Partial Strong Structural Controllability

We often use a structured matrix to denote the sparsity pattern of a numerical matrix [28]. A structured matrix is a matrix whose entries are chosen from the set $\{0,*,\times\}$. The $*$ entries and $\times$ entries are both called indeterminate entries. We use $\{0,*,\times\}^{p\times q}$ to denote set of all $p\times q$ structured matrices with entries from $\{0,*,\times\}$. Moreover, for $\bar{M}\in\{0,*,\times\}^{p\times q}$, ${\bf S}_{\bar{M}}$ is defined as

That is, a $*$ entry can take either zero or nonzero, while a $\times$ entry can take only nonzero value. An $M\in{\bf S}_{\bar{M}}$ is called a realization of $\bar{M}$. We also define two structured matrices $\bar{M}^{*}\in\{0,*\}^{p\times q}$ and $\bar{M}^{\times}\in\{0,\times\}^{p\times q}$ associated with $\bar{M}\in\{0,*,\times\}^{p\times q}$ respectively as

That is, $\bar{M}^{*}$ (resp. $\bar{M}^{\times}$) is obtained from $\bar{M}$ by replacing all of its indeterminate entries with $*$ (resp. $\times$) entries.

The operation ‘+’ between two structured matrices with the same dimension is an entry-wise addition operation so that every indeterminate entry in the sum appears exactly once in one of the addends. We define a generic realization of $\bar{M}$ as the realization whose indeterminate entries are assigned with independent parameters, and call such a realization a generic matrix without specifying the corresponding structured matrix. The generic rank of a generic matrix (or structured matrix), denoted by ${\rm grank}(\cdot)$, is the maximum rank it achieves as the function of its indeterminate parameters. For a generic matrix $M$ and a constant matrix $N$ with the same dimension, $M+\lambda N$ defines a generic matrix pencil [28], which is a matrix-valued polynomial of the independent parameters in $M$ and the variable $\lambda$.

Consider a linear time invariant system as

where $x(t)\in{\mathbb{C}}^{n}$ is the state variable, $u(t)\in{\mathbb{C}}^{m}$ is the input, $A\in{\mathbb{C}}^{n\times n}$ and $B\in{\mathbb{C}}^{n\times m}$ are respectively the state transition matrix and input matrix. For description simplicity, by the pair $(A,B)$ we refer to a system described by (2) that can be either single-input (i.e., $m=1$) or multi-input (i.e., $m>1$), while by $(A,b)$ we refer to a single-input system. Note dynamic systems with complex-valued system matrices and state variables are not rare in the literature [29, 30, 31, 32, 33], either for practical utility or for theoretical studies. Following a similar manner as $(A,B)$ in the real field [23, Chap. 9], it turns out the existence and uniqueness of solutions of system (2) are guaranteed even in the complex domain.

Denote the controllability matrix of system (2) by ${\cal C}(A,B)$, i.e.,

It is known from [29, 30, 31] that $(A,B)$ is controllable, if and only if ${\cal C}(A,B)$ is of full row rank.

Throughout this paper, given $\bar{A}\in\{0,*,\times\}^{n\times n}$, $\bar{B}\in\{0,*,\times\}^{n\times m}$, ${\cal N}_{\times}$ denotes the set of positions of all $\times$ entries in $[\bar{A},\bar{B}]$, i.e., ${\cal N}_{\times}=\{(i,j):[\bar{A},\bar{B}]_{ij}=\times\}$. ${\cal N}_{*}=\{(i,j):[\bar{A},\bar{B}]_{ij}=*\}$ is defined similarly. We use $n_{*}$ and $n_{\times}$ to denote the numbers of $*$ entries and $\times$ entries in $[\bar{A},\bar{B}]$, respectively. Let $p_{1},...,p_{n_{*}}$ and $\bar{p}_{1},...,\bar{p}_{n_{\times}}$ be respectively the parameters for the $*$ entries and $\times$ entries (also called indeterminate parameters) in $[\bar{A},\bar{B}]$. Denote $p=(p_{1},...,p_{n_{*}})$ and $\bar{p}=(\bar{p}_{1},...,\bar{p}_{n_{\times}})$. The following notions are natural extensions of the original ones from the real field to the complex field.

Note that controllability is a generic property in the sense if $(\bar{A},\bar{B})$ is structurally controllable, where $\bar{A}\in\{0,*\}^{n\times n}$, $\bar{B}\in\{0,*\}^{n\times m}$, then $(A,B)$ is controllable for almost all $[A,B]\in{\bf S}_{[\bar{A},\bar{B}]}$. Hereafter, ‘almost all’ means ‘all except for a set of zero Lebesgue measure in the corresponding parameter space’. Motivated by Example 1, as argued in Section 1, the PSSC is formally defined as follows.

Equivalently, $(\bar{A},\bar{B})$ is PSSC, if for all $p\in{\mathbb{C}}^{n_{*}}$ except a set of zero measure in ${\mathbb{C}}^{n_{*}}$ and all $\bar{p}\in{\bar{\mathbb{C}}^{n_{\times}}}$, the corresponding realization of $(\bar{A},\bar{B})$ is controllable. PSSC of $(\bar{A},\bar{B})$ indicates that, if we randomly generate values for the $*$ entries (from a continuous interval), then with probability $1$, the corresponding system will be controllable for all nonzero values of the $\times$ entries. For this reason, we may also call an $*$ entry a generic entry, and the $\times$ one an unspecified entry.

To show PSSC is well-defined, we introduce the concept
  perturbation-tolerant structural controllability (PTSC) from [27], which is defined for a structured system as the property that, for almost all values of its generic entries, there exist no complex values (including zero) for its unspecific entries such that the corresponding system realization is uncontrollable (see Definition 3 of [27] for a precise description). It is proven in [27] that, depending solely on the structure of the structured system, for almost all values of its generic entries, either there exist no values for its unspecific entries that can make the system realization uncontrollable, or there exist such values. Obviously, PTSC is a sufficient condition for the property in Definition 3 to hold. This means, there do exist structured systems satisfying the property in Definition 3.

Further, it is easy to see, if no $*$ entries exist in $(\bar{A},\bar{B})$, then PSSC collapses to SSC, which explains the term ‘partial’ in this terminology. On the other hand, if no $\times$ entries exist in $(\bar{A},\bar{B})$, then PSSC collapses to structural controllability (this also demonstrates that PSSC is well-defined). In this sense, PSSC bridges the gap between structural controllability and SSC. Moreover, it is apparent that for a given $(\bar{A},\bar{B})$, SSC implies PSSC, and PSSC implies structural controllability¹¹1Note as controllability is a generic property [13], if a structured system is structurally controllable, there must exist a controllable realization of it subject to the constraint that an arbitrary subset of its indeterminate entries are nonzero.; while the inverse direction is not necessarily true. The following Example 2 highlights the differences among these concepts.

A (directed) graph ${\cal G}$ is represented by ${\cal G}=(V,E)$, where $V$ is the vertex set and $E\subseteq V\times V$ is the edge set. If there is a path (i.e., a sequence of successive edges) from vertex $v_{j}$ to vertex $v_{i}$, we say $v_{i}$ is reachable from $v_{j}$. A subgraph ${\cal G}_{s}=(V_{s},E_{s})$ of $\cal G$ is a graph such that $V_{s}\subseteq V$ and $E_{s}\subseteq E$. And ${\cal G}_{s}$ is said to be induced by $V_{s}$ if $E_{s}=(V_{s}\times V_{s})\cap E$. For $V_{s}\subseteq V$, ${\cal G}-V_{s}$ denotes the graph obtained from ${\cal G}$ by deleting the vertices in $V_{s}$ together with their incident edges (i.e., ${\cal G}-V_{s}$ is the subgraph of ${\cal G}$ induced by $V\backslash V_{s}$).

A bipartite graph is written as ${\cal G}=(V^{+},V^{-},E)$ with $V^{+}$ and $V^{-}$ being the bipartitions and $E$ the edges [28]. For a $v\in V^{+}\cup V^{-}$, ${\cal N}({\cal G},v)$ denotes the set of neighbors in $\cal G$, i.e., the set of vertices that are connected with $v$ by an edge. A matching of ${\cal G}$ is a set of its edges, no two of which share a common end vertex. The size (or cardinality) of a matching is the number of edges it contains. A vertex is said to be matched by a matching if it is an end vertex of edges in this matching. A maximum matching of $\cal G$ is the matching with the largest size, which value is denoted by ${\rm{mt}}({\cal G})$. An edge of $\cal G$ is said to be admissible if it is contained in some maximum matching of ${\cal G}$. We say a matching ${\cal M}$ covers ${\cal G}$, if ${\cal M}\cap E$ forms a maximum matching of ${\cal G}$. If each edge of $\cal G$ has a non-negative weight, the maximum (minimum) weight maximum matching is the maximum matching of $\cal G$ with the largest (smallest) weight (the weight of a matching is the sum of its edge weights).

In Definition 4, ${\cal G}_{0}$ (if exists) is called the horizontal tail, and ${\cal G}_{\infty}$ the vertical tail. Additionally, if ${\cal G}$ contains only one DM-component, then ${\cal G}$ is called DM-irreducible.

For an $n_{1}\times n_{2}$ matrix $M$, the bipartite graph associated with $M$ is given by ${\cal B}(M)=(V^{+},V^{-},E_{M})$, where $V^{+}=\{v^{+}_{1},...,v^{+}_{n_{1}}\}$ (resp. $V^{-}=\{v^{-}_{1},...,v^{-}_{n_{2}}\}$) corresponds to the rows (columns) of $M$, and $E_{M}=\{(v^{+}_{i},v^{-}_{j}):M_{ij}\neq 0,v^{+}_{i}\in V^{+},v^{-}_{j}\in V^{-}\}$. It is known that, for a generic matrix $M$, ${\rm grank}(M)={\rm mt}({\cal B}(M))$ [28]. For $V_{s}^{+}\subseteq V^{+}$ and $V_{s}^{-}\subseteq V^{-}$, we also write $M[V_{s}^{+},V_{s}^{-}]$ as the submatrix of $M$ whose rows correspond to $V_{s}^{+}$ and columns to $V_{s}^{-}$, where elements in $V_{s}^{+}$ and $V_{s}^{-}$ are ordered. Note such an expression of submatrices is invariant subject to row and column permutations on $M$, that is, upon letting $P\in{\mathbb{R}}^{n_{1}\times n_{1}}$ and $Q\in{\mathbb{R}}^{n_{2}\times n_{2}}$ be two permutation matrices and $M^{\prime}=PMQ$, $M^{\prime}[V^{+}_{s},V^{-}_{s}]$ is the same as $M[V^{+}_{s},V^{-}_{s}]$.

For a structured pair $(\bar{A},\bar{B})$, its associated graph is given by ${\cal G}(\bar{A},\bar{B})=(X,E_{X})$, where $X=\{x_{1},...,x_{n},...,x_{n+m}\}$, $E_{X}=\{(x_{j},x_{i}):[\bar{A},\bar{B}]_{ij}\neq 0\}$. A vertex ${x_{i}\in X}$ is said to be input-reachable, if ${x_{i}}$ is reachable from at least one vertices of $x_{n+1},...,$ $x_{n+m}$ in ${\cal G}(\bar{A},\bar{B})$.

We first give an algebraic condition for PSSC, in terms of determinants of the controllability matrix of the generic realization of $(\bar{A},\bar{b})$.

Although the verification of Theorem 1 is prohibitive for large-scale systems, as computing the determinant of a symbolic matrix has computational complexity increasing exponentially with its dimensions [35], it is theoretically significant in proving some properties of PSSC. Particularly, based on Theorem 1, we have the following two properties of PSSC of single-input systems.

The above proposition reveals for a single-input structured system, if it is not PSSC, then for almost all values of the $*$ entries, there exist nonzero (complex) values for the $\times$ entries so that the realization is uncontrollable; otherwise, for almost all values of the $*$ entries, the corresponding realization is controllable for all nonzero values of the $\times$ entries. This generalizes the generic property embedded in structural controllability. This proposition also explains the motivation behind the definition of PSSC.

The above proposition is beneficial in deriving conditions for PSSC, as it actually transforms the problem of verifying PSSC of $(\bar{A},\bar{b})$ to $n_{\times}$ subproblems of verifying PSSC of $(\bar{A}^{\pi},\bar{b}^{\pi})$ for each $\pi\in{\cal N}_{\times}$.

In this subsection, we give testable necessary and sufficient conditions for single-input systems to be PSSC. Owing to Proposition 2, in the following, we first focus on conditions of PSSC for $(\bar{A}^{\pi},\bar{b}^{\pi})$, i.e. systems that contain only one $\times$ entry, and then on the general systems. Recall from the PBH test [29, 31], an uncontrollable mode for system $(A,B)$ is a $\lambda\in{\mathbb{C}}$ such that ${\rm rank}([A-\lambda I,B])<n$. Inspired by [16, 27], we shall respectively give the conditions for the nonexistence of zero uncontrollable and nonzero uncontrollable modes. Particularly, the following lemma, relating the ‘structure’ of a vector in the left null space of a given matrix to the full row rank of its submatrices, is fundamental to our subsequent derivations.

Proposition 3 gives an algebraic condition for the nonexistence of zero uncontrollable modes. The following proposition gives an equivalent bipartite graph form of Proposition 3. Recall the bipartite graph ${\cal B}([\bar{A},\bar{b}])=(V^{+},V^{-},E_{[\bar{A},\bar{b}]})$ associated with $[\bar{A},\bar{b}]$ is defined in Section 2.3, where $V^{+}=\{v_{1}^{+},...,v_{n}^{+}\}$, $V^{-}=\{v_{1}^{-},...,v_{n+1}^{-}\}$.

We are now presenting the conditions for nonzero uncontrollable modes. Again, suppose $(\bar{A},\bar{b})$ is structurally controllable and $[\bar{A},\bar{b}]$ contains only one $\times$ entry, with its position being $(i,j)$. For a generic realization $(\tilde{A},\tilde{b})$ of $(\bar{A},\bar{b})$, define a generic matrix pencil $H_{\lambda}=[\tilde{A}-\lambda I,\tilde{b}]$, and let $H^{j_{c}}_{\lambda}\doteq H_{\lambda}[J_{n},J_{n+1}\backslash\{j\}]$ and $H^{j}_{\lambda}\doteq H_{\lambda}[J_{n},\{j\}]$. Let ${\cal B}(H_{\lambda})=(V^{+},V^{-},E_{H_{\lambda}})$ be the bipartite graph associated with $H_{\lambda}$, defined as follows: $V^{+}=\{v^{+}_{1},...,v^{+}_{n}\}$, $V^{-}=\{v^{-}_{1},...,v^{-}_{n+1}\}$, and $E_{H_{\lambda}}=E_{I}\cup E_{[\bar{A},\bar{b}]}$, in which $E_{I}=\{(v^{+}_{k},v^{-}_{k}):k=1,...,n\}$ and $E_{[\bar{A},\bar{b}]}=\{(v^{+}_{k},v^{-}_{l}):[\bar{A},\bar{b}]_{kl}\neq 0\}$. An edge $e$ is called a $\lambda$-edge if $e\in{E}_{I}$, and a self-loop if $e\in E_{I}\cap E_{[\bar{A},\bar{b}]}$. ${\cal B}(H_{\lambda}^{j_{c}})$ is the subgraph of ${\cal B}(H_{\lambda})$ induced by $V^{+}\cup V^{-}\backslash\{v^{-}_{j}\}$.

Let ${\cal G}^{j_{c}}_{k}=(V^{+}_{k},V^{-}_{k},E_{k})$ ($k=0,...,d,\infty$) be the DM-components of ${\cal B}(H_{\lambda}^{j_{c}})$. From [27, Lem 4], as $(\bar{A},\bar{b})$ is structurally controllable, we have ${\rm mt}({\cal B}(H_{\lambda}^{j_{c}}))=n$ for $j=1,...,n+1$. Hence, ${\cal G}^{j_{c}}_{0}={\cal G}^{j_{c}}_{\infty}=\emptyset$, $\forall j\in J_{n+1}$. By the correspondence between a matrix and its associated bipartite graph, the DM-decomposition of ${\cal B}(H_{\lambda}^{j_{c}})$ corresponds to that the matrix $H_{\lambda}^{j_{c}}$ is transformed into the block-triangular form via two $n\times n$ permutation matrices $P$ and $Q$ [28]:

where the submatrix $M^{j_{c}}_{k}(\lambda)=H_{\lambda}^{j_{c}}[V^{+}_{k},V^{-}_{k}]$ corresponds to ${\cal G}_{k}$ ($k=1,...,d$). Accordingly, let $M_{\lambda}^{j}\doteq PH_{\lambda}^{j}$.

For each $k\in\{1,...,d\}$, let $\gamma_{\min}({\cal G}^{j_{c}}_{k})$ and $\gamma_{\max}({\cal G}^{j_{c}}_{k})$ be respectively the minimum and maximum numbers of $\lambda$-edges contained in a matching over all maximum matchings of ${\cal G}^{j_{c}}_{k}$. We borrow the boolean function $\gamma_{nz}(\cdot)$ for ${\cal G}_{k}^{j_{c}}$ from [27], which is defined as

The following lemma explains the motivation of introducing $\gamma_{\rm nz}(\cdot)$.

Next, define the set

Moreover, for a vertex $v^{+}_{l}\in V^{+}$, define a set $\Omega^{l}_{j}\subseteq\Omega_{j}$ as

It is clear $\Omega_{j}^{l}=\emptyset$ implies that every maximum matching of ${\cal B}(H_{\lambda}^{j_{c}})-\{v^{+}_{l}\}$ covers $\bigcup\nolimits_{k\in\Omega_{j}}{\cal G}_{k}^{j_{c}}$. As we shall see, $\Omega_{j}^{l}$ is the set of indices of all DM-components associated with which $\det M_{k}^{j_{c}}(\lambda)$ has a nonzero root $z$, such that $M_{z}^{j_{c}}[V^{+}\backslash\{v_{l}^{+}\},V^{-}]$ is of full row rank ($M_{z}^{j_{c}}$ is obtained by substituting $\lambda=z$ into $M_{\lambda}^{j_{c}}$). Hereafter, by saying $M_{\lambda}^{j_{c}}$ (or its sub-matrices) satisfies certain properties, we mean these properties are satisfied for almost all values of the corresponding indeterminate parameters (i.e., they are satisfied generically).