Structural accessibility and structural observability of nonlinear networked systems

TL;DR

This paper extends classical linear notions of controllability and observability to nonlinear networked systems, revealing that nonlinearities can simplify control and observation by reducing the number of variables needed.

Contribution

It introduces and characterizes the concepts of structural accessibility and observability for nonlinear systems, broadening the applicability of network control theory.

Findings

Nonlinearities facilitate control and observation in networked systems.

Reduced number of variables needed for control and measurement.

Enhanced understanding of network structure's role in nonlinear dynamics.

Abstract

The classical notions of structural controllability and structural observability are receiving increasing attention in Network Science, since they provide a mathematical basis to answer how the network structure of a dynamic system affects its controllability and observability properties. However, these two notions are formulated assuming systems with linear dynamics, which significantly limit their applicability. To overcome this limitation, here we introduce and fully characterize the notions "structural accessibility" and "structural observability" for systems with nonlinear dynamics. We show how nonlinearities make easier the problem of controlling and observing networked systems, reducing the number of variables that are necessary to directly control and directly measure. Our results contribute to understanding better the role that the network structure and nonlinearities play in our ability to control and observe complex dynamic systems.

Equations

$$\dot{x}(t)=Ax(t)+Bu(t),y(t)=Cx(t),$$

$$\{\begin{array}{rl}{\dot{x}}_1(t)& =b_{11}{u}_1(t),\\ {\dot{x}}_2(t)& =b_{21}{u}_1(t),\\ y_1(t)& =c_{11}{x}_1(t)+c_{12}{x}_2(t),\end{array}$$

$$\dot{x}(t)=f\left(x(t),u(t)\right),y(t)=h(x(t)),$$

$$\mathfrak{D}(\mathcal{G}):=\{\text{mbox}allmeromorphic\{f,h\}\text{mbox}suchthat{\mathcal{G}}_{f,h}=\mathcal{G}\}.$$

$$\partial {\xi }^{(k)}/\partial u\equiv 0,\forall k\ge 0.$$

$$\mathrm{d}\zeta \notin \text{mbox}{span}_{\mathcal{K}}\{\mathrm{d}y,\mathrm{d}\dot{y},\cdots ,\mathrm{d}{y}^{(k)}\},\forall k\ge 0.$$

$$\{\begin{array}{rl}{\dot{x}}_1(t)& =b_{11}{u}_1(t)+\epsilon ,\\ {\dot{x}}_2(t)& =b_{21}{u}_1(t)+\epsilon u_1^3(t),\\ y_1(t)& =c_{11}{x}_1+c_{12}{x}_2+\epsilon x_1(t)x_2(t),\end{array}$$

$$\left(\begin{array}{c}\partial y_1/\partial x\\ \partial {\dot{y}}_1/\partial x\end{array}\right)=\left(\begin{array}{cc}{c}_{11}+\epsilon x_2& c_{12}+\epsilon x_1\\ \epsilon {\alpha }_1& \epsilon {\alpha }_2\end{array}\right)$$

$$\dot{x}(t)=f(x(t),u(t)),$$

$${\dot{x}}_i=f_i(x,u)+\alpha x_j,$$

$${\dot{x}}_i=f_i(x,u)+\alpha u_j.$$

$${\dot{x}}_i=(f_{T(i)}{)}^{p_i}$$

$$\dot{\phi }=\sum _{i\in \mathsf{S}}\frac{\partial \phi }{\partial x_i}{\dot{x}}_i=\sum _{i\in \mathsf{S}}\frac{\partial \phi }{\partial x_i}{f}_{T(i)}^{p_i}\in T(\mathsf{S}),$$

$${\dot{x}}_i=c_i,i=1,\dots ,d_1;y=x_1{x}_2\cdots x_{d_1},$$

$$\dot{y}=\frac{\mathrm{d}}{\mathrm{d}t}\left(\prod _{i=1}^{d_1}{x}_i\right)=\left(\prod _{i=1}^{d_1}{x}_i\right)\left(\sum _{i=1}^{d_1}\frac{c_i}{x_i}\right)=(y)\left(\sum _{i=1}^{d_1}\frac{c_i}{x_i}\right).$$

$$z^{(k)}=\sum _{i=1}^{d_1}(-1{)}^{k}k!\frac{c_i^{k+1}}{x_i^{k+1}},$$

$$\mathrm{d}{z}^{(k)}=\sum _{i=1}^{d_1}(-1{)}^{k+1}(k+1)!\frac{c_i^{k+1}}{x_i^{k+2}}\mathrm{d}{x}_i.$$

Full text

I Introduction

In a world where complex networks underlie most biological, social and technological systems that shape the human experience [1, 2], one central challenge is finding principles that can help us control and observe complex networked systems. When only the network structure of a dynamical system is known (i.e., a graph of the interactions between its variables), a central theoretical basis for this research program has been the classical notions of “structural controllability” and “structural observability” of linear systems [3]. These two notions characterize the conditions under which almost all linear dynamical systems whose structure matches a given network are controllable or observable, respectively [4]. Linear structural controllability and structural observability thus provide a mathematical formalism for predicting how changes in the network structure of a system impact its controllability and observability properties. For example, linear structural controllability was applied to build and then experimentally validate predictions of how removing different neurons (i.e., removing nodes in the network) affects the locomotion of the round worm C. elegans [5]. Additionally, over the last few years, a central line of research has been characterizing minimal sets of “driver nodes” and “sensor nodes” from which we can efficiently render a complex networked system controllable and observable [3].

The conditions of linear structural controllability and linear structural observability can be stronger than necessary when applied to systems with nonlinear dynamics, resulting in over-conservative predictions. This is because the lack of linear controllability (resp. linear observability) of a nonlinear system cannot be used to predict its lack of controllability (resp. observability). An elementary example of this is a car, which is controllable but not linearly controllable because it cannot move in the direction of the axis defined by its rear wheels. Despite the ubiquity of nonlinear systems in nature and technology, the effects of nonlinearities on our ability to efficiently control and observe complex networked systems remain poorly understood [6, 7, 8, 9, 10].

Given that most systems in nature are expected to contain nonlinearities, in this Note we introduce and characterize the notions of nonlinear “structural accessibility” and nonlinear “structural observability” as counterparts of linear structural controllability and linear structural observability. These two notions we introduce characterize the conditions under which almost all nonlinear systems whose structure matches a given network are locally accessible or locally observable almost everywhere, respectively. Accessibility and observability are nonlinear generalizations of linear controllability and linear observability, which have played a central role in the development of nonlinear control theory [11]. Somewhat counter-intuitively, we show that nonlinearities make significantly easier the problem of controlling or observing a complex networked system. More precisely, our main result proves that the conditions for nonlinear structural accessibility and observability are weaker than the conditions for linear structural controllability and observability. We show this implies that we need smaller sets of driver and sensor nodes when compared to the those necessary for linear structural controllability and linear structural observability.

This Note is organized as follows. Section II summarizes the network characterization of structural controllability and structural observability of linear systems, serving as a comparison point to our results. Section III contains our problem statement and main results. Proofs are collected in Sections IV and V. We end discussing some predictions that our structural accessibility theory offers about the locomotion of C. elegans, and some limitations of our approach.

II Preliminaries

The network or graph of a system with $N$ state variables, $M$ inputs, and $P$ outputs is a directed graph $\mathcal{G}=({\sf X\cup\sf Y\cup\sf U},{\sf A\cup B\cup C})$ containing state nodes ${\sf X}=\{{\sf x}_{1},\cdots,{\sf x}_{N}\}$, output nodes ${\sf Y}=\{{\sf y}_{1},\cdots,{\sf y}_{P}\}$, and input nodes ${\sf U}=\{{\sf u}_{1},\cdots,{\sf u}_{M}\}$, see Fig. 1a. Edges take the form $({\sf x}_{j}\rightarrow{\sf x}_{i})\in{\sf A}$ to denote that the $i$-th state variable directly depends on the $j$-th one, $({\sf x}_{j}\rightarrow{\sf y}_{i})\in{\sf C}$ to denote that the $i$-th measured output directly depends on the $j$-th state variable, and $({\sf u}_{j}\rightarrow{\sf x}_{i})\in{\sf B}$ to denote that the $i$-th state variable directly depends on the $j$-th control input. We allow graphs with empty output or input node sets to represent systems without outputs or inputs, respectively.

In the framework of linear structural controllability and linear structural observability the system dynamics is of course assumed linear. Then the controllability and observability of the set of all linear systems whose structure matches the graph $\mathcal{G}$ is analyzed. More precisely, the system dynamics is assumed to have the form

[TABLE]

where $x(t)\in\mathbb{R}^{N}$, $u(t)\in\mathbb{R}^{M}$ and $y(t)\in\mathbb{R}^{P}$ are the state, input, and output of the system at time $t$, respectively. Here $A=(a_{ij})\in\mathbb{R}^{N\times N}$, $B=(b_{ij})\in\mathbb{R}^{N\times M}$ and $C=(c_{ij})\in\mathbb{R}^{P\times N}$ are matrices of parameters. The structure of Eq. (1) is determined by the zero/non-zero pattern of these three matrices. Thus, given a graph $\mathcal{G}$, the class $\mathfrak{D}_{L}(\mathcal{G})$ of all linear systems whose structure matches $\mathcal{G}$ is defined as all systems (1) such that: $a_{ij}\neq 0$ iff $({\sf x}_{j}\rightarrow{\sf x}_{i})\in{\sf A}$, $b_{ij}\neq 0$ iff $({\sf u}_{j}\rightarrow{\sf x}_{i})\in{\sf B}$, and $c_{ij}\neq 0$ iff $({\sf x}_{j}\rightarrow{\sf y}_{i})\in{\sf C}$. Note that the edges $({\sf x}_{j}\rightarrow{\sf x}_{i})$ and $({\sf u}_{j}\rightarrow{\sf x}_{i})$ are encoded by differential equations. By contrast, the edges $({\sf x}_{j}\rightarrow{\sf y}_{i})$ are encoded by algebraic equations; these output edges have direction because the output map $y=Cx$ is not necessarily one-to-one (e.g., the single output $y=x_{1}+x_{2}$). Thus, the class $\mathfrak{D}_{L}(\mathcal{G})$ describes the set of all linear dynamics that a system can take if its structure coincides with $\mathcal{G}$.

The class $\mathfrak{D}_{L}(\mathcal{G})$ is said structurally controllable (resp. structurally observable) if it contains at least one system that is linearly controllable (resp. linearly observable) [4]. In that case we also say that $\mathcal{G}$ is linearly structurally controllable (resp. linearly structurally observable). It turns out that when one system in $\mathfrak{D}_{L}(\mathcal{G})$ is linearly controllable (resp. linearly observable), then almost all other systems in $\mathfrak{D}_{L}(\mathcal{G})$ are linearly controllable as well (resp. linearly observable) [4]. This means that, if $\mathfrak{D}_{L}(\mathcal{G})$ is structurally controllable (resp. structurally observable), any of its systems is either controllable (resp. observable), or becomes controllable (resp. observable) by an infinitesimal change in the nonzero entries of the matrices $A,B$ and $C$. A central result in the theory of structural linear systems, which can be traced back to the pioneer work of Lin in the 70’s [12], is the following:

Theorem 1**.**

(see, e.g., [4]) $\mathfrak{D}_{L}(\mathcal{G})$ is:

• (i)

structurally controllable iff each state node is the end-node of a path that starts in ${\sf U}$; and there is a disjoint union of cycles and paths starting in $\sf U$ that covers ${\sf X}$.

• (ii)

structurally observable iff each state node is the start-node of a path that ends in ${\sf Y}$; and there is a disjoint union of cycles and paths ending in $\sf Y$ that covers ${\sf X}$.

Recall that a path is a sequence of nodes ${\sf v}_{1}\rightarrow{\sf v}_{2}\rightarrow\cdots\rightarrow{\sf v}_{n}$ where ${\sf v}_{i}\in\sf X\cup Y\cup U$. The start-node of this path is ${\sf v}_{1}$ and its end-node is ${\sf v}_{n}$. A cycle is a path that starts and ends in the same node (i.e., ${\sf v}_{n}={\sf v}_{1}$). Two paths are disjoint if they have disjoint sets of nodes.

Theorem 1 shows that except for a zero-measure set of “singularities,” the graph $\mathcal{G}$ of a linear system determines its controllability and observability properties. Note that for linear structural controllability it is not sufficient that the control inputs propagate their influence through $\mathcal{G}$ to all state nodes. Similarly, for linear structural observability, it is not sufficient that each state node can propagate its state to some output through $\mathcal{G}$. Both notions require that the graph $\mathcal{G}$ contains enough “independent” paths to propagate these effects, encoded by the existence of a disjoint union of cycles and paths that covers all state nodes.

Example 1**.**

For the graph $\mathcal{G}$ of Fig. 1a, the class $\mathfrak{D}_{L}(\mathcal{G})$ contains all linear systems of the form

[TABLE]

with nonzero constants $b_{11},b_{21},c_{11}$ and $c_{12}$. Recall that:

Together with isolated nodes in $\mathcal{G}$, the main obstacle for linear structural controllability is the presence of so-called “dilations” **[12]**. In essence, a dilation consist of two nodes with identical dynamics that are controlled by the same input (top in Fig. 1a). A dilation makes $\mathcal{G}$ not structurally controllable because it is impossible to obtain a disjoint union of paths that covers $\sf X$. For Fig. 1a, all systems in $\mathfrak{D}_{L}(\mathcal{G})$ are uncontrollable because their state is constrained to the plane $b_{11}x_{2}(t)-b_{21}x_{1}(t)=b_{11}x_{2}(0)-b_{21}x_{1}(0)$ for all inputs $u_{1}(t)$ and time $t$ (Fig. 1b).

• 2.

*Analogously, so-called “contractions” in $\mathcal{G}$ are the main obstacle for linear structural observability. In essence, a contraction corresponds to two state nodes that are measured using a single output (bottom in Fig. 1a). Indeed, for Fig. 1a, all systems $\mathfrak{D}_{L}(\mathcal{G})$ are unobservable because using $y_{1}=c_{11}x_{1}+c_{12}x_{2}$ and $k$ of its derivatives $y_{1}^{(k)}=(c_{11}b_{11}+c_{12}b_{12})u_{1}^{(k)}$ it is impossible to infer the value of $x_{1}$ and $x_{2}$ (Fig. 1c). *

Theorem 1 provides a theoretical basis for a very active research line aiming to identify and analyze the “driver” and “sensor” nodes that render a system linearly structurally controllable and linearly structurally observable (see, e.g., [2, 3]). More precisely, consider a graph $\mathcal{G}({\sf X},{\sf A})$ with only state nodes $\sf X$ and edges $({\sf x}_{i}\rightarrow{\sf x}_{j})\in{\sf A}$. Then define:

Definition 1**.**

• (i)

${\sf X}_{D}\subseteq\sf X$* is a set of driver nodes if there exists a set $\sf U$ of input nodes and a set $\sf B$ of edges of the form $({\sf u}_{i}\rightarrow{\sf x}_{j})$ such that: (i) the graph $\mathcal{G}({\sf X\cup\sf U},\sf A\cup B)$ is linearly structurally controllable; and (ii) all and only the driver nodes have incoming edges from the input nodes (i.e., $({\sf u}_{i}\rightarrow{\sf x}_{j})\in{\sf B}$ iff ${\sf x}_{j}\in{\sf X}_{D}$).*

• (ii)

${\sf X}_{S}\subseteq\sf X$* is a set of sensor nodes if there exists a set $\sf Y$ of output nodes and a set $\sf C$ of edges of the form $({\sf x}_{i}\rightarrow{\sf y}_{j})$ such that: (i) the graph $\mathcal{G}({\sf X\cup\sf Y},\sf A\cup C)$ is linearly structurally observable; and (ii) all and only the sensor nodes have outgoing edges to the output nodes (i.e., $({\sf x}_{i}\rightarrow{\sf y}_{j})\in\sf C$ iff ${\sf x}_{i}\in{\sf X}_{S}$).*

A set of driver nodes or sensor nodes is called minimal if it has the minimal cardinality among all sets of driver nodes or sensor nodes, respectively. The conditions in Theorem 1 allows finding a minimal set of driver nodes (resp. a minimal set of sensor nodes) by mapping the satisfaction of these conditions to solving maximum matching problem on the graph $\mathcal{G}$ (resp. $\mathcal{G}^{\intercal}$ obtained from $\mathcal{G}$ by reversing the direction of all its edges), see [3].

III Problem statement and Main results

Here we generalize the analysis of Section II by enlarging the class of dynamics that the system can take to include arbitrary nonlinearities. Specifically, we now consider general nonlinear systems of the form

[TABLE]

where $f:\mathbb{R}^{N}\times\mathbb{R}^{M}\rightarrow\mathbb{R}^{N}$ and $h:\mathbb{R}^{N}\rightarrow\mathbb{R}^{P}$ are arbitrary meromorphic functions of their arguments (i.e., each of their entries is the quotient of analytic functions). The assumption of meromorphic functions is very weak in the sense that it is satisfied by most models in biology, chemistry, ecology, and engineering (see, e.g., Table 1 in Ref. [13]). Recall that meromorphic functions are either identically zero (written as “$\equiv 0$”) or different from zero in an open dense subset of their domain (written as “$\not\equiv 0$”), see [11, Chapter 1]. This property allow us to define:

Definition 2**.**

Given a meromorphic pair $\{f,h\}$, its graph $\mathcal{G}_{f,h}=({\sf X\cup\sf Y\cup\sf U},{\sf A}_{f}\cup{\sf B}_{f}\cup{\sf C}_{h})$ has the edge-set defined as: $({\sf x}_{j}\rightarrow{\sf x}_{i})\in{\sf A}_{f}\Leftrightarrow\partial f_{i}/\partial x_{j}\not\equiv 0$; $({\sf u}_{j}\rightarrow{\sf x}_{i})\in{\sf B}_{f}\Leftrightarrow\partial f_{i}/\partial u_{j}\not\equiv 0$; and $({\sf x}_{j}\rightarrow{\sf y}_{i})\in{\sf C}_{h}\Leftrightarrow\partial h_{i}/\partial x_{j}\not\equiv 0$.

We say that two pairs $\{f,h\}$ and $\{\tilde{f},\tilde{h}\}$ are graph-equivalent if $\mathcal{G}_{f,h}=\mathcal{G}_{\tilde{f},\tilde{h}}$. Since any $\{f,h\}$ is graph-equivalent to itself, graph-equivalence is an equivalence relation. Thus, given a graph $\mathcal{G}$, we can define the equivalence class

[TABLE]

The class $\mathfrak{D}(\mathcal{G})$ represents the set of all nonlinear dynamics that a system can have given that its graph is $\mathcal{G}$. Note that $\mathfrak{D}_{L}(\mathcal{G})\subset\mathfrak{D}(\mathcal{G})$.

As the nonlinear counterparts of linear controllability and linear observability, we consider the concepts of local accessibility and local observability. We will introduce these concepts using the algebraic formalism of Ref. [11]. Consider the field of meromorphic functions $\mathcal{K}$ in the variables $\{x,u,\dot{u},\ddot{u},\cdots,y,\dot{y},\ddot{y},\cdots\}$, and the sets of differential symbols $\mathrm{d}x=(\mathrm{d}x_{1},\cdots,\mathrm{d}x_{N})^{\intercal}$, $\mathrm{d}u=(\mathrm{d}u_{1},\cdots,\mathrm{d}u_{M})^{\intercal}$ and $\mathrm{d}y^{(k)}=(\mathrm{d}y_{1}^{(k)},\cdots,\mathrm{d}y_{P}^{(k)})^{\intercal},k\geq 0$. For a function $\varphi\in\mathcal{K}$, its differential is $\mathrm{d}\varphi=(\partial\varphi/\partial x)^{\intercal}\mathrm{d}x+(\partial\varphi/\partial u)^{\intercal}\mathrm{d}u$. More generally, functions in the vector space spanned over $\mathcal{K}$ by the elements of $\{\mathrm{d}x,\mathrm{d}u,\mathrm{d}y,\cdots,\mathrm{d}y^{(k)}\}$ are called one-forms. We next recall the following notions:

Definition 3**.**

• (i)

An autonomous element of a system is a non-constant meromorphic function $\xi(x)$ such that its $k$-th time derivative $\xi^{(k)}$ is independent of $u$ for all $k\geq 0$, i.e.,

[TABLE]

• (ii)

A hidden element of a system is a non-constant meromorphic function $\zeta(x)$ that is independent of $\{y,\cdots,y^{(k)}\}$ for all $k\geq 0$, i.e.,

[TABLE]

An autonomous element constrains the state of the system to a low-dimensional manifold for all control inputs, just as in an uncontrollable linear system its state is constrained to a hyperplane. A hidden element is an internal variable of the system whose value cannot be inferred from the output, since it cannot be rewritten as a function of the output and its derivatives. A non-constant function that is not a hidden element is called observable.

With these notions a system is called locally accessible (“accessible”, for short) if it does not have autonomous elements, and locally observable (“observable”, for short) if it does not have hidden elements [11]. For linear systems, the lack of autonomous elements is equivalent to linear controllability, and the lack of hidden elements is equivalent to linear observability [11]. For example, all linear systems of Eq. (2) are not controllable because $\xi(x)=b_{11}x_{1}-b_{21}x_{2}$ is an autonomous element for all of them. Indeed $\xi^{(k)}\equiv 0$, which is independent of $u$ for all $k\geq 1$. Similarly, $\zeta(x)=c_{12}x_{1}-c_{11}x_{2}$ is a hidden element for all those linear systems, since $\zeta$ cannot be written as a function of $y_{1}=c_{11}x_{1}+c_{12}x_{2}$ and its derivatives $y_{1}^{(k)}=(c_{11}b_{11}+c_{12}b_{12})u_{1}^{(k-1)}$. Indeed, this happens because no output derivative contains information of the state. In this sense, the above definitions of accessibility and observability provide nonlinear generalizations of linear controllability and linear observability.

In analogy to the definitions of linear structural controllability and observability, we now define:

Definition 4**.**

$\mathfrak{D}(\mathcal{G})$* is:*

• (i)

structurally accessible* if $\mathfrak{D}(\mathcal{G})$ contains at least one system that is accessible.*

• (ii)

structurally observable* if $\mathfrak{D}(\mathcal{G})$ contains at least one system that is observable. *

When $\mathfrak{D}(\mathcal{G})$ is structurally accessible (resp. structurally observable), we also call the graph $\mathcal{G}$ structurally accessible (resp. structurally observable). We also call a particular $f$ structurally accessible if there exists at least one graph-equivalent $\bar{f}$ that is accessible. Similarly, a particular $\{f,h\}$ is structurally observable if there exists at least one graph-equivalent $\{\bar{f},\bar{h}\}$ that is observable.

As in the case of linear systems, in Lemma 1 of Section IV we prove that in a structurally accessible class $\mathfrak{D}(\mathcal{G})$ the subset of accessible systems is open and everywhere dense; furthermore the subset of non-accessible systems is not dense. Similarly, in a structurally observable class $\mathfrak{D}(\mathcal{G})$, we prove in Lemma 4 of Section V that the subset of observable systems is open and everywhere dense; in addition, the subset of non observable systems is not dense. This means that, if $\mathfrak{D}(\mathcal{G})$ is structurally accessible (resp. structurally observable), any of its systems is either accessible (resp. observable) or becomes accessible (resp. observable) by an arbitrarily small change of its dynamics (see example Example 2 below).

Our main result is the following:

Theorem 2**.**

$\mathfrak{D}(\mathcal{G})$* is:*

• (i)

structurally accessible iff each state node is the end-node of a path that starts in ${\sf U}$.

• (ii)

structurally observable iff each state node is the start-node of a path that ends in ${\sf Y}$.

Proof.

See Proposition 2 in Section IV for point (i), and Proposition 3 in Section V for point (ii). ∎

The above Theorem shows that despite the observability of a nonlinear system may depend on which particular inputs are applied to it, its structural observability is independent of the inputs. This happens because removing all edges that connect the inputs to the state variables will not change if condition (ii) of Theorem 2 is satisfied. Indeed, note that including more edges in a graph cannot destroy its structural accessibility or structural observability. Note also that a “duality” similar to the case of linear systems remains: a network is structurally accessible if and only if its “dual network” (with reversed edges and the labels of input and output nodes interchanged) is structurally observable.

In addition and somewhat counterintuitively, Theorem 2 shows that nonlinearities make it easier to “control” and “observe” networked systems because the conditions of Theorem 2 are weaker than those of Theorem 1. We illustrate this point by revisiting Example 1 now considering nonlinear dynamics:

Example 2**.**

For the graph in Fig. 1a, the class $\mathfrak{D}(\mathcal{G})$ contains all nonlinear systems of the form

[TABLE]

with nonzero $b_{11},b_{21}$, and $\varepsilon$. Note that Eq. (4) is an “$\varepsilon$-change” of Eq. (2) because making $\varepsilon=0$ renders Eq. (4) equal to Eq. (2). Note also:

In the dilation of Fig. 1a, the nonlinearities in $\mathfrak{D}(\mathcal{G})$ eliminate the autonomous element that was present in $\mathfrak{D}_{L}(\mathcal{D})$. That is, the function $\xi(x)=b_{21}x_{1}-b_{11}x_{2}$ that was an autonomous element for all linear dynamics of Eq. (2) is no longer an autonomous element for Eq. (4) because $\dot{\xi}=-\varepsilon b_{11}u_{1}^{3}$ depends on $u_{1}$. This proves that $\mathfrak{D}(\mathcal{G})$ is structurally accessible. Indeed, the trajectories of Eq. (4) are no longer constrained to a low-dimensional manifold (Fig. 1d).

• 2.

In the contraction of Fig. 1a, the nonlinearities in $\mathfrak{D}(\mathcal{G})$ also eliminate the hidden element. To see this, compute $\dot{y}_{1}=\alpha_{0}(u_{1})+\varepsilon\alpha_{1}(u_{1})x_{1}+\varepsilon\alpha_{2}(u_{1})x_{2},$ where $\alpha_{0}(u_{1})=c_{11}[b_{11}u_{1}+\varepsilon p_{1}]+c_{12}[b_{21}u_{1}+\varepsilon u_{1}^{3}]$, $\alpha_{1}(u_{1})=b_{21}u_{1}+\varepsilon u_{1}^{3}$ and $\alpha_{2}(u_{1})=b_{11}u_{1}+\varepsilon p_{1}$. Note that $\alpha_{1}\not\equiv 0$ and $\alpha_{2}\not\equiv 0$ for almost all $u_{1}$. Therefore, the Jacobian

[TABLE]

is generically nonsingular. Consequently, from the Implicit Function Theorem, it follows that we can locally infer $x_{1}$ and $x_{2}$ from $y_{1}$ and $\dot{y}_{1}$. Indeed, the function $\zeta=c_{12}x_{1}-c_{11}x_{1}$ that was a hidden element of the linear system of Eq. (2) is no longer a hidden element of Eq. (4). This proves that Eq. (4) is observable (Fig. 1e), and that $\mathfrak{D}(\mathcal{G})$ is structurally observable.

III-A Minimal sets of driver/sensor nodes and input/output nodes.

Consider graph $\mathcal{G}({\sf X},{\sf A})$ consisting of state nodes $\sf X$ and edges $({\sf x}_{i}\rightarrow{\sf x}_{j})\in{\sf A}$. We can extend Definition 1 to nonlinear systems by requiring that a set of driver nodes ${\sf X}_{D}\subseteq\sf X$ renders $\mathcal{G}(\sf X\cup\sf U,\sf A\cup B)$ structurally accessible. Similarly, a set of sensor nodes ${\sf X}_{S}\subseteq\sf X$ must render $\mathcal{G}(\sf X\cup\sf U,\sf A\cup C)$ structurally observable. Then Theorem 2 has the following implication:

Proposition 1**.**

• (i)

A minimal set of driver nodes is given by arbitrarily choosing one node in each root strongly-connected-component of $\mathcal{G}({\sf X},\sf A)$.

• (ii)

A minimal set of sensor nodes is given by arbitrarily choosing one node in each top strongly-connected-component of $\mathcal{G}({\sf X},\sf A)$.

A strongly connected component (SCC) of a graph $\mathcal{G}$ is a maximal subgraph such that there is a directed path in both directions between any two of its nodes [14, pp. 552-557]. A root SCC is an SCC without incoming edges, and a top SCC is an SCC without outgoing edges. Recall that any directed graph can be decomposed into an acyclic graph between its SCCs, with root and top SCCs at the start and end of this graph, respectively [14]. Let $m$ be the number of root SCCs and $p$ the number of top SCCs of $\mathcal{G}({\sf X},\sf A)$. Then a proof of Proposition 1-(i) is obtained from the fact that if a single input node $\sf u$ is connected to one arbitrary node ${\sf x}_{j}$ of each root SCC (i.e., ${\sf u}\rightarrow{\sf x}_{j}$, $j=1,\cdots,m$), the decomposition into the acyclic graph of SCC implies that the graph satisfies condition (i) of Theorem 2. Analogously, a proof of Proposition 1-(ii) is obtained from the fact that if a single output node $\sf y$ is connected with one arbitrary node ${\sf x}_{j}$ of each top SCC (i.e., ${\sf x}_{j}\rightarrow{\sf y}$, $j=1,\cdots,p$), this will yield a graph that satisfies condition (ii) of Theorem 2. An additional consequence of this argument is the following:

Corollary 1**.**

• (i)

The minimal number of driver nodes of any graph is its number of root SCCs, and the minimal number of sensor nodes is its number of top SCCs.

• (ii)

The minimal number of input nodes that renders any graph structurally accessible is always one, and the minimal number of output nodes that renders any graph structurally observable is also one.

The second statement in the above Corollary generalizes the result of Ref. [15] to structural systems and to the case of analyzing observability.

All minimal sets of driver or sensor nodes of arbitrary graphs can be found in linear time, since the SCCs of general graphs can be computed in linear time [14, pp. 35]. For comparison, in the case of linear structural accessibility (resp. linear structural observability), solving the maximum-matching problem to find one set of driver nodes (resp. sensor nodes) takes polynomial time, and identifying all sets of driver nodes (resp. sensor nodes) is intractable for large graphs.

The following two Sections build the proofs for our main results.

IV Proof of the Structural Accessibility Theorem

Given a graph $\mathcal{G}=({\sf X\cup\sf U},\sf A\cup B)$, here we consider the class $\mathfrak{D}(\mathcal{G})$ of all controlled systems

[TABLE]

such that $\mathcal{G}_{f}=\mathcal{G}$.

Our first result shows that accessible systems are “generic” in a structurally accessible class $\mathfrak{D}(\mathcal{G})$, while non-accessible are not (i.e., they are “hard to find”). To establish this result, our argument relies on the notions of the $k$-jet $f_{k}$ of the meromorphic function $f$ —informally defined as taking the first $k$-terms of its Taylor expansion— and the resulting topology that can be constructed —the so-called “Whitney $C^{k}$ topologies”. We refer the reader to [16, Section 2.1] and [17, Chapter II.3] for further details. Specifically, the topology we use is defined from the notion of an open ball of radius $\varepsilon\geq 0$ centered at a meromorphic $f_{0}$. This ball consists of all meromorphic $f$’s for which $\exists k\in\mathbb{N}$ such that the Euclidean distance between the first $\ell$ Taylor coefficients of $f_{0}$ and $f$ is less than $\varepsilon$ for all $\ell\geq k$.

Lemma 1**.**

If $\mathfrak{D}(\mathcal{G})$ is structurally accessible then: (i) the subset of accessible systems is dense everywhere in $\mathfrak{D}(\mathcal{G})$; (ii) the subset of accessible systems is open; and (iii) the subset of non-accessible systems is not dense.

Proof.

Although (iii) $\Rightarrow$ (i) because $\mathfrak{D}(\mathcal{G})$ is the disjoint union of accessible and non-accessible systems, independent proofs of each statement are provided:

• (i)

We show that any $f\in\mathfrak{D}(\mathcal{G})$ has an arbitrarily close neighbor $\tilde{f}\in\mathfrak{D}(\mathcal{G})$ that is accessible (Fig. 2a). Let $f^{*}\in\mathfrak{D}(\mathcal{G})$ be an accessible system in $\mathfrak{D}(\mathcal{G})$ (there is at least one because of the definition of structural accessibility). Define the convex combination $f_{\lambda}=\lambda f^{*}+(1-\lambda)f$. Note that $f_{\lambda}\in\mathfrak{D}(\mathcal{G})$ for almost all $\lambda\in[0,1]$. Note also that for $\lambda=1$ we have $f_{1}=f^{*}$, implying that $f_{1}$ is accessible. Consequently, due to the generic properties of meromorphic functions and the Accessibility rank condition [11], the family of systems $\{f_{\lambda}\}$ are accessible for almost all $\lambda\in[0,1]$. Therefore, for any $\varepsilon>0$, there exists $\lambda^{*}>0$ such that $\lambda^{*}<\varepsilon$ and $f_{\lambda^{*}}$ is accessible. Thus, $f_{\lambda^{*}}$ is an $\varepsilon$-neighbor of $f$ which is accessible, completing the proof.

• (ii)

We prove that any accessible $f\in\mathfrak{D}(\mathcal{G})$ has a neighborhood consisting only of accessible systems. Since $f$ is meromorphic, we can rewrite this function as the Taylor-expansion $f(x,u)=\alpha_{0}(x)+\sum_{i=1}^{\infty}\alpha_{i}(x)u^{i}$ with $\alpha_{i}\in\mathcal{K}$. Note that the accessibility of $f$ implies there exists a $k\in\mathbb{\mathbb{missing}}N$ such that the $k$-jet $f_{k}(x,u):=\alpha_{0}(x)+\sum_{i=1}^{k}\alpha_{i}(x)u^{i}$ is accessible. Indeed, since $f$ is accessible there cannot be autonomous elements $\xi\in\mathcal{K}$, implying that $\mathrm{d}\xi$ is not orthogonal to at least some $\alpha_{k}$, $k\in\mathbb{N}$. This implies that no (non-constant) $\xi\in\mathcal{K}$ can be an autonomous element for $f_{k}$, making the $k$-jet $f_{k}$ accessible.

Recall that this $k$-jet represents the first $k$ terms of the Taylor expansion of $f$, implying we can associate $f_{k}$ to a point in $\mathbb{R}^{K}$ for some $K$ that depends on $k$ (right in Fig. 2b). Next we regard $f_{k}$ as a polynomial function of its Taylor coefficients, so that the generic properties of meromorphic functions imply that $f_{k}$ has a neighborhood $N_{f_{k}}\subseteq\mathbb{R}^{K}$ of accessible systems. All $\tilde{f}\in\mathfrak{D}(\mathcal{G})$ such that their $k$-jets $\tilde{f}_{k}$ satisfy $\tilde{f}_{k}\in N_{f_{k}}$ will form the open neighborhood of $f$ of accessible systems.

• (iii)

We prove by contradiction, assuming that $\mathfrak{D}(\mathcal{G})$ is structurally accessible but that it contains an open set $\mathfrak{N}$ such that non-accessible systems are dense on it (pink in Fig. 2c). Since $\mathfrak{D}(\mathcal{G})$ is structurally accessible and accessible systems are dense due to Lemma 1-(i), then $\mathfrak{N}$ contains at least one accessible system $f$ (blue in Fig. 2c). Now choose $k\geq 0$ large enough such that the $k$-jet $f_{k}$ of the accessible system $f$ is accessible. The $k$-jets $\tilde{f}_{k}$ of all non-accessible systems $\tilde{f}$’s remain non-accessible. Since the $f_{k}$ and the $\tilde{f}_{k}$’s represent the first $k$ terms of the Taylor expansion of $f$ and the $\tilde{f}$’s, we can associate each of them to a point in $\mathbb{R}^{K}$ corresponding to the value of the first $k$ coefficients of their Taylor expansion (here again $K$ is some constant that depends on $k$). Since $\mathfrak{N}$ is a neighborhood of $f$, all its elements are mapped to a corresponding neighborhood of $f_{k}$ in $\mathbb{R}^{K}$ such that the points corresponding to non-accessible systems are dense (Fig. 2c). Considering now that $f_{k}$ is accessible and that it is a polynomial function of its Taylor coefficients, the generic properties of meromorphic functions imply that there exists a neighborhood $N_{f_{k}}\subseteq\mathbb{R}^{K}$ of $f_{k}$ such that all its corresponding elements are accessible (blue neighborhood in Fig. 2c). This gives the desired contradiction, since it contradicts the fact that the non-accessible systems were dense.

∎

The next result allows us to analyze the structural accessibility of a graph from its spanning subgraphs, which will be instrumental for the proof of the main result. Recall that a subgraph $\tilde{\mathcal{G}}$ of $\mathcal{G}$ is spanning when $\tilde{\mathcal{G}}$ includes all nodes of $\mathcal{G}$.

Lemma 2**.**

Let $\tilde{\mathcal{G}}\subseteq\mathcal{G}$ be any spanning subgraph of $\mathcal{G}$. If $\mathfrak{D}(\tilde{\mathcal{G}})$ is structurally accessible then $\mathfrak{D}({\mathcal{G}})$ is also structurally accessible.

Proof.

Since $\mathfrak{D}(\tilde{\mathcal{G}})$ is structurally accessible, it contains one system $\dot{x}=f(x,u)$ which is accessible. Notice that starting from $\tilde{\mathcal{G}}$, we can recover $\mathcal{G}$ by adding some edges. Suppose that the edge ${\sf x}_{j}\rightarrow{\sf x}_{i}$ is added to $\tilde{\mathcal{G}}$ to obtain $\mathcal{G}$. Then $\mathfrak{D}(\mathcal{G})$ contains the systems

[TABLE]

for any constant $\alpha\neq 0$. Similarly, if the edge ${\sf u}_{j}\rightarrow{\sf x}_{i}$ is added $\mathfrak{D}(\mathcal{G})$ contains the systems

[TABLE]

For $\alpha=0$ the systems of Eqs. (6) or (7) are accessible. Additionally, their right-hand side is a meromorphic function of $\alpha$. Thus, due to the generic properties of meromorphic functions [11], both systems are accessible for almost all $\alpha\in\mathbb{R}$. Therefore, the class $\mathfrak{D}(\mathcal{G})$ is structurally accessible. Repeating the same argument for all other edges completes the proof. ∎

Now consider a meromorphic function $\varphi(x,u):\mathbb{R}^{N}\times\mathbb{R}^{M}\rightarrow\mathbb{R}^{N}$ and a subset of nodes ${\sf V}\subseteq{\sf X\cup\sf U}$. We write $\varphi\in{\sf S}$ if $\varphi(x)$ depends on all variables $v_{i}$ for all ${\sf v}_{i}\in{\sf V}$. With this notation, an autonomous element of Eq. (5) is a non-constant meromorphic function $\varphi(x)$ such that $\varphi^{(k)}\not\in{\sf U}$ for all $k\geq 0$.

Example 3**.**

For the graph of Fig. 1a with the linear dynamics of Eq. (2) we have that $\xi=b_{11}x_{2}-b_{21}x_{1}$ satisfies $\xi^{(k)}=0$ for all $k\geq 1$. Thus we have that $\xi^{(k)}\not\in\sf U$ for all $k$ and hence $\xi$ is an autonomous element.

Next, for a set $\sf S$ of state nodes, define its “tail-set” $T(\sf S)\subseteq{\sf V}\cup{\sf U}$ as all nodes which point to $\sf S$ (Fig. 3a). We denote $T^{k}({\sf S}):=T(T^{k-1}(\sf S))$.

Example 4**.**

Consider a graph $\mathcal{G}$ which is a (connected) directed tree with each state node ${\sf x}_{i}$ having a single incoming edge, and rooted at a single input node ${\sf u}_{1}$ (Fig. 3a). Note that the state nodes can be organized into $L$ layers according to the distance they have to the input node, with the first layer being all state nodes with distance one. Consider the polynomial dynamics

[TABLE]

where $f_{T(i)}=x_{T(i)}$ if ${\sf x}_{i}$ is in layer $\geq 2$, and $f_{T(i)}=u_{1}$ otherwise. The vector $p=(p_{1},\cdots,p_{N})\in\mathbb{N}_{+}^{N}$ contains $N$ different integers with $\min_{k}p_{k}$ large enough.

For this graph $\mathcal{G}$ and the dynamics of Eq. (8), any non-constant meromorphic function $\varphi\in{\sf S}$ satisfies $\dot{\varphi}\in T({\sf S})$ for any $\sf S\subseteq\sf X$. Namely, if $\varphi$ depends on $\{x_{i},\cdots,x_{k}\}$, then $\dot{\varphi}$ depends on all variables $\{f_{T(i)},\cdots,f_{T(k)}\}$. To show this, just note that

[TABLE]

and that no term can cancel out in the sums because they have different exponents.

This observation allow us to prove that this system is accessible. Indeed, take any $\sf S\subseteq\sf X$ and any non-constant meromorphic function $\varphi\in\sf S$. Since all state nodes are the end-node of a $\sf U$-rooted path, there exists a finite $k$ such that ${\sf u}_{1}\in T^{k}(\sf S)$. Since $\varphi^{(k)}\in T^{k}({\sf S})$, this implies that $\varphi$ cannot be an autonomous element.

Combining Example 4 with Lemma 2, we have actually proved the following result:

Lemma 3**.**

Assume that $\mathcal{G}$ is spanned by a disjoint union of directed trees rooted at $\sf U$, with each state node having a single incoming edge. Then $\mathfrak{D}(\mathcal{G})$ is structurally accessible.

We now have all the ingredients for proving our main result:

Proposition 2**.**

$\mathfrak{D}(\mathcal{G})$* is structurally accessible iff each state node is the end-node of a path that starts in ${\sf U}$.*

Proof.

• ($\Leftarrow$)

By contradiction. If there is a state node ${\sf x}_{i}$ that is not the end-node of any ${\sf U}$-rooted path, then $x_{i}$ itself is an autonomous element.

• ($\Rightarrow$)

Since each state node is the end-node of a $\sf U$-rooted path, note we can always obtain a spanning subgraph $\tilde{\mathcal{G}}$ of $\mathcal{G}$ such that: (i) it is a disjoint union of (connected) directed trees rooted at $\sf U$; (ii) each state node has a single incoming edge (Fig. 2b). By Lemma 3, the class $\mathfrak{D}({\mathcal{G}})$ is structurally accessible.

∎

Remark 1**.**

Note that in the trivial cases of an empty graph (i.e., a graph without nodes) or a graph without state nodes (i.e., the underlying system has no dynamics), applying Definition 4 yields that both graphs are structurally accessible because the set of autonomous element is empty.

Remark 2**.**

Note that, even if $\mathcal{G}$ is linearly structurally controllable, this does not imply that all nonlinear systems with graph $\mathcal{G}$ are accessible. For example, the graph corresponding to the system $\dot{x}_{1}=x_{2}+x_{3}u$, $\dot{x}_{2}=-x_{1}$, and $\dot{x}_{3}=-x_{1}u$ is linearly structurally controllable. Yet, this nonlinear system is not accessible because $\xi=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$ is an autonomous element.

Remark 3**.**

Note that restricting the system dynamics of Eq. (5) to be affine in the control input changes the graph conditions for structural accessibility. In such case, graphs that contains “pure dilations of the control input” as in Fig.1a are not structurally accesible because those subgraphs only admit linear dynamics

V Proof of the Structural Observability Theorem

We start with the following observation:

Lemma 4**.**

• (i)

If $\mathfrak{D}(\mathcal{G})$ is structurally observable then the subset of observable systems is open and dense everywhere in $\mathfrak{D}(\mathcal{G})$; furthermore, the subset of non observable systems is not dense.

• (ii)

Let $\tilde{\mathcal{G}}\subseteq\mathcal{G}$ be any spanning subgraph of $\mathcal{G}$. If $\mathfrak{D}(\tilde{\mathcal{G}})$ is structurally observable then $\mathfrak{D}(\mathcal{G})$ is also structurally observable.

Proof.

A proof for item (i) follows using the exact same argument as in the proof of Lemma 1. Similarly, item (ii) follows using the same argument as in the proof of Lemma 2. ∎

We next prove the structural observability of a particular class of graphs:

Lemma 5**.**

Suppose that $\mathcal{G}(\sf X\cup Y,A\cup C)$ is a (connected) directed tree topped at a single output node $\sf y$, with each state node having a single outgoing edge. Then $\mathfrak{D}(\mathcal{G})$ is structurally observable.

Proof.

From the structure of the graph we can order its nodes by layers, where nodes with distance $k$ to the output $\sf y$ belong to the $k$-th layer (Fig. 4a). We will prove the claim by induction in the number of layers:

• (i)

For one layer, denote its nodes by $\{{\sf x}_{1},\cdots,{\sf x}_{d_{1}}\}$ where $d_{1}$ is the number of nodes. One particular dynamics admissible for this graph is

[TABLE]

with $c_{i}$ some non-zero constants. In the following we show that Eq. (9) is observable by proving that the span of $\mathrm{d}y$ and its derivatives $\mathrm{d}y^{(k)}$ equals $\mathop{\sf span}_{\mathcal{K}}\mathrm{d}x$. If $d_{1}=1$ the claim follows directly, because there is only one state variable $x_{1}$ and $y=x_{1}$ renders it observable. Consider now that $d_{1}>1$. From direct calculation we obtain the identity:

[TABLE]

The variable $z:=\dot{y}/y$ is observable from $y$. Therefore, the system of Eq. (9) will be observable if the span of $\mathrm{d}z$ and its derivatives $\mathrm{d}z^{(k)}$ equals $\mathop{\sf span}_{\mathcal{K}}\mathrm{d}x$. Note that

[TABLE]

so its differential is

[TABLE]

Taking $k=d_{1}+1$, the set $\{\mathrm{d}z,\cdots,\mathrm{d}z^{(d_{1}+1)}\}$ will contain the functions $\{1/x_{i}^{2},\cdots,1/x_{i}^{d_{i}+3}\}$, $i=1,\cdots,d_{1}$, whose span is $\mathop{\sf span}_{\mathcal{K}}\mathrm{d}x$. This proves that the system of Eq. (9) is observable, and thus that a graph $\mathcal{G}$ with one layer is locally observable.

• (ii)

For the induction step, we show that if a graph $\mathcal{G}$ with $L$ layers is structurally observable, then a graph with $L+1$ layers is also structurally observable. By definition, the nodes in the $(L+1)$-th layer are only connected to nodes in the $L$-th layer. Furthermore, they are connected in the same way as nodes in the first layer are connected to the output node (Fig. 4a). Therefore, the argument in point (i) with $y$ replaced by the corresponding node in the $L$-th layer implies that the nodes in the $(L+1)$-th layer are observable. This completes the proof.

∎

The final result follows by decomposing the graph into disjoint trees topped at the output nodes:

Proposition 3**.**

$\mathfrak{D}(\mathcal{G})$* is structurally observable iff each state node is the start-node of a path that ends in ${\sf Y}$.*

Proof.

• ($\Leftarrow$)

By contradiction. If there is a state node ${\sf x}_{i}$ that is not the start-node of any ${\sf Y}$-topped path, then $x_{i}$ itself is a hidden element.

• ($\Rightarrow$)

Since each state node is the start-node of a $\sf Y$-topped path, note we can always obtain a spanning subgraph $\tilde{\mathcal{G}}$ of $\mathcal{G}$ such that: (i) it is a disjoint union of (connected) directed trees topped at $\sf Y$; (ii) each state node has a single outgoing edge (Fig. 4b). By Lemma 5, $\mathfrak{D}(\tilde{\mathcal{G}})$ is structurally observable. Since $\tilde{\mathcal{G}}\subseteq\mathcal{G}$ is a spanning subgraph, Lemma 4-(ii) implies that $\mathfrak{D}(\mathcal{G})$ is structurally observable.

∎

Remark 4**.**

In analogy to Remark 1, in the trivial cases of an empty graph (i.e., a graph without nodes) or a graph without state nodes (i.e., the underlying system has no dynamics), applying Definition 4 yields that both graphs are structurally observable because the set of hidden elements is empty.

VI Discussion and Concluding Remarks

The notions of structural accessibility and structural observability that we have introduced and characterized are nonlinear counterparts of the notions of linear structural controllability and linear structural observability.

We next discuss some testable predictions offered by our theory. In a recent study of the locomotion of the worm C. elegans, the ablation of the neuron PDB was found to generate a dilation in the nervous system connectome that decreased its (output) structural linear controllability [5]. This loss of linear controllability was suggested to imply that the worm lost some “directions” in which it is was able to move, which were experimentally confirmed by a decreased ability to produce some specific motion patterns (quantified by a decrease in certain so-called “eigenworms”). Assuming that the nervous system of the C. elegans is an arbitrary nonlinear system instead of a linear system, our theory implies that the dilation caused by ablating PBD cannot decrease the structural accessibility of the C. elegans connectome. Namely, the nervous system of an ablated worm can reach the same set of states as those of normal worms using perhaps different “longer” trajectories (e.g., by using different paths in the connectome that yield different combinations of “eigenworms”). Thus, our structural accessibility theory predicts that PDB ablated worms can still adopt each body pose that a non-ablated worm can adopt. More generally, we predict that the ability of a worm to adopt a body pose is preserved as long as the ablated interneurons do not fully disconnect an input (i.e., a sensory neuron) or an output (i.e., a motor neuron).

We emphasize that more detailed predictions for the impact of the network structure on the controllability or observability properties can be obtained when the class of dynamics that the system can take is better known —such as neuronal, ecological, gene regulatory, or epidemic systems, see e.g., [13]. Such an analysis would provide graph conditions for structural accessibility and structural local observability that are “between” those of Theorem 1 (i.e., linear systems), and those of Theorem 2 (i.e., arbitrary nonlinear systems). Indeed, note that the conditions of Theorem 2 are always necessary, but they may not be sufficient when we restrict the system dynamics to belong to a particular class. For example, in [18] and [19], we analyzed the structural accessibility and structural local observability properties for the particular class of nonlinear dynamics found in ecosystems. In this analysis, we found that the conditions for structural accessibility and structural local observability for ecological dynamics are indeed stronger than those of Theorem 2.

Finally, our results provide a broader perspective of what we can deduce about the controllability or observability properties of a system from knowing only its interconnection network. We have shown that if the control inputs can reach all state nodes through a path in the network, then there exists some admissible system dynamics that is accessible. Similarly, if all state nodes can reach an output through a path in the network, then there exists some admissible system dynamics that is locally observable. These two facts suggest that the interconnection network only encodes the essential information of the controllability and observability properties of complex systems.

Acknowledgements. M.T.A. gratefully acknowledges the financial support from CONACyT, México, and LS2N, France. We also thank Yang-Yu Liu and Yu Kawano for insightful comments about preliminary versions of this paper.