Generic Diagonalizability, Structural Functional Observability and Output Controllability

Abstract

This paper investigates the structural functional observability (SFO) and structural output controllability (SOC) of a class of systems with generically diagonalizable state matrices and explores the associated minimal sensor and actuator placement problems. The verification of SOC and the corresponding sensor and actuator placement problems, i.e., the problems of determining the minimum number of outputs and inputs required to achieve SFO and SOC, respectively, are yet open for general systems, which motivates our focus on a class of systems enabling polynomial-time solutions. In this line, we first define and characterize generically diagonalizable systems, referring to structured systems for which almost all realizations of the state matrices are diagonalizable. We then develop computationally efficient criteria for SFO and SOC within the context of generically diagonalizable systems. Our work expands the class of systems amenable to polynomial-time SOC verification. Thanks to the simplicity of the obtained criteria, we derive closed-form solutions for determining the minimal sensor placement to achieve SFO and the minimal actuator deployment to achieve SOC in such systems, along with efficient weighted maximum matching based and weighted maximum flow based algorithms. For more general systems to achieve SFO, an upper bound is given by identifying a non-decreasing property of SFO with respect to a specific class of edge additions, which is shown to be optimal under certain circumstances.

Figures (5)

• Figure 1: (a) and (b): Examples of structurally diagonalizable graphs. Bold edges correspond to a maximum matching ${\mathcal{M}}$, associated with which ${\mathcal{G}}({\mathcal{M}})=(X,{\mathcal{M}})$ is a union of disjoint cycles and isolated vertices. (c) and (d): Examples of structurally non-diagonalizable graphs. Each SCC is in a box, while the subgraphs in blue are induced by a specific set of SCCs. In (c), the subgraph induced by $\{x_3,x_4\}$ is acyclic, and thus is structurally non-diagonalizable (Corollary \ref{['corollay-acyclic']}). In (d), the subgraph induced by $\{x_1,x_3,x_4,x_5\}$ is structurally non-diagonalizable as any maximum matching of it cannot correspond to disjoint cycles and isolated vertices.
• Figure 2: ${\mathcal{G}}(\bar{A}, \bar{B}, \bar{C})$ and ${\mathcal{D}}(\bar{A}_r,\bar{B},\bar{C})$ in Example \ref{['illustrate-example']}. Vertices in blue correspond to input-unreachable vertices in ${\mathcal{G}}(\bar{A}, \bar{B})$. Bold red edges constitute a linking of size $2$.
• Figure 3: Example of the procedure of Algorithm \ref{['alg1']} applied to the system digraph ${\mathcal{G}}(\bar{A})$ in (a), where $\bar{A}$ is generically diagonalizable. Nodes in gray represent functional states. In (b), the number on each edge is the cost assigned to it (see (\ref{['cost']})), and bold red edges form a minimum weight maximum matching ${\mathcal{M}}$ of ${\mathcal{B}}(\bar{A})$. Accordingly, $X_S=\{x_2,x_4\}$ and $X_F^u=\{x_6\}$, resulting in $p^*=1$. In (c), the dotted red edges represent the obtained solution.
• Figure 4: Illustration of Algorithms \ref{['alg2']} and \ref{['alg3']} applied to the system digraph ${\mathcal{G}}(\bar{A})$ in (a). Nodes in gray represent functional states, and in red represent sensors. In (a), bold red edges form a maximum matching ${\mathcal{M}}$ that does not right match $x_3\in X_F$ in ${\mathcal{B}}(\bar{A})$, resulting in $p^*=1$ via Algorithm \ref{['alg1']}. In (b), the dotted red edges represent the solution returned by Algorithm \ref{['alg2']}. Subfigure (c) presents the digraph corresponding to the bipartite graph $\tilde{\mathcal{B}}(\bar{A}, \bar{I}_{X_F})$, where partial return edges from $Y$ to $X$ are omitted for simplicity. The number on each edge is the cost assigned to it via (\ref{['edge_cost']}), where zero costs are omitted for edges not belonging to $E_{XX}\cup E_{XY}$. Bold (solid and dotted) red edges correspond to a maximum weighted maximum matching ${\mathcal{H}}^*$ of $\tilde{\mathcal{B}}(\bar{A}, \bar{I}_{X_F})$ with weight $4+4+3+3=14$. Accordingly, $X_{{\mathcal{H}}^*}=\{x_3,x_4\}$, and there are two output stems $(x_1,x_2,x_3,y_1)$ and $(x_4,y_2)$ associated with it in ${\mathcal{G}}(\bar{A}, \bar{I}_{X_F})$. Algorithm \ref{['alg3']} then returns the solution of placing two dedicated sensors at $x_3,x_4$.
• Figure 5: Illustration of Algorithm \ref{['alg4']}. Subfigure (a) gives the digraph ${\mathcal{G}}(\bar{A}, \bar{C})$. Subfigure (b) depicts the corresponding flow network ${\mathcal{F}}(\bar{A}, \bar{C})$ constructed in step 1 of Algorithm \ref{['alg4']}. All edges have capacity one, dotted red edges have cost one, and the rest have cost zero. Bold red edges form a minimum cost maximum flow $f^*$ with total cost one, where each edge has flow $1$. Associated with it, $X^{f_1^*}=\{x_2\}$ and $X^{f_2^*}=\{x_2,x_4\}$.

Theorems & Definitions (38)

• Definition 1: Modern_Control_Ogata,Modern_Control_Ogata
• Definition 2: fernando2010functional2,fernando2010functional2
• Lemma 1
• Definition 3: zhang2023functional,zhang2023functional
• Example 1
• Lemma 2: Prop 2, zhang2023functional,zhang2023functional
• Definition 4: Generic diagonalizability
• Lemma 3: hosoe1979irreducibility,hosoe1979irreducibility
• Lemma 4
• Theorem 1