_{2}), \mu_{k}(i_{1},i_{2})) \big| B_{kj} \neq 0 \right\} \\[3pt] \qquad \ \left\{ (\mu_{j}(i_{1},i_{2}), \mu_{k}(i_{1},i_{2}))\big | A^{1}_{ji} \neq 0 \right\} \cup \\[6pt] \qquad \ \left\{ (x_{j}(i_{1},i_{2}), x_{k}(i_{1},i_{2}))\big | A^{2}_{ji} \neq 0 \right\} \cup \\[6pt] \qquad \ \left\{(x_{j}(i_{1},i_{2}), y_{k}(i_{1},i_{2}))\big | C_{kj} \neq 0 \right\} \end{array} \end{align} (4.6) with   \begin{align*} A=& \left[ \begin{array}{@{}cc@{}} A_{1} & A_{0}+A_{1}A_{2} \\ I & A_{2} \end{array}\right] \textrm{where} \left[ \begin{array}{@{}cc@{}} A^{1} & 0 \\ 0 & A^{2} \end{array}\right] \quad \textrm{is the diagonal bloc matrix of A} \\ & \left[ \begin{array}{@{}c@{}} B_{1} \\ B_{2} \end{array}\right] = \left[ \begin{array}{@{}c@{}} B \\ 0 \end{array}\right] \ \textrm{and} \ \left[ \begin{array}{@{}cc@{}} C_{1} & C_{2} \end{array}\right] = \left[ \begin{array}{@{}cc@{}} 0 & C \end{array}\right]\!. \end{align*} 4.4. Main Properties of Two-Dimensional Disjoint-Union Digraph associated with Structural Two-Dimensional System Since the two-dimensional digraph is the disjoint union of two sub-digraphs (see Figs. 3 and 4), we can then use this property for studying the partial structural controllability of the two-dimensional systems, where the sub-digraph G(h) corresponds to the horizontal part of two-dimensional system respectively the sub-digraph G(v) corresponds to the vertical part of two-dimensional system. If the conditions of the structural-controllability are satisfied for only G(h) (respectively G(v)), we obtain a partial horizontal structural-controllability respectively partial vertical structural-controllability. Example 4.2 We consider the example of Givone–Roesser bi-dimensional graph, This example illustrates that we can locally analyse the Givone–Roesser two-dimensional graph i.e. we can study only the structural controllability of horizontal sub-graph respectively vertical sub-graph. Fig. 3. View largeDownload slide Horizontal sub-graph. Fig. 3. View largeDownload slide Horizontal sub-graph. Fig. 4. View largeDownload slide Vertical sub-graph. Fig. 4. View largeDownload slide Vertical sub-graph. Lemma 4.1 G is the disjoint union of Gh and Gv, and is input-connectable if and only if Gh and Gv are input connectable. Proof. Necessary The states vertex are either horizontal or vertical, if the state vertex is horizontal, $${x^{h}_{n}} \in G^{h}$$, and given that Gh is input connectables, there is a path from at least one of the input vertices to $${x^{h}_{n}} \in G^{h}$$, in the same way if the state vertex is vertical, $${x^{v}_{m}} \in G^{h}$$, there is a path from at least one of the input vertices to $${x^{v}_{m}} \in G^{v}$$. Sufficient Obviously, if G is input connectable, then Gh and Gv are by definition input connectable, because G = Gh ∪ Gv. Lemma 4.2 Knowing that $$G(\left [Q_{1} \right ]) = G^{h}(\left [Q_{1} \right ]) \cup G^{v}(\left [Q_{1} \right ])$$. If for each horizontal state vertex in $$G^{h}(\left [Q_{1} \right ])$$ and for each vertical state vertex in $$G^{v}(\left [Q_{1} \right ])$$ there is at least one path from one of the m1 input vertices to $${x^{h}_{n}}$$, and there is at least one path from one of the m2 input vertices to $${x^{v}_{m}}$$, then there exist respectively for each state vertex in $$G(\left [Q_{1} \right ])$$ there is at least one path from one of the mi for i = 1, 2, with m = m1 + m2 input vertices to the chosen state vertex. Proof. The states vertex are either horizontal or vertical, if the state vertex is horizontal, $${x^{h}_{i}} \in G^{h}(\left [Q_{1} \right ])$$, with $$i \in \overline{1,m_{1}}$$, then, there is at least a path from one of the m1 input vertices to $${x^{h}_{i}}$$, in the same way if the state vertex is vertical, $${x^{v}_{i}} \in G^{v}(\left [Q_{1} \right ])$$, then, in $$G(\left [Q_{1} \right ])$$ there is at least one path from one of the mi for i = 1, 2, with m = m1 + m2 input vertices to the chosen state vertex. Lemma 4.3 If there is at least one cycle family of width n1 in $$G^{h}(\left [Q_{1} \right ])$$, and least one cycle family of width n2 in $$G^{v}(\left [Q_{1} \right ])$$, then there exist respectively at least one cycle family of width ni for i = 1, 2, with n = n1 + n2 in $$G(\left [Q_{1} \right ])=G^{h}(\left [Q_{1} \right ]) \cup G^{v}(\left [Q_{1} \right ])$$. Proof. Knowing that the number of state vertices in $$G^{h}(\left [Q_{1} \right ])$$ is n1 and the number of state vertices in $$G^{v}(\left [Q_{1} \right ])$$ is n2. If there is at least one cycle family of width n1 respectively n2, then, by definition the cycle family is in $$G^{h}(\left [Q_{1} \right ])$$ respectively in $$G^{v}(\left [Q_{1} \right ])$$, so there exist respectively at least one cycle family of width ni for i = 1, 2, with n = n1 + n2 in $$G(\left [Q_{1} \right ])$$. Remark 4 The fundamental notion that is used in this lemmas is the principal property of the two-dimensional digraph, who is the disjoint union of two sub-graph; the proof of lemma 4.1, 4.2 and 4.3 is founded on the idea that there exists a horizontal (respectively vertical) sub-graph. 4.5. Generalization of the Criteria of Structural-Controllability for the Structural Two-dimensional Systems The previous three lemmas are necessary to prove the criteria of structural controllability of the two-dimensional linear systems. Theorem 4.4 A class of system characterized by the n × (n + m) structure matrix pair [A, B] is structural-controllable if and only if the structure matrix pair [A1, B1] and [A2, B2] are structural-controllable. Proof. Necessary If for each state vertex in $$G([{Q^{h}_{1}}])$$ respectively $$G([{Q^{v}_{1}}])$$ there is at least one path from one of the m1 input respectively m2 input vertices to the chosen state vertices, then, according to the Lemma 4.2, there exist for each state vertex in $$G([Q_{1}])= G([{Q^{h}_{1}}]) \cup G([{Q^{v}_{1}}])$$, at least one path from one of the m = m1 + m2 input vertices to the chosen state vertex. If there is at least one cycle family of width n1 in $$G([{Q^{h}_{1}}])$$ and n2 in $$G([{Q^{v}_{1}}])$$, then, according to the Lemma 4.3, there exist at least one family of width n = n1 + n2 in $$G([Q_{1}])= G([{Q^{h}_{1}}]) \cup G([{Q^{v}_{1}}])$$. Sufficient It is evident that if the system is structural controllable, then for each sub-system characterized by the ni + (ni + mi) structure matrix pair [Ai, Bi] is structural controllable, especially for the [A1, B1] and [A2, B2]. Theorem 4.5 A class of system characterized by the n × (n + m) structure matrix pair [A, B] is structural-controllable if and only if the structure matrix [A1, B1] corresponding to Gh([Q1]) and [A2, B2] corresponding to Gv([Q1]) are structural-controllable. Proof. Necessary If [A1, B1] and [A2, B2] are structural-controllable then, Gh([Q1]) and Gv([Q1]) are input-connectable, by using the lemma 4.1, G([Q1]) is input-connectable. If [A1, B1] and [A2, B2] are structural-controllable, then, s-rank [A1, B1] = n1 and s-rank [A2, B2] = n2, and given that the bi-dimensional graph is defined as a disjoint union of two sub-graph s-rank [A, B] = n1 + n2 = n. Sufficient It is evident that if the system is structural-controllable, then for each sub system characterized by the ni + (ni + mi) structure matrix the pair [Ai, Bi] is structural controllable, especially for the [A1, B1] and [A2, B2]. We finely deduce the following result: Corollary 2 A class of system characterized by the structure matrix pair [A, B] is strongly structurally controllable if and only if [A1, B1]. and [A2, B2]. are strongly structurally controllable. 5. Illustrative examples We illustrate previous results concerning structural-controllability of two-dimensional linear models by some examples. Example 5.1 We consider a linear two-dimensional structured model with two horizontal states, a vertical state, two horizontal inputs, a vertical input, a horizontal output and a vertical output.   $$ A = \left[ \begin{array}{@{}ccc@{}} 0 & 0 & 0 \\ l_{1} & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\!, \,\, B = \left[ \begin{array}{@{}ccc@{}} l_{3} & 0 & 0 \\ 0 & l_{4} & 0 \\ 0 & 0 & l_{5} \end{array}\right]\!,\ \textrm{and}\ C = \left[ \begin{array}{@{}ccc@{}} 0 & l_{6} & 0 \\ 0 & 0 & l_{7} \end{array}\right] $$ This Structural two-dimensional model given by Fig. 5 is structurally controllable by applying the Theorem 4.4. In fact, the conditions of structural controllability characterized by Lemma 4.2 and Lemma 4.3 are satisfied in the horizontal part and vertical part of the graph. If we put $$ E = \left [ \begin{array}{@{}ccc@{}} l_{7} & 0 & 0 \\ 0 & l_{8} & 0 \\ 0 & 0 & l_{9} \end{array}\right ]$$ then, its G[Q1] graph is characterized by Fig. 6 as follows. In this graph (Fig. 7) we consider just input vertex and the state vertex by Theorem 4.4 the structural two-dimensional model is structurally controllable. Fig. 5. View largeDownload slide Illustrative diagraph of structural-controllability of a two-dimensional linear model. Fig. 5. View largeDownload slide Illustrative diagraph of structural-controllability of a two-dimensional linear model. Fig. 6. View largeDownload slide Illustrative diagraph of structural-controllability of a two-dimensional linear model with static state feedback. Fig. 6. View largeDownload slide Illustrative diagraph of structural-controllability of a two-dimensional linear model with static state feedback. Fig. 7. View largeDownload slide Illustrative diagraph of partial structural-controllability of a two-dimensional linear model. Fig. 7. View largeDownload slide Illustrative diagraph of partial structural-controllability of a two-dimensional linear model. Example 5.2 This example illustrates the partial structural controllability, we consider here the structural graph of certain structural two-dimensional linear system. We remark that the vertical sub-graph is not structural controllable, but the horizontal sub-graph is structural controllable, then we can conclude that the two-dimensional linear model is horizontal-partial structural controllable as defined in the sub-section 4.4. 6. Conclusion In the studies by Fornasini & Valcher (1997) and Fornasini & Valcher (1998) E. Fornasini et al. consider the two-dimensional directed graph of pair matrix (A, B) such as x(i1 + 1, i2 + 1) = Ax(i1 + 1, i2) + Bx(i1, i2 + 1), and study characteristic polynomials of irreducible matrix pairs (A, B), respectively, premitivity of positivity matrix pair (A, B) among other by the graph theoretic description. They define the associated directed graph D*(A, B) with arcs of two different kinds, namely, A-arcs and B-arcs; there are A-arcs from vertex vi to vertex vj if (vi, vj) is in A, and B-arcs if (vi, vj) is in B. To study the characterizations of global reachability of two-dimensional structured systems Pereira et al. (2013). introduce the shift operators σ1x(i1, i2) = x(i1 + 1, i2) and σ2x(i1, i2) = x(i1, i2 + 1) to rewrite the Fornasini–Marchesini model as one-dimensional model, and define the directed graph as the superposition of two sub-graphs and eliminating the repeated edges to use results that exist in one dimension. In this paper, we have studied the structural approach of two-dimensional systems by focusing on a classical problem. 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